diff --git a/PFR-declarations/PFR.ApproxHomPFR.jsonl b/PFR-declarations/PFR.ApproxHomPFR.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..427187d3a13cb540c7f07a4673b14182b668fac0 --- /dev/null +++ b/PFR-declarations/PFR.ApproxHomPFR.jsonl @@ -0,0 +1 @@ +{"name":"approx_hom_pfr","declaration":"/-- Let $G, G'$ be finite abelian $2$-groups.\nLet $f : G \\to G'$ be a function, and suppose that there are at least\n$|G|^2 / K$ pairs $(x,y) \\in G^2$ such that $$ f(x+y) = f(x) + f(y).$$\nThen there exists a homomorphism $\\phi : G \\to G'$ and a constant $c \\in G'$ such that\n$f(x) = \\phi(x)+c$ for at least $|G| / (2 ^ {172} * K ^ {146})$ values of $x \\in G$. -/\ntheorem approx_hom_pfr {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (f : G → G') (K : ℝ) (hK : K > 0) (hf : ↑(Nat.card ↑{x | f (x.1 + x.2) = f x.1 + f x.2}) ≥ ↑(Nat.card G) ^ 2 / K) : ∃ φ c, ↑(Nat.card ↑{x | f x = φ x + c}) ≥ ↑(Nat.card G) / (2 ^ 172 * K ^ 146)"} diff --git a/PFR-declarations/PFR.Endgame.jsonl b/PFR-declarations/PFR.Endgame.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0163c3032380d8d02fe2a45e7d9a1e0f0152f8e8 --- /dev/null +++ b/PFR-declarations/PFR.Endgame.jsonl @@ -0,0 +1,19 @@ +{"name":"hV","declaration":"theorem hV {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : H[X₁' + X₂] = H[X₁ + X₂']"} +{"name":"independenceCondition5","declaration":"theorem independenceCondition5 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁, X₁', X₂ + X₂'] MeasureTheory.volume"} +{"name":"independenceCondition3","declaration":"theorem independenceCondition3 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁', X₂, X₁ + X₂'] MeasureTheory.volume"} +{"name":"independenceCondition4","declaration":"theorem independenceCondition4 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₁', X₁ + X₂'] MeasureTheory.volume"} +{"name":"sum_dist_diff_le","declaration":"/-- $$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\ntheorem sum_dist_diff_le {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : d[p.X₀₁ # X₁ + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # X₁ + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₁ # X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] +\n (d[p.X₀₂ # X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂])) +\n (d[p.X₀₁ # X₁' + X₁ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] +\n (d[p.X₀₂ # X₁' + X₁ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂])) ≤\n (6 - 3 * p.η) * d[X₁ # X₂] + 3 * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'])"} +{"name":"independenceCondition2","declaration":"theorem independenceCondition2 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₁, X₁' + X₂'] MeasureTheory.volume"} +{"name":"construct_good'","declaration":"theorem construct_good' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) (μ : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] : d[X₁ # X₂] ≤\n I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n p.η / 3 *\n (I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n (d[p.X₀₁ ; MeasureTheory.volume # T₁ ; μ] - d[p.X₀₁ # X₁] +\n (d[p.X₀₂ ; MeasureTheory.volume # T₁ ; μ] - d[p.X₀₂ # X₂])) +\n (d[p.X₀₁ ; MeasureTheory.volume # T₂ ; μ] - d[p.X₀₁ # X₁] +\n (d[p.X₀₂ ; MeasureTheory.volume # T₂ ; μ] - d[p.X₀₂ # X₂])) +\n (d[p.X₀₁ ; MeasureTheory.volume # T₃ ; μ] - d[p.X₀₁ # X₁] +\n (d[p.X₀₂ ; MeasureTheory.volume # T₃ ; μ] - d[p.X₀₂ # X₂])))"} +{"name":"I₃_eq","declaration":"/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\ntheorem I₃_eq {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] = I[X₁ + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂']"} +{"name":"cond_c_eq_integral","declaration":"theorem cond_c_eq_integral {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Y : Ω' → G} {Z : Ω' → G} (hY : Measurable Y) (hZ : Measurable Z) : d[p.X₀₁ # Y | Z] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # Y | Z] - d[p.X₀₂ # X₂]) =\n ∫ (x : G),\n (fun z =>\n d[p.X₀₁ ; MeasureTheory.volume # Y ; ProbabilityTheory.cond MeasureTheory.volume (Z ⁻¹' {z})] - d[p.X₀₁ # X₁] +\n (d[p.X₀₂ ; MeasureTheory.volume # Y ; ProbabilityTheory.cond MeasureTheory.volume (Z ⁻¹' {z})] -\n d[p.X₀₂ # X₂]))\n x ∂MeasureTheory.Measure.map Z MeasureTheory.volume"} +{"name":"independenceCondition1","declaration":"theorem independenceCondition1 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁, X₂, X₁' + X₂'] MeasureTheory.volume"} +{"name":"construct_good","declaration":"/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n-\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\n\nThen there exist random variables $T'_1, T'_2$ such that\n\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X _1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2])$$\n\nis at most\n\n$$\\delta + \\frac{\\eta}{3} \\biggl( \\delta + \\sum_{i=1}^2 \\sum_{j = 1}^3\n (d[X^0_i;T_j] - d[X^0_i; X_i]) \\biggr).$$\n-/\ntheorem construct_good {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] +\n p.η / 3 *\n (I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] + (d[p.X₀₁ # T₁] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₁] - d[p.X₀₂ # X₂])) +\n (d[p.X₀₁ # T₂] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂] - d[p.X₀₂ # X₂])) +\n (d[p.X₀₁ # T₃] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₃] - d[p.X₀₂ # X₂])))"} +{"name":"independenceCondition6","declaration":"theorem independenceCondition6 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₂', X₁' + X₁] MeasureTheory.volume"} +{"name":"sum_condMutual_le","declaration":"/-- $$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\ntheorem sum_condMutual_le {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] + I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] +\n I[X₁' + X₁ : X₁ + X₂|X₁ + X₂ + X₁' + X₂'] ≤\n 6 * p.η * d[X₁ # X₂] - (1 - 5 * p.η) / (1 - p.η) * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'])"} +{"name":"cond_construct_good","declaration":"theorem cond_construct_good {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) {R : Ω' → G} (hR : Measurable R) : d[X₁ # X₂] ≤\n I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] +\n p.η / 3 *\n (I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] +\n (d[p.X₀₁ # T₁ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₁ | R] - d[p.X₀₂ # X₂])) +\n (d[p.X₀₁ # T₂ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂ | R] - d[p.X₀₂ # X₂])) +\n (d[p.X₀₁ # T₃ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₃ | R] - d[p.X₀₂ # X₂])))"} +{"name":"tau_strictly_decreases_aux","declaration":"/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases_aux {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) (hpη : p.η = 1 / 9) : d[X₁ # X₂] = 0"} +{"name":"sum_uvw_eq_zero","declaration":"/-- $U+V+W=0$. -/\ntheorem sum_uvw_eq_zero {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] {Ω : Type u_4} (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) : X₁ + X₂ + (X₁' + X₂) + (X₁' + X₁) = 0"} +{"name":"delta'_eq_integral","declaration":"theorem delta'_eq_integral {G : Type u_1} [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} {R : Ω' → G} : I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] =\n ∫ (x : G),\n (fun r =>\n I[T₁ : T₂ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})] +\n I[T₂ : T₃ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})] +\n I[T₃ : T₁ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})])\n x ∂MeasureTheory.Measure.map R MeasureTheory.volume"} +{"name":"construct_good_prelim","declaration":"/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\nThen there exist random variables $T'_1, T'_2$ such that\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X_1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2]) $$\nis at most\n$$ \\delta + \\eta ( d[X^0_1;T_1]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2]-d[X^0_2;X_2]) $$\n$$ + \\tfrac12 \\eta I[T_1: T_3] + \\tfrac12 \\eta I[T_2: T_3].$$\n-/\ntheorem construct_good_prelim {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] + p.η * (d[p.X₀₁ # T₁] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂] - d[p.X₀₂ # X₂])) +\n p.η * (I[T₁ : T₃] + I[T₂ : T₃]) / 2"} +{"name":"hU","declaration":"theorem hU {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : H[X₁ + X₂] = H[X₁' + X₂']"} diff --git a/PFR-declarations/PFR.EntropyPFR.jsonl b/PFR-declarations/PFR.EntropyPFR.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1c1b8d53e9c659812c2968652bc1c593b61326ca --- /dev/null +++ b/PFR-declarations/PFR.EntropyPFR.jsonl @@ -0,0 +1,3 @@ +{"name":"entropic_PFR_conjecture","declaration":"/-- `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\nsubgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 11 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 9) : ∃ H Ω mΩ U,\n MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n Measurable U ∧\n ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 11 * d[p.X₀₁ # p.X₀₂]"} +{"name":"tau_strictly_decreases","declaration":"/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\\tau[X'_1;X'_2] < \\tau[X_1;X_2]$.\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {Ω : Type u_3} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) {X₁ : Ω → G} {X₂ : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) (hpη : p.η = 1 / 9) : d[X₁ # X₂] = 0"} +{"name":"entropic_PFR_conjecture'","declaration":"theorem entropic_PFR_conjecture' {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 9) : ∃ H Ω mΩ U,\n ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧\n d[p.X₀₁ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] ∧ d[p.X₀₂ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂]"} diff --git a/PFR-declarations/PFR.Fibring.jsonl b/PFR-declarations/PFR.Fibring.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f8f0c13bcf47f57f52efc4bb52b3e96fbd8c61a9 --- /dev/null +++ b/PFR-declarations/PFR.Fibring.jsonl @@ -0,0 +1,7 @@ +{"name":"sum_of_rdist_eq","declaration":"/-- Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables.\n Then\n$$d[Y_1-Y_3; Y_2-Y_4] + d[Y_1|Y_1-Y_3; Y_2|Y_2-Y_4] $$\n$$ + I[Y_1-Y_2 : Y_2 - Y_4 | Y_1-Y_2-Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$$\n-/\ntheorem sum_of_rdist_eq {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Fin 4 → Ω → G) (h_indep : ProbabilityTheory.iIndepFun (fun x => hG) Y μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : d[Y 0 ; μ # Y 1 ; μ] + d[Y 2 ; μ # Y 3 ; μ] =\n d[Y 0 - Y 2 ; μ # Y 1 - Y 3 ; μ] + d[Y 0 | Y 0 - Y 2 ; μ # Y 1 | Y 1 - Y 3 ; μ] +\n I[Y 0 - Y 1 : Y 1 - Y 3|Y 0 - Y 1 - Y 2 + Y 3;μ]"} +{"name":"rdist_of_hom_le","declaration":"/-- \\[d[X;Y]\\geq d[\\pi(X);\\pi(Y)].\\] -/\ntheorem rdist_of_hom_le {H : Type u_1} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H] [MeasurableSingletonClass H] {H' : Type u_2} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H'] [MeasurableSingletonClass H'] (π : H →+ H') {Ω : Type u_3} {Ω' : Type u_4} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {Z_1 : Ω → H} {Z_2 : Ω' → H} (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] : d[⇑π ∘ Z_1 ; μ # ⇑π ∘ Z_2 ; μ'] ≤ d[Z_1 ; μ # Z_2 ; μ']"} +{"name":"rdist_le_sum_fibre","declaration":"theorem rdist_le_sum_fibre {H : Type u_1} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H] [MeasurableSingletonClass H] {H' : Type u_2} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H'] [MeasurableSingletonClass H'] (π : H →+ H') {Ω : Type u_3} {Ω' : Type u_4} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {Z_1 : Ω → H} {Z_2 : Ω' → H} (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] : d[⇑π ∘ Z_1 ; μ # ⇑π ∘ Z_2 ; μ'] + d[Z_1 | ⇑π ∘ Z_1 ; μ # Z_2 | ⇑π ∘ Z_2 ; μ'] ≤ d[Z_1 ; μ # Z_2 ; μ']"} +{"name":"sum_of_rdist_eq_step_condMutualInfo","declaration":"/-- The conditional mutual information step of `sum_of_rdist_eq` -/\ntheorem sum_of_rdist_eq_step_condMutualInfo {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin 4 → Ω → G} (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : I[⟨Y 0 - Y 1, Y 2 - Y 3⟩ : ⟨Y 0 - Y 2, Y 1 - Y 3⟩|Y 0 - Y 1 - (Y 2 - Y 3);μ] =\n I[Y 0 - Y 1 : Y 1 - Y 3|Y 0 - Y 1 - Y 2 + Y 3;μ]"} +{"name":"rdist_of_indep_eq_sum_fibre","declaration":"/-- If $Z_1, Z_2$ are independent, then $d[Z_1; Z_2]$ is equal to\n$$ d[\\pi(Z_1);\\pi(Z_2)] + d[Z_1|\\pi(Z_1); Z_2 |\\pi(Z_2)]$$\nplus\n$$I( Z_1 - Z_2 : (\\pi(Z_1), \\pi(Z_2)) | \\pi(Z_1 - Z_2) ).$$\n-/\ntheorem rdist_of_indep_eq_sum_fibre {H : Type u_1} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H] [MeasurableSingletonClass H] {H' : Type u_2} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H'] [MeasurableSingletonClass H'] (π : H →+ H') {Ω : Type u_3} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Z_1 : Ω → H} {Z_2 : Ω → H} (h : ProbabilityTheory.IndepFun Z_1 Z_2 μ) (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] : d[Z_1 ; μ # Z_2 ; μ] =\n d[⇑π ∘ Z_1 ; μ # ⇑π ∘ Z_2 ; μ] + d[Z_1 | ⇑π ∘ Z_1 ; μ # Z_2 | ⇑π ∘ Z_2 ; μ] +\n I[Z_1 - Z_2 : ⟨⇑π ∘ Z_1, ⇑π ∘ Z_2⟩|⇑π ∘ (Z_1 - Z_2);μ]"} +{"name":"sum_of_rdist_eq_step_condRuzsaDist","declaration":"/-- The conditional Ruzsa Distance step of `sum_of_rdist_eq` -/\ntheorem sum_of_rdist_eq_step_condRuzsaDist {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin 4 → Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun x => hG) Y μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : d[⟨Y 0, Y 2⟩ | Y 0 - Y 2 ; μ # ⟨Y 1, Y 3⟩ | Y 1 - Y 3 ; μ] = d[Y 0 | Y 0 - Y 2 ; μ # Y 1 | Y 1 - Y 3 ; μ]"} +{"name":"sum_of_rdist_eq_char_2","declaration":"/-- Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables.\n Then\n$$d[Y_1+Y_3; Y_2+Y_4] + d[Y_1|Y_1+Y_3; Y_2|Y_2+Y_4] $$\n$$ + I[Y_1+Y_2 : Y_2 + Y_4 | Y_1+Y_2+Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$$\n-/\ntheorem sum_of_rdist_eq_char_2 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [ElementaryAddCommGroup G 2] (Y : Fin 4 → Ω → G) (h_indep : ProbabilityTheory.iIndepFun (fun x => hG) Y μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : d[Y 0 ; μ # Y 1 ; μ] + d[Y 2 ; μ # Y 3 ; μ] =\n d[Y 0 + Y 2 ; μ # Y 1 + Y 3 ; μ] + d[Y 0 | Y 0 + Y 2 ; μ # Y 1 | Y 1 + Y 3 ; μ] +\n I[Y 0 + Y 1 : Y 1 + Y 3|Y 0 + Y 1 + Y 2 + Y 3;μ]"} diff --git a/PFR-declarations/PFR.FirstEstimate.jsonl b/PFR-declarations/PFR.FirstEstimate.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1a41ca2ce6ffa0c467a9761e8dea34c7e8155e53 --- /dev/null +++ b/PFR-declarations/PFR.FirstEstimate.jsonl @@ -0,0 +1,9 @@ +{"name":"first_estimate","declaration":"/-- We have $I_1 \\leq 2 \\eta k$ -/\ntheorem first_estimate {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] ≤ 2 * p.η * d[X₁ # X₂]"} +{"name":"diff_rdist_le_1","declaration":"/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\ntheorem diff_rdist_le_1 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂' : Measurable X₂') (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ d[X₁ # X₂] / 2 + H[X₂] / 4 - H[X₁] / 4"} +{"name":"condRuzsaDist_of_sums_ge","declaration":"/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\ntheorem condRuzsaDist_of_sums_ge {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] ≥\n d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂])"} +{"name":"diff_rdist_le_2","declaration":"/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\ntheorem diff_rdist_le_2 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ d[X₁ # X₂] / 2 + H[X₁] / 4 - H[X₂] / 4"} +{"name":"diff_rdist_le_3","declaration":"/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n\\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\ntheorem diff_rdist_le_3 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂' : Measurable X₂') (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ d[X₁ # X₂] / 2 + H[X₁] / 4 - H[X₂] / 4"} +{"name":"rdist_of_sums_ge","declaration":"/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\ntheorem rdist_of_sums_ge {G : Type u_1} [addgroup : AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ + X₂' # X₂ + X₁'] ≥\n d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂])"} +{"name":"rdist_add_rdist_add_condMutual_eq","declaration":"/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\ntheorem rdist_add_rdist_add_condMutual_eq {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] + I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] = 2 * d[X₁ # X₂]"} +{"name":"ent_ofsum_le","declaration":"/-- $$\\mathbb{H}[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} \\mathbb{H}[X_1]+\\tfrac{1}{2} \\mathbb{H}[X_2] + (2 + \\eta) k - I_1.$$\n-/\ntheorem ent_ofsum_le {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : H[X₁ + X₂ + X₁' + X₂'] ≤ H[X₁] / 2 + H[X₂] / 2 + (2 + p.η) * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂']"} +{"name":"diff_rdist_le_4","declaration":"/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n\\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\ntheorem diff_rdist_le_4 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ d[X₁ # X₂] / 2 + H[X₂] / 4 - H[X₁] / 4"} diff --git a/PFR-declarations/PFR.ForMathlib.CompactProb.jsonl b/PFR-declarations/PFR.ForMathlib.CompactProb.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a7c65dd53eebc5bad3df2a00aacc9c49ad3faeda --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.CompactProb.jsonl @@ -0,0 +1,9 @@ +{"name":"continuous_pmf_apply","declaration":"theorem continuous_pmf_apply {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] (i : X) : Continuous fun μ => (fun s => (↑↑↑μ s).toNNReal) {i}"} +{"name":"probabilityMeasureEquivStdSimplex_symm_coe_apply","declaration":"theorem probabilityMeasureEquivStdSimplex_symm_coe_apply {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] (p : ↑(stdSimplex ℝ X)) : ↑(probabilityMeasureEquivStdSimplex.symm p) =\n Finset.sum Finset.univ fun i => ENNReal.ofReal (↑p i) • MeasureTheory.Measure.dirac i"} +{"name":"tendsto_lintegral_of_forall_of_finite","declaration":"theorem tendsto_lintegral_of_forall_of_finite {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] [Finite X] {ι : Type u_2} {L : Filter ι} (μs : ι → MeasureTheory.Measure X) (μ : MeasureTheory.Measure X) (f : BoundedContinuousFunction X NNReal) (h : ∀ (x : X), Filter.Tendsto (fun i => ↑↑(μs i) {x}) L (nhds (↑↑μ {x}))) : Filter.Tendsto (fun i => ∫⁻ (x : X), ↑(f x) ∂μs i) L (nhds (∫⁻ (x : X), ↑(f x) ∂μ))"} +{"name":"continuous_pmf_apply'","declaration":"theorem continuous_pmf_apply' {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] (i : X) : Continuous fun μ => (↑μ).real {i}"} +{"name":"probabilityMeasureEquivStdSimplex_coe_apply","declaration":"theorem probabilityMeasureEquivStdSimplex_coe_apply {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] (μ : MeasureTheory.ProbabilityMeasure X) (i : X) : ↑(probabilityMeasureEquivStdSimplex μ) i = ↑((fun s => (↑↑↑μ s).toNNReal) {i})"} +{"name":"probabilityMeasureEquivStdSimplex","declaration":"/-- The canonical bijection between the set of probability measures on a fintype and the set of\nnonnegative functions on the points adding up to one. -/\ndef probabilityMeasureEquivStdSimplex {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] : MeasureTheory.ProbabilityMeasure X ≃ ↑(stdSimplex ℝ X)"} +{"name":"probabilityMeasureHomeoStdSimplex","declaration":"/-- The canonical homeomorphism between the space of probability measures on a finite space and the\nstandard simplex. -/\ndef probabilityMeasureHomeoStdSimplex {X : Type u_1} [MeasurableSpace X] [Fintype X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] : MeasureTheory.ProbabilityMeasure X ≃ₜ ↑(stdSimplex ℝ X)"} +{"name":"instCompactSpaceProbabilityMeasureInstTopologicalSpaceProbabilityMeasure","declaration":"/-- This is still true when `X` is a metrizable compact space, but the proof requires Riesz\nrepresentation theorem.\nTODO: remove once the general version is proved. -/\ninstance instCompactSpaceProbabilityMeasureInstTopologicalSpaceProbabilityMeasure {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [Finite X] [DiscreteTopology X] [BorelSpace X] : CompactSpace (MeasureTheory.ProbabilityMeasure X)"} +{"name":"instSecondCountableTopologyProbabilityMeasureInstTopologicalSpaceProbabilityMeasure","declaration":"instance instSecondCountableTopologyProbabilityMeasureInstTopologicalSpaceProbabilityMeasure {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [Finite X] [DiscreteTopology X] [BorelSpace X] : SecondCountableTopology (MeasureTheory.ProbabilityMeasure X)"} diff --git a/PFR-declarations/PFR.ForMathlib.Elementary.jsonl b/PFR-declarations/PFR.ForMathlib.Elementary.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c1b54f7d3eb1a087cd7ff04e06bdc0db72191f0f --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Elementary.jsonl @@ -0,0 +1,25 @@ +{"name":"ElementaryAddCommGroup.mk","declaration":"ctor ElementaryAddCommGroup.mk {G : Type u_1} [AddCommGroup G] {p : outParam ℕ} (orderOf_of_ne : ∀ {x : G}, x ≠ 0 → addOrderOf x = p) : ElementaryAddCommGroup G p"} +{"name":"ElementaryAddCommGroup.finite_closure","declaration":"theorem ElementaryAddCommGroup.finite_closure {G : Type u_1} [AddCommGroup G] {n : ℕ} [ElementaryAddCommGroup G (n + 1)] {A : Set G} (h : Set.Finite A) : Set.Finite ↑(AddSubgroup.closure A)"} +{"name":"ElementaryAddCommGroup.char_smul_eq_zero'","declaration":"theorem ElementaryAddCommGroup.char_smul_eq_zero' {p : ℕ} {Γ : Type u_1} [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] (x : Γ) (k : ℤ) : (k * ↑p) • x = 0"} +{"name":"ElementaryAddCommGroup.two_le_char_of_ne_zero","declaration":"theorem ElementaryAddCommGroup.two_le_char_of_ne_zero {p : ℕ} {Γ : Type u_1} [NeZero p] [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] {x : Γ} (x_ne_zero : x ≠ 0) : 2 ≤ p"} +{"name":"ElementaryAddCommGroup.quotient_group","declaration":"theorem ElementaryAddCommGroup.quotient_group {G : Type u_1} [AddCommGroup G] {p : ℕ} (hp : Nat.Prime p) {H : AddSubgroup G} (hH : ∀ (x : G), p • x ∈ H) : ElementaryAddCommGroup (G ⧸ H) p"} +{"name":"ElementaryAddCommGroup.of_torsion","declaration":"theorem ElementaryAddCommGroup.of_torsion {G : Type u_1} [AddCommGroup G] {p : ℕ} (hp : Nat.Prime p) (h : ∀ (x : G), p • x = 0) : ElementaryAddCommGroup G p"} +{"name":"ElementaryAddCommGroup.char_ne_one_of_ne_zero","declaration":"theorem ElementaryAddCommGroup.char_ne_one_of_ne_zero {p : ℕ} {Γ : Type u_1} [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] {x : Γ} (x_ne_zero : x ≠ 0) : p ≠ 1"} +{"name":"ElementaryAddCommGroup.sum_add_sum_eq_sum","declaration":"theorem ElementaryAddCommGroup.sum_add_sum_eq_sum {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) (y : G) (z : G) : x + y + (y + z) = x + z"} +{"name":"ElementaryAddCommGroup.torsion","declaration":"theorem ElementaryAddCommGroup.torsion {G : Type u_1} [AddCommGroup G] (p : ℕ) [elem : ElementaryAddCommGroup G p] (x : G) : p • x = 0"} +{"name":"ElementaryAddCommGroup.sub_eq_add","declaration":"theorem ElementaryAddCommGroup.sub_eq_add {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) (y : G) : x - y = x + y"} +{"name":"ElementaryAddCommGroup.ofModule","declaration":"/-- A vector space over Z/p is an elementary abelian p-group. -/\ntheorem ElementaryAddCommGroup.ofModule {G : Type u_1} {p : ℕ} [AddCommGroup G] [Module (ZMod p) G] [Fact (Nat.Prime p)] : ElementaryAddCommGroup G p"} +{"name":"ElementaryAddCommGroup.char_smul_eq_zero","declaration":"theorem ElementaryAddCommGroup.char_smul_eq_zero {p : ℕ} {Γ : Type u_1} [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] (x : Γ) : p • x = 0"} +{"name":"ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero","declaration":"theorem ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) (y : G) (z : G) : x + y + (y + z) + (z + x) = 0"} +{"name":"ElementaryAddCommGroup.exists_subgroup_subset_card_le","declaration":"/-- In an elementary abelian $p$-group, every finite subgroup $H$ contains a further subgroup of\ncardinality between $k$ and $pk$, if $k \\leq |H|$.-/\ntheorem ElementaryAddCommGroup.exists_subgroup_subset_card_le {G : Type u_1} {p : ℕ} (hp : Nat.Prime p) [AddCommGroup G] [h : ElementaryAddCommGroup G p] {k : ℕ} (H : AddSubgroup G) (hk : k ≤ Nat.card ↥H) (h'k : k ≠ 0) : ∃ H', Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' ∧ H' ≤ H"} +{"name":"ElementaryAddCommGroup.neg_eq_self","declaration":"theorem ElementaryAddCommGroup.neg_eq_self {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) : -x = x"} +{"name":"ElementaryAddCommGroup.Int.mod_eq","declaration":"theorem ElementaryAddCommGroup.Int.mod_eq (n : ℤ) (m : ℤ) : n % m = n - n / m * m"} +{"name":"ElementaryAddCommGroup.orderOf_of_ne","declaration":"def ElementaryAddCommGroup.orderOf_of_ne {G : Type u_1} [AddCommGroup G] {p : outParam ℕ} [self : ElementaryAddCommGroup G p] {x : G} (hx : x ≠ 0) : addOrderOf x = p"} +{"name":"ElementaryAddCommGroup.module","declaration":"instance ElementaryAddCommGroup.module {G : Type u_1} {n : ℕ} [AddCommGroup G] [ElementaryAddCommGroup G n] : Module (ZMod n) G"} +{"name":"ElementaryAddCommGroup.instElementaryAddCommGroupForAllAddCommGroup","declaration":"instance ElementaryAddCommGroup.instElementaryAddCommGroupForAllAddCommGroup (Ω : Type u_1) (Γ : Type u_2) (p : ℕ) [NeZero p] [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] : ElementaryAddCommGroup (Ω → Γ) p"} +{"name":"ElementaryAddCommGroup.subgroup","declaration":"theorem ElementaryAddCommGroup.subgroup {G : Type u_1} [AddCommGroup G] {n : ℕ} [ElementaryAddCommGroup G n] (H : AddSubgroup G) : ElementaryAddCommGroup (↥H) n"} +{"name":"ElementaryAddCommGroup.mem_periodicPts","declaration":"theorem ElementaryAddCommGroup.mem_periodicPts {p : ℕ} {Γ : Type u_1} [NeZero p] [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] {x : Γ} (y : Γ) : y ∈ Function.periodicPts fun z => x + z"} +{"name":"ElementaryAddCommGroup.add_self","declaration":"theorem ElementaryAddCommGroup.add_self {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) : x + x = 0"} +{"name":"ElementaryAddCommGroup.exists_finsupp","declaration":"theorem ElementaryAddCommGroup.exists_finsupp {G : Type u_1} [AddCommGroup G] {n : ℕ} [ElementaryAddCommGroup G (n + 1)] {A : Set G} {x : G} (hx : x ∈ Submodule.span ℤ A) : ∃ μ, (Finsupp.sum μ fun a r => ZMod.cast r • ↑a) = x"} +{"name":"ElementaryAddCommGroup","declaration":"/-- An elementary `p`-group, i.e., a commutative additive group in which every nonzero element has\norder exactly `p`. -/\nclass ElementaryAddCommGroup (G : Type u_1) [AddCommGroup G] (p : outParam ℕ) : Prop"} +{"name":"ElementaryAddCommGroup.instElementaryAddCommGroupOfNatNatInstOfNatNat","declaration":"instance ElementaryAddCommGroup.instElementaryAddCommGroupOfNatNatInstOfNatNat {G : Type u_1} [AddCommGroup G] [Module (ZMod 2) G] : ElementaryAddCommGroup G 2"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.Basic.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..bf791dfb354f59d861459e031329900db7b5e70f --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.Basic.jsonl @@ -0,0 +1,111 @@ +{"name":"ProbabilityTheory.entropy_const","declaration":"/-- The entropy of any constant is zero. -/\ntheorem ProbabilityTheory.entropy_const {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (c : S) : H[fun x => c ; μ] = 0"} +{"name":"ProbabilityTheory.condEntropy_le_entropy","declaration":"/-- $$ H[X|Y] ≤ H[X] $$ -/\ntheorem ProbabilityTheory.condEntropy_le_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] ≤ H[X ; μ]"} +{"name":"ProbabilityTheory.condMutualInfo_eq_kernel_mutualInfo","declaration":"/-- The conditional mutual information agrees with the information of the conditional kernel.\n-/\ntheorem ProbabilityTheory.condMutualInfo_eq_kernel_mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : I[X : Y|Z;μ] = Ik[ProbabilityTheory.condDistrib (⟨X, Y⟩) Z μ , MeasureTheory.Measure.map Z μ]"} +{"name":"ProbabilityTheory.condMutualInfo_eq_integral_mutualInfo","declaration":"theorem ProbabilityTheory.condMutualInfo_eq_integral_mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} : I[X : Y|Z;μ] = ∫ (x : U), (fun z => I[X : Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]) x ∂MeasureTheory.Measure.map Z μ"} +{"name":"ProbabilityTheory.«termI[_:_|_]»","declaration":"/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\ndef ProbabilityTheory.«termI[_:_|_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.mutualInfo","declaration":"/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\ndef ProbabilityTheory.mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"} +{"name":"ProbabilityTheory.entropy_pair_eq_add","declaration":"/-- $H[X, Y] = H[X] + H[Y]$ if and only if $X, Y$ are independent. -/\ntheorem ProbabilityTheory.entropy_pair_eq_add {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] = H[X ; μ] + H[Y ; μ] ↔ ProbabilityTheory.IndepFun X Y μ"} +{"name":"ProbabilityTheory.«termI[_:_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_|_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.condEntropy_le_log_card","declaration":"/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\ntheorem ProbabilityTheory.condEntropy_le_log_card {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [Fintype S] (X : Ω → S) (Y : Ω → T) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : H[X | Y ; μ] ≤ Real.log ↑(Fintype.card S)"} +{"name":"ProbabilityTheory.condMutualInfo_comm","declaration":"/-- $I[X : Y | Z] = I[Y : X | Z]$. -/\ntheorem ProbabilityTheory.condMutualInfo_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (Z : Ω → U) (μ : MeasureTheory.Measure Ω) : I[X : Y|Z;μ] = I[Y : X|Z;μ]"} +{"name":"ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy'","declaration":"/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\ntheorem ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : I[X : Y ; μ] = H[Y ; μ] - H[Y | X ; μ]"} +{"name":"ProbabilityTheory.entropy_add_entropy_sub_mutualInfo","declaration":"theorem ProbabilityTheory.entropy_add_entropy_sub_mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : H[X ; μ] + H[Y ; μ] - I[X : Y ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_le_log_card_of_mem","declaration":"/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\ntheorem ProbabilityTheory.entropy_le_log_card_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {A : Finset S} {μ : MeasureTheory.Measure Ω} {X : Ω → S} (hX : Measurable X) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ A) : H[X ; μ] ≤ Real.log ↑(Nat.card { x // x ∈ A })"} +{"name":"ProbabilityTheory.entropy_of_comp_eq_of_comp","declaration":"/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\nother, then they have the same entropy. Can be used as a substitute for\n`entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\ntheorem ProbabilityTheory.entropy_of_comp_eq_of_comp {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : S → T) (g : T → S) (h1 : Y = f ∘ X) (h2 : X = g ∘ Y) [FiniteRange X] [FiniteRange Y] : H[X ; μ] = H[Y ; μ]"} +{"name":"ProbabilityTheory.«termH[_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.entropy_eq_sum_finset'","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finset' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {A : Finset S} (hA : ↑↑(MeasureTheory.Measure.map X μ) (↑A)ᶜ = 0) : H[X ; μ] = Finset.sum A fun x => Real.negMulLog ((MeasureTheory.Measure.map X μ).real {x})"} +{"name":"ProbabilityTheory.condEntropy_eq_sum_prod","declaration":"/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\ntheorem ProbabilityTheory.condEntropy_eq_sum_prod {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] {X : Ω → S} [MeasurableSingletonClass T] (hX : Measurable X) {Y : Ω → T} (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] =\n Finset.sum (FiniteRange.toFinset X ×ˢ FiniteRange.toFinset Y) fun p =>\n (↑↑(MeasureTheory.Measure.map Y μ) {p.2}).toReal *\n Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {p.2}))) {p.1}).toReal"} +{"name":"ProbabilityTheory.entropy_eq_sum","declaration":"/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\ntheorem ProbabilityTheory.entropy_eq_sum {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : H[X ; μ] = ∑' (x : S), Real.negMulLog (↑↑(MeasureTheory.Measure.map X μ) {x}).toReal"} +{"name":"ProbabilityTheory.mutualInfo_const","declaration":"/-- The mutual information with a constant is always zero. -/\ntheorem ProbabilityTheory.mutualInfo_const {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} (hX : Measurable X) (c : T) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] : I[X : fun x => c ; μ] = 0"} +{"name":"ProbabilityTheory.condEntropy_eq_sum","declaration":"/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\ntheorem ProbabilityTheory.condEntropy_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hY : Measurable Y) [FiniteRange Y] : H[X | Y ; μ] =\n Finset.sum (FiniteRange.toFinset Y) fun y => (↑↑(MeasureTheory.Measure.map Y μ) {y}).toReal * H[X | Y ← y ; μ]"} +{"name":"ProbabilityTheory.const_of_nonpos_entropy","declaration":"/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\ntheorem ProbabilityTheory.const_of_nonpos_entropy {S : Type uS} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {X : Ω → S} (hX : Measurable X) [FiniteRange X] (hent : H[X ; μ] ≤ 0) : ∃ s, μ.real (X ⁻¹' {s}) = 1"} +{"name":"ProbabilityTheory.IdentDistrib.mutualInfo_eq","declaration":"/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\ntheorem ProbabilityTheory.IdentDistrib.mutualInfo_eq {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] {X : Ω → S} {Y : Ω → T} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_1} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} {X' : Ω' → S} {Y' : Ω' → T} (hXY : ProbabilityTheory.IdentDistrib (⟨X, Y⟩) (⟨X', Y'⟩) μ μ') : I[X : Y ; μ] = I[X' : Y' ; μ']"} +{"name":"ProbabilityTheory.«termH[_|_;_]»","declaration":"/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\ndef ProbabilityTheory.«termH[_|_;_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.condEntropy_comm","declaration":"/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\ntheorem ProbabilityTheory.condEntropy_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y⟩ | Z ; μ] = H[⟨Y, X⟩ | Z ; μ]"} +{"name":"ProbabilityTheory.entropy_nonneg","declaration":"/-- Entropy is always non-negative. -/\ntheorem ProbabilityTheory.entropy_nonneg {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : 0 ≤ H[X ; μ]"} +{"name":"ProbabilityTheory.«termI[_:_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_|_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.entropy_eq_sum'","declaration":"theorem ProbabilityTheory.entropy_eq_sum' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : H[X ; μ] = ∑' (x : S), Real.negMulLog ((MeasureTheory.Measure.map X μ).real {x})"} +{"name":"ProbabilityTheory.condEntropy_of_injective","declaration":"/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S → U$ is\ninjective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\nThus for instance $H[X-Y|Y] = H[X|Y]$. -/\ntheorem ProbabilityTheory.condEntropy_of_injective {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass U] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : T → S → U) (hf : ∀ (t : T), Function.Injective (f t)) [FiniteRange Y] : H[fun ω => f (Y ω) (X ω) | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.condMutualInfo_eq_sum'","declaration":"/-- A variant of `condMutualInfo_eq_sum` when `Z` has finite codomain. -/\ntheorem ProbabilityTheory.condMutualInfo_eq_sum' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hZ : Measurable Z) [Fintype U] : I[X : Y|Z;μ] =\n Finset.sum Finset.univ fun z => (↑↑μ (Z ⁻¹' {z})).toReal * I[X : Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy'","declaration":"/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[Y ; μ] - I[X : Y ; μ] = H[Y | X ; μ]"} +{"name":"ProbabilityTheory.«termI[_:_;_]»","declaration":"/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\ndef ProbabilityTheory.«termI[_:_;_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.IsUniform.entropy_eq'","declaration":"/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\ntheorem ProbabilityTheory.IsUniform.entropy_eq' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {H : Set S} [Finite ↑H] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : ProbabilityTheory.IsUniform H X μ) (hX' : Measurable X) : H[X ; μ] = Real.log ↑(Nat.card ↑H)"} +{"name":"ProbabilityTheory.«termH[_|_←_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_|_←_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.«termH[_;_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_;_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.entropy_congr","declaration":"/-- Two variables that agree almost everywhere, have the same entropy. -/\ntheorem ProbabilityTheory.entropy_congr {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {μ : MeasureTheory.Measure Ω} {X : Ω → S} {X' : Ω → S} (h : X =ᶠ[MeasureTheory.Measure.ae μ] X') : H[X ; μ] = H[X' ; μ]"} +{"name":"ProbabilityTheory.condEntropy_zero_measure","declaration":"/-- Any random variable on a zero measure space has zero conditional entropy. -/\ntheorem ProbabilityTheory.condEntropy_zero_measure {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) : H[X | Y ; 0] = 0"} +{"name":"ProbabilityTheory.IdentDistrib.entropy_eq","declaration":"/-- Two variables that have the same distribution, have the same entropy. -/\ntheorem ProbabilityTheory.IdentDistrib.entropy_eq {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_1} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} {X' : Ω' → S} (h : ProbabilityTheory.IdentDistrib X X' μ μ') : H[X ; μ] = H[X' ; μ']"} +{"name":"ProbabilityTheory.condEntropy","declaration":"/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\ndef ProbabilityTheory.condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"} +{"name":"ProbabilityTheory.entropy_eq_sum_finiteRange","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finiteRange {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] : H[X ; μ] = Finset.sum (FiniteRange.toFinset X) fun x => Real.negMulLog (↑↑(MeasureTheory.Measure.map X μ) {x}).toReal"} +{"name":"ProbabilityTheory.chain_rule''","declaration":"/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\ntheorem ProbabilityTheory.chain_rule'' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] = H[⟨X, Y⟩ ; μ] - H[Y ; μ]"} +{"name":"ProbabilityTheory.IndepFun.condEntropy_eq_entropy","declaration":"theorem ProbabilityTheory.IndepFun.condEntropy_eq_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} {μ : MeasureTheory.Measure Ω} (h : ProbabilityTheory.IndepFun X Y μ) (hX : Measurable X) (hY : Measurable Y) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] = H[X ; μ]"} +{"name":"ProbabilityTheory.«termH[_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.«termH[_|_←_;_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_|_←_;_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.IsUniform.entropy_eq","declaration":"/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\ntheorem ProbabilityTheory.IsUniform.entropy_eq {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] (H : Finset S) (X : Ω → S) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : ProbabilityTheory.IsUniform (↑H) X μ) (hX' : Measurable X) : H[X ; μ] = Real.log ↑(Nat.card { x // x ∈ H })"} +{"name":"ProbabilityTheory.«termI[_:_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.cond_chain_rule","declaration":"/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\ntheorem ProbabilityTheory.cond_chain_rule {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, Y⟩ | Z ; μ] = H[Y | Z ; μ] + H[X | ⟨Y, Z⟩ ; μ]"} +{"name":"ProbabilityTheory.mutualInfo_def","declaration":"theorem ProbabilityTheory.mutualInfo_def {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : I[X : Y ; μ] = H[X ; μ] + H[Y ; μ] - H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.IndepFun.mutualInfo_eq_zero","declaration":"/-- **Alias** of the reverse direction of `ProbabilityTheory.mutualInfo_eq_zero`.\n\n---\n\n$I[X : Y] = 0$ iff $X, Y$ are independent. -/\ntheorem ProbabilityTheory.IndepFun.mutualInfo_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : ProbabilityTheory.IndepFun X Y μ → I[X : Y ; μ] = 0"} +{"name":"ProbabilityTheory.entropy_eq_sum_finiteRange'","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finiteRange' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] : H[X ; μ] = Finset.sum (FiniteRange.toFinset X) fun x => Real.negMulLog ((MeasureTheory.Measure.map X μ).real {x})"} +{"name":"ProbabilityTheory.entropy_comp_le","declaration":"/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\ntheorem ProbabilityTheory.entropy_comp_le {Ω : Type uΩ} {S : Type uS} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable U] [Nonempty S] [Nonempty U] [MeasurableSpace S] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass U] {X : Ω → S} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (f : S → U) [FiniteRange X] : H[f ∘ X ; μ] ≤ H[X ; μ]"} +{"name":"ProbabilityTheory.«termH[_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.map_prod_comap_swap","declaration":"/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\ntheorem ProbabilityTheory.map_prod_comap_swap {Ω : Type uΩ} {S : Type uS} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace U] {Z : Ω → U} {X : Ω → S} (hX : Measurable X) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) : MeasureTheory.Measure.comap Prod.swap (MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ) =\n MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ"} +{"name":"ProbabilityTheory.entropy_eq_kernel_entropy","declaration":"/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\ntheorem ProbabilityTheory.entropy_eq_kernel_entropy {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : H[X ; μ] = Hk[ProbabilityTheory.kernel.const Unit (MeasureTheory.Measure.map X μ) , MeasureTheory.Measure.dirac ()]"} +{"name":"ProbabilityTheory.IndepFun.entropy_pair_eq_add","declaration":"/-- If $X, Y$ are independent, then $H[X, Y] = H[X] + H[Y]$. -/\ntheorem ProbabilityTheory.IndepFun.entropy_pair_eq_add {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : ProbabilityTheory.IndepFun X Y μ → H[⟨X, Y⟩ ; μ] = H[X ; μ] + H[Y ; μ]"} +{"name":"ProbabilityTheory.condMutualInfo_nonneg","declaration":"/-- Conditional information is non-nonegative. -/\ntheorem ProbabilityTheory.condMutualInfo_nonneg {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (Z : Ω → U) (μ : MeasureTheory.Measure Ω) [FiniteRange X] [FiniteRange Y] : 0 ≤ I[X : Y|Z;μ]"} +{"name":"ProbabilityTheory.condMutualInfo_eq'","declaration":"/-- $$ I[X : Y| Z] = H[X| Z] - H[X|Y, Z].$$ -/\ntheorem ProbabilityTheory.condMutualInfo_eq' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : I[X : Y|Z;μ] = H[X | Z ; μ] - H[X | ⟨Y, Z⟩ ; μ]"} +{"name":"ProbabilityTheory.«termI[_:_|_;_]»","declaration":"/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\ndef ProbabilityTheory.«termI[_:_|_;_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.«termH[_|_←_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_←_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.condEntropy_eq_kernel_entropy","declaration":"/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\ntheorem ProbabilityTheory.condEntropy_eq_kernel_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] [FiniteRange Y] : H[X | Y ; μ] = Hk[ProbabilityTheory.condDistrib X Y μ , MeasureTheory.Measure.map Y μ]"} +{"name":"ProbabilityTheory.chain_rule","declaration":"/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\ntheorem ProbabilityTheory.chain_rule {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] = H[Y ; μ] + H[X | Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_eq_sum_sum","declaration":"/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\ntheorem ProbabilityTheory.condEntropy_eq_sum_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] {X : Ω → S} [MeasurableSingletonClass T] (hX : Measurable X) {Y : Ω → T} (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] =\n Finset.sum (FiniteRange.toFinset Y) fun y =>\n Finset.sum (FiniteRange.toFinset X) fun x =>\n (↑↑(MeasureTheory.Measure.map Y μ) {y}).toReal *\n Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {y}))) {x}).toReal"} +{"name":"ProbabilityTheory.condMutualInfo_def","declaration":"theorem ProbabilityTheory.condMutualInfo_def {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (X : Ω → S) (Y : Ω → T) (Z : Ω → U) (μ : MeasureTheory.Measure Ω) : I[X : Y|Z;μ] =\n ∫ (x : U), (fun z => H[X | Z ← z ; μ] + H[Y | Z ← z ; μ] - H[⟨X, Y⟩ | Z ← z ; μ]) x ∂MeasureTheory.Measure.map Z μ"} +{"name":"ProbabilityTheory.prob_ge_exp_neg_entropy'","declaration":"/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\ntheorem ProbabilityTheory.prob_ge_exp_neg_entropy' {S : Type uS} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (X : Ω → S) (hX : Measurable X) [FiniteRange X] : ∃ s, Real.exp (-H[X ; μ]) ≤ μ.real (X ⁻¹' {s})"} +{"name":"ProbabilityTheory.condMutualInfo_of_inj_map","declaration":"/-- If $f(Z, X)$ is injective for each fixed $Z$, then $I[f(Z, X) : Y| Z] = I[X : Y| Z]$.-/\ntheorem ProbabilityTheory.condMutualInfo_of_inj_map {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) {V : Type u_1} [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V] [Countable V] (f : U → S → V) (hf : ∀ (t : U), Function.Injective (f t)) [FiniteRange Z] : I[fun ω => f (Z ω) (X ω) : Y|Z;μ] = I[X : Y|Z;μ]"} +{"name":"ProbabilityTheory.mutualInfo_eq_zero","declaration":"/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\ntheorem ProbabilityTheory.mutualInfo_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : I[X : Y ; μ] = 0 ↔ ProbabilityTheory.IndepFun X Y μ"} +{"name":"ProbabilityTheory.condMutualInfo","declaration":"/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\ndef ProbabilityTheory.condMutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (X : Ω → S) (Y : Ω → T) (Z : Ω → U) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"} +{"name":"ProbabilityTheory.entropy_submodular","declaration":"/-- $H[X | Y, Z] \\leq H[X | Z]$ -/\ntheorem ProbabilityTheory.entropy_submodular {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[X | ⟨Y, Z⟩ ; μ] ≤ H[X | Z ; μ]"} +{"name":"ProbabilityTheory.entropy_comp_of_injective","declaration":"/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\ntheorem ProbabilityTheory.entropy_comp_of_injective {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} (μ : MeasureTheory.Measure Ω) (hX : Measurable X) (f : S → T) (hf : Function.Injective f) : H[f ∘ X ; μ] = H[X ; μ]"} +{"name":"ProbabilityTheory.entropy_prod_comp","declaration":"/-- $H[X, f(X)] = H[X]$.-/\ntheorem ProbabilityTheory.entropy_prod_comp {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) (f : S → T) : H[⟨X, f ∘ X⟩ ; μ] = H[X ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy","declaration":"/-- $H[X] - I[X : Y] = H[X | Y]$. -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X ; μ] - I[X : Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.«termH[_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.IdentDistrib.condEntropy_eq","declaration":"/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\ntheorem ProbabilityTheory.IdentDistrib.condEntropy_eq {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} {Ω' : Type u_1} [MeasurableSpace Ω'] {X : Ω → S} {Y : Ω → T} {μ' : MeasureTheory.Measure Ω'} {X' : Ω' → S} {Y' : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y') (h : ProbabilityTheory.IdentDistrib (⟨X, Y⟩) (⟨X', Y'⟩) μ μ') [FiniteRange X] [FiniteRange Y] [FiniteRange X'] [FiniteRange Y'] : H[X | Y ; μ] = H[X' | Y' ; μ']"} +{"name":"ProbabilityTheory.«termH[_|_]»","declaration":"/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\ndef ProbabilityTheory.«termH[_|_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.entropy_eq_log_card","declaration":"/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\ntheorem ProbabilityTheory.entropy_eq_log_card {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} [Fintype S] (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [hμ : NeZero μ] [MeasureTheory.IsFiniteMeasure μ] : H[X ; μ] = Real.log ↑(Fintype.card S) ↔\n ∀ (s : S), ↑↑(MeasureTheory.Measure.map X μ) {s} = ↑↑μ Set.univ / ↑(Fintype.card S)"} +{"name":"ProbabilityTheory.cond_chain_rule'","declaration":"/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\ntheorem ProbabilityTheory.cond_chain_rule' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, Y⟩ | Z ; μ] = H[X | Z ; μ] + H[Y | ⟨X, Z⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_def","declaration":"/-- Entropy of a random variable agrees with entropy of its distribution. -/\ntheorem ProbabilityTheory.entropy_def {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : H[X ; μ] = Hm[MeasureTheory.Measure.map X μ]"} +{"name":"ProbabilityTheory.chain_rule'","declaration":"/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\ntheorem ProbabilityTheory.chain_rule' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] = H[X ; μ] + H[Y | X ; μ]"} +{"name":"ProbabilityTheory.condEntropy_eq_sum_fintype","declaration":"/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\ntheorem ProbabilityTheory.condEntropy_eq_sum_fintype {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hY : Measurable Y) [Fintype T] : H[X | Y ; μ] = Finset.sum Finset.univ fun y => (↑↑μ (Y ⁻¹' {y})).toReal * H[X | Y ← y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_eq_zero","declaration":"theorem ProbabilityTheory.condEntropy_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (t : T) (ht : (↑↑(MeasureTheory.Measure.map Y μ) {t}).toReal = 0) : H[X | Y ← t ; μ] = 0"} +{"name":"ProbabilityTheory.entropy_triple_add_entropy_le","declaration":"/-- The submodularity inequality:\n$$ H[X, Y, Z] + H[Z] \\leq H[X, Z] + H[Y, Z].$$ -/\ntheorem ProbabilityTheory.entropy_triple_add_entropy_le {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, ⟨Y, Z⟩⟩ ; μ] + H[Z ; μ] ≤ H[⟨X, Z⟩ ; μ] + H[⟨Y, Z⟩ ; μ]"} +{"name":"ProbabilityTheory.condEntropy_prod_eq_of_indepFun","declaration":"theorem ProbabilityTheory.condEntropy_prod_eq_of_indepFun {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [Fintype T] [Fintype U] [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] (h : ProbabilityTheory.IndepFun (⟨X, Y⟩) Z μ) : H[X | ⟨Y, Z⟩ ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.«termI[_:_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.entropy_sub_condEntropy","declaration":"/-- $$ H[X] - H[X|Y] = I[X : Y] $$ -/\ntheorem ProbabilityTheory.entropy_sub_condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[X ; μ] - H[X | Y ; μ] = I[X : Y ; μ]"} +{"name":"ProbabilityTheory.entropy_zero_measure","declaration":"/-- Any variable on a zero measure space has zero entropy. -/\ntheorem ProbabilityTheory.entropy_zero_measure {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) : H[X ; 0] = 0"} +{"name":"ProbabilityTheory.condMutualInfo_eq_zero","declaration":"/-- $I[X : Y| Z]=0$ iff $X, Y$ are conditionally independent over $Z$. -/\ntheorem ProbabilityTheory.condMutualInfo_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} [MeasurableSingletonClass S] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : I[X : Y|Z;μ] = 0 ↔ ProbabilityTheory.CondIndepFun X Y Z μ"} +{"name":"ProbabilityTheory.entropy_le_log_card","declaration":"/-- Entropy is at most the logarithm of the cardinality of the range. -/\ntheorem ProbabilityTheory.entropy_le_log_card {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : H[X ; μ] ≤ Real.log ↑(Fintype.card S)"} +{"name":"ProbabilityTheory.condEntropy_comp_ge","declaration":"/-- Data-processing inequality for the conditional entropy:\n$$ H[Y|f(X)] \\geq H[Y|X]$$\nTo upgrade this to equality, see `condEntropy_of_injective'` -/\ntheorem ProbabilityTheory.condEntropy_comp_ge {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass S] [MeasurableSingletonClass T] [FiniteRange X] [FiniteRange Y] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : S → U) : H[Y | f ∘ X ; μ] ≥ H[Y | X ; μ]"} +{"name":"ProbabilityTheory.mutualInfo_comm","declaration":"/-- $I[X : Y] = I[Y : X]$. -/\ntheorem ProbabilityTheory.mutualInfo_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : I[X : Y ; μ] = I[Y : X ; μ]"} +{"name":"ProbabilityTheory.mutualInfo_nonneg","declaration":"/-- Mutual information is non-negative. -/\ntheorem ProbabilityTheory.mutualInfo_nonneg {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [FiniteRange X] [FiniteRange Y] : 0 ≤ I[X : Y ; μ]"} +{"name":"ProbabilityTheory.ent_of_cond_indep","declaration":"/-- If $X, Y$ are conditionally independent over $Z$, then $H[X, Y, Z] = H[X, Z] + H[Y, Z] - H[Z]$. -/\ntheorem ProbabilityTheory.ent_of_cond_indep {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} [MeasurableSingletonClass S] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.CondIndepFun X Y Z μ) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, ⟨Y, Z⟩⟩ ; μ] = H[⟨X, Z⟩ ; μ] + H[⟨Y, Z⟩ ; μ] - H[Z ; μ]"} +{"name":"ProbabilityTheory.condEntropy_nonneg","declaration":"/-- Conditional entropy is non-negative. -/\ntheorem ProbabilityTheory.condEntropy_nonneg {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : 0 ≤ H[X | Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_comp_self","declaration":"/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\ntheorem ProbabilityTheory.condEntropy_comp_self {Ω : Type uΩ} {S : Type uS} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable U] [Nonempty S] [Nonempty U] [MeasurableSpace S] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass U] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) {f : S → U} (hf : Measurable f) [FiniteRange X] : H[X | f ∘ X ; μ] = H[X ; μ] - H[f ∘ X ; μ]"} +{"name":"ProbabilityTheory.condEntropy_of_injective'","declaration":"/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\ntheorem ProbabilityTheory.condEntropy_of_injective' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass S] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : T → U) (hf : Function.Injective f) (hfY : Measurable (f ∘ Y)) [FiniteRange X] [FiniteRange Y] : H[X | f ∘ Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.prob_ge_exp_neg_entropy","declaration":"/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\ntheorem ProbabilityTheory.prob_ge_exp_neg_entropy {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) [hX' : FiniteRange X] : ∃ s, ↑↑(MeasureTheory.Measure.map X μ) {s} ≥ ↑↑μ Set.univ * ↑(Real.toNNReal (Real.exp (-H[X ; μ])))"} +{"name":"ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy","declaration":"/-- $I[X : Y] = H[X] - H[X|Y]$. -/\ntheorem ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : I[X : Y ; μ] = H[X ; μ] - H[X | Y ; μ]"} +{"name":"ProbabilityTheory.condMutualInfo_eq_sum","declaration":"theorem ProbabilityTheory.condMutualInfo_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hZ : Measurable Z) [FiniteRange Z] : I[X : Y|Z;μ] =\n Finset.sum (FiniteRange.toFinset Z) fun z =>\n (↑↑μ (Z ⁻¹' {z})).toReal * I[X : Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]"} +{"name":"ProbabilityTheory.entropy_assoc","declaration":"/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\ntheorem ProbabilityTheory.entropy_assoc {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) : H[⟨X, ⟨Y, Z⟩⟩ ; μ] = H[⟨⟨X, Y⟩, Z⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_eq_sum_finset","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finset {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {A : Finset S} (hA : ↑↑(MeasureTheory.Measure.map X μ) (↑A)ᶜ = 0) : H[X ; μ] = Finset.sum A fun x => Real.negMulLog (↑↑(MeasureTheory.Measure.map X μ) {x}).toReal"} +{"name":"ProbabilityTheory.entropy","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.entropy {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"} +{"name":"ProbabilityTheory.entropy_comm","declaration":"/-- $H[X, Y] = H[Y, X]$. -/\ntheorem ProbabilityTheory.entropy_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y⟩ ; μ] = H[⟨Y, X⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_pair_le_add","declaration":"/-- Subadditivity of entropy. -/\ntheorem ProbabilityTheory.entropy_pair_le_add {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] ≤ H[X ; μ] + H[Y ; μ]"} +{"name":"ProbabilityTheory.entropy_cond_eq_sum","declaration":"/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\ntheorem ProbabilityTheory.entropy_cond_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (y : T) : H[X | Y ← y ; μ] =\n ∑' (x : S), Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {y}))) {x}).toReal"} +{"name":"ProbabilityTheory.condEntropy_comp_of_injective","declaration":"/-- A weaker version of the above lemma in which f is independent of Y. -/\ntheorem ProbabilityTheory.condEntropy_comp_of_injective {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass S] [MeasurableSingletonClass U] (μ : MeasureTheory.Measure Ω) (hX : Measurable X) (f : S → U) (hf : Function.Injective f) : H[f ∘ X | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_def","declaration":"theorem ProbabilityTheory.condEntropy_def {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : H[X | Y ; μ] = ∫ (x : T), (fun y => H[X | Y ← y ; μ]) x ∂MeasureTheory.Measure.map Y μ"} +{"name":"ProbabilityTheory.«termI[_:_]»","declaration":"/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\ndef ProbabilityTheory.«termI[_:_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.condEntropy_prod_eq_sum","declaration":"theorem ProbabilityTheory.condEntropy_prod_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_1} {X : Ω → S} {Y : Ω → T} {Z : Ω → T'} [MeasurableSpace T'] [MeasurableSingletonClass T'] (μ : MeasureTheory.Measure Ω) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsFiniteMeasure μ] [Fintype T] [Fintype T'] : H[X | ⟨Y, Z⟩ ; μ] =\n Finset.sum Finset.univ fun z => (↑↑μ (Z ⁻¹' {z})).toReal * H[X | Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]"} +{"name":"ProbabilityTheory.«termH[_|_←_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_←_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.«termH[_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.entropy_cond_eq_sum_finiteRange","declaration":"theorem ProbabilityTheory.entropy_cond_eq_sum_finiteRange {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (y : T) [FiniteRange X] : H[X | Y ← y ; μ] =\n Finset.sum (FiniteRange.toFinset X) fun x =>\n Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {y}))) {x}).toReal"} +{"name":"ProbabilityTheory.condMutualInfo_eq","declaration":"/-- $$ I[X : Y| Z] = H[X| Z] + H[Y| Z] - H[X, Y| Z].$$ -/\ntheorem ProbabilityTheory.condMutualInfo_eq {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : I[X : Y|Z;μ] = H[X | Z ; μ] + H[Y | Z ; μ] - H[⟨X, Y⟩ | Z ; μ]"} +{"name":"ProbabilityTheory.condEntropy_two_eq_kernel_entropy","declaration":"theorem ProbabilityTheory.condEntropy_two_eq_kernel_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {Z : Ω → U} {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Y] [FiniteRange Z] : H[X | ⟨Y, Z⟩ ; μ] =\n Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.condDistrib (fun a => (Y a, X a)) Z μ) ,\n MeasureTheory.Measure.compProd (MeasureTheory.Measure.map Z μ)\n (ProbabilityTheory.kernel.fst (ProbabilityTheory.condDistrib (fun a => (Y a, X a)) Z μ))]"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.Group.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.Group.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..94f32af8a31e3b4cee7a4466323c17c35ed32cd1 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.Group.jsonl @@ -0,0 +1,56 @@ +{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_add","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X + Y | Z]$$-/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_add {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} [FiniteRange Z] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X + Y | Z ; μ]"} +{"name":"ProbabilityTheory.entropy_div_left","declaration":"/-- $H[Y / X, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y / X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_neg","declaration":"/-- If $X$ is $G$-valued, then $H[-X]=H[X]$.-/\ntheorem ProbabilityTheory.entropy_neg {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) : H[-X ; μ] = H[X ; μ]"} +{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_sub","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X - Y | Z]$$-/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_sub {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X - Y | Z ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_left","declaration":"/-- $H[Y - X, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y - X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X + Y ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X * Y ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X + Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_sub_left","declaration":"/-- $$H[Y - X | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_sub_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y - X | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_add_right","declaration":"/-- $$H[X + Y | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_add_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X + Y | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_mul","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_mul {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X * Y ; μ]"} +{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_div","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X / Y]$$ -/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_div {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X / Y ; μ]"} +{"name":"ProbabilityTheory.entropy_mul_left","declaration":"/-- $H[Y * X, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y * X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_neg_left","declaration":"/-- $H[-X, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_neg_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨-X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.condEntropy_add_left","declaration":"/-- $$H[Y + X | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_add_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y + X | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X - Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X - Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_mul_right","declaration":"/-- $$H[X * Y | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_mul_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X * Y | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_right'","declaration":"/-- $H[X, Y - X] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y - X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_div_left'","declaration":"/-- $H[X / Y, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X / Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_add","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_add {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X + Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_mul_left","declaration":"/-- $$H[Y * X | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_mul_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y * X | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.max_entropy_le_entropy_mul","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X * Y]$$. -/\ntheorem ProbabilityTheory.max_entropy_le_entropy_mul {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X * Y ; μ]"} +{"name":"ProbabilityTheory.max_entropy_le_entropy_add","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.max_entropy_le_entropy_add {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X + Y ; μ]"} +{"name":"ProbabilityTheory.entropy_add_const","declaration":"theorem ProbabilityTheory.entropy_add_const {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (c : G) : H[X + fun x => c ; μ] = H[X ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X / Y ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X / Y ; μ]"} +{"name":"ProbabilityTheory.mutualInfo_add_right","declaration":"/-- $I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.-/\ntheorem ProbabilityTheory.mutualInfo_add_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun X Y μ) : I[X : X + Y ; μ] = H[X + Y ; μ] - H[Y ; μ]"} +{"name":"ProbabilityTheory.entropy_mul_right","declaration":"/-- $H[X, X * Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X * Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_add_left","declaration":"/-- $H[Y + X, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y + X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.mutualInfo_mul_right","declaration":"/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\ntheorem ProbabilityTheory.mutualInfo_mul_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun X Y μ) : I[X : X * Y ; μ] = H[X * Y ; μ] - H[Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_sub_right","declaration":"/-- $$H[X - Y | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_sub_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X - Y | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.max_entropy_le_entropy_sub","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X - Y]$$.-/\ntheorem ProbabilityTheory.max_entropy_le_entropy_sub {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X - Y ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_comm","declaration":"/-- $$H[X - Y] = H[Y - X]$$-/\ntheorem ProbabilityTheory.entropy_sub_comm {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) : H[X - Y ; μ] = H[Y - X ; μ]"} +{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_div","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X / Y | Z]$$ -/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_div {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X / Y | Z ; μ]"} +{"name":"ProbabilityTheory.entropy_add_right","declaration":"/-- $H[X, X + Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X + Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_mul_const","declaration":"theorem ProbabilityTheory.entropy_mul_const {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (c : G) : H[X * fun x => c ; μ] = H[X ; μ]"} +{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_mul","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X * Y | Z]$$ -/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_mul {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} [FiniteRange Z] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X * Y | Z ; μ]"} +{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_sub","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X - Y]$$-/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_sub {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X - Y ; μ]"} +{"name":"ProbabilityTheory.entropy_div_right","declaration":"/-- $H[X, X / Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X / Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_inv_left","declaration":"/-- $H[X⁻¹, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_inv_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X⁻¹, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_div_comm","declaration":"/-- $$H[X / Y] = H[Y / X]$$ -/\ntheorem ProbabilityTheory.entropy_div_comm {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) : H[X / Y ; μ] = H[Y / X ; μ]"} +{"name":"ProbabilityTheory.entropy_neg_right","declaration":"/-- $H[X, -Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_neg_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, -Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_add_right'","declaration":"/-- $H[X, Y + X] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y + X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_mul_left'","declaration":"/-- $H[X * Y, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X * Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.max_entropy_le_entropy_div","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X / Y]$$. -/\ntheorem ProbabilityTheory.max_entropy_le_entropy_div {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X / Y ; μ]"} +{"name":"ProbabilityTheory.condEntropy_div_left","declaration":"/-- $$H[Y / X | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_div_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y / X | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.entropy_add_left'","declaration":"/-- $H[X + Y, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X + Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_inv_right","declaration":"/-- $H[X, Y⁻¹] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_inv_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y⁻¹⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_right","declaration":"/-- $H[X, X - Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X - Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X - Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X - Y ; μ]"} +{"name":"ProbabilityTheory.entropy_div_right'","declaration":"/-- $H[X, Y / X] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y / X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_inv","declaration":"/-- If $X$ is $G$-valued, then $H[X⁻¹]=H[X]$. -/\ntheorem ProbabilityTheory.entropy_inv {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) : H[X⁻¹ ; μ] = H[X ; μ]"} +{"name":"ProbabilityTheory.entropy_mul_right'","declaration":"/-- $H[X, Y * X] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y * X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_left'","declaration":"/-- $H[X - Y, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X - Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"} +{"name":"ProbabilityTheory.condEntropy_div_right","declaration":"/-- $$H[X / Y | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_div_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X / Y | Y ; μ] = H[X | Y ; μ]"} +{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X * Y ; μ]"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Basic.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..78fcc68d231f163e684caca361ed4eb69ff6e3e1 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Basic.jsonl @@ -0,0 +1,39 @@ +{"name":"ProbabilityTheory.kernel.entropy_const","declaration":"theorem ProbabilityTheory.kernel.entropy_const {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (ν : MeasureTheory.Measure S) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.const T ν , μ] = (↑↑μ Set.univ).toReal * Hm[ν]"} +{"name":"ProbabilityTheory.kernel.entropy_compProd","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η (MeasureTheory.Measure.compProd μ κ)) : Hk[ProbabilityTheory.kernel.compProd κ η , μ] = Hk[κ , μ] + Hk[η , MeasureTheory.Measure.compProd μ κ]"} +{"name":"ProbabilityTheory.kernel.entropy_prodMkRight","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.prodMkRight S η , MeasureTheory.Measure.compProd μ κ] = Hk[η , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_le","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_le {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] ≤ Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.aefiniteKernelSupport_condDistrib","declaration":"theorem ProbabilityTheory.kernel.aefiniteKernelSupport_condDistrib {Ω : Type u_1} {S : Type u_2} {T : Type u_3} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.condDistrib X Y μ) (MeasureTheory.Measure.map Y μ)"} +{"name":"ProbabilityTheory.kernel.entropy_nonneg","declaration":"theorem ProbabilityTheory.kernel.entropy_nonneg {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) : 0 ≤ Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_prodMkLeft_unit","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkLeft_unit {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (κ : ↥(ProbabilityTheory.kernel T S)) {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] : Hk[ProbabilityTheory.kernel.prodMkLeft Unit κ , MeasureTheory.Measure.map (Prod.mk ()) μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.chain_rule'","declaration":"theorem ProbabilityTheory.kernel.chain_rule' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] =\n Hk[ProbabilityTheory.kernel.snd κ , μ] +\n Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.swapRight κ) ,\n MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.snd κ)]"} +{"name":"ProbabilityTheory.kernel.finiteSupport_of_compProd'","declaration":"/-- Composing a finitely supported measure with a finitely supported kernel gives a finitely supported kernel. -/\ntheorem ProbabilityTheory.kernel.finiteSupport_of_compProd' {S : Type u_2} {T : Type u_3} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.compProd μ κ)"} +{"name":"ProbabilityTheory.kernel.entropy_comap","declaration":"theorem ProbabilityTheory.kernel.entropy_comap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) (f : T' → T) (hf : MeasurableEmbedding f) (hf_range : Set.range f =ᶠ[MeasureTheory.Measure.ae μ] Set.univ) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.comap f μ)] (hfμ : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.comap f μ)) : Hk[ProbabilityTheory.kernel.comap κ f ⋯ , MeasureTheory.Measure.comap f μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.«termHk[_,_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.kernel.«termHk[_,_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.kernel.entropy_comap_swap","declaration":"theorem ProbabilityTheory.kernel.entropy_comap_swap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] [Nonempty T'] (κ : ↥(ProbabilityTheory.kernel (T' × T) S)) {μ : MeasureTheory.Measure (T' × T)} [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] : Hk[ProbabilityTheory.kernel.comap κ Prod.swap ⋯ , MeasureTheory.Measure.comap Prod.swap μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.chain_rule","declaration":"theorem ProbabilityTheory.kernel.chain_rule {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] [hU : Nonempty U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] =\n Hk[ProbabilityTheory.kernel.fst κ , μ] +\n Hk[ProbabilityTheory.kernel.condKernel κ , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ)]"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_const","declaration":"/-- Constant kernels with finite support, have finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_const {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (ν : MeasureTheory.Measure S) [ProbabilityTheory.FiniteSupport ν] : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.const T ν)"} +{"name":"ProbabilityTheory.kernel.entropy_map_of_injective","declaration":"theorem ProbabilityTheory.kernel.entropy_map_of_injective {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) {f : S → U} (hf : Function.Injective f) (hmes : Measurable f) : Hk[ProbabilityTheory.kernel.map κ f hmes , μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_prod","declaration":"theorem ProbabilityTheory.kernel.entropy_prod {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : Hk[ProbabilityTheory.kernel.prod κ η , μ] = Hk[κ , μ] + Hk[η , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_map_le","declaration":"/-- Data-processing inequality for the kernel entropy. -/\ntheorem ProbabilityTheory.kernel.entropy_map_le {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] (f : S → U) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.map κ f ⋯ , μ] ≤ Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_deterministic","declaration":"theorem ProbabilityTheory.kernel.entropy_deterministic {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (f : T → S) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.deterministic f ⋯ , μ] = 0"} +{"name":"ProbabilityTheory.kernel.entropy_fst_le","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_le {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] ≤ Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_zero_measure","declaration":"theorem ProbabilityTheory.kernel.entropy_zero_measure {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) : Hk[κ , 0] = 0"} +{"name":"ProbabilityTheory.kernel.entropy_swapRight","declaration":"theorem ProbabilityTheory.kernel.entropy_swapRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.swapRight κ , μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.prodMKLeft_unit_equiv","declaration":"/-- Measurable equivalence with the product with the one-point space `Unit`.-/\ndef ProbabilityTheory.kernel.prodMKLeft_unit_equiv (T : Type u_5) [MeasurableSpace T] : Unit × T ≃ᵐ T"} +{"name":"ProbabilityTheory.kernel.entropy_prodMkLeft","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkLeft {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] {η : ↥(ProbabilityTheory.kernel T U)} {ν : MeasureTheory.Measure S} [MeasureTheory.IsProbabilityMeasure ν] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : Hk[ProbabilityTheory.kernel.prodMkLeft S η , MeasureTheory.Measure.prod ν μ] = Hk[η , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_of_map_eq_of_map","declaration":"theorem ProbabilityTheory.kernel.entropy_of_map_eq_of_map {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] (f : S → U) (g : U → S) (h1 : η = ProbabilityTheory.kernel.map κ f ⋯) (h2 : κ = ProbabilityTheory.kernel.map η g ⋯) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : Hk[κ , μ] = Hk[η , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_zero_kernel","declaration":"theorem ProbabilityTheory.kernel.entropy_zero_kernel {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (μ : MeasureTheory.Measure T) : Hk[0 , μ] = 0"} +{"name":"ProbabilityTheory.kernel.entropy_le_log_card","declaration":"theorem ProbabilityTheory.kernel.entropy_le_log_card {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) [Fintype S] [MeasureTheory.IsProbabilityMeasure μ] : Hk[κ , μ] ≤ Real.log ↑(Fintype.card S)"} +{"name":"ProbabilityTheory.kernel.FiniteSupport.comap_equiv","declaration":"theorem ProbabilityTheory.kernel.FiniteSupport.comap_equiv {T : Type u_3} [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] {μ : MeasureTheory.Measure T} (f : T' ≃ᵐ T) [ProbabilityTheory.FiniteSupport μ] : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.comap (⇑f) μ)"} +{"name":"ProbabilityTheory.kernel.entropy","declaration":"/-- Entropy of a kernel with respect to a measure. -/\ndef ProbabilityTheory.kernel.entropy {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) : ℝ"} +{"name":"ProbabilityTheory.kernel.«termHk[_,_]»","declaration":"/-- Entropy of a kernel with respect to a measure. -/\ndef ProbabilityTheory.kernel.«termHk[_,_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.kernel.entropy_compProd_aux","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd_aux {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : Hk[ProbabilityTheory.kernel.compProd κ η , μ] =\n Hk[κ , μ] + ∫ (x : T), (fun t => Hk[ProbabilityTheory.kernel.comap η (Prod.mk t) ⋯ , κ t]) x ∂μ"} +{"name":"ProbabilityTheory.kernel.entropy_compProd'","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : Hk[ProbabilityTheory.kernel.compProd κ η , μ] = Hk[κ , μ] + Hk[η , MeasureTheory.Measure.compProd μ κ]"} +{"name":"ProbabilityTheory.kernel.finiteSupport_of_compProd","declaration":"theorem ProbabilityTheory.kernel.finiteSupport_of_compProd {S : Type u_2} {T : Type u_3} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.compProd μ κ)"} +{"name":"ProbabilityTheory.kernel.entropy_prodMkRight'","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkRight' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] {η : ↥(ProbabilityTheory.kernel T U)} {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] {ν : MeasureTheory.Measure S} [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : Hk[ProbabilityTheory.kernel.prodMkRight S η , MeasureTheory.Measure.prod μ ν] = Hk[η , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_comap_equiv","declaration":"theorem ProbabilityTheory.kernel.entropy_comap_equiv {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] (κ : ↥(ProbabilityTheory.kernel T S)) {μ : MeasureTheory.Measure T} (f : T' ≃ᵐ T) [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] : Hk[ProbabilityTheory.kernel.comap κ ⇑f ⋯ , MeasureTheory.Measure.comap (⇑f) μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_congr","declaration":"theorem ProbabilityTheory.kernel.entropy_congr {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] {μ : MeasureTheory.Measure T} {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T S)} (h : ⇑κ =ᶠ[MeasureTheory.Measure.ae μ] ⇑η) : Hk[κ , μ] = Hk[η , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_map_swap","declaration":"theorem ProbabilityTheory.kernel.entropy_map_swap {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ Prod.swap ⋯ , μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_compProd_deterministic","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd_deterministic {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsFiniteMeasure μ] (f : T × S → U) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.deterministic f ⋯) , μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_compProd_deterministic_of_injective","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_compProd_deterministic_of_injective {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) {f : T × S → U} (hf : ∀ (t : T), Function.Injective fun x => f (t, x)) (hmes : Measurable f) : Hk[ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.deterministic f hmes)) ,\n μ] =\n Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_eq_integral_sum","declaration":"theorem ProbabilityTheory.kernel.entropy_eq_integral_sum {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) : Hk[κ , μ] = ∫ (x : T), (fun y => ∑' (x : S), Real.negMulLog (↑↑(κ y) {x}).toReal) x ∂μ"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Group.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Group.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3dd66ea7abe70d9d7d34a73c56fdfcb5f42d7c0c --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Group.jsonl @@ -0,0 +1,30 @@ +{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_div","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_div_comm","declaration":"theorem ProbabilityTheory.kernel.entropy_div_comm {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ] =\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 / p.1) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_div","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"} +{"name":"measureEntropy_inv","declaration":"theorem measureEntropy_inv {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (μ : MeasureTheory.Measure G) : Hm[MeasureTheory.Measure.map (fun x => x⁻¹) μ] = Hm[μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_div_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_div_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 / p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_add_sum","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_add_sum {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 + p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_sub","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"} +{"name":"measureEntropy_sub_comm","declaration":"theorem measureEntropy_sub_comm {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (μ : MeasureTheory.Measure (G × G)) : Hm[MeasureTheory.Measure.map (fun p => p.1 - p.2) μ] = Hm[MeasureTheory.Measure.map (fun p => p.2 - p.1) μ]"} +{"name":"ProbabilityTheory.kernel.entropy_inv","declaration":"theorem ProbabilityTheory.kernel.entropy_inv {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun x => x⁻¹) ⋯ , μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_sub","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"} +{"name":"measureEntropy_neg","declaration":"theorem measureEntropy_neg {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (μ : MeasureTheory.Measure G) : Hm[MeasureTheory.Measure.map (fun x => -x) μ] = Hm[μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_sub_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_sub_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 - p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul'","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_neg","declaration":"theorem ProbabilityTheory.kernel.entropy_neg {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun x => -x) ⋯ , μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_sub","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_mul_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_mul_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 * p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add'","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul'","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add'","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_sub_comm","declaration":"theorem ProbabilityTheory.kernel.entropy_sub_comm {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ] =\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 - p.1) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add'","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"} +{"name":"measureEntropy_div_comm","declaration":"theorem measureEntropy_div_comm {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (μ : MeasureTheory.Measure (G × G)) : Hm[MeasureTheory.Measure.map (fun p => p.1 / p.2) μ] = Hm[MeasureTheory.Measure.map (fun p => p.2 / p.1) μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul'","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_div","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.MutualInfo.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.MutualInfo.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..96b05fa43618b383338e2124d9900b871db45a31 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.MutualInfo.jsonl @@ -0,0 +1,27 @@ +{"name":"ProbabilityTheory.kernel.compProd_assoc","declaration":"theorem ProbabilityTheory.kernel.compProd_assoc {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSpace V] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S × U) V)) [ProbabilityTheory.IsMarkovKernel η] : ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.compProd ξ κ) η)\n ⇑MeasurableEquiv.prodAssoc ⋯ =\n ProbabilityTheory.kernel.compProd ξ\n (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.comap η ⇑MeasurableEquiv.prodAssoc ⋯))"} +{"name":"ProbabilityTheory.kernel.entropy_triple_add_entropy_le'","declaration":"/-- The submodularity inequality:\n$$ H[X,Y,Z] + H[X] \\leq H[X,Z] + H[X,Y].$$ -/\ntheorem ProbabilityTheory.kernel.entropy_triple_add_entropy_le' {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {κ : ↥(ProbabilityTheory.kernel T (S × U × V))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] + Hk[ProbabilityTheory.kernel.fst κ , μ] ≤\n Hk[ProbabilityTheory.kernel.deleteMiddle κ , μ] + Hk[ProbabilityTheory.kernel.deleteRight κ , μ]"} +{"name":"ProbabilityTheory.kernel.Measure.compProd_compProd''","declaration":"theorem ProbabilityTheory.kernel.Measure.compProd_compProd'' {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] : MeasureTheory.Measure.compProd (MeasureTheory.Measure.compProd μ ξ) κ =\n MeasureTheory.Measure.comap (⇑MeasurableEquiv.prodAssoc)\n (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ))"} +{"name":"ProbabilityTheory.kernel.entropy_condKernel_compProd_triple","declaration":"theorem ProbabilityTheory.kernel.entropy_condKernel_compProd_triple {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S × U) V)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] : Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.compProd ξ κ) η) ,\n MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ)] =\n Hk[η , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ)]"} +{"name":"ProbabilityTheory.kernel.«termIk[_,_]»","declaration":"/-- Mutual information of a kernel into a product space with respect to a measure. -/\ndef ProbabilityTheory.kernel.«termIk[_,_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.kernel.mutualInfo_nonneg","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_nonneg {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : 0 ≤ Ik[κ , μ]"} +{"name":"ProbabilityTheory.kernel.mutualInfo_eq_fst_sub","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_eq_fst_sub {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Ik[κ , μ] =\n Hk[ProbabilityTheory.kernel.fst κ , μ] -\n Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.swapRight κ) ,\n MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.snd κ)]"} +{"name":"ProbabilityTheory.kernel.entropy_reverse","declaration":"theorem ProbabilityTheory.kernel.entropy_reverse {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {κ : ↥(ProbabilityTheory.kernel T (S × U × V))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.reverse κ , μ] = Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.mutualInfo_nonneg'","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_nonneg' {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : 0 ≤ Ik[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_compProd_triple_add_entropy_le","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd_triple_add_entropy_le {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {ξ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel ξ] {κ : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S × U) V)} [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ (MeasureTheory.Measure.compProd μ ξ)) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η\n (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ))) (hξ : ProbabilityTheory.kernel.AEFiniteKernelSupport ξ μ) : Hk[ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.compProd ξ κ) η , μ] + Hk[ξ , μ] ≤\n Hk[ProbabilityTheory.kernel.compProd ξ\n (ProbabilityTheory.kernel.snd\n (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.comap η ⇑MeasurableEquiv.prodAssoc ⋯))) ,\n μ] +\n Hk[ProbabilityTheory.kernel.compProd ξ κ , μ]"} +{"name":"ProbabilityTheory.kernel.mutualInfo_prod","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_prod {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : Ik[ProbabilityTheory.kernel.prod κ η , μ] = 0"} +{"name":"ProbabilityTheory.kernel.mutualInfo_swapRight","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_swapRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Ik[ProbabilityTheory.kernel.swapRight κ , μ] = Ik[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_condKernel_le_entropy_snd","declaration":"theorem ProbabilityTheory.kernel.entropy_condKernel_le_entropy_snd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.condKernel κ , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ)] ≤\n Hk[ProbabilityTheory.kernel.snd κ , μ]"} +{"name":"ProbabilityTheory.kernel.mutualInfo_zero_measure","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_zero_measure {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) : Ik[κ , 0] = 0"} +{"name":"ProbabilityTheory.kernel.mutualInfo_def","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_def {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Ik[κ , μ] = Hk[ProbabilityTheory.kernel.fst κ , μ] + Hk[ProbabilityTheory.kernel.snd κ , μ] - Hk[κ , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_of_injective","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_of_injective {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Nonempty T] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {V : Type u_5} [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (f : S × U → V) (hfi : ∀ (x : S), Function.Injective fun y => f (x, y)) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ f ⋯ , μ]"} +{"name":"ProbabilityTheory.kernel.Measure.compProd_compProd'","declaration":"theorem ProbabilityTheory.kernel.Measure.compProd_compProd' {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] : MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ) =\n MeasureTheory.Measure.comap (⇑MeasurableEquiv.prodAssoc.symm)\n (MeasureTheory.Measure.compProd (MeasureTheory.Measure.compProd μ ξ) κ)"} +{"name":"ProbabilityTheory.kernel.mutualInfo_congr","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_congr {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {η : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} (h : ⇑κ =ᶠ[MeasureTheory.Measure.ae μ] ⇑η) : Ik[κ , μ] = Ik[η , μ]"} +{"name":"ProbabilityTheory.kernel.entropy_triple_add_entropy_le","declaration":"/-- The submodularity inequality:\n$$ H[X,Y,Z] + H[Z] \\leq H[X,Z] + H[Y,Z].$$ -/\ntheorem ProbabilityTheory.kernel.entropy_triple_add_entropy_le {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] (κ : ↥(ProbabilityTheory.kernel T (S × U × V))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] + Hk[ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.snd κ) , μ] ≤\n Hk[ProbabilityTheory.kernel.deleteMiddle κ , μ] + Hk[ProbabilityTheory.kernel.snd κ , μ]"} +{"name":"ProbabilityTheory.kernel.«termIk[_,_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.kernel.«termIk[_,_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.kernel.mutualInfo_compProd","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η (MeasureTheory.Measure.compProd μ κ)) : Ik[ProbabilityTheory.kernel.compProd κ η , μ] =\n Hk[κ , μ] + Hk[ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.compProd κ η) , μ] -\n Hk[ProbabilityTheory.kernel.compProd κ η , μ]"} +{"name":"ProbabilityTheory.kernel.mutualInfo_eq_snd_sub","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_eq_snd_sub {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Ik[κ , μ] =\n Hk[ProbabilityTheory.kernel.snd κ , μ] -\n Hk[ProbabilityTheory.kernel.condKernel κ , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ)]"} +{"name":"ProbabilityTheory.kernel.Measure.compProd_compProd","declaration":"theorem ProbabilityTheory.kernel.Measure.compProd_compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] : MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ) =\n MeasureTheory.Measure.map (⇑MeasurableEquiv.prodAssoc)\n (MeasureTheory.Measure.compProd (MeasureTheory.Measure.compProd μ ξ) κ)"} +{"name":"ProbabilityTheory.kernel.entropy_condKernel_le_entropy_fst","declaration":"theorem ProbabilityTheory.kernel.entropy_condKernel_le_entropy_fst {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.swapRight κ) ,\n MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.snd κ)] ≤\n Hk[ProbabilityTheory.kernel.fst κ , μ]"} +{"name":"ProbabilityTheory.kernel.mutualInfo_zero_kernel","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_zero_kernel {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) : Ik[0 , μ] = 0"} +{"name":"ProbabilityTheory.kernel.mutualInfo","declaration":"/-- Mutual information of a kernel into a product space with respect to a measure. -/\ndef ProbabilityTheory.kernel.mutualInfo {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : ℝ"} +{"name":"ProbabilityTheory.kernel.entropy_submodular_compProd","declaration":"theorem ProbabilityTheory.kernel.entropy_submodular_compProd {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {ξ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel ξ] {κ : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S × U) V)} [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ (MeasureTheory.Measure.compProd μ ξ)) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η\n (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ))) (hξ : ProbabilityTheory.kernel.AEFiniteKernelSupport ξ μ) : Hk[η , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ)] ≤\n Hk[ProbabilityTheory.kernel.snd\n (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.comap η ⇑MeasurableEquiv.prodAssoc ⋯)) ,\n MeasureTheory.Measure.compProd μ ξ]"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.RuzsaDist.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.RuzsaDist.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f73fc4ea86aadd8336aee738ffb62efeed52bfcd --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.Kernel.RuzsaDist.jsonl @@ -0,0 +1,20 @@ +{"name":"ProbabilityTheory.kernel.rdist_zero_kernel_left","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_kernel_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[0 ; μ # η ; ν] = -Hk[η , ν] / 2"} +{"name":"ProbabilityTheory.kernel.rdist","declaration":"/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\ndef ProbabilityTheory.kernel.rdist {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (μ : MeasureTheory.Measure T) (ν : MeasureTheory.Measure T') : ℝ"} +{"name":"ProbabilityTheory.kernel.rdist_zero_left","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (ν' : MeasureTheory.Measure T') : dk[κ ; 0 # η ; ν'] = 0"} +{"name":"ProbabilityTheory.kernel.rdist_symm","declaration":"theorem ProbabilityTheory.kernel.rdist_symm {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] = dk[η ; ν # κ ; μ]"} +{"name":"ProbabilityTheory.kernel.rdist_dirac_zero_left","declaration":"theorem ProbabilityTheory.kernel.rdist_dirac_zero_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[ProbabilityTheory.kernel.const T (MeasureTheory.Measure.dirac 0) ; μ # η ; ν] = Hk[η , ν] / 2"} +{"name":"ProbabilityTheory.kernel.abs_sub_entropy_le_rdist","declaration":"theorem ProbabilityTheory.kernel.abs_sub_entropy_le_rdist {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η ν) : |Hk[κ , μ] - Hk[η , ν]| ≤ 2 * dk[κ ; μ # η ; ν]"} +{"name":"ProbabilityTheory.kernel.rdist_zero_kernel_right","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_kernel_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} [ProbabilityTheory.IsFiniteKernel κ] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # 0 ; ν] = -Hk[κ , μ] / 2"} +{"name":"ProbabilityTheory.kernel.rdist_zero_right","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (μ : MeasureTheory.Measure T) : dk[κ ; μ # η ; 0] = 0"} +{"name":"ProbabilityTheory.kernel.rdist_nonneg","declaration":"theorem ProbabilityTheory.kernel.rdist_nonneg {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η ν) : 0 ≤ dk[κ ; μ # η ; ν]"} +{"name":"ProbabilityTheory.kernel.ent_of_diff_le","declaration":"/-- The **improved entropic Ruzsa triangle inequality**. -/\ntheorem ProbabilityTheory.kernel.ent_of_diff_le {T : Type u_1} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ] ≤\n Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.fst κ) η)\n (fun p => p.1 - p.2) ⋯ ,\n μ] +\n Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod η (ProbabilityTheory.kernel.snd κ))\n (fun p => p.1 - p.2) ⋯ ,\n μ] -\n Hk[η , μ]"} +{"name":"ProbabilityTheory.kernel.«termDk[_;_#_;_]»","declaration":"/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\ndef ProbabilityTheory.kernel.«termDk[_;_#_;_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.kernel.«termDk[_;_#_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.kernel.«termDk[_;_#_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.kernel.ruzsa_triangle_aux","declaration":"theorem ProbabilityTheory.kernel.ruzsa_triangle_aux {T : Type u_1} {G : Type u_4} [MeasurableSpace T] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] : ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.2 - p.1.2) ⋯ =\n ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod η (ProbabilityTheory.kernel.snd κ)) (fun p => p.1 - p.2) ⋯"} +{"name":"ProbabilityTheory.kernel.rdist_eq","declaration":"theorem ProbabilityTheory.kernel.rdist_eq {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] =\n ∫ (x : T × T'),\n (fun p => Hm[MeasureTheory.Measure.map (fun x => x.1 - x.2) (MeasureTheory.Measure.prod (κ p.1) (η p.2))])\n x ∂MeasureTheory.Measure.prod μ ν -\n Hk[κ , μ] / 2 -\n Hk[η , ν] / 2"} +{"name":"ProbabilityTheory.kernel.rdist_eq'","declaration":"theorem ProbabilityTheory.kernel.rdist_eq' {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] =\n Hk[ProbabilityTheory.kernel.map\n (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T' κ)\n (ProbabilityTheory.kernel.prodMkLeft T η))\n (fun x => x.1 - x.2) ⋯ ,\n MeasureTheory.Measure.prod μ ν] -\n Hk[κ , μ] / 2 -\n Hk[η , ν] / 2"} +{"name":"ProbabilityTheory.kernel.rdist_dirac_zero_right","declaration":"theorem ProbabilityTheory.kernel.rdist_dirac_zero_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} [ProbabilityTheory.IsFiniteKernel κ] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # ProbabilityTheory.kernel.const T' (MeasureTheory.Measure.dirac 0) ; ν] = Hk[κ , μ] / 2"} +{"name":"ProbabilityTheory.kernel.rdistm","declaration":"/-- The Rusza distance between two measures, defined as `H[X - Y] - H[X]/2 - H[Y]/2` where `X`\nand `Y` are independent variables distributed according to the two measures. -/\ndef ProbabilityTheory.kernel.rdistm {G : Type u_4} [MeasurableSpace G] [AddCommGroup G] (μ : MeasureTheory.Measure G) (ν : MeasureTheory.Measure G) : ℝ"} +{"name":"ProbabilityTheory.kernel.rdist_triangle_aux1","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle_aux1 {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace T''] [MeasurableSingletonClass T''] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] : Hk[ProbabilityTheory.kernel.map\n (ProbabilityTheory.kernel.prod\n (ProbabilityTheory.kernel.prodMkRight T' (ProbabilityTheory.kernel.prodMkRight T'' κ))\n (ProbabilityTheory.kernel.prodMkLeft (T × T'') η))\n (fun p => p.1 - p.2) ⋯ ,\n MeasureTheory.Measure.prod (MeasureTheory.Measure.prod μ μ'') μ'] =\n Hk[ProbabilityTheory.kernel.map\n (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T' κ)\n (ProbabilityTheory.kernel.prodMkLeft T η))\n (fun x => x.1 - x.2) ⋯ ,\n MeasureTheory.Measure.prod μ μ']"} +{"name":"ProbabilityTheory.kernel.rdist_triangle_aux2","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle_aux2 {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace T''] [MeasurableSingletonClass T''] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] (η : ↥(ProbabilityTheory.kernel T' G)) (ξ : ↥(ProbabilityTheory.kernel T'' G)) [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.IsMarkovKernel ξ] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] : Hk[ProbabilityTheory.kernel.map\n (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkLeft (T × T'') η)\n (ProbabilityTheory.kernel.prodMkRight T' (ProbabilityTheory.kernel.prodMkLeft T ξ)))\n (fun p => p.1 - p.2) ⋯ ,\n MeasureTheory.Measure.prod (MeasureTheory.Measure.prod μ μ'') μ'] =\n Hk[ProbabilityTheory.kernel.map\n (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T'' η)\n (ProbabilityTheory.kernel.prodMkLeft T' ξ))\n (fun x => x.1 - x.2) ⋯ ,\n MeasureTheory.Measure.prod μ' μ'']"} +{"name":"ProbabilityTheory.kernel.rdist_triangle","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable T''] [MeasurableSpace T''] [MeasurableSingletonClass T''] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (ξ : ↥(ProbabilityTheory.kernel T'' G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.IsMarkovKernel ξ] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) (hξ : ProbabilityTheory.kernel.FiniteKernelSupport ξ) : dk[κ ; μ # ξ ; μ''] ≤ dk[κ ; μ # η ; μ'] + dk[η ; μ' # ξ ; μ'']"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.Measure.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.Measure.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..4898705f5b83f8116cf641401b96dfbc98b62fac --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.Measure.jsonl @@ -0,0 +1,55 @@ +{"name":"ProbabilityTheory.measureEntropy_nonneg","declaration":"theorem ProbabilityTheory.measureEntropy_nonneg {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) : 0 ≤ Hm[μ]"} +{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Finset.sum A fun s => Real.negMulLog (↑↑μ {s}).toReal"} +{"name":"ProbabilityTheory.finiteSupport_zero","declaration":"instance ProbabilityTheory.finiteSupport_zero {S : Type u_2} [MeasurableSpace S] : ProbabilityTheory.FiniteSupport 0"} +{"name":"ProbabilityTheory.«termHm[_]»","declaration":"/-- Entropy of a measure on a finite measurable space.\n\nWe normalize the measure by `(μ Set.univ)⁻¹` to extend the entropy definition to finite measures.\nWhat we really want to do is deal with `μ=0` or `IsProbabilityMeasure μ`, but we don't have\na typeclass for that (we could create one though).\nThe added complexity in the expression is not an issue because if `μ` is a probability measure,\na call to `simp` will simplify `(μ Set.univ)⁻¹ • μ` to `μ`. -/\ndef ProbabilityTheory.«termHm[_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.finiteSupport_of_prod","declaration":"instance ProbabilityTheory.finiteSupport_of_prod {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] {μ : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ] {ν : MeasureTheory.Measure T} [MeasureTheory.SigmaFinite ν] [ProbabilityTheory.FiniteSupport ν] : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.prod μ ν)"} +{"name":"ProbabilityTheory.measureEntropy","declaration":"/-- Entropy of a measure on a finite measurable space.\n\nWe normalize the measure by `(μ Set.univ)⁻¹` to extend the entropy definition to finite measures.\nWhat we really want to do is deal with `μ=0` or `IsProbabilityMeasure μ`, but we don't have\na typeclass for that (we could create one though).\nThe added complexity in the expression is not an issue because if `μ` is a probability measure,\na call to `simp` will simplify `(μ Set.univ)⁻¹ • μ` to `μ`. -/\ndef ProbabilityTheory.measureEntropy {S : Type u_2} [MeasurableSpace S] (μ : autoParam (MeasureTheory.Measure S) _auto✝) : ℝ"} +{"name":"ProbabilityTheory.integrable_of_finiteSupport","declaration":"/-- The countability hypothesis can probably be dropped here. Proof is unwieldy and can probably\nbe golfed. -/\ntheorem ProbabilityTheory.integrable_of_finiteSupport {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (μ : MeasureTheory.Measure S) [ProbabilityTheory.FiniteSupport μ] {β : Type u_5} [NormedAddCommGroup β] [MeasureTheory.IsFiniteMeasure μ] [Countable S] {f : S → β} : MeasureTheory.Integrable f μ"} +{"name":"ProbabilityTheory.measureEntropy_zero","declaration":"theorem ProbabilityTheory.measureEntropy_zero {S : Type u_2} [MeasurableSpace S] : Hm[0] = 0"} +{"name":"ProbabilityTheory.measureEntropy_prod","declaration":"/-- An ambitious goal would be to replace FiniteSupport with finite entropy. -/\ntheorem ProbabilityTheory.measureEntropy_prod {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure S} {ν : MeasureTheory.Measure T} [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] : Hm[MeasureTheory.Measure.prod μ ν] = Hm[μ] + Hm[ν]"} +{"name":"ProbabilityTheory.FiniteSupport","declaration":"/-- A measure has finite support if there exsists a finite set whose complement has zero measure. -/\nclass ProbabilityTheory.FiniteSupport {S : Type u_2} [MeasurableSpace S] (μ : autoParam (MeasureTheory.Measure S) _auto✝) : Prop"} +{"name":"ProbabilityTheory.«termIm[_]»","declaration":"/-- The mutual information between the marginals of a measure on a product space. -/\ndef ProbabilityTheory.«termIm[_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.finiteSupport_of_mul","declaration":"instance ProbabilityTheory.finiteSupport_of_mul {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ] (c : ENNReal) : ProbabilityTheory.FiniteSupport (c • μ)"} +{"name":"ProbabilityTheory.measureMutualInfo_swap","declaration":"theorem ProbabilityTheory.measureMutualInfo_swap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure (S × T)) : Im[MeasureTheory.Measure.map Prod.swap μ] = Im[μ]"} +{"name":"ProbabilityTheory.full_measure_of_finiteRange","declaration":"/-- duplicate of `FiniteRange.null_of_compl` -/\ntheorem ProbabilityTheory.full_measure_of_finiteRange {Ω : Type u_1} {S : Type u_2} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} {X : Ω → S} (hX : Measurable X) [hX' : FiniteRange X] : ↑↑(MeasureTheory.Measure.map X μ) (↑(FiniteRange.toFinset X))ᶜ = 0"} +{"name":"ProbabilityTheory.finiteSupport_of_finiteRange","declaration":"instance ProbabilityTheory.finiteSupport_of_finiteRange {Ω : Type u_1} {S : Type u_2} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} {X : Ω → S} [hX' : FiniteRange X] : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.map X μ)"} +{"name":"ProbabilityTheory.FiniteSupport.finite","declaration":"def ProbabilityTheory.FiniteSupport.finite {S : Type u_2} [MeasurableSpace S] {μ : autoParam (MeasureTheory.Measure S) _auto✝} [self : ProbabilityTheory.FiniteSupport μ] : ∃ A, ↑↑μ (↑A)ᶜ = 0"} +{"name":"ProbabilityTheory.measureEntropy_univ_smul","declaration":"theorem ProbabilityTheory.measureEntropy_univ_smul {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} : Hm[(↑↑μ Set.univ)⁻¹ • μ] = Hm[μ]"} +{"name":"ProbabilityTheory.measureMutualInfo_univ_smul","declaration":"theorem ProbabilityTheory.measureMutualInfo_univ_smul {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace U] (μ : MeasureTheory.Measure (S × U)) : Im[(↑↑μ Set.univ)⁻¹ • μ] = Im[μ]"} +{"name":"ProbabilityTheory.FiniteEntropy","declaration":"/-- TODO: replace FiniteSupport hypotheses in these files with FiniteEntropy hypotheses. -/\ndef ProbabilityTheory.FiniteEntropy {S : Type u_2} [MeasurableSpace S] (μ : autoParam (MeasureTheory.Measure S) _auto✝) : Prop"} +{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = ∑' (s : S), Real.negMulLog (↑↑μ {s}).toReal"} +{"name":"ProbabilityTheory.measureMutualInfo_eq_zero_iff","declaration":"theorem ProbabilityTheory.measureMutualInfo_eq_zero_iff {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure (S × U)} [ProbabilityTheory.FiniteSupport μ] [MeasureTheory.IsProbabilityMeasure μ] : Im[μ] = 0 ↔\n ∀ (p : S × U),\n μ.real {p} = (MeasureTheory.Measure.map Prod.fst μ).real {p.1} * (MeasureTheory.Measure.map Prod.snd μ).real {p.2}"} +{"name":"ProbabilityTheory.measureEntropy_le_log_card_of_mem","declaration":"theorem ProbabilityTheory.measureEntropy_le_log_card_of_mem {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {A : Finset S} (μ : MeasureTheory.Measure S) (hμA : ↑↑μ (↑A)ᶜ = 0) : Hm[μ] ≤ Real.log ↑(Nat.card { x // x ∈ A })"} +{"name":"ProbabilityTheory.measureMutualInfo_prod","declaration":"theorem ProbabilityTheory.measureMutualInfo_prod {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure S} {ν : MeasureTheory.Measure T} [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] : Im[MeasureTheory.Measure.prod μ ν] = 0"} +{"name":"MeasureTheory.Measure.support","declaration":"def MeasureTheory.Measure.support {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [hμ : ProbabilityTheory.FiniteSupport μ] : Finset S"} +{"name":"ProbabilityTheory.finiteSupport_of_dirac","declaration":"instance ProbabilityTheory.finiteSupport_of_dirac {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (x : S) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.dirac x)"} +{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite'","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite' {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Finset.sum A fun s => Real.negMulLog (μ.real {s})"} +{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq_aux","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq_aux {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (μ : MeasureTheory.Measure S) [Fintype S] [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), ↑↑μ {s} = ↑(↑(Fintype.card S))⁻¹"} +{"name":"ProbabilityTheory.FiniteSupport.mk","declaration":"ctor ProbabilityTheory.FiniteSupport.mk {S : Type u_2} [MeasurableSpace S] {μ : autoParam (MeasureTheory.Measure S) _auto✝} (finite : ∃ A, ↑↑μ (↑A)ᶜ = 0) : ProbabilityTheory.FiniteSupport μ"} +{"name":"ProbabilityTheory.measureEntropy_le_card_aux","declaration":"theorem ProbabilityTheory.measureEntropy_le_card_aux {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure S} [MeasureTheory.IsProbabilityMeasure μ] (A : Finset S) (hμ : ↑↑μ (↑A)ᶜ = 0) : Hm[μ] ≤ Real.log ↑A.card"} +{"name":"ProbabilityTheory.measureEntropy_of_not_isFiniteMeasure","declaration":"theorem ProbabilityTheory.measureEntropy_of_not_isFiniteMeasure {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} (h : ¬MeasureTheory.IsFiniteMeasure μ) : Hm[μ] = 0"} +{"name":"ProbabilityTheory.Measure.ext_iff_measureReal_singleton_finiteSupport","declaration":"theorem ProbabilityTheory.Measure.ext_iff_measureReal_singleton_finiteSupport {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ1] [ProbabilityTheory.FiniteSupport μ2] [MeasureTheory.IsFiniteMeasure μ1] [MeasureTheory.IsFiniteMeasure μ2] : μ1 = μ2 ↔ ∀ (x : S), μ1.real {x} = μ2.real {x}"} +{"name":"ProbabilityTheory.finiteSupport_of_fintype","declaration":"instance ProbabilityTheory.finiteSupport_of_fintype {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [Fintype S] : ProbabilityTheory.FiniteSupport μ"} +{"name":"ProbabilityTheory.measureMutualInfo_def","declaration":"theorem ProbabilityTheory.measureMutualInfo_def {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (μ : MeasureTheory.Measure (S × T)) : Im[μ] = Hm[MeasureTheory.Measure.map Prod.fst μ] + Hm[MeasureTheory.Measure.map Prod.snd μ] - Hm[μ]"} +{"name":"ProbabilityTheory.measureMutualInfo","declaration":"/-- The mutual information between the marginals of a measure on a product space. -/\ndef ProbabilityTheory.measureMutualInfo {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (μ : autoParam (MeasureTheory.Measure (S × T)) _auto✝) : ℝ"} +{"name":"ProbabilityTheory.measureEntropy_def","declaration":"theorem ProbabilityTheory.measureEntropy_def {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) : Hm[μ] = ∑' (s : S), Real.negMulLog (↑↑((↑↑μ Set.univ)⁻¹ • μ) {s}).toReal"} +{"name":"ProbabilityTheory.measure_compl_support","declaration":"theorem ProbabilityTheory.measure_compl_support {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [hμ : ProbabilityTheory.FiniteSupport μ] : ↑↑μ (↑(MeasureTheory.Measure.support μ))ᶜ = 0"} +{"name":"ProbabilityTheory.measureEntropy_le_log_card","declaration":"theorem ProbabilityTheory.measureEntropy_le_log_card {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype S] (μ : MeasureTheory.Measure S) : Hm[μ] ≤ Real.log ↑(Fintype.card S)"} +{"name":"ProbabilityTheory.Measure.ext_iff_singleton_finiteSupport","declaration":"/-- This generalizes Measure.ext_iff_singleton in MeasureReal -/\ntheorem ProbabilityTheory.Measure.ext_iff_singleton_finiteSupport {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ1] [ProbabilityTheory.FiniteSupport μ2] : μ1 = μ2 ↔ ∀ (x : S), ↑↑μ1 {x} = ↑↑μ2 {x}"} +{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq_aux","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq_aux {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype S] (μ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), μ.real {s} = (↑(Fintype.card S))⁻¹"} +{"name":"ProbabilityTheory.measureMutualInfo_of_not_isFiniteMeasure","declaration":"theorem ProbabilityTheory.measureMutualInfo_of_not_isFiniteMeasure {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace U] {μ : MeasureTheory.Measure (S × U)} (h : ¬MeasureTheory.IsFiniteMeasure μ) : Im[μ] = 0"} +{"name":"ProbabilityTheory.measureEntropy_def_finite'","declaration":"theorem ProbabilityTheory.measureEntropy_def_finite' {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) : Hm[μ] = Finset.sum A fun s => Real.negMulLog (((μ.real Set.univ)⁻¹ • μ.real) {s})"} +{"name":"ProbabilityTheory.measureEntropy_map_of_injective","declaration":"theorem ProbabilityTheory.measureEntropy_map_of_injective {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure S) (f : S → T) (hf_m : Measurable f) (hf : Function.Injective f) : Hm[MeasureTheory.Measure.map f μ] = Hm[μ]"} +{"name":"ProbabilityTheory.measureEntropy_comap_equiv","declaration":"theorem ProbabilityTheory.measureEntropy_comap_equiv {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure S) (f : T ≃ᵐ S) : Hm[MeasureTheory.Measure.comap (⇑f) μ] = Hm[μ]"} +{"name":"ProbabilityTheory.measureEntropy_comap","declaration":"theorem ProbabilityTheory.measureEntropy_comap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure S) (f : T → S) (hf : MeasurableEmbedding f) (hf_range : Set.range f =ᶠ[MeasureTheory.Measure.ae μ] Set.univ) : Hm[MeasureTheory.Measure.comap f μ] = Hm[μ]"} +{"name":"ProbabilityTheory.measureEntropy_dirac","declaration":"theorem ProbabilityTheory.measureEntropy_dirac {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (x : S) : Hm[MeasureTheory.Measure.dirac x] = 0"} +{"name":"ProbabilityTheory.measureMutualInfo_zero_measure","declaration":"theorem ProbabilityTheory.measureMutualInfo_zero_measure {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] : Im[0] = 0"} +{"name":"ProbabilityTheory.measureMutualInfo_nonneg_aux","declaration":"/-- An ambitious goal would be to replace FiniteSupport with finite entropy. Proof is long and slow; needs to be optimized -/\ntheorem ProbabilityTheory.measureMutualInfo_nonneg_aux {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure (S × U)} [ProbabilityTheory.FiniteSupport μ] [MeasureTheory.IsProbabilityMeasure μ] : 0 ≤ Im[μ] ∧\n (Im[μ] = 0 ↔\n ∀ (p : S × U),\n μ.real {p} =\n (MeasureTheory.Measure.map Prod.fst μ).real {p.1} * (MeasureTheory.Measure.map Prod.snd μ).real {p.2})"} +{"name":"ProbabilityTheory.measureEntropy_def'","declaration":"theorem ProbabilityTheory.measureEntropy_def' {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) : Hm[μ] = ∑' (s : S), Real.negMulLog (((μ.real Set.univ)⁻¹ • μ.real) {s})"} +{"name":"ProbabilityTheory.measureEntropy_def_finite","declaration":"theorem ProbabilityTheory.measureEntropy_def_finite {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) : Hm[μ] = Finset.sum A fun s => Real.negMulLog (↑↑((↑↑μ Set.univ)⁻¹ • μ) {s}).toReal"} +{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure'","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure' {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = ∑' (s : S), Real.negMulLog (μ.real {s})"} +{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [MeasurableSingletonClass S] [Fintype S] [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), μ.real {s} = μ.real Set.univ / ↑(Fintype.card S)"} +{"name":"ProbabilityTheory.integral_congr_finiteSupport","declaration":"theorem ProbabilityTheory.integral_congr_finiteSupport {Ω : Type u_1} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {G : Type u_5} [MeasurableSingletonClass Ω] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] {f : Ω → G} {g : Ω → G} [ProbabilityTheory.FiniteSupport μ] [MeasureTheory.IsFiniteMeasure μ] (hfg : ∀ (x : Ω), ↑↑μ {x} ≠ 0 → f x = g x) : ∫ (x : Ω), f x ∂μ = ∫ (x : Ω), g x ∂μ"} +{"name":"ProbabilityTheory.measureMutualInfo_nonneg","declaration":"theorem ProbabilityTheory.measureMutualInfo_nonneg {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure (S × U)} [ProbabilityTheory.FiniteSupport μ] : 0 ≤ Im[μ]"} +{"name":"ProbabilityTheory.finiteSupport_of_comp","declaration":"theorem ProbabilityTheory.finiteSupport_of_comp {Ω : Type u_1} {S : Type u_2} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} [ProbabilityTheory.FiniteSupport μ] {X : Ω → S} (hX : Measurable X) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.map X μ)"} +{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [MeasurableSingletonClass S] [Fintype S] [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), ↑↑μ {s} = ↑↑μ Set.univ / ↑(Fintype.card S)"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.RuzsaDist.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.RuzsaDist.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7270f4bef58fa8476cd04ff84879435ee07ea448 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.RuzsaDist.jsonl @@ -0,0 +1,75 @@ +{"name":"ent_of_diff_le","declaration":"/-- The **improved entropic Ruzsa triangle inequality**. -/\ntheorem ent_of_diff_le {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] (X : Ω → G) (Y : Ω → G) (Z : Ω → G) (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun (⟨X, Y⟩) Z μ) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[X - Y ; μ] ≤ H[X - Z ; μ] + H[Z - Y ; μ] - H[Z ; μ]"} +{"name":"condRuzsaDist'_eq_sum","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\ntheorem condRuzsaDist'_eq_sum {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange W] : d[X ; μ # Y | W ; μ'] =\n Finset.sum (FiniteRange.toFinset W) fun w =>\n (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"} +{"name":"«termD[_|_#_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_|_#_|_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"condRuzsaDist'_of_copy","declaration":"theorem condRuzsaDist'_of_copy {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {Ω''' : Type u_4} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [mΩ''' : MeasurableSpace Ω'''] {μ''' : MeasureTheory.Measure Ω'''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → G) {Y : Ω' → G} (hY : Measurable Y) {W : Ω' → T} (hW : Measurable W) (X' : Ω'' → G) {Y' : Ω''' → G} (hY' : Measurable Y') {W' : Ω''' → T} (hW' : Measurable W') [MeasureTheory.IsFiniteMeasure μ'] [MeasureTheory.IsFiniteMeasure μ'''] (h1 : ProbabilityTheory.IdentDistrib X X' μ μ'') (h2 : ProbabilityTheory.IdentDistrib (⟨Y, W⟩) (⟨Y', W'⟩) μ' μ''') [FiniteRange W] [FiniteRange W'] : d[X ; μ # Y | W ; μ'] = d[X' ; μ'' # Y' | W' ; μ''']"} +{"name":"condRuzsaDist_def","declaration":"theorem condRuzsaDist_def {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] : d[X | Z ; μ # Y | W ; μ'] =\n dk[ProbabilityTheory.condDistrib X Z μ ; MeasureTheory.Measure.map Z μ # ProbabilityTheory.condDistrib Y W μ' ;\n MeasureTheory.Measure.map W μ']"} +{"name":"continuous_rdist_restrict_probabilityMeasure","declaration":"/-- Ruzsa distance depends continuously on the measure. -/\ntheorem continuous_rdist_restrict_probabilityMeasure {G : Type u_5} [hG : MeasurableSpace G] [AddCommGroup G] [Countable G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] : Continuous fun μ => d[id ; ↑μ.1 # id ; ↑μ.2]"} +{"name":"«termD[_;_#_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_;_#_|_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"condRuzsaDist_le'_prod","declaration":"theorem condRuzsaDist_le'_prod {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {Z : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange W] [FiniteRange Z] : d[X ; μ # Y | ⟨W, Z⟩ ; μ'] ≤ d[X ; μ # Y | Z ; μ'] + I[Y : W|Z;μ'] / 2"} +{"name":"rdist_add_const'","declaration":"/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\ntheorem rdist_add_const' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (c : G) (c' : G) (hX : Measurable X) (hY : Measurable Y) : d[X + fun x => c ; μ # Y + fun x => c' ; μ'] = d[X ; μ # Y ; μ']"} +{"name":"condRuzsaDist_symm","declaration":"/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\ntheorem condRuzsaDist_symm {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [Nonempty S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} (hZ : Measurable Z) (hW : Measurable W) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [FiniteRange Z] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ'] = d[Y | W ; μ' # X | Z ; μ]"} +{"name":"entropy_sub_entropy_eq_condRuzsaDist_add","declaration":"theorem entropy_sub_entropy_eq_condRuzsaDist_add {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : H[Y + Z ; μ'] - H[Y ; μ'] = d[Y ; μ' # Z ; μ'] + H[Z ; μ'] / 2 - H[Y ; μ'] / 2"} +{"name":"condRuzsaDist_diff_le''","declaration":"theorem condRuzsaDist_diff_le'' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y | Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y + Z ; μ'] - H[Z ; μ']) / 2"} +{"name":"condRuszaDist_zero_left","declaration":"theorem condRuszaDist_zero_left {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] : d[X | Z ; 0 # Y | W ; μ'] = 0"} +{"name":"ent_of_proj_le","declaration":"/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\ntheorem ent_of_proj_le {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {UH : Ω' → G} [FiniteRange X] [FiniteRange UH] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hU : Measurable UH) {H : AddSubgroup G} [Finite ↥H] (hunif : ProbabilityTheory.IsUniform (↑H) UH μ') : H[⇑(QuotientAddGroup.mk' H) ∘ X ; μ] ≤ 2 * d[X ; μ # UH ; μ']"} +{"name":"ProbabilityTheory.IndepFun.rdist_eq","declaration":"/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\ntheorem ProbabilityTheory.IndepFun.rdist_eq {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {X : Ω → G} [MeasureTheory.IsFiniteMeasure μ] {Y : Ω → G} (h : ProbabilityTheory.IndepFun X Y μ) (hX : Measurable X) (hY : Measurable Y) : d[X ; μ # Y ; μ] = H[X - Y ; μ] - H[X ; μ] / 2 - H[Y ; μ] / 2"} +{"name":"kaimanovich_vershik'","declaration":"/-- A version of the **Kaimanovich-Vershik inequality** with some variables negated. -/\ntheorem kaimanovich_vershik' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableAdd₂ G] [Countable G] {X : Ω → G} {Y : Ω → G} {Z : Ω → G} (h : ProbabilityTheory.iIndepFun (fun x => hG) ![X, Y, Z] μ) (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : H[X - (Y + Z) ; μ] - H[X - Y ; μ] ≤ H[Y + Z ; μ] - H[Y ; μ]"} +{"name":"comparison_of_ruzsa_distances","declaration":"theorem comparison_of_ruzsa_distances {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y + Z ; μ'] - H[Y ; μ']) / 2 ∧\n (ElementaryAddCommGroup G 2 → H[Y + Z ; μ'] - H[Y ; μ'] = d[Y ; μ' # Z ; μ'] + H[Z ; μ'] / 2 - H[Y ; μ'] / 2)"} +{"name":"«termD[_|_;_#_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_|_;_#_|_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ent_bsg","declaration":"/-- The **entropic Balog-Szemerédi-Gowers inequality**. Let $A, B$ be $G$-valued random variables on\n$\\Omega$, and set $Z := A+B$. Then\n$$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]. $$\nTODO: remove the hypothesis of `Fintype G` from here and from `condIndep_copies'` -/\ntheorem ent_bsg {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] {A : Ω → G} {B : Ω → G} (hA : Measurable A) (hB : Measurable B) [Fintype G] : ∫ (x : G),\n (fun z => d[A ; ProbabilityTheory.cond μ ((A + B) ⁻¹' {z}) # B ; ProbabilityTheory.cond μ ((A + B) ⁻¹' {z})])\n x ∂MeasureTheory.Measure.map (A + B) μ ≤\n 3 * I[A : B ; μ] + 2 * H[A + B ; μ] - H[A ; μ] - H[B ; μ]"} +{"name":"rdist_zero_eq_half_ent","declaration":"/-- $$ d[X ; 0] = H[X] / 2 $$ -/\ntheorem rdist_zero_eq_half_ent {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] {X : Ω → G} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : d[X ; μ # fun x => 0 ; μ'] = H[X ; μ] / 2"} +{"name":"condRuzsaDist_of_const","declaration":"/-- Conditioning by a constant does not affect Ruzsa distance. -/\ntheorem condRuzsaDist_of_const {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} (hX : Measurable X) (Y : Ω' → G) (W : Ω' → T) (c : S) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [FiniteRange W] : d[X | fun x => c ; μ # Y | W ; μ'] = d[X ; μ # Y | W ; μ']"} +{"name":"«termD[_;_#_;_]»","declaration":"/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\ndef «termD[_;_#_;_]» : Lean.ParserDescr"} +{"name":"condRuzsaDist'_of_inj_map'","declaration":"theorem condRuzsaDist'_of_inj_map' {Ω : Type u_1} {Ω'' : Type u_3} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [elem : ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ''] {A : Ω'' → G} {B : Ω → G} {C : Ω → G} (hA : Measurable A) (hB : Measurable B) (hC : Measurable C) [FiniteRange A] [FiniteRange B] [FiniteRange C] : d[A ; μ'' # B | B + C ; μ] = d[A ; μ'' # C | B + C ; μ]"} +{"name":"rdist_add_const","declaration":"/-- Adding a constant to a random variable does not change the Rusza distance. -/\ntheorem rdist_add_const {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] {c : G} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) : d[X ; μ # Y + fun x => c ; μ'] = d[X ; μ # Y ; μ']"} +{"name":"diff_ent_le_rdist","declaration":"/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\ntheorem diff_ent_le_rdist {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) : |H[X ; μ] - H[Y ; μ']| ≤ 2 * d[X ; μ # Y ; μ']"} +{"name":"«termD[_#_|_]»","declaration":"/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\ndef «termD[_#_|_]» : Lean.ParserDescr"} +{"name":"condRuzsaDist_eq_sum","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\ntheorem condRuzsaDist_eq_sum {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange Z] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ'] =\n Finset.sum (FiniteRange.toFinset Z) fun z =>\n Finset.sum (FiniteRange.toFinset W) fun w =>\n (↑↑μ (Z ⁻¹' {z})).toReal * (↑↑μ' (W ⁻¹' {w})).toReal *\n d[X ; ProbabilityTheory.cond μ (Z ⁻¹' {z}) # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"} +{"name":"condRuzsaDist'_of_indep","declaration":"/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\ntheorem condRuzsaDist'_of_indep {Ω : Type u_1} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω → G} {W : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun X (⟨Y, W⟩) μ) [FiniteRange W] : d[X ; μ # Y | W ; μ] = H[X - Y | W ; μ] - H[X ; μ] / 2 - H[Y | W ; μ] / 2"} +{"name":"condRuzsaDist'_eq_integral","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\ntheorem condRuzsaDist'_eq_integral {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → G) {Y : Ω' → G} {W : Ω' → T} (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange W] : d[X ; μ # Y | W ; μ'] =\n ∫ (x : T), (fun w => d[X ; μ # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]) x ∂MeasureTheory.Measure.map W μ'"} +{"name":"rdist_nonneg","declaration":"/-- $$ d[X ; Y] \\geq 0$$ -/\ntheorem rdist_nonneg {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) : 0 ≤ d[X ; μ # Y ; μ']"} +{"name":"ProbabilityTheory.IdentDistrib.rdist_eq","declaration":"/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\ntheorem ProbabilityTheory.IdentDistrib.rdist_eq {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {Ω''' : Type u_4} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [mΩ''' : MeasurableSpace Ω'''] {μ''' : MeasureTheory.Measure Ω'''} [hG : MeasurableSpace G] [AddCommGroup G] {X : Ω → G} {Y : Ω' → G} {X' : Ω'' → G} {Y' : Ω''' → G} (hX : ProbabilityTheory.IdentDistrib X X' μ μ'') (hY : ProbabilityTheory.IdentDistrib Y Y' μ' μ''') : d[X ; μ # Y ; μ'] = d[X' ; μ'' # Y' ; μ''']"} +{"name":"condRuzsaDist_comp_right","declaration":"theorem condRuzsaDist_comp_right {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_7} [Fintype T] [Fintype T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasureTheory.IsFiniteMeasure μ'] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (e : T → T') (hY : Measurable Y) (hW : Measurable W) (he : Measurable e) (h'e : Function.Injective e) : d[X ; μ # Y | e ∘ W ; μ'] = d[X ; μ # Y | W ; μ']"} +{"name":"diff_ent_le_rdist'","declaration":"/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\ntheorem diff_ent_le_rdist' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) [FiniteRange Y] : H[X - Y ; μ] - H[X ; μ] ≤ 2 * d[X ; μ # Y ; μ]"} +{"name":"«termD[_|_#_|_]»","declaration":"/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\ndef «termD[_|_#_|_]» : Lean.ParserDescr"} +{"name":"condRuzsaDist_le'","declaration":"theorem condRuzsaDist_le' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) [FiniteRange X] [FiniteRange Y] [FiniteRange W] : d[X ; μ # Y | W ; μ'] ≤ d[X ; μ # Y ; μ'] + I[Y : W ; μ'] / 2"} +{"name":"condRuszaDist_zero_right","declaration":"theorem condRuszaDist_zero_right {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : d[X | Z ; μ # Y | W ; 0] = 0"} +{"name":"condRuzsaDist_eq_sum'","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\ntheorem condRuzsaDist_eq_sum' {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [Fintype S] [Fintype T] : d[X | Z ; μ # Y | W ; μ'] =\n Finset.sum Finset.univ fun z =>\n Finset.sum Finset.univ fun w =>\n (↑↑μ (Z ⁻¹' {z})).toReal * (↑↑μ' (W ⁻¹' {w})).toReal *\n d[X ; ProbabilityTheory.cond μ (Z ⁻¹' {z}) # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"} +{"name":"continuous_entropy_restrict_probabilityMeasure","declaration":"theorem continuous_entropy_restrict_probabilityMeasure {G : Type u_5} [hG : MeasurableSpace G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] : Continuous fun μ => H[id ; ↑μ]"} +{"name":"condRuzsaDist'_of_inj_map","declaration":"theorem condRuzsaDist'_of_inj_map {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [elem : ElementaryAddCommGroup G 2] {X : Ω → G} {B : Ω → G} {C : Ω → G} (hX : Measurable X) (hB : Measurable B) (hC : Measurable C) (h_indep : ProbabilityTheory.IndepFun X (⟨B, C⟩) μ) [FiniteRange X] [FiniteRange B] [FiniteRange C] : d[X ; μ # B | B + C ; μ] = d[X ; μ # C | B + C ; μ]"} +{"name":"rdist_def","declaration":"/-- Explicit formula for the Ruzsa distance. -/\ntheorem rdist_def {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [AddCommGroup G] (X : Ω → G) (Y : Ω' → G) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') : d[X ; μ # Y ; μ'] =\n H[fun x => x.1 - x.2 ; MeasureTheory.Measure.prod (MeasureTheory.Measure.map X μ) (MeasureTheory.Measure.map Y μ')] -\n H[X ; μ] / 2 -\n H[Y ; μ'] / 2"} +{"name":"condRuzsaDist'","declaration":"/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\ndef condRuzsaDist' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace T] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsFiniteMeasure μ'] : ℝ"} +{"name":"condRuzsaDist_diff_ofsum_le","declaration":"theorem condRuzsaDist_diff_ofsum_le {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} {Z' : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (hZ' : Measurable Z') (h : ProbabilityTheory.iIndepFun (fun x => hG) ![Y, Z, Z'] μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange Z'] : d[X ; μ # Y + Z | Y + Z + Z' ; μ'] - d[X ; μ # Y ; μ'] ≤\n (H[Y + Z + Z' ; μ'] + H[Y + Z ; μ'] - H[Y ; μ'] - H[Z' ; μ']) / 2"} +{"name":"rdist_symm","declaration":"/-- $$ d[X ; Y] = d[Y ; X]$$ -/\ntheorem rdist_symm {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'] : d[X ; μ # Y ; μ'] = d[Y ; μ' # X ; μ]"} +{"name":"rdist_of_inj","declaration":"/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\ntheorem rdist_of_inj {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] {X : Ω → G} {Y : Ω' → G} {H : Type u_7} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H] [MeasurableSub₂ H] [MeasurableAdd₂ H] [Countable H] (hX : Measurable X) (hY : Measurable Y) (φ : G →+ H) (hφ : Function.Injective ⇑φ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : d[⇑φ ∘ X ; μ # ⇑φ ∘ Y ; μ'] = d[X ; μ # Y ; μ']"} +{"name":"condRuzsaDist'_eq_sum'","declaration":"/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\ntheorem condRuzsaDist'_eq_sum' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [Fintype T] : d[X ; μ # Y | W ; μ'] =\n Finset.sum Finset.univ fun w => (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"} +{"name":"condRuzsaDist_diff_le'","declaration":"theorem condRuzsaDist_diff_le' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ d[Y ; μ' # Z ; μ'] / 2 + H[Z ; μ'] / 4 - H[Y ; μ'] / 4"} +{"name":"condRuzsaDist_of_indep","declaration":"/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\ntheorem condRuzsaDist_of_indep {S : Type u_7} {Ω : Type u_1} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω → G} {W : Ω → T} (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun (⟨X, Z⟩) (⟨Y, W⟩) μ) [FiniteRange Z] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ] = H[X - Y | ⟨Z, W⟩ ; μ] - H[X | Z ; μ] / 2 - H[Y | W ; μ] / 2"} +{"name":"kaimanovich_vershik","declaration":"/-- The **Kaimanovich-Vershik inequality**. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$ -/\ntheorem kaimanovich_vershik {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableAdd₂ G] [Countable G] {X : Ω → G} {Y : Ω → G} {Z : Ω → G} (h : ProbabilityTheory.iIndepFun (fun x => hG) ![X, Y, Z] μ) (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : H[X + Y + Z ; μ] - H[X + Y ; μ] ≤ H[Y + Z ; μ] - H[Y ; μ]"} +{"name":"condRuzsaDist_diff_le","declaration":"/-- Let $X, Y, Z$ be random variables taking values in some abelian group, and with $Y, Z$\nindependent. Then we have\n$$d[X ; Y + Z] -d[X ; Y] \\leq \\tfrac{1}{2} (H[Y+ Z] - H[Y])$$\n$$= \\tfrac{1}{2} d[Y ; Z] + \\tfrac{1}{4} H[Z] - \\tfrac{1}{4} H[Y]$$\nand\n$$d[X ; Y|Y+ Z] - d[X ; Y] \\leq \\tfrac{1}{2} \\bigl(H[Y+ Z] - H[Z]\\bigr)$$\n$$= \\tfrac{1}{2} d[Y ; Z] + \\tfrac{1}{4} H[Y] - \\tfrac{1}{4} H[Z]$$\n-/\ntheorem condRuzsaDist_diff_le {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y + Z ; μ'] - H[Y ; μ']) / 2"} +{"name":"«termD[_#_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_#_|_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"condRuzsaDist'_zero_right","declaration":"theorem condRuzsaDist'_zero_right {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace T] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) : d[X ; μ # Y | W ; 0] = 0"} +{"name":"continuous_measureEntropy_probabilityMeasure","declaration":"/-- Entropy depends continuously on the measure. -/\ntheorem continuous_measureEntropy_probabilityMeasure {Ω : Type u_7} [Fintype Ω] [TopologicalSpace Ω] [DiscreteTopology Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] : Continuous fun μ => Hm[↑μ]"} +{"name":"«termD[_|_;_#_|_;_]»","declaration":"/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\ndef «termD[_|_;_#_|_;_]» : Lean.ParserDescr"} +{"name":"condRuzsaDist","declaration":"/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\ndef condRuzsaDist {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) [MeasureTheory.IsFiniteMeasure μ] (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsFiniteMeasure μ'] : ℝ"} +{"name":"condRuzsaDist'_prod_eq_sum'","declaration":"/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\ntheorem condRuzsaDist'_prod_eq_sum' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {W' : Ω' → T} (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W) [MeasureTheory.IsFiniteMeasure μ'] [Fintype T] : d[X ; μ # Y | ⟨W', W⟩ ; μ'] =\n Finset.sum Finset.univ fun w => (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y | W' ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"} +{"name":"rdist_eq_rdistm","declaration":"/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\ntheorem rdist_eq_rdistm {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [AddCommGroup G] {X : Ω → G} {Y : Ω' → G} : d[X ; μ # Y ; μ'] = ProbabilityTheory.kernel.rdistm (MeasureTheory.Measure.map X μ) (MeasureTheory.Measure.map Y μ')"} +{"name":"rdist_triangle","declaration":"/-- The **entropic Ruzsa triangle inequality** -/\ntheorem rdist_triangle {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} {Z : Ω'' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [hμ : MeasureTheory.IsProbabilityMeasure μ] [hμ' : MeasureTheory.IsProbabilityMeasure μ'] [hμ'' : MeasureTheory.IsProbabilityMeasure μ''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : d[X ; μ # Z ; μ''] ≤ d[X ; μ # Y ; μ'] + d[Y ; μ' # Z ; μ'']"} +{"name":"«termD[_;_#_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_;_#_;_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"condRuzsaDist_nonneg","declaration":"theorem condRuzsaDist_nonneg {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [Nonempty S] [MeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} (hX : Measurable X) [FiniteRange X] {Z : Ω → S} (hZ : Measurable Z) [FiniteRange Z] {Y : Ω' → G} (hY : Measurable Y) [FiniteRange Y] {W : Ω' → T} (hW : Measurable W) [FiniteRange W] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : 0 ≤ d[X | Z ; μ # Y | W ; μ']"} +{"name":"condRuszaDist_prod_eq_of_indepFun","declaration":"theorem condRuszaDist_prod_eq_of_indepFun {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {W' : Ω' → T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) (hW' : Measurable W') (h : ProbabilityTheory.IndepFun (⟨Y, W⟩) W' μ') [MeasureTheory.IsProbabilityMeasure μ'] [Fintype T] : d[X ; μ # Y | ⟨W, W'⟩ ; μ'] = d[X ; μ # Y | W ; μ']"} +{"name":"«termD[_#_]»","declaration":"/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\ndef «termD[_#_]» : Lean.ParserDescr"} +{"name":"condRuzsaDist'_prod_eq_sum","declaration":"theorem condRuzsaDist'_prod_eq_sum {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {W' : Ω' → T} (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W) [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange W] [FiniteRange W'] : d[X ; μ # Y | ⟨W', W⟩ ; μ'] =\n Finset.sum (FiniteRange.toFinset W) fun w =>\n (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y | W' ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"} +{"name":"condRuzsaDist'_def","declaration":"/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\ntheorem condRuzsaDist'_def {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace T] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] : d[X ; μ # Y | W ; μ'] =\n dk[ProbabilityTheory.kernel.const Unit (MeasureTheory.Measure.map X μ) ; MeasureTheory.Measure.dirac () #\n ProbabilityTheory.condDistrib Y W μ' ; MeasureTheory.Measure.map W μ']"} +{"name":"rdist_le_avg_ent","declaration":"/-- $d[X;Y] ≤ H[X]/2 + H[Y]/2$. -/\ntheorem rdist_le_avg_ent {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : d[X ; μ # Y ; μ'] ≤ (H[X ; μ] + H[Y ; μ']) / 2"} +{"name":"condRuzsaDist_diff_le'''","declaration":"theorem condRuzsaDist_diff_le''' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y | Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ d[Y ; μ' # Z ; μ'] / 2 + H[Y ; μ'] / 4 - H[Z ; μ'] / 4"} +{"name":"condRuzsaDist_le","declaration":"/-- Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian\ngroup. Then $$d[X | Z ; Y | W] \\leq d[X ; Y] + \\tfrac{1}{2} I[X : Z] + \\tfrac{1}{2} I[Y : W]$$ -/\ntheorem condRuzsaDist_le {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [Nonempty S] (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) [FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ'] ≤ d[X ; μ # Y ; μ'] + I[X : Z ; μ] / 2 + I[Y : W ; μ'] / 2"} +{"name":"continuous_rdist_restrict_probabilityMeasure₁'","declaration":"/-- Ruzsa distance depends continuously on the second measure. -/\ntheorem continuous_rdist_restrict_probabilityMeasure₁' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [AddCommGroup G] [Countable G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] (X : Ω → G) (P : autoParam (MeasureTheory.Measure Ω) _auto✝) [MeasureTheory.IsProbabilityMeasure P] (X_mble : Measurable X) : Continuous fun μ => d[X ; P # id ; ↑μ]"} +{"name":"«termD[_;_#_|_;_]»","declaration":"/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\ndef «termD[_;_#_|_;_]» : Lean.ParserDescr"} +{"name":"condRuzsaDist_of_inj_map","declaration":"theorem condRuzsaDist_of_inj_map {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {G' : Type u_7} [Countable G'] [AddCommGroup G'] [MeasurableSpace G'] [MeasurableSingletonClass G'] [MeasureTheory.IsProbabilityMeasure μ] (Y : Fin 4 → Ω → G) (h_indep : ProbabilityTheory.IndepFun (⟨Y 0, Y 2⟩) (⟨Y 1, Y 3⟩) μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) (π : G × G →+ G') (hπ : ∀ (h : G), Function.Injective fun g => π (g, h)) [FiniteRange (Y 2)] [FiniteRange (Y 3)] : d[⇑π ∘ ⟨Y 0, Y 2⟩ | Y 2 ; μ # ⇑π ∘ ⟨Y 1, Y 3⟩ | Y 3 ; μ] = d[Y 0 | Y 2 ; μ # Y 1 | Y 3 ; μ]"} +{"name":"condRuzsaDist_of_copy","declaration":"/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\ntheorem condRuzsaDist_of_copy {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {Ω''' : Type u_4} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [mΩ''' : MeasurableSpace Ω'''] {μ''' : MeasureTheory.Measure Ω'''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} (hX : Measurable X) {Z : Ω → S} (hZ : Measurable Z) {Y : Ω' → G} (hY : Measurable Y) {W : Ω' → T} (hW : Measurable W) {X' : Ω'' → G} (hX' : Measurable X') {Z' : Ω'' → S} (hZ' : Measurable Z') {Y' : Ω''' → G} (hY' : Measurable Y') {W' : Ω''' → T} (hW' : Measurable W') [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'] [MeasureTheory.IsFiniteMeasure μ''] [MeasureTheory.IsFiniteMeasure μ'''] (h1 : ProbabilityTheory.IdentDistrib (⟨X, Z⟩) (⟨X', Z'⟩) μ μ'') (h2 : ProbabilityTheory.IdentDistrib (⟨Y, W⟩) (⟨Y', W'⟩) μ' μ''') [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] : d[X | Z ; μ # Y | W ; μ'] = d[X' | Z' ; μ'' # Y' | W' ; μ''']"} +{"name":"rdist_eq_rdist_id_map","declaration":"/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\ntheorem rdist_eq_rdist_id_map {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [AddCommGroup G] {X : Ω → G} {Y : Ω' → G} : d[X ; μ # Y ; μ'] = d[id ; MeasureTheory.Measure.map X μ # id ; MeasureTheory.Measure.map Y μ']"} +{"name":"continuous_rdist_restrict_probabilityMeasure₁","declaration":"theorem continuous_rdist_restrict_probabilityMeasure₁ {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [AddCommGroup G] [Countable G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] (X : Ω → G) (P : autoParam (MeasureTheory.Measure Ω) _auto✝) [MeasureTheory.IsProbabilityMeasure P] (X_mble : Measurable X) : Continuous fun μ => d[id ; MeasureTheory.Measure.map X P # id ; ↑μ]"} +{"name":"«termD[_#_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_#_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"rdist","declaration":"/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\ndef rdist {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [AddCommGroup G] (X : Ω → G) (Y : Ω' → G) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) : ℝ"} +{"name":"diff_ent_le_rdist''","declaration":"/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\ntheorem diff_ent_le_rdist'' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) [FiniteRange Y] : H[X - Y ; μ] - H[Y ; μ] ≤ 2 * d[X ; μ # Y ; μ]"} diff --git a/PFR-declarations/PFR.ForMathlib.Entropy.RuzsaSetDist.jsonl b/PFR-declarations/PFR.ForMathlib.Entropy.RuzsaSetDist.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..02227e041661e6b5a0a95b82079cf6a525e56d1e --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Entropy.RuzsaSetDist.jsonl @@ -0,0 +1,18 @@ +{"name":"MeasureTheory.Measure.discreteUniform_apply","declaration":"/-- The usual formula for the discrete uniform measure applied to an arbitrary set. -/\ntheorem MeasureTheory.Measure.discreteUniform_apply {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] (A : Set S) : ↑↑(MeasureTheory.Measure.discreteUniform H) A = ↑(Nat.card ↑(A ∩ H)) / ↑(Nat.card ↑H)"} +{"name":"ProbabilityTheory.rdist_set_eq_rdist","declaration":"/-- Relating Ruzsa distance between sets to Ruzsa distance between random variables -/\ntheorem ProbabilityTheory.rdist_set_eq_rdist {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] {A : Set G} {B : Set G} [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] {Ω : Type u_2} {Ω' : Type u_3} [mΩ : MeasureTheory.MeasureSpace Ω] [mΩ' : MeasureTheory.MeasureSpace Ω'] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} (hμ : MeasureTheory.IsProbabilityMeasure μ) (hμ' : MeasureTheory.IsProbabilityMeasure μ') {UA : Ω → G} {UB : Ω' → G} (hUA : ProbabilityTheory.IsUniform A UA μ) (hUB : ProbabilityTheory.IsUniform B UB μ') (hUA_mes : Measurable UA) (hUB_mes : Measurable UB) : dᵤ[A # B] = d[UA ; μ # UB ; μ']"} +{"name":"ProbabilityTheory.rdist_set_nonneg","declaration":"/-- Ruzsa distance between sets is nonnegative. -/\ntheorem ProbabilityTheory.rdist_set_nonneg {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] : 0 ≤ dᵤ[A # B]"} +{"name":"MeasureTheory.Measure.map_discreteUniform_of_inj","declaration":"/-- injective map of discrete uniform is discrete uniform -/\ntheorem MeasureTheory.Measure.map_discreteUniform_of_inj {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] {T : Type u_2} [MeasurableSpace T] [MeasurableSingletonClass T] {f : S → T} (hmes : Measurable f) (hf : Function.Injective f) : MeasureTheory.Measure.map f (MeasureTheory.Measure.discreteUniform H) = MeasureTheory.Measure.discreteUniform (f '' H)"} +{"name":"ProbabilityTheory.rdist_set_le","declaration":"/-- Ruzsa distance between sets is controlled by the doubling constant. -/\ntheorem ProbabilityTheory.rdist_set_le {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] (hA : Set.Nonempty A) (hB : Set.Nonempty B) : dᵤ[A # B] ≤ Real.log ↑(Nat.card ↑(A - B)) - Real.log ↑(Nat.card ↑A) / 2 - Real.log ↑(Nat.card ↑B) / 2"} +{"name":"MeasureTheory.Measure.isUniform_iff_uniform_dist","declaration":"/-- A random variable is uniform iff its distribution is. -/\ntheorem MeasureTheory.Measure.isUniform_iff_uniform_dist {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] [Nonempty ↑H] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [Countable S] (hμ : MeasureTheory.IsProbabilityMeasure μ) {U : Ω → S} (hU : Measurable U) : ProbabilityTheory.IsUniform H U μ ↔ MeasureTheory.Measure.map U μ = MeasureTheory.Measure.discreteUniform H"} +{"name":"ProbabilityTheory.«termDᵤ[_#_]»","declaration":"def ProbabilityTheory.«termDᵤ[_#_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.«termDᵤ[_#_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termDᵤ[_#_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"ProbabilityTheory.rdist_set_add_const","declaration":"/-- Ruzsa distance between sets is translation invariant. -/\ntheorem ProbabilityTheory.rdist_set_add_const {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] (c : G) (c' : G) : dᵤ[A + {c} # B + {c'}] = dᵤ[A # B]"} +{"name":"ProbabilityTheory.rdist_set_triangle","declaration":"/-- Ruzsa distance between sets obeys the triangle inequality. -/\ntheorem ProbabilityTheory.rdist_set_triangle {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) (C : Set G) [Finite ↑A] [Finite ↑B] [Finite ↑C] [Nonempty ↑A] [Nonempty ↑B] [Nonempty ↑C] : dᵤ[A # C] ≤ dᵤ[A # B] + dᵤ[B # C]"} +{"name":"MeasureTheory.Measure.discreteUniform_apply'","declaration":"/-- Variant of `discreteUniform_apply' using real-valued measures. -/\ntheorem MeasureTheory.Measure.discreteUniform_apply' {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] (A : Set S) : (MeasureTheory.Measure.discreteUniform H).real A = ↑(Nat.card ↑(A ∩ H)) / ↑(Nat.card ↑H)"} +{"name":"MeasureTheory.Measure.discreteUniform.isProbabilityMeasure","declaration":"instance MeasureTheory.Measure.discreteUniform.isProbabilityMeasure {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] [Nonempty ↑H] : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.discreteUniform H)"} +{"name":"MeasureTheory.Measure.discreteUniform","declaration":"/-- In practice one would also impose the conditions `MeasurableSingletonClass S`, `Finite H` and `Nonempty H` before attempting to use this definition. -/\ndef MeasureTheory.Measure.discreteUniform {S : Type u_1} [MeasurableSpace S] (H : Set S) : MeasureTheory.Measure S"} +{"name":"ProbabilityTheory.rdist_set_of_inj","declaration":"/-- Ruzsa distance between sets is preserved by injective homomorphisms. -/\ntheorem ProbabilityTheory.rdist_set_of_inj {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] {H : Type u_2} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H] [Countable H] {φ : G →+ H} (hφ : Function.Injective ⇑φ) : dᵤ[⇑φ '' A # ⇑φ '' B] = dᵤ[A # B]"} +{"name":"ProbabilityTheory.rdist_set_symm","declaration":"/-- Ruzsa distance between sets is symmetric. -/\ntheorem ProbabilityTheory.rdist_set_symm {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] : dᵤ[A # B] = dᵤ[B # A]"} +{"name":"MeasureTheory.Measure.discreteUniform_of_infinite","declaration":"/-- The uniform distribution on an infinite set vanishes by definition. -/\ntheorem MeasureTheory.Measure.discreteUniform_of_infinite {S : Type u_1} [MeasurableSpace S] (H : Set S) (h : Set.Infinite H) : MeasureTheory.Measure.discreteUniform H = 0"} +{"name":"ProbabilityTheory.entropy_of_discreteUniform","declaration":"/-- The entropy of a uniform measure is the log of the cardinality of its support. -/\ntheorem ProbabilityTheory.entropy_of_discreteUniform {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] [Nonempty ↑H] : Hm[MeasureTheory.Measure.discreteUniform H] = Real.log ↑(Nat.card ↑H)"} +{"name":"ProbabilityTheory.rdist_set","declaration":"/-- The Ruzsa distance between two subsets `A`, `B` of a group `G` is defined to be the Ruzsa distance between their uniform probability distributions. Is only intended for use when `A`, `B` are finite and non-empty. -/\ndef ProbabilityTheory.rdist_set {G : Type u_1} [MeasurableSpace G] [AddCommGroup G] (A : Set G) (B : Set G) : ℝ"} diff --git a/PFR-declarations/PFR.ForMathlib.FiniteMeasureComponent.jsonl b/PFR-declarations/PFR.ForMathlib.FiniteMeasureComponent.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d3e141d5ad0bdbc41b1d17c7244990cbdc897f3f --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.FiniteMeasureComponent.jsonl @@ -0,0 +1,2 @@ +{"name":"continuous_finiteMeasure_apply_of_isClopen","declaration":"/-- The measure of any connected component depends continuously on the `FiniteMeasure`. -/\ntheorem continuous_finiteMeasure_apply_of_isClopen {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} (s_clopen : IsClopen s) : Continuous fun μ => (↑μ).real s"} +{"name":"continuous_probabilityMeasure_apply_of_isClopen","declaration":"/-- The probability of any connected component depends continuously on the `ProbabilityMeasure`. -/\ntheorem continuous_probabilityMeasure_apply_of_isClopen {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} (s_clopen : IsClopen s) : Continuous fun μ => (↑μ).real s"} diff --git a/PFR-declarations/PFR.ForMathlib.FiniteMeasureProd.jsonl b/PFR-declarations/PFR.ForMathlib.FiniteMeasureProd.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..457a8d88af69855bffcd80831b830766cb335502 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.FiniteMeasureProd.jsonl @@ -0,0 +1,25 @@ +{"name":"MeasureTheory.ProbabilityMeasure.map_prod_map","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_prod_map {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) : MeasureTheory.ProbabilityMeasure.prod (MeasureTheory.ProbabilityMeasure.map μ ⋯)\n (MeasureTheory.ProbabilityMeasure.map ν ⋯) =\n MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯"} +{"name":"MeasureTheory.FiniteMeasure.prod","declaration":"/-- The binary product of finite measures. -/\ndef MeasureTheory.FiniteMeasure.prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure (α × β)"} +{"name":"MeasureTheory.ProbabilityMeasure.prod_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set α) (t : Set β) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) (s ×ˢ t) =\n (fun s => (↑↑↑μ s).toNNReal) s * (fun s => (↑↑↑ν s).toNNReal) t"} +{"name":"MeasureTheory.ProbabilityMeasure.prod_swap","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_swap {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ =\n MeasureTheory.ProbabilityMeasure.prod ν μ"} +{"name":"MeasureTheory.FiniteMeasure.measure_ae_null_of_prod_null","declaration":"theorem MeasureTheory.FiniteMeasure.measure_ae_null_of_prod_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {s : Set (α × β)} (h : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = 0) : (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} +{"name":"MeasureTheory.ProbabilityMeasure.toMeasure_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.toMeasure_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : ↑(MeasureTheory.ProbabilityMeasure.prod μ ν) = MeasureTheory.Measure.prod ↑μ ↑ν"} +{"name":"MeasureTheory.FiniteMeasure.mass_prod","declaration":"theorem MeasureTheory.FiniteMeasure.mass_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.mass (MeasureTheory.FiniteMeasure.prod μ ν) =\n MeasureTheory.FiniteMeasure.mass μ * MeasureTheory.FiniteMeasure.mass ν"} +{"name":"MeasureTheory.FiniteMeasure.prod_swap","declaration":"theorem MeasureTheory.FiniteMeasure.prod_swap {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.swap = MeasureTheory.FiniteMeasure.prod ν μ"} +{"name":"MeasureTheory.FiniteMeasure.map_fst_prod","declaration":"theorem MeasureTheory.FiniteMeasure.map_fst_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.fst =\n (fun s => (↑↑↑ν s).toNNReal) Set.univ • μ"} +{"name":"MeasureTheory.ProbabilityMeasure.map_fst_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_fst_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ = μ"} +{"name":"MeasureTheory.ProbabilityMeasure.prod_apply_null","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {s : Set (α × β)} (hs : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s = 0 ↔\n (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} +{"name":"MeasureTheory.FiniteMeasure.prod_apply","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = (∫⁻ (x : α), ↑↑↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal"} +{"name":"MeasureTheory.FiniteMeasure.prod_apply_symm","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply_symm {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s =\n (∫⁻ (y : β), ↑↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal"} +{"name":"MeasureTheory.FiniteMeasure.prod_prod","declaration":"theorem MeasureTheory.FiniteMeasure.prod_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set α) (t : Set β) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) (s ×ˢ t) =\n (fun s => (↑↑↑μ s).toNNReal) s * (fun s => (↑↑↑ν s).toNNReal) t"} +{"name":"MeasureTheory.FiniteMeasure.prod_apply_null","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {s : Set (α × β)} (hs : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = 0 ↔\n (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} +{"name":"MeasureTheory.ProbabilityMeasure.map_snd_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_snd_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ = ν"} +{"name":"MeasureTheory.ProbabilityMeasure.prod","declaration":"/-- The binary product of probability measures. -/\ndef MeasureTheory.ProbabilityMeasure.prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure (α × β)"} +{"name":"MeasureTheory.ProbabilityMeasure.prod_apply_symm","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply_symm {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s =\n (∫⁻ (y : β), ↑↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal"} +{"name":"MeasureTheory.FiniteMeasure.map_snd_prod","declaration":"theorem MeasureTheory.FiniteMeasure.map_snd_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.snd =\n (fun s => (↑↑↑μ s).toNNReal) Set.univ • ν"} +{"name":"MeasureTheory.ProbabilityMeasure.measure_ae_null_of_prod_null","declaration":"theorem MeasureTheory.ProbabilityMeasure.measure_ae_null_of_prod_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {s : Set (α × β)} (h : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s = 0) : (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} +{"name":"MeasureTheory.ProbabilityMeasure.prod_apply","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s =\n (∫⁻ (x : α), ↑↑↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal"} +{"name":"MeasureTheory.FiniteMeasure.map_prod_map","declaration":"theorem MeasureTheory.FiniteMeasure.map_prod_map {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) : MeasureTheory.FiniteMeasure.prod (MeasureTheory.FiniteMeasure.map μ f) (MeasureTheory.FiniteMeasure.map ν g) =\n MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) (Prod.map f g)"} +{"name":"MeasureTheory.FiniteMeasure.zero_prod","declaration":"theorem MeasureTheory.FiniteMeasure.zero_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.prod 0 ν = 0"} +{"name":"MeasureTheory.FiniteMeasure.toMeasure_prod","declaration":"theorem MeasureTheory.FiniteMeasure.toMeasure_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : ↑(MeasureTheory.FiniteMeasure.prod μ ν) = MeasureTheory.Measure.prod ↑μ ↑ν"} +{"name":"MeasureTheory.FiniteMeasure.prod_zero","declaration":"theorem MeasureTheory.FiniteMeasure.prod_zero {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) : MeasureTheory.FiniteMeasure.prod μ 0 = 0"} diff --git a/PFR-declarations/PFR.ForMathlib.FiniteRange.jsonl b/PFR-declarations/PFR.ForMathlib.FiniteRange.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..bb522a36290acba809c70474e7a5e5d19ae50f6d --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.FiniteRange.jsonl @@ -0,0 +1,24 @@ +{"name":"FiniteRange.pow","declaration":"/-- A function of finite range raised to a constant power, has finite range. -/\ninstance FiniteRange.pow {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Group G] [hX : FiniteRange X] (c : ℤ) : FiniteRange (X ^ c)"} +{"name":"FiniteRange.finite","declaration":"def FiniteRange.finite {Ω : Type u_1} {G : Type u_2} {X : Ω → G} [self : FiniteRange X] : Set.Finite (Set.range X)"} +{"name":"FiniteRange.mk","declaration":"ctor FiniteRange.mk {Ω : Type u_1} {G : Type u_2} {X : Ω → G} (finite : Set.Finite (Set.range X)) : FiniteRange X"} +{"name":"instFiniteRangeComp_1","declaration":"/-- If X has finite range, then X of any function has finite range. -/\ninstance instFiniteRangeComp_1 {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_3} (X : Ω → G) (f : Ω' → Ω) [hX : FiniteRange X] : FiniteRange (X ∘ f)"} +{"name":"FiniteRange.toFinset","declaration":"/-- The range of a finite range map, as a finset. -/\ndef FiniteRange.toFinset {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Finset G"} +{"name":"FiniteRange.mem","declaration":"theorem FiniteRange.mem {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [FiniteRange X] (ω : Ω) : X ω ∈ FiniteRange.toFinset X"} +{"name":"FiniteRange.sub","declaration":"/-- The difference of functions of finite range, has finite range.-/\ninstance FiniteRange.sub {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [AddGroup G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X - Y)"} +{"name":"instFiniteRange_1","declaration":"/-- Constants have finite range -/\ninstance instFiniteRange_1 {Ω : Type u_1} {G : Type u_2} (c : G) : FiniteRange fun x => c"} +{"name":"instFiniteRangeProdProd","declaration":"/-- If X, Y have finite range, then so does the pair ⟨X, Y⟩. -/\ninstance instFiniteRangeProdProd {Ω : Type u_1} {G : Type u_2} {H : Type u_3} (X : Ω → G) (Y : Ω → H) [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (⟨X, Y⟩)"} +{"name":"FiniteRange.null_of_compl","declaration":"theorem FiniteRange.null_of_compl {Ω : Type u_1} {G : Type u_2} [MeasurableSpace Ω] [MeasurableSpace G] [MeasurableSingletonClass G] (μ : MeasureTheory.Measure Ω) (X : Ω → G) [FiniteRange X] : ↑↑(MeasureTheory.Measure.map X μ) (↑(FiniteRange.toFinset X))ᶜ = 0"} +{"name":"FiniteRange.range","declaration":"theorem FiniteRange.range {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Set.range X = ↑(FiniteRange.toFinset X)"} +{"name":"FiniteRange.nsmul","declaration":"/-- The multiple of a function of finite range by a constant, has finite range.-/\ninstance FiniteRange.nsmul {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [AddGroup G] [hX : FiniteRange X] (c : ℤ) : FiniteRange (c • X)"} +{"name":"FiniteRange.sum","declaration":"/-- The sum of functions of finite range, has finite range.-/\ninstance FiniteRange.sum {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [AddGroup G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X + Y)"} +{"name":"FiniteRange.neg","declaration":"/-- The negation of a function of finite range, has finite range.-/\ninstance FiniteRange.neg {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [AddGroup G] [hX : FiniteRange X] : FiniteRange (-X)"} +{"name":"FiniteRange","declaration":"/-- The property of having a finite range. -/\nclass FiniteRange {Ω : Type u_1} {G : Type u_2} (X : Ω → G) : Prop"} +{"name":"FiniteRange.fintype","declaration":"/-- fintype structure on the range of a finite range map. -/\ndef FiniteRange.fintype {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Fintype ↑(Set.range X)"} +{"name":"instFiniteRangeComp","declaration":"/-- If X has finite range, then any function of X has finite range. -/\ninstance instFiniteRangeComp {Ω : Type u_1} {G : Type u_2} {H : Type u_3} (X : Ω → G) (f : G → H) [hX : FiniteRange X] : FiniteRange (f ∘ X)"} +{"name":"FiniteRange.mem_iff","declaration":"theorem FiniteRange.mem_iff {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [FiniteRange X] (x : G) : x ∈ FiniteRange.toFinset X ↔ ∃ ω, X ω = x"} +{"name":"instFiniteRange","declaration":"/-- If the codomain of X is finite, then X has finite range. -/\ninstance instFiniteRange {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Fintype G] : FiniteRange X"} +{"name":"finiteRange_of_finset","declaration":"/-- Functions ranging in a Finset have finite range -/\ntheorem finiteRange_of_finset {Ω : Type u_1} {G : Type u_2} (f : Ω → G) (A : Finset G) (h : ∀ (ω : Ω), f ω ∈ A) : FiniteRange f"} +{"name":"FiniteRange.full","declaration":"theorem FiniteRange.full {Ω : Type u_1} {G : Type u_2} [MeasurableSpace Ω] [MeasurableSpace G] [MeasurableSingletonClass G] {X : Ω → G} (hX : Measurable X) [FiniteRange X] (μ : MeasureTheory.Measure Ω) : ↑↑(MeasureTheory.Measure.map X μ) ↑(FiniteRange.toFinset X) = ↑↑μ Set.univ"} +{"name":"FiniteRange.div","declaration":"/-- The quotient of two functions with finite range, has finite range. -/\ninstance FiniteRange.div {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [Group G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X / Y)"} +{"name":"FiniteRange.prod","declaration":"/-- The product of functions of finite range, has finite range. -/\ninstance FiniteRange.prod {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [Group G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X * Y)"} +{"name":"FiniteRange.inv","declaration":"/-- The inverse of a function of finite range, has finite range.-/\ninstance FiniteRange.inv {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Group G] [hX : FiniteRange X] : FiniteRange X⁻¹"} diff --git a/PFR-declarations/PFR.ForMathlib.Graph.jsonl b/PFR-declarations/PFR.ForMathlib.Graph.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..8b9cfffd0b700e80894a12c0503d71370c7715e8 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Graph.jsonl @@ -0,0 +1,11 @@ +{"name":"Set.graph_comp","declaration":"theorem Set.graph_comp {A : Type u_3} {B : Type u_4} {C : Type u_5} {f : A → B} (g : B → C) : Set.graph (g ∘ f) = (fun p => (p.1, g p.2)) '' Set.graph f"} +{"name":"Set.image_snd_graph","declaration":"theorem Set.image_snd_graph {G : Type u_1} {G' : Type u_2} {f : G → G'} : Prod.snd '' Set.graph f = f '' Set.univ"} +{"name":"Set.graph_nonempty","declaration":"theorem Set.graph_nonempty {G : Type u_1} {G' : Type u_2} [Nonempty G] (f : G → G') : Set.Nonempty (Set.graph f)"} +{"name":"Set.graph","declaration":"def Set.graph {G : Type u_1} {G' : Type u_2} (f : G → G') : Set (G × G')"} +{"name":"Set.fst_injOn_graph","declaration":"theorem Set.fst_injOn_graph {G : Type u_1} {G' : Type u_2} (f : G → G') : Set.InjOn Prod.fst (Set.graph f)"} +{"name":"Set.mem_graph","declaration":"theorem Set.mem_graph {G : Type u_1} {G' : Type u_2} {f : G → G'} (x : G × G') : x ∈ Set.graph f ↔ f x.1 = x.2"} +{"name":"Set.image_fst_graph","declaration":"theorem Set.image_fst_graph {G : Type u_1} {G' : Type u_2} {f : G → G'} : Prod.fst '' Set.graph f = Set.univ"} +{"name":"Set.graph_def","declaration":"theorem Set.graph_def {G : Type u_1} {G' : Type u_2} (f : G → G') : Set.graph f = {x | ∃ x_1, (x_1, f x_1) = x}"} +{"name":"Set.card_graph","declaration":"theorem Set.card_graph {G : Type u_1} {G' : Type u_2} (f : G → G') : Nat.card ↑(Set.graph f) = Nat.card G"} +{"name":"Set.graph_add","declaration":"theorem Set.graph_add {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddCommGroup G'] {f : G →+ G'} {c : G × G'} : (fun x => c + x) '' Set.graph ⇑f = {x | ∃ g, (g, f g + (c.2 - f c.1)) = x}"} +{"name":"Set.equiv_filter_graph","declaration":"theorem Set.equiv_filter_graph {G : Type u_3} {G' : Type u_4} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [DecidableEq G] [DecidableEq G'] (f : G → G') : let A := Set.Finite.toFinset ⋯;\n{ x //\n x ∈\n Finset.filter\n (fun x =>\n match x with\n | (a, a') => a + a' ∈ A)\n (A ×ˢ A) } ≃\n ↑{x | f (x.1 + x.2) = f x.1 + f x.2}"} diff --git a/PFR-declarations/PFR.ForMathlib.MeasureReal.jsonl b/PFR-declarations/PFR.ForMathlib.MeasureReal.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..8247077a09c1305e95e315a06833755a5f6dc018 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.MeasureReal.jsonl @@ -0,0 +1,74 @@ +{"name":"sum_measure_preimage_singleton'","declaration":"/-- Variant of `sum_measure_preimage_singleton` using real numbers rather than extended nonnegative\nreals. -/\ntheorem sum_measure_preimage_singleton' {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {T : Type u} [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] {Y : Ω → T} (hY : Measurable Y) : (Finset.sum Finset.univ fun y => (↑↑μ (Y ⁻¹' {y})).toReal) = 1"} +{"name":"MeasureTheory.measureReal_symmDiff_le","declaration":"theorem MeasureTheory.measureReal_symmDiff_le {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t u : Set α),\n autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (symmDiff s u) ≤ μ.real (symmDiff s t) + μ.real (symmDiff t u)"} +{"name":"Finset.sum_toReal_measure_singleton","declaration":"theorem Finset.sum_toReal_measure_singleton {S : Type u_1} {s : Finset S} : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S)\n [inst : MeasureTheory.IsFiniteMeasure μ], (Finset.sum s fun x_1 => (↑↑μ {x_1}).toReal) = (↑↑μ ↑s).toReal"} +{"name":"MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal","declaration":"/-- Pigeonhole principle for measure spaces: if `s` is a `Finset` and\n`∑ i in s, μ.real (t i) > μ.real univ`, then one of the intersections `t i ∩ t j` is not empty. -/\ntheorem MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal {α : Type u_1} {ι : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, MeasurableSet (t i)) (H : μ.real Set.univ < Finset.sum s fun i => μ.real (t i)) : ∃ i ∈ s, ∃ j ∈ s, ∃ (_ : i ≠ j), Set.Nonempty (t i ∩ t j)"} +{"name":"MeasureTheory.measureReal_le_measureReal_union_left","declaration":"theorem MeasureTheory.measureReal_le_measureReal_union_left {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n autoParam (↑↑μ t ≠ ⊤) _auto✝ → μ.real s ≤ μ.real (s ∪ t)"} +{"name":"sum_toReal_measure_singleton","declaration":"theorem sum_toReal_measure_singleton {S : Type u_1} [Fintype S] : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S)\n [inst : MeasureTheory.IsFiniteMeasure μ],\n (Finset.sum Finset.univ fun x_1 => (↑↑μ {x_1}).toReal) = (↑↑μ Set.univ).toReal"} +{"name":"MeasureTheory.measure_symmDiff_eq_top","declaration":"theorem MeasureTheory.measure_symmDiff_eq_top {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}, ↑↑μ s ≠ ⊤ → ↑↑μ t = ⊤ → ↑↑μ (symmDiff s t) = ⊤"} +{"name":"MeasureTheory.measureReal_mono","declaration":"theorem MeasureTheory.measureReal_mono {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n s₁ ⊆ s₂ → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real s₁ ≤ μ.real s₂"} +{"name":"MeasureTheory.measureReal_symmDiff_eq","declaration":"theorem MeasureTheory.measureReal_symmDiff_eq {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n MeasurableSet s →\n MeasurableSet t →\n autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (symmDiff s t) = μ.real (s \\ t) + μ.real (t \\ s)"} +{"name":"MeasureTheory.measureReal_eq_measureReal_of_between_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_of_between_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},\n s₁ ⊆ s₂ →\n s₂ ⊆ s₃ →\n μ.real (s₃ \\ s₁) = 0 → autoParam (↑↑μ (s₃ \\ s₁) ≠ ⊤) _auto✝ → μ.real s₁ = μ.real s₂ ∧ μ.real s₂ = μ.real s₃"} +{"name":"MeasureTheory.measureReal_empty","declaration":"theorem MeasureTheory.measureReal_empty {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α}, μ.real ∅ = 0"} +{"name":"MeasureTheory.nonempty_inter_of_measureReal_lt_add'","declaration":"/-- If two sets `s` and `t` are included in a set `u` of finite measure,\nand `μ.real s + μ.real t > μ.real u`, then `s` intersects `t`.\nVersion assuming that `s` is measurable. -/\ntheorem MeasureTheory.nonempty_inter_of_measureReal_lt_add' {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s : Set α} {t : Set α} {u : Set α} (hs : MeasurableSet s) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ.real u < μ.real s + μ.real t) (hu : autoParam (↑↑μ u ≠ ⊤) _auto✝) : Set.Nonempty (s ∩ t)"} +{"name":"MeasureTheory.measureReal_union_null_iff","declaration":"theorem MeasureTheory.measureReal_union_null_iff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ →\n autoParam (↑↑μ s₂ ≠ ⊤) _auto✝¹ → (μ.real (s₁ ∪ s₂) = 0 ↔ μ.real s₁ = 0 ∧ μ.real s₂ = 0)"} +{"name":"MeasureTheory.measureReal_nonneg","declaration":"theorem MeasureTheory.measureReal_nonneg {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}, 0 ≤ μ.real s"} +{"name":"MeasureTheory.Measure.real","declaration":"/-- The real-valued version of a measure. Maps infinite measure sets to zero. Use as `μ.real s`. -/\ndef MeasureTheory.Measure.real {α : Type u_1} : {x : MeasurableSpace α} → MeasureTheory.Measure α → Set α → ℝ"} +{"name":"MeasureTheory.measureReal_union_congr_of_subset","declaration":"theorem MeasureTheory.measureReal_union_congr_of_subset {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ t₁ t₂ : Set α},\n s₁ ⊆ s₂ →\n μ.real s₂ ≤ μ.real s₁ →\n t₁ ⊆ t₂ →\n μ.real t₂ ≤ μ.real t₁ →\n autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → autoParam (↑↑μ t₂ ≠ ⊤) _auto✝¹ → μ.real (s₁ ∪ t₁) = μ.real (s₂ ∪ t₂)"} +{"name":"MeasureTheory.measureReal_zero","declaration":"theorem MeasureTheory.measureReal_zero {α : Type u_1} : ∀ {x : MeasurableSpace α} (s : Set α), 0.real s = 0"} +{"name":"MeasureTheory.measureReal_union_le","declaration":"theorem MeasureTheory.measureReal_union_le {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s₁ s₂ : Set α), μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂"} +{"name":"MeasureTheory.measureReal_union₀'","declaration":"theorem MeasureTheory.measureReal_union₀' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n MeasureTheory.NullMeasurableSet s μ →\n MeasureTheory.AEDisjoint μ s t →\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) = μ.real s + μ.real t"} +{"name":"MeasureTheory.measureReal_biUnion_finset","declaration":"theorem MeasureTheory.measureReal_biUnion_finset {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_3} {s : Finset ι} {f : ι → Set α},\n Set.PairwiseDisjoint (↑s) f →\n (∀ b ∈ s, MeasurableSet (f b)) →\n autoParam (∀ b ∈ s, ↑↑μ (f b) ≠ ⊤) _auto✝ → μ.real (⋃ b ∈ s, f b) = Finset.sum s fun p => μ.real (f p)"} +{"name":"MeasureTheory.IsProbabilityMeasure.measureReal_univ","declaration":"theorem MeasureTheory.IsProbabilityMeasure.measureReal_univ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsProbabilityMeasure μ],\n μ.real Set.univ = 1"} +{"name":"MeasureTheory.measureReal_inter_add_diff₀","declaration":"theorem MeasureTheory.measureReal_inter_add_diff₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n MeasureTheory.NullMeasurableSet t μ → autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real (s ∩ t) + μ.real (s \\ t) = μ.real s"} +{"name":"measureReal_preimage_snd_singleton_eq_sum","declaration":"theorem measureReal_preimage_snd_singleton_eq_sum {S : Type u_1} {T : Type u_2} [Fintype S] : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] {x_1 : MeasurableSpace T}\n [inst : MeasurableSingletonClass T] (μ : MeasureTheory.Measure (S × T)) [inst : MeasureTheory.IsFiniteMeasure μ]\n (y : T), μ.real (Prod.snd ⁻¹' {y}) = Finset.sum Finset.univ fun x_2 => μ.real {(x_2, y)}"} +{"name":"MeasureTheory.measureReal_add_measureReal_compl₀","declaration":"theorem MeasureTheory.measureReal_add_measureReal_compl₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsFiniteMeasure μ] {s : Set α},\n MeasureTheory.NullMeasurableSet s μ → μ.real s + μ.real sᶜ = μ.real Set.univ"} +{"name":"MeasureTheory.measureReal_diff","declaration":"theorem MeasureTheory.measureReal_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n s₂ ⊆ s₁ → MeasurableSet s₂ → autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → μ.real (s₁ \\ s₂) = μ.real s₁ - μ.real s₂"} +{"name":"MeasureTheory.measureReal_iUnion_fintype","declaration":"theorem MeasureTheory.measureReal_iUnion_fintype {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : Fintype β] {f : β → Set α},\n Pairwise (Disjoint on f) →\n (∀ (i : β), MeasurableSet (f i)) →\n autoParam (∀ (i : β), ↑↑μ (f i) ≠ ⊤) _auto✝ → μ.real (⋃ b, f b) = Finset.sum Finset.univ fun p => μ.real (f p)"} +{"name":"MeasureTheory.measureReal_diff_null'","declaration":"theorem MeasureTheory.measureReal_diff_null' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n μ.real (s₁ ∩ s₂) = 0 → autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → μ.real (s₁ \\ s₂) = μ.real s₁"} +{"name":"MeasureTheory.measureReal_smul_apply","declaration":"theorem MeasureTheory.measureReal_smul_apply {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (c : ENNReal), (c • μ).real s = c.toReal • μ.real s"} +{"name":"Finset.sum_measure_singleton","declaration":"theorem Finset.sum_measure_singleton {S : Type u_1} {s : Finset S} : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S),\n (Finset.sum s fun x_1 => ↑↑μ {x_1}) = ↑↑μ ↑s"} +{"name":"MeasureTheory.measureReal_union_add_inter₀","declaration":"theorem MeasureTheory.measureReal_union_add_inter₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n MeasureTheory.NullMeasurableSet t μ →\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"} +{"name":"MeasureTheory.nonempty_of_measureReal_ne_zero","declaration":"theorem MeasureTheory.nonempty_of_measureReal_ne_zero {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}, μ.real s ≠ 0 → Set.Nonempty s"} +{"name":"Mathlib.Meta.Positivity.evalMeasureReal","declaration":"/-- Extension for the `positivity` tactic: applications of `μ.real` are nonnegative. -/\ndef Mathlib.Meta.Positivity.evalMeasureReal : Mathlib.Meta.Positivity.PositivityExt"} +{"name":"MeasureTheory.measureReal_mono_null","declaration":"theorem MeasureTheory.measureReal_mono_null {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n s₁ ⊆ s₂ → μ.real s₂ = 0 → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real s₁ = 0"} +{"name":"measureReal_preimage_fst_singleton_eq_sum","declaration":"theorem measureReal_preimage_fst_singleton_eq_sum {S : Type u_1} {T : Type u_2} : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] [inst : Fintype T] {x_1 : MeasurableSpace T}\n [inst_1 : MeasurableSingletonClass T] (μ : MeasureTheory.Measure (S × T)) [inst_2 : MeasureTheory.IsFiniteMeasure μ]\n (x_2 : S), μ.real (Prod.fst ⁻¹' {x_2}) = Finset.sum Finset.univ fun y => μ.real {(x_2, y)}"} +{"name":"MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},\n s₁ ⊆ s₂ → s₂ ⊆ s₃ → μ.real (s₃ \\ s₁) = 0 → autoParam (↑↑μ (s₃ \\ s₁) ≠ ⊤) _auto✝ → μ.real s₁ = μ.real s₂"} +{"name":"MeasureTheory.sum_measureReal_preimage_singleton","declaration":"/-- If `s` is a `Finset`, then the measure of its preimage can be found as the sum of measures\nof the fibers `f ⁻¹' {y}`. -/\ntheorem MeasureTheory.sum_measureReal_preimage_singleton {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) {f : α → β},\n (∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) →\n autoParam (∀ a ∈ s, ↑↑μ (f ⁻¹' {a}) ≠ ⊤) _auto✝ → (Finset.sum s fun b => μ.real (f ⁻¹' {b})) = μ.real (f ⁻¹' ↑s)"} +{"name":"MeasureTheory.measureReal_union_null","declaration":"theorem MeasureTheory.measureReal_union_null {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n μ.real s₁ = 0 → μ.real s₂ = 0 → μ.real (s₁ ∪ s₂) = 0"} +{"name":"MeasureTheory.measureReal_union_add_inter₀'","declaration":"theorem MeasureTheory.measureReal_union_add_inter₀' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n MeasureTheory.NullMeasurableSet s μ →\n ∀ (t : Set α),\n autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"} +{"name":"MeasureTheory.measureReal_union₀","declaration":"theorem MeasureTheory.measureReal_union₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n MeasureTheory.NullMeasurableSet t μ →\n MeasureTheory.AEDisjoint μ s t →\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) = μ.real s + μ.real t"} +{"name":"MeasureTheory.measureReal_univ_pos","declaration":"theorem MeasureTheory.measureReal_univ_pos {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsFiniteMeasure μ] [inst : NeZero μ],\n 0 < μ.real Set.univ"} +{"name":"sum_measure_singleton","declaration":"theorem sum_measure_singleton {S : Type u_1} [Fintype S] : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S),\n (Finset.sum Finset.univ fun x_1 => ↑↑μ {x_1}) = ↑↑μ Set.univ"} +{"name":"MeasureTheory.measureReal_iUnion_fintype_le","declaration":"theorem MeasureTheory.measureReal_iUnion_fintype_le {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : Fintype β] (f : β → Set α),\n μ.real (⋃ b, f b) ≤ Finset.sum Finset.univ fun p => μ.real (f p)"} +{"name":"MeasureTheory.measureReal_univ_ne_zero","declaration":"theorem MeasureTheory.measureReal_univ_ne_zero {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsFiniteMeasure μ] [inst : NeZero μ],\n μ.real Set.univ ≠ 0"} +{"name":"MeasureTheory.measure_diff_eq_top","declaration":"theorem MeasureTheory.measure_diff_eq_top {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}, ↑↑μ s = ⊤ → ↑↑μ t ≠ ⊤ → ↑↑μ (s \\ t) = ⊤"} +{"name":"MeasureTheory.measureReal_congr","declaration":"/-- If two sets are equal modulo a set of measure zero, then `μ.real s = μ.real t`. -/\ntheorem MeasureTheory.measureReal_congr {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n s =ᶠ[MeasureTheory.Measure.ae μ] t → μ.real s = μ.real t"} +{"name":"MeasureTheory.measure_ne_top_of_subset","declaration":"theorem MeasureTheory.measure_ne_top_of_subset {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}, s ⊆ t → ↑↑μ t ≠ ⊤ → ↑↑μ s ≠ ⊤"} +{"name":"MeasureTheory.measureReal_union","declaration":"theorem MeasureTheory.measureReal_union {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n Disjoint s₁ s₂ →\n MeasurableSet s₂ →\n autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝¹ → μ.real (s₁ ∪ s₂) = μ.real s₁ + μ.real s₂"} +{"name":"MeasureTheory.measureReal_le_measureReal_union_right","declaration":"theorem MeasureTheory.measureReal_le_measureReal_union_right {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real t ≤ μ.real (s ∪ t)"} +{"name":"MeasureTheory.map_measureReal_apply","declaration":"theorem MeasureTheory.map_measureReal_apply {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β},\n Measurable f → ∀ {s : Set β}, MeasurableSet s → (MeasureTheory.Measure.map f μ).real s = μ.real (f ⁻¹' s)"} +{"name":"MeasureTheory.measureReal_compl","declaration":"theorem MeasureTheory.measureReal_compl {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [inst : MeasureTheory.IsFiniteMeasure μ],\n MeasurableSet s → μ.real sᶜ = μ.real Set.univ - μ.real s"} +{"name":"MeasureTheory.measureReal_diff'","declaration":"theorem MeasureTheory.measureReal_diff' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n MeasurableSet t →\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s \\ t) = μ.real (s ∪ t) - μ.real t"} +{"name":"MeasureTheory.measureReal_union_add_inter'","declaration":"theorem MeasureTheory.measureReal_union_add_inter' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n MeasurableSet s →\n ∀ (t : Set α),\n autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"} +{"name":"MeasureTheory.measureReal_union'","declaration":"theorem MeasureTheory.measureReal_union' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n Disjoint s₁ s₂ →\n MeasurableSet s₁ →\n autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝¹ → μ.real (s₁ ∪ s₂) = μ.real s₁ + μ.real s₂"} +{"name":"MeasureTheory.measureReal_def","declaration":"theorem MeasureTheory.measureReal_def {α : Type u_1} : ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : Set α), μ.real s = (↑↑μ s).toReal"} +{"name":"MeasureTheory.measureReal_eq_measureReal_of_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_of_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n s ⊆ t → μ.real (t \\ s) = 0 → autoParam (↑↑μ (t \\ s) ≠ ⊤) _auto✝ → μ.real s = μ.real t"} +{"name":"MeasureTheory.measureReal_prod_prod","declaration":"theorem MeasureTheory.measureReal_prod_prod {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β}\n [inst_1 : MeasureTheory.SigmaFinite ν] (s : Set α) (t : Set β),\n (MeasureTheory.Measure.prod μ ν).real (s ×ˢ t) = μ.real s * ν.real t"} +{"name":"MeasureTheory.Measure.ext_iff_singleton","declaration":"/-- Generalized in Measure.ext_iff_singleton_finiteSupport at Entropy.Measure -/\ntheorem MeasureTheory.Measure.ext_iff_singleton {S : Type u_3} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} : μ1 = μ2 ↔ ∀ (x : S), ↑↑μ1 {x} = ↑↑μ2 {x}"} +{"name":"MeasureTheory.measureReal_diff_null","declaration":"theorem MeasureTheory.measureReal_diff_null {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n μ.real s₂ = 0 → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real (s₁ \\ s₂) = μ.real s₁"} +{"name":"MeasureTheory.measureReal_biUnion_finset_le","declaration":"theorem MeasureTheory.measureReal_biUnion_finset_le {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) (f : β → Set α),\n μ.real (⋃ b ∈ s, f b) ≤ Finset.sum s fun p => μ.real (f p)"} +{"name":"MeasureTheory.measureReal_eq_measureReal_larger_of_between_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_larger_of_between_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},\n s₁ ⊆ s₂ → s₂ ⊆ s₃ → μ.real (s₃ \\ s₁) = 0 → autoParam (↑↑μ (s₃ \\ s₁) ≠ ⊤) _auto✝ → μ.real s₂ = μ.real s₃"} +{"name":"MeasureTheory.le_measureReal_diff","declaration":"theorem MeasureTheory.le_measureReal_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real s₁ - μ.real s₂ ≤ μ.real (s₁ \\ s₂)"} +{"name":"MeasureTheory.measureReal_biUnion_finset₀","declaration":"theorem MeasureTheory.measureReal_biUnion_finset₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_3} {s : Finset ι} {f : ι → Set α},\n Set.Pairwise (↑s) (MeasureTheory.AEDisjoint μ on f) →\n (∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ) →\n autoParam (∀ b ∈ s, ↑↑μ (f b) ≠ ⊤) _auto✝ → μ.real (⋃ b ∈ s, f b) = Finset.sum s fun p => μ.real (f p)"} +{"name":"MeasureTheory.measureReal_add_diff","declaration":"theorem MeasureTheory.measureReal_add_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n MeasurableSet s →\n ∀ (t : Set α),\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real s + μ.real (t \\ s) = μ.real (s ∪ t)"} +{"name":"MeasureTheory.Finset.sum_realMeasure_singleton","declaration":"/-- If `s` is a `Finset`, then the sums of the real measures of the singletons in the set is the\nreal measure of the set. -/\ntheorem MeasureTheory.Finset.sum_realMeasure_singleton {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasurableSingletonClass α]\n [inst : MeasureTheory.IsFiniteMeasure μ] (s : Finset α), (Finset.sum s fun b => μ.real {b}) = μ.real ↑s"} +{"name":"MeasureTheory.sum_measureReal_le_measureReal_univ","declaration":"theorem MeasureTheory.sum_measureReal_le_measureReal_univ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_3} [inst : MeasureTheory.IsFiniteMeasure μ]\n {s : Finset ι} {t : ι → Set α},\n (∀ i ∈ s, MeasurableSet (t i)) → Set.PairwiseDisjoint (↑s) t → (Finset.sum s fun i => μ.real (t i)) ≤ μ.real Set.univ"} +{"name":"MeasureTheory.measureReal_eq_zero_iff","declaration":"theorem MeasureTheory.measureReal_eq_zero_iff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → (μ.real s = 0 ↔ ↑↑μ s = 0)"} +{"name":"MeasureTheory.measureReal_diff_le_iff_le_add","declaration":"theorem MeasureTheory.measureReal_diff_le_iff_le_add {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n MeasurableSet s → s ⊆ t → ∀ (ε : ℝ), autoParam (↑↑μ t ≠ ⊤) _auto✝ → (μ.real (t \\ s) ≤ ε ↔ μ.real t ≤ μ.real s + ε)"} +{"name":"MeasureTheory.measureReal_add_measureReal_compl","declaration":"theorem MeasureTheory.measureReal_add_measureReal_compl {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [inst : MeasureTheory.IsFiniteMeasure μ],\n MeasurableSet s → μ.real s + μ.real sᶜ = μ.real Set.univ"} +{"name":"MeasureTheory.measureReal_union_add_inter","declaration":"theorem MeasureTheory.measureReal_union_add_inter {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n MeasurableSet t →\n autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"} +{"name":"MeasureTheory.measureReal_diff_add_inter","declaration":"theorem MeasureTheory.measureReal_diff_add_inter {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n MeasurableSet t → autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real (s \\ t) + μ.real (s ∩ t) = μ.real s"} +{"name":"MeasureTheory.measureReal_inter_add_diff","declaration":"theorem MeasureTheory.measureReal_inter_add_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n MeasurableSet t → autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real (s ∩ t) + μ.real (s \\ t) = μ.real s"} +{"name":"MeasureTheory.ext_iff_measureReal_singleton","declaration":"theorem MeasureTheory.ext_iff_measureReal_singleton {S : Type u_3} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} [MeasureTheory.IsFiniteMeasure μ1] [MeasureTheory.IsFiniteMeasure μ2] : μ1 = μ2 ↔ ∀ (x : S), μ1.real {x} = μ2.real {x}"} +{"name":"MeasureTheory.measureReal_diff_lt_of_lt_add","declaration":"theorem MeasureTheory.measureReal_diff_lt_of_lt_add {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n MeasurableSet s → s ⊆ t → ∀ (ε : ℝ), μ.real t < μ.real s + ε → autoParam (↑↑μ t ≠ ⊤) _auto✝ → μ.real (t \\ s) < ε"} +{"name":"MeasureTheory.nonempty_inter_of_measureReal_lt_add","declaration":"/-- If two sets `s` and `t` are included in a set `u` of finite measure,\nand `μ.real s + μ.real t > μ.real u`, then `s` intersects `t`.\nVersion assuming that `t` is measurable. -/\ntheorem MeasureTheory.nonempty_inter_of_measureReal_lt_add {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s : Set α} {t : Set α} {u : Set α} (ht : MeasurableSet t) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ.real u < μ.real s + μ.real t) (hu : autoParam (↑↑μ u ≠ ⊤) _auto✝) : Set.Nonempty (s ∩ t)"} diff --git a/PFR-declarations/PFR.ForMathlib.Pair.jsonl b/PFR-declarations/PFR.ForMathlib.Pair.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6dd626b71e04d86a2ef1452365b3baf3eefa381f --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Pair.jsonl @@ -0,0 +1,3 @@ +{"name":"prod","declaration":"/-- The pair of two random variables -/\ndef prod {Ω : Type u_1} {S : Type u_2} {T : Type u_3} (X : Ω → S) (Y : Ω → T) (ω : Ω) : S × T"} +{"name":"«term⟨_,_⟩».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «term⟨_,_⟩».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"«term⟨_,_⟩»","declaration":"/-- The pair of two random variables -/\ndef «term⟨_,_⟩» : Lean.ParserDescr"} diff --git a/PFR-declarations/PFR.ForMathlib.ProbabilityMeasureProdCont.jsonl b/PFR-declarations/PFR.ForMathlib.ProbabilityMeasureProdCont.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0899eefc589d3b81ac94a69591ff356bb68c8121 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.ProbabilityMeasureProdCont.jsonl @@ -0,0 +1,4 @@ +{"name":"MeasureTheory.ProbabilityMeasure.continuous_prod_of_finite","declaration":"/-- The product of two probability measures on finite spaces depend continuously on the two\nprobability measures.\nTODO: In Mathlib, this should be done on all separable metrizable spaces. -/\ntheorem MeasureTheory.ProbabilityMeasure.continuous_prod_of_finite {α : Type u_1} {β : Type u_2} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] [Finite β] [TopologicalSpace β] [DiscreteTopology β] [MeasurableSpace β] [BorelSpace β] : Continuous fun x =>\n match x with\n | (μ, ν) => MeasureTheory.ProbabilityMeasure.prod μ ν"} +{"name":"MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_apply_tendsto","declaration":"/-- Probability measures on a finite space tend to a limit if and only if the probability masses\nof all points tend to the corresponding limits. -/\ntheorem MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_apply_tendsto {ι : Type u_1} {α : Type u_2} {L : Filter ι} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] (μs : ι → MeasureTheory.ProbabilityMeasure α) (μ : MeasureTheory.ProbabilityMeasure α) : Filter.Tendsto μs L (nhds μ) ↔\n ∀ (a : α), Filter.Tendsto (fun x => (fun s => (↑↑↑(μs x) s).toNNReal) {a}) L (nhds ((fun s => (↑↑↑μ s).toNNReal) {a}))"} +{"name":"MeasureTheory.t1Space_probabilityMeasure_of_finite","declaration":"instance MeasureTheory.t1Space_probabilityMeasure_of_finite {α : Type u_1} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] : T1Space (MeasureTheory.ProbabilityMeasure α)"} +{"name":"MeasureTheory.ProbabilityMeasure.tendsto_prod_of_tendsto_of_tendsto","declaration":"/-- If probability measures on two finite spaces tend to limits, then the products of them\non the product space tend to the product of the limits.\nTODO: In Mathlib, this should be done on all separable metrizable spaces. -/\ntheorem MeasureTheory.ProbabilityMeasure.tendsto_prod_of_tendsto_of_tendsto {ι : Type u_1} {L : Filter ι} {α : Type u_2} {β : Type u_3} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] [Finite β] [TopologicalSpace β] [DiscreteTopology β] [MeasurableSpace β] [BorelSpace β] (μs : ι → MeasureTheory.ProbabilityMeasure α) (μ : MeasureTheory.ProbabilityMeasure α) (μs_lim : Filter.Tendsto μs L (nhds μ)) (νs : ι → MeasureTheory.ProbabilityMeasure β) (ν : MeasureTheory.ProbabilityMeasure β) (νs_lim : Filter.Tendsto νs L (nhds ν)) : Filter.Tendsto (fun i => MeasureTheory.ProbabilityMeasure.prod (μs i) (νs i)) L\n (nhds (MeasureTheory.ProbabilityMeasure.prod μ ν))"} diff --git a/PFR-declarations/PFR.ForMathlib.Summable.jsonl b/PFR-declarations/PFR.ForMathlib.Summable.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..481df35279f8a13681be6a9f5359d427288df404 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Summable.jsonl @@ -0,0 +1,2 @@ +{"name":"tsum_of_not_summable","declaration":"/-- Currently not needed. -/\ntheorem tsum_of_not_summable {S : Type u_1} {f : S → ℝ} (hf : ∀ (s : S), 0 ≤ f s) (hsum : ¬Summable f) : ∑' (s : S), ENNReal.ofReal (f s) = ⊤"} +{"name":"tsum_eq_toReal_tsum_ofReal","declaration":"/-- Currently not needed. -/\ntheorem tsum_eq_toReal_tsum_ofReal {S : Type u_1} {f : S → ℝ} (hf : ∀ (s : S), 0 ≤ f s) : ∑' (s : S), f s = (∑' (s : S), ENNReal.ofReal (f s)).toReal"} diff --git a/PFR-declarations/PFR.ForMathlib.Uniform.jsonl b/PFR-declarations/PFR.ForMathlib.Uniform.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..34ad0a97d01dbea006a0cfd6d54d38a9ff04db69 --- /dev/null +++ b/PFR-declarations/PFR.ForMathlib.Uniform.jsonl @@ -0,0 +1,23 @@ +{"name":"ProbabilityTheory.exists_isUniform_measureSpace","declaration":"/-- Uniform distributions exist, version giving a measure space -/\ntheorem ProbabilityTheory.exists_isUniform_measureSpace {S : Type u} [MeasurableSpace S] [MeasurableSingletonClass S] (H : Finset S) (h : H.Nonempty) : ∃ Ω mΩ U,\n MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ (∀ (ω : Ω), U ω ∈ H) ∧ FiniteRange U"} +{"name":"ProbabilityTheory.IsUniform.mk","declaration":"ctor ProbabilityTheory.IsUniform.mk {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {H : Set S} {X : Ω → S} {μ : autoParam (MeasureTheory.Measure Ω) _auto✝} (eq_of_mem : ∀ (x y : S), x ∈ H → y ∈ H → ↑↑μ (X ⁻¹' {x}) = ↑↑μ (X ⁻¹' {y})) (measure_preimage_compl : ↑↑μ (X ⁻¹' Hᶜ) = 0) : ProbabilityTheory.IsUniform H X μ"} +{"name":"ProbabilityTheory.IsUniform.measureReal_preimage","declaration":"/-- $\\mathbb{P}(U_H \\in H') = \\dfrac{|H' \\cap H|}{|H|}$ -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) (H' : Set S) : μ.real (X ⁻¹' H') = μ.real Set.univ * ↑(Nat.card ↑(H' ∩ ↑H)) / ↑(Nat.card { x // x ∈ H })"} +{"name":"ProbabilityTheory.exists_isUniform","declaration":"/-- Uniform distributions exist. -/\ntheorem ProbabilityTheory.exists_isUniform {S : Type uS} [MeasurableSpace S] [MeasurableSingletonClass S] (H : Finset S) (h : H.Nonempty) : ∃ Ω mΩ X μ,\n MeasureTheory.IsProbabilityMeasure μ ∧\n Measurable X ∧ ProbabilityTheory.IsUniform (↑H) X μ ∧ (∀ (ω : Ω), X ω ∈ H) ∧ FiniteRange X"} +{"name":"ProbabilityTheory.IsUniform.of_identDistrib","declaration":"/-- A copy of a uniform random variable is also uniform.-/\ntheorem ProbabilityTheory.IsUniform.of_identDistrib {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} {Ω' : Type u_1} [MeasurableSpace Ω'] (h : ProbabilityTheory.IsUniform H X μ) {X' : Ω' → S} {μ' : MeasureTheory.Measure Ω'} (h' : ProbabilityTheory.IdentDistrib X X' μ μ') (hH : MeasurableSet H) : ProbabilityTheory.IsUniform H X' μ'"} +{"name":"ProbabilityTheory.IsUniform.nonempty_preimage_of_mem","declaration":"theorem ProbabilityTheory.IsUniform.nonempty_preimage_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [NeZero μ] {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : Set.Nonempty (X ⁻¹' {s})"} +{"name":"ProbabilityTheory.IsUniform.full_measure","declaration":"theorem ProbabilityTheory.IsUniform.full_measure {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) (hX : Measurable X) : ↑↑(MeasureTheory.Measure.map X μ) H = ↑↑μ Set.univ"} +{"name":"ProbabilityTheory.IsUniform.comp","declaration":"/-- The image of a uniform random variable under an injective map is uniform on the image. -/\ntheorem ProbabilityTheory.IsUniform.comp {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [DecidableEq T] {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) {f : S → T} (hf : Function.Injective f) : ProbabilityTheory.IsUniform (↑(Finset.image f H)) (f ∘ X) μ"} +{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_of_mem","declaration":"/-- A \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : μ.real (X ⁻¹' {s}) = 1 / ↑(Nat.card { x // x ∈ H })"} +{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_of_nmem","declaration":"/-- Another \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_of_nmem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) {s : S} (hs : s ∉ H) : μ.real (X ⁻¹' {s}) = 0"} +{"name":"ProbabilityTheory.exists_isUniform_measureSpace'","declaration":"/-- Uniform distributions exist, version with a Finite set rather than a Finset and giving a measure space -/\ntheorem ProbabilityTheory.exists_isUniform_measureSpace' {S : Type u} [MeasurableSpace S] [MeasurableSingletonClass S] (H : Set S) [Finite ↑H] [Nonempty ↑H] : ∃ Ω mΩ U,\n MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n Measurable U ∧ ProbabilityTheory.IsUniform H U MeasureTheory.volume ∧ (∀ (ω : Ω), U ω ∈ H) ∧ FiniteRange U"} +{"name":"ProbabilityTheory.IsUniform.restrict","declaration":"/-- If $X$ is uniform w.r.t. $\\mu$ on $H$, then $X$ is uniform w.r.t. $\\mu$ conditioned by\n$H'$ on $H' \\cap H$. -/\ntheorem ProbabilityTheory.IsUniform.restrict {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) (hX : Measurable X) (H' : Set S) : ProbabilityTheory.IsUniform (H' ∩ H) X (ProbabilityTheory.cond μ (X ⁻¹' H'))"} +{"name":"ProbabilityTheory.IsUniform.ae_mem","declaration":"/-- A uniform random variable on H almost surely takes values in H. -/\ntheorem ProbabilityTheory.IsUniform.ae_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ H"} +{"name":"ProbabilityTheory.IsUniform.measure_preimage_of_mem","declaration":"/-- A \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : ↑↑μ (X ⁻¹' {s}) = ↑↑μ Set.univ / ↑(Nat.card { x // x ∈ H })"} +{"name":"ProbabilityTheory.IsUniform.nonempty","declaration":"/-- Uniform random variables only exist for non-empty sets H. -/\ntheorem ProbabilityTheory.IsUniform.nonempty {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) [hμ : NeZero μ] : H.Nonempty"} +{"name":"ProbabilityTheory.IsUniform.measure_preimage_of_nmem","declaration":"/-- Another \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage_of_nmem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) {s : S} (hs : s ∉ H) : ↑↑μ (X ⁻¹' {s}) = 0"} +{"name":"ProbabilityTheory.IsUniform.measure_preimage_compl","declaration":"def ProbabilityTheory.IsUniform.measure_preimage_compl {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {H : Set S} {X : Ω → S} {μ : autoParam (MeasureTheory.Measure Ω) _auto✝} (self : ProbabilityTheory.IsUniform H X μ) : ↑↑μ (X ⁻¹' Hᶜ) = 0"} +{"name":"ProbabilityTheory.IsUniform.eq_of_mem","declaration":"def ProbabilityTheory.IsUniform.eq_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {H : Set S} {X : Ω → S} {μ : autoParam (MeasureTheory.Measure Ω) _auto✝} (self : ProbabilityTheory.IsUniform H X μ) (x : S) (y : S) : x ∈ H → y ∈ H → ↑↑μ (X ⁻¹' {x}) = ↑↑μ (X ⁻¹' {y})"} +{"name":"ProbabilityTheory.IdentDistrib.of_isUniform","declaration":"theorem ProbabilityTheory.IdentDistrib.of_isUniform {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} {H : Set S} {Ω' : Type u_1} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [Finite ↑H] {X : Ω → S} {X' : Ω' → S} (hX : Measurable X) (hX' : Measurable X') (hX_unif : ProbabilityTheory.IsUniform H X μ) (hX'_unif : ProbabilityTheory.IsUniform H X' μ') : ProbabilityTheory.IdentDistrib X X' μ μ'"} +{"name":"ProbabilityTheory.IsUniform","declaration":"/-- The assertion that the law of $X$ is the uniform probability measure on a finite set $H$.\nWhile in applications $H$ will be non-empty finite set, $X$ measurable, and and $μ$ a probability\nmeasure, it could be technically convenient to have a definition that works even without these\nhypotheses. (For instance, `isUniform` would be well-defined, but false, for infinite `H`) -/\nstructure ProbabilityTheory.IsUniform {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] (H : Set S) (X : Ω → S) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : Prop"} +{"name":"ProbabilityTheory.IsUniform.measure_preimage_ne_zero","declaration":"/-- $\\mathbb{P}(U_H \\in H') \\neq 0$ if $H'$ intersects $H$ and the measure is non-zero. -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage_ne_zero {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} [NeZero μ] (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) (H' : Set S) [Nonempty ↑(H' ∩ ↑H)] : ↑↑μ (X ⁻¹' H') ≠ 0"} +{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_of_mem'","declaration":"theorem ProbabilityTheory.IsUniform.measureReal_preimage_of_mem' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : (MeasureTheory.Measure.map X μ).real {s} = 1 / ↑(Nat.card { x // x ∈ H })"} +{"name":"ProbabilityTheory.IsUniform.measure_preimage","declaration":"/-- $\\mathbb{P}(U_H \\in H') = \\dfrac{|H' \\cap H|}{|H|}$ -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) (H' : Set S) : ↑↑μ (X ⁻¹' H') = ↑↑μ Set.univ * ↑(Nat.card ↑(H' ∩ ↑H)) / ↑(Nat.card { x // x ∈ H })"} diff --git a/PFR-declarations/PFR.HomPFR.jsonl b/PFR-declarations/PFR.HomPFR.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6eee7c053cf9c78d8e1ac6482e004da58d3aa691 --- /dev/null +++ b/PFR-declarations/PFR.HomPFR.jsonl @@ -0,0 +1,3 @@ +{"name":"hahn_banach","declaration":"/-- Let $H_0$ be a subgroup of $G$. Then every homomorphism $\\phi: H_0 \\to G'$ can be extended to a\nhomomorphism $\\tilde \\phi: G \\to G'$. -/\ntheorem hahn_banach {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [AddCommGroup G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (H₀ : AddSubgroup G) (φ : ↥H₀ →+ G') : ∃ φ', ∀ (x : ↥H₀), φ x = φ' ↑x"} +{"name":"homomorphism_pfr","declaration":"/-- Let $f: G \\to G'$ be a function, and let $S$ denote the set\n$$ S := \\{ f(x+y)-f(x)-f(y): x,y \\in G \\}.$$\nThen there exists a homomorphism $\\phi: G \\to G'$ such that\n$$ |\\{f(x) - \\phi(x)\\}| \\leq |S|^{12}. $$ -/\ntheorem homomorphism_pfr {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (f : G → G') (S : Set G') (hS : ∀ (x y : G), f (x + y) - f x - f y ∈ S) : ∃ φ T, Nat.card ↑T ≤ Nat.card ↑S ^ 12 ∧ ∀ (x : G), f x - φ x ∈ T"} +{"name":"goursat","declaration":"/-- Let $H$ be a subgroup of $G \\times G'$. Then there exists a subgroup $H_0$ of $G$, a\nsubgroup $H_1$ of $G'$, and a homomorphism $\\phi: G \\to G'$ such that\n$$ H := \\{ (x, \\phi(x) + y): x \\in H_0, y \\in H_1 \\}.$$\nIn particular, $|H| = |H_0| |H_1|$. -/\ntheorem goursat {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [AddCommGroup G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (H : AddSubgroup (G × G')) : ∃ H₀ H₁ φ, (∀ (x : G × G'), x ∈ H ↔ x.1 ∈ H₀ ∧ x.2 - φ x.1 ∈ H₁) ∧ Nat.card ↥H = Nat.card ↥H₀ * Nat.card ↥H₁"} diff --git a/PFR-declarations/PFR.HundredPercent.jsonl b/PFR-declarations/PFR.HundredPercent.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..07d43a94920c132c6f238fd8fa146cb86fe20c5a --- /dev/null +++ b/PFR-declarations/PFR.HundredPercent.jsonl @@ -0,0 +1,7 @@ +{"name":"mem_symmGroup","declaration":"theorem mem_symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] {X : Ω → G} (hX : Measurable X) {x : G} : x ∈ symmGroup X hX ↔ ProbabilityTheory.IdentDistrib X (fun ω => X ω + x) MeasureTheory.volume MeasureTheory.volume"} +{"name":"exists_isUniform_of_rdist_eq_zero","declaration":"/-- If $d[X_1;X_2]=0$, then there exists a subgroup $H \\leq G$ such that\n$d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality. -/\ntheorem exists_isUniform_of_rdist_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] {Ω' : Type u_3} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X' : Ω' → G} (hX : Measurable X) (hX' : Measurable X') (hdist : d[X # X'] = 0) : ∃ H U, Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[X # U] = 0 ∧ d[X' # U] = 0"} +{"name":"sub_mem_symmGroup","declaration":"/-- If $d[X ;X]=0$, and $x,y \\in G$ are such that $P[X=x], P[X=y]>0$,\nthen $x-y \\in \\mathrm{Sym}[X]$. -/\ntheorem sub_mem_symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) {x : G} {y : G} (hx : ↑↑MeasureTheory.volume (X ⁻¹' {x}) ≠ 0) (hy : ↑↑MeasureTheory.volume (X ⁻¹' {y}) ≠ 0) : x - y ∈ symmGroup X hX"} +{"name":"exists_isUniform_of_rdist_self_eq_zero","declaration":"/-- If $d[X ;X]=0$, then there exists a subgroup $H \\leq G$ such that $d[X ;U_H] = 0$. -/\ntheorem exists_isUniform_of_rdist_self_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) : ∃ H U, Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[X # U] = 0"} +{"name":"symmGroup","declaration":"/-- The symmetry group Sym of $X$: the set of all $h ∈ G$ such that $X + h$ has an identical\ndistribution to $X$. -/\ndef symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] (X : Ω → G) (hX : Measurable X) : AddSubgroup G"} +{"name":"ProbabilityTheory.IdentDistrib.symmGroup_eq","declaration":"theorem ProbabilityTheory.IdentDistrib.symmGroup_eq {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] {X : Ω → G} {Ω' : Type u_3} [MeasureTheory.MeasureSpace Ω'] {X' : Ω' → G} (hX : Measurable X) (hX' : Measurable X') (h : ProbabilityTheory.IdentDistrib X X' MeasureTheory.volume MeasureTheory.volume) : symmGroup X hX = symmGroup X' hX'"} +{"name":"isUniform_sub_const_of_rdist_eq_zero","declaration":"/-- If `d[X # X] = 0`, then `X - x₀` is the uniform distribution on the subgroup of `G`\nstabilizing the distribution of `X`, for any `x₀` of positive probability. -/\ntheorem isUniform_sub_const_of_rdist_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) {x₀ : G} (hx₀ : ↑↑MeasureTheory.volume (X ⁻¹' {x₀}) ≠ 0) : ProbabilityTheory.IsUniform (↑(symmGroup X hX)) (fun ω => X ω - x₀) MeasureTheory.volume"} diff --git a/PFR-declarations/PFR.ImprovedPFR.jsonl b/PFR-declarations/PFR.ImprovedPFR.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..09795aa54994a53f61812550a3a08675176cc8ee --- /dev/null +++ b/PFR-declarations/PFR.ImprovedPFR.jsonl @@ -0,0 +1,20 @@ +{"name":"entropic_PFR_conjecture_improv","declaration":"/-- `entropic_PFR_conjecture_improv`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\nsubgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture_improv {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 8) : ∃ H Ω mΩ U,\n MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n Measurable U ∧\n ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 10 * d[p.X₀₁ # p.X₀₂]"} +{"name":"gen_ineq_aux1","declaration":"theorem gen_ineq_aux1 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] ≤\n d[Y # Z₁] + (d[Z₁ # Z₂] + d[Z₁ # Z₃] + d[Z₂ # Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 2 +\n (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₁]) / 4"} +{"name":"tau_strictly_decreases_aux'","declaration":"/-- Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X₁', X₂'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) (hp : 8 * p.η < 1) : d[X₁ # X₂] = 0"} +{"name":"entropic_PFR_conjecture_improv'","declaration":"/-- `entropic_PFR_conjecture_improv'`: For two $G$-valued random variables $X^0_1, X^0_2$, there is\nsome subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$., and\nd[X^0_1; U_H] and d[X^0_2; U_H] are at most 5/2 * d[X^0_1;X^0_2] -/\ntheorem entropic_PFR_conjecture_improv' {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 8) : ∃ H Ω mΩ U,\n MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n Measurable U ∧\n ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧\n d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 10 * d[p.X₀₁ # p.X₀₂] ∧\n d[p.X₀₁ # U] ≤ 11 / 2 * d[p.X₀₁ # p.X₀₂] ∧ d[p.X₀₂ # U] ≤ 11 / 2 * d[p.X₀₁ # p.X₀₂]"} +{"name":"dist_diff_bound_1","declaration":"theorem dist_diff_bound_1 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁] +\n (d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) ≤\n (16 * d[X₁ # X₂] + 6 * d[X₁ # X₁] + 2 * d[X₂ # X₂]) / 4 + (H[X₁ + X₁'] - H[X₂ + X₂']) / 4 +\n (H[X₂ | X₂ + X₂'] - H[X₁ | X₁ + X₁']) / 4"} +{"name":"PFR_conjecture_improv'","declaration":"/-- Corollary of `PFR_conjecture_improv` in which the ambient group is not required to be finite\n(but) then $H$ and $c$ are finite. -/\ntheorem PFR_conjecture_improv' {G : Type u_3} [AddCommGroup G] [ElementaryAddCommGroup G 2] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (Afin : Set.Finite A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, Set.Finite c ∧ Set.Finite ↑H ∧ ↑(Nat.card ↑c) < 2 * K ^ 11 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"} +{"name":"PFR_conjecture_improv_aux","declaration":"/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^6 |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^10$. -/\ntheorem PFR_conjecture_improv_aux {G : Type u_1} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c,\n ↑(Nat.card ↑c) ≤ K ^ 6 * ↑(Nat.card ↑A) ^ (1 / 2) * ↑(Nat.card ↑↑H) ^ (-1 / 2) ∧\n ↑(Nat.card ↥H) ≤ K ^ 10 * ↑(Nat.card ↑A) ∧ ↑(Nat.card ↑A) ≤ K ^ 10 * ↑(Nat.card ↥H) ∧ A ⊆ c + ↑H"} +{"name":"tau_strictly_decreases'","declaration":"theorem tau_strictly_decreases' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) (hp : 8 * p.η < 1) : d[X₁ # X₂] = 0"} +{"name":"gen_ineq_00","declaration":"/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\ntheorem gen_ineq_00 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] - d[Y # Z₁] ≤\n (d[Z₁ # Z₂] + 2 * d[Z₁ # Z₃] + d[Z₂ # Z₄]) / 4 +\n (d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 4 +\n (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₃] + H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃]) / 8"} +{"name":"tau_minimizer_exists_rdist_eq_zero","declaration":"/-- For `p.η ≤ 1/8`, there exist τ-minimizers `X₁, X₂` at zero Rusza distance. For `p.η < 1/8`,\nall minimizers are fine, by `tau_strictly_decreases'`. For `p.η = 1/8`, we use a limit of\nminimizers for `η < 1/8`, which exists by compactness. -/\ntheorem tau_minimizer_exists_rdist_eq_zero {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) : ∃ Ω mΩ X₁ X₂,\n Measurable X₁ ∧\n Measurable X₂ ∧ MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧ tau_minimizes p X₁ X₂ ∧ d[X₁ # X₂] = 0"} +{"name":"construct_good_improved''","declaration":"/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\ntheorem construct_good_improved'' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] {X₁ : Ω → G} {X₂ : Ω → G} (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_6} [MeasurableSpace Ω'] (μ : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n p.η / 6 *\n (d[p.X₀₁ ; MeasureTheory.volume # T₁ | T₂ ; μ] - d[p.X₀₁ # X₁] +\n (d[p.X₀₁ ; MeasureTheory.volume # T₁ | T₃ ; μ] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ ; MeasureTheory.volume # T₂ | T₁ ; μ] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ ; MeasureTheory.volume # T₂ | T₃ ; μ] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ ; MeasureTheory.volume # T₃ | T₁ ; μ] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ ; MeasureTheory.volume # T₃ | T₂ ; μ] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₂ ; MeasureTheory.volume # T₁ | T₂ ; μ] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ ; MeasureTheory.volume # T₁ | T₃ ; μ] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ ; MeasureTheory.volume # T₂ | T₁ ; μ] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ ; MeasureTheory.volume # T₂ | T₃ ; μ] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ ; MeasureTheory.volume # T₃ | T₁ ; μ] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ ; MeasureTheory.volume # T₃ | T₂ ; μ] - d[p.X₀₂ # X₂]))"} +{"name":"construct_good_prelim'","declaration":"/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ]$. -/\ntheorem construct_good_prelim' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] {X₁ : Ω → G} {X₂ : Ω → G} (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] +\n p.η * (d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂]))"} +{"name":"gen_ineq_10","declaration":"/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\ntheorem gen_ineq_10 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₃ + Z₄ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] - d[Y # Z₁] ≤\n (d[Z₁ # Z₂] + 2 * d[Z₁ # Z₃] + d[Z₂ # Z₄]) / 4 +\n (d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 4 +\n (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₃] + H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃]) / 8"} +{"name":"PFR_conjecture_improv","declaration":"/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{11$} cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture_improv {G : Type u_1} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, ↑(Nat.card ↑c) < 2 * K ^ 11 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"} +{"name":"averaged_construct_good","declaration":"/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\ntheorem averaged_construct_good {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ # X₂] ≤\n I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] + I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] +\n I[X₁' + X₁ : X₁ + X₂|X₁ + X₂ + X₁' + X₂'] +\n p.η / 6 *\n (d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁] +\n (d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂] +\n (d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂])))"} +{"name":"averaged_final","declaration":"theorem averaged_final {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : d[X₁ # X₂] ≤\n 6 * p.η * d[X₁ # X₂] -\n (1 - 5 * p.η) / (1 - p.η) * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂']) +\n p.η / 6 * (8 * d[X₁ # X₂] + 2 * (d[X₁ # X₁] + d[X₂ # X₂]))"} +{"name":"construct_good_improved'","declaration":"/-- In fact $k$ is at most\n$$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\ntheorem construct_good_improved' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] {X₁ : Ω → G} {X₂ : Ω → G} (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] +\n p.η / 6 *\n (d[p.X₀₁ # T₁ | T₂] - d[p.X₀₁ # X₁] + (d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # T₂ | T₁] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # T₂ | T₃] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # T₃ | T₁] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₁ # T₃ | T₂] - d[p.X₀₁ # X₁]) +\n (d[p.X₀₂ # T₁ | T₂] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # T₁ | T₃] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # T₂ | T₁] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # T₃ | T₁] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # T₃ | T₂] - d[p.X₀₂ # X₂]))"} +{"name":"dist_diff_bound_2","declaration":"theorem dist_diff_bound_2 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂] +\n (d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n (d[p.X₀₂ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) ≤\n (16 * d[X₁ # X₂] + 6 * d[X₂ # X₂] + 2 * d[X₁ # X₁]) / 4 + (H[X₂ + X₂'] - H[X₁ + X₁']) / 4 +\n (H[X₁ | X₁ + X₁'] - H[X₂ | X₂ + X₂']) / 4"} +{"name":"gen_ineq_01","declaration":"/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\ntheorem gen_ineq_01 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₂ + Z₄, Z₁ + Z₂ + Z₃ + Z₄⟩] - d[Y # Z₁] ≤\n (d[Z₁ # Z₂] + 2 * d[Z₁ # Z₃] + d[Z₂ # Z₄]) / 4 +\n (d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 4 +\n (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₃] + H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃]) / 8"} +{"name":"gen_ineq_aux2","declaration":"theorem gen_ineq_aux2 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] ≤\n d[Y # Z₁] + (d[Z₁ # Z₃] + d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄]) / 2 +\n (H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃] + H[Z₁] - H[Z₃]) / 4"} diff --git a/PFR-declarations/PFR.Main.jsonl b/PFR-declarations/PFR.Main.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d275c75b6461bcdbc9de72333108f94dff9f0cb2 --- /dev/null +++ b/PFR-declarations/PFR.Main.jsonl @@ -0,0 +1,9 @@ +{"name":"PFR_conjecture_pos_aux","declaration":"/-- Record positivity results that are useful in the proof of PFR. -/\ntheorem PFR_conjecture_pos_aux {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A - A)) ≤ K * ↑(Nat.card ↑A)) : 0 < ↑(Nat.card ↑A) ∧ 0 < ↑(Nat.card ↑(A - A)) ∧ 0 < K"} +{"name":"PFR_conjecture","declaration":"/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{12}$ cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] [ElementaryAddCommGroup G 2] [Fintype G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, ↑(Nat.card ↑c) < 2 * K ^ 12 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"} +{"name":"PFR_conjecture_pos_aux'","declaration":"theorem PFR_conjecture_pos_aux' {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : 0 < ↑(Nat.card ↑A) ∧ 0 < ↑(Nat.card ↑(A + A)) ∧ 0 < K"} +{"name":"rdist_le_of_isUniform_of_card_add_le","declaration":"/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le {G : Type u_1} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A - A)) ≤ K * ↑(Nat.card ↑A)) {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U₀ : Ω → G} (U₀unif : ProbabilityTheory.IsUniform A U₀ MeasureTheory.volume) (U₀meas : Measurable U₀) : d[U₀ # U₀] ≤ Real.log K"} +{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_sub","declaration":"/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A ∩ (B + x)) / #A ⬝ #B`. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_sub {G : Type u_1} {Ω : Type u_2} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Finset G} {B : Finset G} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U : Ω → G} {V : Ω → G} (Uunif : ProbabilityTheory.IsUniform (↑A) U MeasureTheory.volume) (Umeas : Measurable U) (Vunif : ProbabilityTheory.IsUniform (↑B) V MeasureTheory.volume) (Vmeas : Measurable V) (hindep : ProbabilityTheory.IndepFun U V MeasureTheory.volume) (x : G) : MeasureTheory.volume.real ((U - V) ⁻¹' {x}) =\n ↑(Nat.card ↑(↑A ∩ (↑B + {x}))) / (↑(Nat.card { x // x ∈ A }) * ↑(Nat.card { x // x ∈ B }))"} +{"name":"sumset_eq_sub","declaration":"theorem sumset_eq_sub {G : Type u_1} [AddCommGroup G] {A : Set G} [ElementaryAddCommGroup G 2] : A + A = A - A"} +{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_sub_zero","declaration":"/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A ∩ B) / #A ⬝ #B`. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_sub_zero {G : Type u_1} {Ω : Type u_2} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Finset G} {B : Finset G} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U : Ω → G} {V : Ω → G} (Uunif : ProbabilityTheory.IsUniform (↑A) U MeasureTheory.volume) (Umeas : Measurable U) (Vunif : ProbabilityTheory.IsUniform (↑B) V MeasureTheory.volume) (Vmeas : Measurable V) (hindep : ProbabilityTheory.IndepFun U V MeasureTheory.volume) : MeasureTheory.volume.real ((U - V) ⁻¹' {0}) =\n ↑(Nat.card ↑(↑A ∩ ↑B)) / (↑(Nat.card { x // x ∈ A }) * ↑(Nat.card { x // x ∈ B }))"} +{"name":"PFR_conjecture_aux","declaration":"/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\ntheorem PFR_conjecture_aux {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] [ElementaryAddCommGroup G 2] [Fintype G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c,\n ↑(Nat.card ↑c) ≤ K ^ (13 / 2) * ↑(Nat.card ↑A) ^ (1 / 2) * ↑(Nat.card ↑↑H) ^ (-1 / 2) ∧\n ↑(Nat.card ↥H) ≤ K ^ 11 * ↑(Nat.card ↑A) ∧ ↑(Nat.card ↑A) ≤ K ^ 11 * ↑(Nat.card ↥H) ∧ A ⊆ c + ↑H"} +{"name":"PFR_conjecture'","declaration":"/-- Corollary of `PFR_conjecture` in which the ambient group is not required to be finite (but) then\n$H$ and $c$ are finite. -/\ntheorem PFR_conjecture' {G : Type u_2} [AddCommGroup G] [ElementaryAddCommGroup G 2] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (Afin : Set.Finite A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, Set.Finite c ∧ Set.Finite ↑H ∧ ↑(Nat.card ↑c) < 2 * K ^ 12 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"} diff --git a/PFR-declarations/PFR.Mathlib.Data.Fin.VecNotation.jsonl b/PFR-declarations/PFR.Mathlib.Data.Fin.VecNotation.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/PFR-declarations/PFR.Mathlib.Data.Set.Pointwise.SMul.jsonl b/PFR-declarations/PFR.Mathlib.Data.Set.Pointwise.SMul.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..9dcb9456fd0dda5f30e4eba27217b284e24cb72a --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Data.Set.Pointwise.SMul.jsonl @@ -0,0 +1,6 @@ +{"name":"Set.singleton_add'","declaration":"theorem Set.singleton_add' {α : Type u_1} [Add α] (a : α) (s : Set α) : {a} + s = a +ᵥ s"} +{"name":"Set.vadd_sub_vadd_comm","declaration":"theorem Set.vadd_sub_vadd_comm {α : Type u_1} [AddCommGroup α] (a : α) (s : Set α) (b : α) (t : Set α) : a +ᵥ s - (b +ᵥ t) = a - b +ᵥ (s - t)"} +{"name":"Set.singleton_mul'","declaration":"theorem Set.singleton_mul' {α : Type u_1} [Mul α] (a : α) (s : Set α) : {a} * s = a • s"} +{"name":"Set.smul_div_smul_comm","declaration":"theorem Set.smul_div_smul_comm {α : Type u_1} [CommGroup α] (a : α) (s : Set α) (b : α) (t : Set α) : a • s / b • t = (a / b) • (s / t)"} +{"name":"Set.mul_singleton'","declaration":"theorem Set.mul_singleton' {α : Type u_1} [Mul α] (s : Set α) (a : α) : s * {a} = MulOpposite.op a • s"} +{"name":"Set.add_singleton'","declaration":"theorem Set.add_singleton' {α : Type u_1} [Add α] (s : Set α) (a : α) : s + {a} = AddOpposite.op a +ᵥ s"} diff --git a/PFR-declarations/PFR.Mathlib.GroupTheory.Subgroup.Pointwise.jsonl b/PFR-declarations/PFR.Mathlib.GroupTheory.Subgroup.Pointwise.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..da3d60f894efde9a5a1244c84b16dd7db16a48e5 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.GroupTheory.Subgroup.Pointwise.jsonl @@ -0,0 +1,6 @@ +{"name":"AddSubgroupClass.coe_sub_coe","declaration":"theorem AddSubgroupClass.coe_sub_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [SubtractionMonoid G] [AddSubgroupClass S G] (H : S) : ↑H - ↑H = ↑H"} +{"name":"AddSubgroupClass.neg_coe","declaration":"theorem AddSubgroupClass.neg_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [SubtractionMonoid G] [AddSubgroupClass S G] (H : S) : -↑H = ↑H"} +{"name":"AddSubgroupClass.coe_add_coe","declaration":"theorem AddSubgroupClass.coe_add_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [SubNegMonoid G] [AddSubgroupClass S G] (H : S) : ↑H + ↑H = ↑H"} +{"name":"SubgroupClass.inv_coe","declaration":"theorem SubgroupClass.inv_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [DivisionMonoid G] [SubgroupClass S G] (H : S) : (↑H)⁻¹ = ↑H"} +{"name":"SubgroupClass.coe_div_coe","declaration":"theorem SubgroupClass.coe_div_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [DivisionMonoid G] [SubgroupClass S G] (H : S) : ↑H / ↑H = ↑H"} +{"name":"SubgroupClass.coe_mul_coe","declaration":"theorem SubgroupClass.coe_mul_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [DivInvMonoid G] [SubgroupClass S G] (H : S) : ↑H * ↑H = ↑H"} diff --git a/PFR-declarations/PFR.Mathlib.GroupTheory.Torsion.jsonl b/PFR-declarations/PFR.Mathlib.GroupTheory.Torsion.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..be47de8dd9c0b0bc18ed1da82cd728f419fe44cc --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.GroupTheory.Torsion.jsonl @@ -0,0 +1 @@ +{"name":"AddMonoid.IsTorsionFree.noZeroNsmulDivisors","declaration":"/-- See note [reducible non-instances]. -/\ndef AddMonoid.IsTorsionFree.noZeroNsmulDivisors {M : Type u_1} [AddMonoid M] (hM : AddMonoid.IsTorsionFree M) : NoZeroSMulDivisors ℕ M"} diff --git a/PFR-declarations/PFR.Mathlib.LinearAlgebra.Basis.VectorSpace.jsonl b/PFR-declarations/PFR.Mathlib.LinearAlgebra.Basis.VectorSpace.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f22f01bbc861dcaa5c80efe532b1bb4e7c03164f --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.LinearAlgebra.Basis.VectorSpace.jsonl @@ -0,0 +1 @@ +{"name":"Submodule.exists_equiv_fst_sndModFst","declaration":"/-- Given a submodule $E$ of $B \\times F$, there is an equivalence $f : E \\to B' \\times F'$\ngiven by the projections $E \\to B$ and $E \\to F$ \"modulo\" $φ : B \\to F$. -/\ntheorem Submodule.exists_equiv_fst_sndModFst {B : Type u_1} {F : Type u_2} {R : Type u_3} [DivisionRing R] [AddCommGroup B] [AddCommGroup F] [Module R B] [Module R F] (E : Submodule R (B × F)) : ∃ B' F' f φ,\n (∀ (x : ↥E), ↑(f x).1 = (↑x).1 ∧ ↑(f x).2 = (↑x).2 - φ (↑x).1) ∧\n ∀ (x₁ : ↥B') (x₂ : ↥F'), ↑((LinearEquiv.symm f) (x₁, x₂)) = (↑x₁, ↑x₂ + φ ↑x₁)"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Pi.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Pi.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7b04d2b3e0e32ebf3104e0e39df7c1e1b77bf38a --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Pi.jsonl @@ -0,0 +1,5 @@ +{"name":"MeasureTheory.Measure.pi_pi_finset","declaration":"theorem MeasureTheory.Measure.pi_pi_finset {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (t : Finset ι) (s : (i : ι) → Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Set.pi (↑t) s) = Finset.prod t fun i => ↑↑(μ i) (s i)"} +{"name":"MeasureTheory.Measure.instIsProbabilityMeasureForAllPiPi","declaration":"instance MeasureTheory.Measure.instIsProbabilityMeasureForAllPiPi {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.pi μ)"} +{"name":"MeasureTheory.Measure.pi_eval_preimage","declaration":"theorem MeasureTheory.Measure.pi_eval_preimage {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (i : ι) (s : Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Function.eval i ⁻¹' s) = ↑↑(μ i) s"} +{"name":"MeasureTheory.Measure.map_eval_pi","declaration":"theorem MeasureTheory.Measure.map_eval_pi {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (i : ι) : MeasureTheory.Measure.map (Function.eval i) (MeasureTheory.Measure.pi μ) = μ i"} +{"name":"MeasureTheory.Measure.pi_pi_set","declaration":"theorem MeasureTheory.Measure.pi_pi_set {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (t : Set ι) [DecidablePred fun x => x ∈ t] (s : (i : ι) → Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Set.pi t s) =\n Finset.prod (Finset.filter (fun x => x ∈ t) Finset.univ) fun i => ↑↑(μ i) (s i)"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3d22f2b0d9b3e11bfbbd1eef77f0ee006c5c97e2 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic.jsonl @@ -0,0 +1,3 @@ +{"name":"MeasureTheory.Measure.map_prod_comap_swap","declaration":"/-- The law of $(X, Z)$ is the image of the law of $(Z,X)$.-/\ntheorem MeasureTheory.Measure.map_prod_comap_swap {Ω : Type u_1} {α : Type u_2} {γ : Type u_4} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace γ] {X : Ω → α} {Z : Ω → γ} (hX : Measurable X) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) : MeasureTheory.Measure.comap Prod.swap (MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ) =\n MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ"} +{"name":"MeasureTheory.Measure.prod_apply_singleton","declaration":"theorem MeasureTheory.Measure.prod_apply_singleton {α : Type u_5} {β : Type u_6} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure β)\n [inst : MeasureTheory.SigmaFinite ν] (x_2 : α × β),\n ↑↑(MeasureTheory.Measure.prod μ ν) {x_2} = ↑↑μ {x_2.1} * ↑↑ν {x_2.2}"} +{"name":"MeasureTheory.Measure.prod_of_full_measure_finset","declaration":"theorem MeasureTheory.Measure.prod_of_full_measure_finset {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SigmaFinite ν] {A : Finset α} {B : Finset β} (hA : ↑↑μ (↑A)ᶜ = 0) (hB : ↑↑ν (↑B)ᶜ = 0) : ↑↑(MeasureTheory.Measure.prod μ ν) (↑(A ×ˢ B))ᶜ = 0"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Bochner.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Bochner.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..25b83af5032bda5c07663c301460cbbe1f88df53 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Bochner.jsonl @@ -0,0 +1 @@ +{"name":"MeasureTheory.integral_eq_sum","declaration":"theorem MeasureTheory.integral_eq_sum {α : Type u_1} {E : Type u_2} [MeasurableSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasurableSingletonClass α] [Fintype α] (f : α → E) : ∫ (x : α), f x ∂μ = Finset.sum Finset.univ fun x => (↑↑μ {x}).toReal • f x"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Lebesgue.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Lebesgue.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..494f1d8587766b24de26b8603a5eb5f12201bd09 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Lebesgue.jsonl @@ -0,0 +1,5 @@ +{"name":"MeasureTheory.lintegral_eq_sum'","declaration":"theorem MeasureTheory.lintegral_eq_sum' {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) {s : Finset α} (hA : ↑↑μ (↑s)ᶜ = 0) (f : α → ENNReal) : ∫⁻ (x : α), f x ∂μ = Finset.sum s fun x => f x * ↑↑μ {x}"} +{"name":"MeasureTheory.lintegral_eq_single","declaration":"theorem MeasureTheory.lintegral_eq_single {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (a : α) (f : α → ENNReal) (ha : ∀ (b : α), b ≠ a → f b = 0) : ∫⁻ (x : α), f x ∂μ = f a * ↑↑μ {a}"} +{"name":"MeasureTheory.lintegral_eq_sum","declaration":"theorem MeasureTheory.lintegral_eq_sum {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (f : α → ENNReal) [Fintype α] : ∫⁻ (x : α), f x ∂μ = Finset.sum Finset.univ fun x => ↑↑μ {x} * f x"} +{"name":"MeasureTheory.lintegral_eq_zero_of_ae_zero","declaration":"theorem MeasureTheory.lintegral_eq_zero_of_ae_zero {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ENNReal} {E : Set α} (hE : ↑↑μ Eᶜ = 0) (hf : ∀ x ∈ E, f x = 0) (hmes : MeasurableSet E) : ∫⁻ (x : α), f x ∂μ = 0"} +{"name":"MeasureTheory.lintegral_eq_sum_countable","declaration":"theorem MeasureTheory.lintegral_eq_sum_countable {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (f : α → ENNReal) [Countable α] : ∫⁻ (x : α), f x ∂μ = ∑' (x : α), ↑↑μ {x} * f x"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.SetIntegral.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.SetIntegral.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a7813c55ede7237bac51c5aa5ca9236d139c68b5 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.SetIntegral.jsonl @@ -0,0 +1,2 @@ +{"name":"MeasureTheory.setIntegral_eq_sum","declaration":"theorem MeasureTheory.setIntegral_eq_sum {α : Type u_1} {E : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (s : Finset α) (f : α → E) : ∫ (x : α) in ↑s, f x ∂μ = Finset.sum s fun x => (↑↑μ {x}).toReal • f x"} +{"name":"MeasureTheory.integral_eq_sum'","declaration":"theorem MeasureTheory.integral_eq_sum' {α : Type u_1} {E : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] {s : Finset α} (hs : ↑↑μ (↑s)ᶜ = 0) (f : α → E) : ∫ (x : α), f x ∂μ = Finset.sum s fun x => (↑↑μ {x}).toReal • f x"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..73562978f2ee4d95f9a9732b0b78799d3e882379 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic.jsonl @@ -0,0 +1,2 @@ +{"name":"prodMKLeft_unit_equiv","declaration":"/-- Measurable equivalence with the product with the one-point space `Unit`.-/\ndef prodMKLeft_unit_equiv (α : Type u_4) [MeasurableSpace α] : Unit × α ≃ᵐ α"} +{"name":"prodAssoc","declaration":"/-- Canonical bijection between `(α × β) × γ` and `α × β × γ`. -/\ndef prodAssoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} : {x : MeasurableSpace α} → {x_1 : MeasurableSpace β} → {x_2 : MeasurableSpace γ} → (α × β) × γ ≃ᵐ α × β × γ"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.MeasureSpace.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.MeasureSpace.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..9372bd5387da504b065ce9c3e20ee3269d3c54c4 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.MeasureSpace.jsonl @@ -0,0 +1 @@ +{"name":"MeasureTheory.full_measure_of_null_compl","declaration":"theorem MeasureTheory.full_measure_of_null_compl {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {μ : MeasureTheory.Measure α} {A : Finset α} (hA : ↑↑μ (↑A)ᶜ = 0) : ↑↑μ ↑A = ↑↑μ Set.univ"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.NullMeasurable.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.NullMeasurable.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3a1d385e2be50211de95571845c1da8297098f7a --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.NullMeasurable.jsonl @@ -0,0 +1,4 @@ +{"name":"MeasureTheory.measure_preimage_snd_singleton_eq_sum","declaration":"theorem MeasureTheory.measure_preimage_snd_singleton_eq_sum {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype α] (μ : MeasureTheory.Measure (α × β)) (y : β) : ↑↑μ (Prod.snd ⁻¹' {y}) = Finset.sum Finset.univ fun x => ↑↑μ {(x, y)}"} +{"name":"MeasureTheory.measure_preimage_snd_singleton_eq_sum_countable","declaration":"theorem MeasureTheory.measure_preimage_snd_singleton_eq_sum_countable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Countable α] (μ : MeasureTheory.Measure (α × β)) (y : β) : ↑↑μ (Prod.snd ⁻¹' {y}) = ∑' (x : α), ↑↑μ {(x, y)}"} +{"name":"MeasureTheory.measure_preimage_fst_singleton_eq_sum_countable","declaration":"theorem MeasureTheory.measure_preimage_fst_singleton_eq_sum_countable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Countable β] (μ : MeasureTheory.Measure (α × β)) (x : α) : ↑↑μ (Prod.fst ⁻¹' {x}) = ∑' (y : β), ↑↑μ {(x, y)}"} +{"name":"MeasureTheory.measure_preimage_fst_singleton_eq_sum","declaration":"theorem MeasureTheory.measure_preimage_fst_singleton_eq_sum {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] (μ : MeasureTheory.Measure (α × β)) (x : α) : ↑↑μ (Prod.fst ⁻¹' {x}) = Finset.sum Finset.univ fun y => ↑↑μ {(x, y)}"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7873eb8f11fd3a120b72196e1a24084e5e7280cd --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure.jsonl @@ -0,0 +1,6 @@ +{"name":"lintegral_indicatorBCF","declaration":"theorem lintegral_indicatorBCF {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) {s : Set α} (s_clopen : IsClopen s) (s_mble : MeasurableSet s) : ∫⁻ (x : α), ENNReal.ofReal ((indicatorBCF s_clopen) x) ∂μ = ↑↑μ s"} +{"name":"integral_indicatorBCF","declaration":"theorem integral_indicatorBCF {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) {s : Set α} (s_clopen : IsClopen s) (s_mble : MeasurableSet s) : ∫ (x : α), (indicatorBCF s_clopen) x ∂μ = (↑↑μ s).toReal"} +{"name":"continuous_integral_finiteMeasure","declaration":"theorem continuous_integral_finiteMeasure {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) : Continuous fun μ => ∫ (x : α), f x ∂↑μ"} +{"name":"indicatorBCF_apply","declaration":"theorem indicatorBCF_apply {α : Type u_1} [TopologicalSpace α] {s : Set α} (s_clopen : IsClopen s) (x : α) : (indicatorBCF s_clopen) x = Set.indicator s (fun x => 1) x"} +{"name":"indicatorBCF","declaration":"/-- The indicator function of a clopen set, as a bounded continuous function. -/\ndef indicatorBCF {α : Type u_1} [TopologicalSpace α] {s : Set α} (s_clopen : IsClopen s) : BoundedContinuousFunction α ℝ"} +{"name":"continuous_integral_probabilityMeasure","declaration":"theorem continuous_integral_probabilityMeasure {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) : Continuous fun μ => ∫ (x : α), f x ∂↑μ"} diff --git a/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.Typeclasses.jsonl b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.Typeclasses.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..be1293e4fc5396d3cb65be898dba961452e276aa --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.Typeclasses.jsonl @@ -0,0 +1 @@ +{"name":"MeasureTheory.IsFiniteMeasure_comap_equiv","declaration":"instance MeasureTheory.IsFiniteMeasure_comap_equiv {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} (f : β ≃ᵐ α) [MeasureTheory.IsFiniteMeasure μ] : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.comap (⇑f) μ)"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.ConditionalProbability.jsonl b/PFR-declarations/PFR.Mathlib.Probability.ConditionalProbability.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/PFR-declarations/PFR.Mathlib.Probability.IdentDistrib.jsonl b/PFR-declarations/PFR.Mathlib.Probability.IdentDistrib.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0dc5dc1363f92cae954d6657263436c1d54a4cde --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.IdentDistrib.jsonl @@ -0,0 +1,28 @@ +{"name":"ProbabilityTheory.identDistrib_map","declaration":"/-- A random variable is identically distributed to its pullbacks. -/\ntheorem ProbabilityTheory.identDistrib_map {Ω : Type u_5} {α : Type u_7} {β : Type u_9} {mΩ : MeasurableSpace Ω} [MeasurableSpace α] [MeasurableSpace β] {X : Ω → α} (hX : Measurable X) {f : α → β} (hf : Measurable f) (μ : MeasureTheory.Measure Ω) : ProbabilityTheory.IdentDistrib f (f ∘ X) (MeasureTheory.Measure.map X μ) μ"} +{"name":"ProbabilityTheory.identDistrib_of_finiteRange","declaration":"/-- If `X` has identical distribution to `X₀`, and `X₀` has finite range, then `X` is almost everywhere equivalent to a random variable of finite range. -/\ntheorem ProbabilityTheory.identDistrib_of_finiteRange {Ω : Type u_11} {Ω₀ : Type u_12} {S : Type u_13} [MeasurableSpace Ω] [MeasurableSpace Ω₀] [MeasurableSpace S] [MeasurableSingletonClass S] [hS : Nonempty S] {μ : MeasureTheory.Measure Ω} {μ₀ : MeasureTheory.Measure Ω₀} {X₀ : Ω₀ → S} [FiniteRange X₀] {X : Ω → S} (hX : Measurable X) (hi : ProbabilityTheory.IdentDistrib X₀ X μ₀ μ) : ∃ X', Measurable X' ∧ FiniteRange X' ∧ X' =ᶠ[MeasureTheory.Measure.ae μ] X"} +{"name":"ProbabilityTheory.IdentDistrib.snd_id","declaration":"/-- The second projection in a product space with measure `μ.prod ν` is distributed like `ν`. -/\ntheorem ProbabilityTheory.IdentDistrib.snd_id {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] : ProbabilityTheory.IdentDistrib Prod.snd id (MeasureTheory.Measure.prod μ ν) ν"} +{"name":"ProbabilityTheory.identDistrib_ulift_self","declaration":"theorem ProbabilityTheory.identDistrib_ulift_self {Ω : Type u_5} {α : Type u_7} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasurableSpace α] {X : Ω → α} (hX : Measurable X) : ProbabilityTheory.IdentDistrib X (X ∘ ULift.down) μ (MeasureTheory.Measure.comap ULift.down μ)"} +{"name":"ProbabilityTheory.independent_copies4_nondep","declaration":"/-- A version with exactly 4 random variables that have the same codomain.\nIt's unfortunately incredibly painful to prove this from the general case. -/\ntheorem ProbabilityTheory.independent_copies4_nondep {α : Type u} [mS : MeasurableSpace α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} {Ω₄ : Type u_4} [mΩ₁ : MeasurableSpace Ω₁] [mΩ₂ : MeasurableSpace Ω₂] [mΩ₃ : MeasurableSpace Ω₃] [mΩ₄ : MeasurableSpace Ω₄] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} {X₄ : Ω₄ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) (hX₄ : Measurable X₄) (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) (μ₄ : MeasureTheory.Measure Ω₄) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] [hμ₄ : MeasureTheory.IsProbabilityMeasure μ₄] : ∃ A mA μA X₁' X₂' X₃' X₄',\n MeasureTheory.IsProbabilityMeasure μA ∧\n ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃', X₄'] μA ∧\n Measurable X₁' ∧\n Measurable X₂' ∧\n Measurable X₃' ∧\n Measurable X₄' ∧\n ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧\n ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃ ∧ ProbabilityTheory.IdentDistrib X₄' X₄ μA μ₄"} +{"name":"ProbabilityTheory.identDistrib_comp_snd","declaration":"/-- A random variable is identically distributed to its lift to a product space (in the second factor). -/\ntheorem ProbabilityTheory.identDistrib_comp_snd {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] {X : Ω → α} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.SigmaFinite μ] [MeasureTheory.IsProbabilityMeasure μ'] : ProbabilityTheory.IdentDistrib (X ∘ Prod.snd) X (MeasureTheory.Measure.prod μ' μ) μ"} +{"name":"ProbabilityTheory.identDistrib_comp_left","declaration":"/-- A function is identically distributed to itself composed with a measurable embedding of conull\nrange. -/\ntheorem ProbabilityTheory.identDistrib_comp_left {α : Type u_1} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → γ} {i : δ → α} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range i) (hf : Measurable f) : ProbabilityTheory.IdentDistrib (f ∘ i) f (MeasureTheory.Measure.comap i μ) μ"} +{"name":"ProbabilityTheory.independent_copies_finiteRange","declaration":"/-- A version of `independent_copies` that guarantees that the copies have `FiniteRange` if the original variables do. -/\ntheorem ProbabilityTheory.independent_copies_finiteRange {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] [MeasurableSpace β] {X : Ω → α} {Y : Ω' → β} (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] [MeasurableSingletonClass α] [Nonempty α] [MeasurableSingletonClass β] [Nonempty β] (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : ∃ ν X' Y',\n MeasureTheory.IsProbabilityMeasure ν ∧\n Measurable X' ∧\n Measurable Y' ∧\n ProbabilityTheory.IndepFun X' Y' ν ∧\n ProbabilityTheory.IdentDistrib X' X ν μ ∧\n ProbabilityTheory.IdentDistrib Y' Y ν μ' ∧ FiniteRange X' ∧ FiniteRange Y'"} +{"name":"ProbabilityTheory.IdentDistrib.cond","declaration":"theorem ProbabilityTheory.IdentDistrib.cond {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {f' : α → γ} {g : β → γ} {g' : β → γ} {s : Set γ} (hs : MeasurableSet s) (hf' : Measurable f') (hg' : Measurable g') (hfg : ProbabilityTheory.IdentDistrib (fun a => (f a, f' a)) (fun b => (g b, g' b)) μ ν) : ProbabilityTheory.IdentDistrib f g (ProbabilityTheory.cond μ (f' ⁻¹' s)) (ProbabilityTheory.cond ν (g' ⁻¹' s))"} +{"name":"ProbabilityTheory.independent_copies","declaration":"/-- For $X, Y$ random variables, one can find independent copies $X', Y'$ of $X, Y$. -/\ntheorem ProbabilityTheory.independent_copies {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] [MeasurableSpace β] {X : Ω → α} {Y : Ω' → β} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : ∃ ν X' Y',\n MeasureTheory.IsProbabilityMeasure ν ∧\n Measurable X' ∧\n Measurable Y' ∧\n ProbabilityTheory.IndepFun X' Y' ν ∧\n ProbabilityTheory.IdentDistrib X' X ν μ ∧ ProbabilityTheory.IdentDistrib Y' Y ν μ'"} +{"name":"ProbabilityTheory.independent_copies3_nondep_finiteRange","declaration":"/-- A version of `independent_copies3_nondep` that guarantees that the copies have `FiniteRange` if the original variables do. -/\ntheorem ProbabilityTheory.independent_copies3_nondep_finiteRange {α : Type u} [mS : MeasurableSpace α] [MeasurableSingletonClass α] [Nonempty α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] [MeasurableSpace Ω₃] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) [FiniteRange X₁] [FiniteRange X₂] [FiniteRange X₃] (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] : ∃ A mA μA X₁' X₂' X₃',\n MeasureTheory.IsProbabilityMeasure μA ∧\n ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃'] μA ∧\n Measurable X₁' ∧\n Measurable X₂' ∧\n Measurable X₃' ∧\n ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧\n ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃ ∧ FiniteRange X₁' ∧ FiniteRange X₂' ∧ FiniteRange X₃'"} +{"name":"ProbabilityTheory.IdentDistrib.mul","declaration":"theorem ProbabilityTheory.IdentDistrib.mul {Ω : Type u_5} {Ω' : Type u_6} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ω'} {f : Ω → β} {g : Ω → β} {f' : Ω' → β} {g' : Ω' → β} [Mul β] [MeasurableMul₂ β] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (hff' : ProbabilityTheory.IdentDistrib f f' μ ν) (hgg' : ProbabilityTheory.IdentDistrib g g' μ ν) (h : ProbabilityTheory.IndepFun f g μ) (h' : ProbabilityTheory.IndepFun f' g' ν) : ProbabilityTheory.IdentDistrib (f * g) (f' * g') μ ν"} +{"name":"ProbabilityTheory.independent_copies'","declaration":"/-- Let $X_i : \\Omega_i \\to S_i$ be random variables for $i=1,\\dots,k$.\nThen there exist jointly independent random variables $X'_i : \\Omega' \\to S_i$ for $i=1,\\dots,k$\nsuch that each $X'_i$ is a copy of $X_i$. -/\ntheorem ProbabilityTheory.independent_copies' {I : Type u} [Fintype I] {α : I → Type u'} [mS : (i : I) → MeasurableSpace (α i)] {Ω : I → Type v} [mΩ : (i : I) → MeasurableSpace (Ω i)] (X : (i : I) → Ω i → α i) (hX : ∀ (i : I), Measurable (X i)) (μ : (i : I) → MeasureTheory.Measure (Ω i)) [∀ (i : I), MeasureTheory.IsProbabilityMeasure (μ i)] : ∃ A mA μA X',\n MeasureTheory.IsProbabilityMeasure μA ∧\n ProbabilityTheory.iIndepFun mS X' μA ∧\n ∀ (i : I), Measurable (X' i) ∧ ProbabilityTheory.IdentDistrib (X' i) (X i) μA (μ i)"} +{"name":"ProbabilityTheory.identDistrib_id_left","declaration":"theorem ProbabilityTheory.identDistrib_id_left {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {X : α → β} (hX : AEMeasurable X μ) : ProbabilityTheory.IdentDistrib id X (MeasureTheory.Measure.map X μ) μ"} +{"name":"ProbabilityTheory.IdentDistrib.add","declaration":"theorem ProbabilityTheory.IdentDistrib.add {Ω : Type u_5} {Ω' : Type u_6} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ω'} {f : Ω → β} {g : Ω → β} {f' : Ω' → β} {g' : Ω' → β} [Add β] [MeasurableAdd₂ β] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (hff' : ProbabilityTheory.IdentDistrib f f' μ ν) (hgg' : ProbabilityTheory.IdentDistrib g g' μ ν) (h : ProbabilityTheory.IndepFun f g μ) (h' : ProbabilityTheory.IndepFun f' g' ν) : ProbabilityTheory.IdentDistrib (f + g) (f' + g') μ ν"} +{"name":"ProbabilityTheory.identDistrib_of_sum","declaration":"/-- To show identical distribution of two random variables on a mixture of probability measures, it suffices to do so on each non-trivial component. -/\ntheorem ProbabilityTheory.identDistrib_of_sum {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] {T : Type u_11} {X : Ω → α} {Y : Ω' → α} [Fintype T] {μ : T → MeasureTheory.Measure Ω} {μ' : T → MeasureTheory.Measure Ω'} {w : T → ENNReal} (hX : Measurable X) (hY : Measurable Y) (h_ident : ∀ (y : T), w y ≠ 0 → ProbabilityTheory.IdentDistrib X Y (μ y) (μ' y)) : ProbabilityTheory.IdentDistrib X Y (Finset.sum Finset.univ fun y => w y • μ y)\n (Finset.sum Finset.univ fun y => w y • μ' y)"} +{"name":"ProbabilityTheory.identDistrib_id","declaration":"theorem ProbabilityTheory.identDistrib_id {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} : ProbabilityTheory.IdentDistrib id id μ ν ↔ μ = ν"} +{"name":"ProbabilityTheory.independent_copies3_nondep","declaration":"/-- A version with exactly 3 random variables that have the same codomain.\nIt's unfortunately incredibly painful to prove this from the general case. -/\ntheorem ProbabilityTheory.independent_copies3_nondep {α : Type u} [mS : MeasurableSpace α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] [MeasurableSpace Ω₃] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] : ∃ A mA μA X₁' X₂' X₃',\n MeasureTheory.IsProbabilityMeasure μA ∧\n ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃'] μA ∧\n Measurable X₁' ∧\n Measurable X₂' ∧\n Measurable X₃' ∧\n ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧ ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃"} +{"name":"ProbabilityTheory.independent_copies4_nondep_finiteRange","declaration":"/-- A version of `independent_copies4_nondep` that guarantees that the copies have `FiniteRange` if the original variables do. -/\ntheorem ProbabilityTheory.independent_copies4_nondep_finiteRange {α : Type u} [mS : MeasurableSpace α] [MeasurableSingletonClass α] [Nonempty α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} {Ω₄ : Type u_4} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] [MeasurableSpace Ω₃] [MeasurableSpace Ω₄] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} {X₄ : Ω₄ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) (hX₄ : Measurable X₄) [FiniteRange X₁] [FiniteRange X₂] [FiniteRange X₃] [FiniteRange X₄] (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) (μ₄ : MeasureTheory.Measure Ω₄) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] [hμ₄ : MeasureTheory.IsProbabilityMeasure μ₄] : ∃ A mA μA X₁' X₂' X₃' X₄',\n MeasureTheory.IsProbabilityMeasure μA ∧\n ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃', X₄'] μA ∧\n Measurable X₁' ∧\n Measurable X₂' ∧\n Measurable X₃' ∧\n Measurable X₄' ∧\n ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧\n ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃ ∧\n ProbabilityTheory.IdentDistrib X₄' X₄ μA μ₄ ∧\n FiniteRange X₁' ∧ FiniteRange X₂' ∧ FiniteRange X₃' ∧ FiniteRange X₄'"} +{"name":"ProbabilityTheory.identDistrib_id_right","declaration":"theorem ProbabilityTheory.identDistrib_id_right {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {X : α → β} (hX : AEMeasurable X μ) : ProbabilityTheory.IdentDistrib X id μ (MeasureTheory.Measure.map X μ)"} +{"name":"ProbabilityTheory.IdentDistrib.comp_left","declaration":"/-- Composing identically distributed functions with a measurable embedding of conull range\ngives identically distributed functions. -/\ntheorem ProbabilityTheory.IdentDistrib.comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {i : δ → α} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range i) (hf : Measurable f) (hfg : ProbabilityTheory.IdentDistrib f g μ ν) : ProbabilityTheory.IdentDistrib (f ∘ i) g (MeasureTheory.Measure.comap i μ) ν"} +{"name":"ProbabilityTheory.identDistrib_comp_fst","declaration":"/-- A random variable is identically distributed to its lift to a product space (in the first factor). -/\ntheorem ProbabilityTheory.identDistrib_comp_fst {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] {X : Ω → α} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ'] : ProbabilityTheory.IdentDistrib (X ∘ Prod.fst) X (MeasureTheory.Measure.prod μ μ') μ"} +{"name":"ProbabilityTheory.independent_copies_two","declaration":"/-- For $X, Y$ random variables, one can find independent copies $X', Y'$ of $X, Y$. Version\nformulated in spaces with a canonical measures. -/\ntheorem ProbabilityTheory.independent_copies_two {α : Type u_7} {β : Type u_9} [MeasurableSpace α] [MeasurableSpace β] {Ω : Type u} {Ω' : Type v} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X : Ω → α} {Y : Ω' → β} (hX : Measurable X) (hY : Measurable Y) : ∃ Ω'' m'' X' Y',\n MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n Measurable X' ∧\n Measurable Y' ∧\n ProbabilityTheory.IndepFun X' Y' MeasureTheory.volume ∧\n ProbabilityTheory.IdentDistrib X' X MeasureTheory.volume MeasureTheory.volume ∧\n ProbabilityTheory.IdentDistrib Y' Y MeasureTheory.volume MeasureTheory.volume"} +{"name":"ProbabilityTheory.identDistrib_iff","declaration":"theorem ProbabilityTheory.identDistrib_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (f : α → γ) (g : β → γ) (μ : autoParam (MeasureTheory.Measure α) _auto✝) (ν : autoParam (MeasureTheory.Measure β) _auto✝) : ProbabilityTheory.IdentDistrib f g μ ν ↔\n AEMeasurable f μ ∧ AEMeasurable g ν ∧ MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map g ν"} +{"name":"ProbabilityTheory.IdentDistrib.comp_right","declaration":"/-- Composing identically distributed functions with a measurable embedding of conull range\ngives identically distributed functions. -/\ntheorem ProbabilityTheory.IdentDistrib.comp_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {i : δ → β} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : β) ∂ν, a ∈ Set.range i) (hg : Measurable g) (hfg : ProbabilityTheory.IdentDistrib f g μ ν) : ProbabilityTheory.IdentDistrib f (g ∘ i) μ (MeasureTheory.Measure.comap i ν)"} +{"name":"ProbabilityTheory.IdentDistrib.prod_mk","declaration":"theorem ProbabilityTheory.IdentDistrib.prod_mk {Ω : Type u_5} {Ω' : Type u_6} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ω'} {f : Ω → β} {g : Ω → β} {f' : Ω' → β} {g' : Ω' → β} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (hff' : ProbabilityTheory.IdentDistrib f f' μ ν) (hgg' : ProbabilityTheory.IdentDistrib g g' μ ν) (h : ProbabilityTheory.IndepFun f g μ) (h' : ProbabilityTheory.IndepFun f' g' ν) : ProbabilityTheory.IdentDistrib (fun x => (f x, g x)) (fun x => (f' x, g' x)) μ ν"} +{"name":"ProbabilityTheory.identDistrib_comp_right","declaration":"/-- A function is identically distributed to itself composed with a measurable embedding of conull\nrange. -/\ntheorem ProbabilityTheory.identDistrib_comp_right {α : Type u_1} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → γ} {i : δ → α} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range i) (hf : Measurable f) : ProbabilityTheory.IdentDistrib f (f ∘ i) μ (MeasureTheory.Measure.comap i μ)"} +{"name":"ProbabilityTheory.IdentDistrib.fst_id","declaration":"/-- The first projection in a product space with measure `μ.prod ν` is distributed like `μ`. -/\ntheorem ProbabilityTheory.IdentDistrib.fst_id {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure ν] : ProbabilityTheory.IdentDistrib Prod.fst id (MeasureTheory.Measure.prod μ ν) μ"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.Independence.Basic.jsonl b/PFR-declarations/PFR.Mathlib.Probability.Independence.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..bc930dee8c95843ed1b4f98e60abd3e74b9c5dbc --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.Independence.Basic.jsonl @@ -0,0 +1,28 @@ +{"name":"ProbabilityTheory.iIndepFun_iff'","declaration":"theorem ProbabilityTheory.iIndepFun_iff' {Ω : Type u_1} {ι : Type u_2} [MeasurableSpace Ω] {β : ι → Type u_11} (m : (i : ι) → MeasurableSpace (β i)) (f : (i : ι) → Ω → β i) (μ : MeasureTheory.Measure Ω) : ProbabilityTheory.iIndepFun m f μ ↔\n ∀ (s : Finset ι) ⦃f' : ι → Set Ω⦄,\n (∀ (i : ι), MeasurableSet (f' i)) → ↑↑μ (⋂ i ∈ s, f' i) = Finset.prod s fun i => ↑↑μ (f' i)"} +{"name":"ProbabilityTheory.EventuallyEq.finite_iInter","declaration":"/-- The new Mathlib tool `Finset.eventuallyEq_iInter` will supersede this result. -/\ntheorem ProbabilityTheory.EventuallyEq.finite_iInter {ι : Type u_14} {α : Type u_2} {l : Filter α} (s : Finset ι) {E : ι → Set α} {F : ι → Set α} (h : ∀ i ∈ s, E i =ᶠ[l] F i) : ⋂ i ∈ s, E i =ᶠ[l] ⋂ i ∈ s, F i"} +{"name":"ProbabilityTheory.iIndepFun.pi","declaration":"/-- If a family of functions `(i, j) ↦ f i j` is independent, then the family of function tuples\n`i ↦ (f i j)ⱼ` is independent. -/\ntheorem ProbabilityTheory.iIndepFun.pi {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_5} {κ : ι → Type u_6} [(i : ι) → Fintype (κ i)] {α : (i : ι) → κ i → Type u_7} {f : (i : ι) → (j : κ i) → Ω → α i j} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] (f_meas : ∀ (i : ι) (j : κ i), Measurable (f i j)) (hf : ProbabilityTheory.iIndepFun (fun ij => m ij.fst ij.snd) (fun ij => f ij.fst ij.snd) μ) : ProbabilityTheory.iIndepFun (fun i => MeasurableSpace.pi) (fun i ω j => f i j ω) μ"} +{"name":"ProbabilityTheory.iIndepFun.neg","declaration":"theorem ProbabilityTheory.iIndepFun.neg {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (i : ι) [Neg (α i)] [MeasurableNeg (α i)] [DecidableEq ι] (h : ProbabilityTheory.iIndepFun n f μ) : ProbabilityTheory.iIndepFun n (Function.update f i (-f i)) μ"} +{"name":"ProbabilityTheory.IndepFun.comp_right","declaration":"/-- Composing independent functions with a measurable embedding of conull range gives independent\nfunctions. -/\ntheorem ProbabilityTheory.IndepFun.comp_right {α : Type u_11} {Ω : Type u_1} {β : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_10} [MeasurableSpace Ω'] [MeasurableSpace α] [MeasurableSpace β] {f : Ω → α} {g : Ω → β} {i : Ω' → Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : Ω) ∂μ, a ∈ Set.range i) (hf : Measurable f) (hg : Measurable g) (hfg : ProbabilityTheory.IndepFun f g μ) : ProbabilityTheory.IndepFun (f ∘ i) (g ∘ i) (MeasureTheory.Measure.comap i μ)"} +{"name":"ProbabilityTheory.IndepFun.measureReal_inter_preimage_eq_mul","declaration":"theorem ProbabilityTheory.IndepFun.measureReal_inter_preimage_eq_mul {Ω : Type u_1} {β : Type u_10} {β' : Type u_11} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} (h : ProbabilityTheory.IndepFun f g μ) {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) : μ.real (f ⁻¹' s ∩ g ⁻¹' t) = μ.real (f ⁻¹' s) * μ.real (g ⁻¹' t)"} +{"name":"Finset.prod_univ_prod","declaration":"theorem Finset.prod_univ_prod {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [CommMonoid β] (f : (i : ι) → κ i → β) : (Finset.prod Finset.univ fun ij => f ij.fst ij.snd) =\n Finset.prod Finset.univ fun i => Finset.prod Finset.univ fun j => f i j"} +{"name":"ProbabilityTheory.iIndepFun_reindex_iff","declaration":"theorem ProbabilityTheory.iIndepFun_reindex_iff {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (g : ι' ≃ ι) : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' ⇑g) ((fun {x} => f) ∘' ⇑g) μ ↔ ProbabilityTheory.iIndepFun n f μ"} +{"name":"ProbabilityTheory.iIndepFun.reindex_of_injective","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_of_injective {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (h : ProbabilityTheory.iIndepFun n f μ) (g : ι' → ι) (hg : Function.Injective g) : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' g) ((fun {x} => f) ∘' g) μ"} +{"name":"ProbabilityTheory.iIndepFun.reindex_symm","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_symm {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (g : ι' ≃ ι) (h : ProbabilityTheory.iIndepFun n f μ) : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' ⇑g) ((fun {x} => f) ∘' ⇑g) μ"} +{"name":"ProbabilityTheory.indepFun_const","declaration":"/-- Random variables are always independent of constants. -/\ntheorem ProbabilityTheory.indepFun_const {α : Type u_11} {Ω : Type u_1} {β : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} [MeasurableSpace α] [MeasurableSpace β] [MeasureTheory.IsProbabilityMeasure μ] (c : α) : ProbabilityTheory.IndepFun f (fun x => c) μ"} +{"name":"ProbabilityTheory.iIndepFun.pi'","declaration":"/-- If a family of functions `(i, j) ↦ f i j` is independent, then the family of function tuples\n`i ↦ (f i j)ⱼ` is independent. -/\ntheorem ProbabilityTheory.iIndepFun.pi' {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_5} {κ : ι → Type u_6} [(i : ι) → Fintype (κ i)] {α : (i : ι) → κ i → Type u_7} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] {f : (ij : (i : ι) × κ i) → Ω → α ij.fst ij.snd} (f_meas : ∀ (i : (i : ι) × κ i), Measurable (f i)) (hf : ProbabilityTheory.iIndepFun (fun ij => m ij.fst ij.snd) f μ) : ProbabilityTheory.iIndepFun (fun _i => MeasurableSpace.pi) (fun i ω j => f { fst := i, snd := j } ω) μ"} +{"name":"Finset.sum_univ_sum","declaration":"theorem Finset.sum_univ_sum {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [AddCommMonoid β] (f : (i : ι) → κ i → β) : (Finset.sum Finset.univ fun ij => f ij.fst ij.snd) =\n Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => f i j"} +{"name":"Finset.sum_univ_sum'","declaration":"theorem Finset.sum_univ_sum' {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [AddCommMonoid β] (f : (i : ι) × κ i → β) : (Finset.sum Finset.univ fun ij => f ij) =\n Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => f { fst := i, snd := j }"} +{"name":"ProbabilityTheory.IndepFun.symm'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.IndepFun.symm' {γ : Type u_14} {β : Type u_15} {Ω : Type u_16} : ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace β} {x_2 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β}\n {g : Ω → γ}, ProbabilityTheory.IndepFun f g μ → ProbabilityTheory.IndepFun g f μ"} +{"name":"ProbabilityTheory.kernel.IndepFun.symm'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.kernel.IndepFun.symm' {Ω : Type u_14} {α : Type u_15} {β : Type u_16} {γ : Type u_17} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {x_3 : MeasurableSpace γ}\n {κ : ↥(ProbabilityTheory.kernel α Ω)} {f : Ω → β} {g : Ω → γ} {μ : MeasureTheory.Measure α},\n ProbabilityTheory.kernel.IndepFun f g κ μ → ProbabilityTheory.kernel.IndepFun g f κ μ"} +{"name":"ProbabilityTheory.IndepFun.measure_inter_preimage_eq_mul","declaration":"theorem ProbabilityTheory.IndepFun.measure_inter_preimage_eq_mul {Ω : Type u_1} {β : Type u_10} {β' : Type u_11} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} (h : ProbabilityTheory.IndepFun f g μ) {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) : ↑↑μ (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑μ (f ⁻¹' s) * ↑↑μ (g ⁻¹' t)"} +{"name":"ProbabilityTheory.iIndepFun.ae_eq","declaration":"/-- TODO: a kernel version of this theorem-/\ntheorem ProbabilityTheory.iIndepFun.ae_eq {Ω : Type u_13} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_14} {β : ι → Type u_15} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} {g : (i : ι) → Ω → β i} (hf_Indep : ProbabilityTheory.iIndepFun m f μ) (hfg : ∀ (i : ι), f i =ᶠ[MeasureTheory.Measure.ae μ] g i) : ProbabilityTheory.iIndepFun m g μ"} +{"name":"Finset.prod_univ_prod'","declaration":"theorem Finset.prod_univ_prod' {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [CommMonoid β] (f : (i : ι) × κ i → β) : (Finset.prod Finset.univ fun ij => f ij) =\n Finset.prod Finset.univ fun i => Finset.prod Finset.univ fun j => f { fst := i, snd := j }"} +{"name":"ProbabilityTheory.measurable_sigmaCurry","declaration":"theorem ProbabilityTheory.measurable_sigmaCurry {ι : Type u_1} {κ : ι → Type u_3} {α : (i : ι) → κ i → Type u_4} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] : Measurable Sigma.curry"} +{"name":"ProbabilityTheory.IndepFun.ae_eq'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.IndepFun.ae_eq' {β : Type u_11} {β' : Type u_12} {Ω : Type u_13} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f : Ω → β} {f' : Ω → β} {g : Ω → β'} {g' : Ω → β'} (hfg : ProbabilityTheory.IndepFun f g μ) (hf : f =ᶠ[MeasureTheory.Measure.ae μ] f') (hg : g =ᶠ[MeasureTheory.Measure.ae μ] g') : ProbabilityTheory.IndepFun f' g' μ"} +{"name":"ProbabilityTheory.iIndepFun.prod","declaration":"theorem ProbabilityTheory.iIndepFun.prod {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_8} {ι' : Type u_9} {α : ι → Type u_10} {n : (i : ι) → MeasurableSpace (α i)} {f : (i : ι) → Ω → α i} {hf : ∀ (i : ι), Measurable (f i)} {ST : ι' → Finset ι} (hS : Pairwise (Disjoint on ST)) (h : ProbabilityTheory.iIndepFun n f μ) : let β := fun k => (i : { x // x ∈ ST k }) → α ↑i;\nProbabilityTheory.iIndepFun (fun k => MeasurableSpace.pi) (fun k x i => f (↑i) x) μ"} +{"name":"ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map'","declaration":"theorem ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map' {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f : Ω → β} {g : Ω → β'} [MeasureTheory.IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : ProbabilityTheory.IndepFun f g μ ↔\n MeasureTheory.Measure.map (fun ω => (f ω, g ω)) μ =\n MeasureTheory.Measure.prod (MeasureTheory.Measure.map f μ) (MeasureTheory.Measure.map g μ)"} +{"name":"ProbabilityTheory.iIndepFun.reindex","declaration":"theorem ProbabilityTheory.iIndepFun.reindex {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (g : ι' ≃ ι) (h : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' ⇑g) ((fun {x} => f) ∘' ⇑g) μ) : ProbabilityTheory.iIndepFun n f μ"} +{"name":"ProbabilityTheory.iIndepFun.comp","declaration":"theorem ProbabilityTheory.iIndepFun.comp {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ι → Type u_13} {β : ι → Type u_14} [n : (i : ι) → MeasurableSpace (α i)] [m : (i : ι) → MeasurableSpace (β i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (h : ProbabilityTheory.iIndepFun n f μ) (g : (i : ι) → α i → β i) (hg : ∀ (i : ι), Measurable (g i)) : ProbabilityTheory.iIndepFun m (fun i => g i ∘ f i) μ"} +{"name":"ProbabilityTheory.indepFun_fst_snd","declaration":"theorem ProbabilityTheory.indepFun_fst_snd {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_10} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : ProbabilityTheory.IndepFun Prod.fst Prod.snd (MeasureTheory.Measure.prod μ μ')"} +{"name":"ProbabilityTheory.iIndepFun_iff_pi_map_eq_map","declaration":"theorem ProbabilityTheory.iIndepFun_iff_pi_map_eq_map {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_11} {β : ι → Type u_12} [Fintype ι] (f : (x : ι) → Ω → β x) [m : (x : ι) → MeasurableSpace (β x)] [MeasureTheory.IsProbabilityMeasure μ] (hf : ∀ (x : ι), Measurable (f x)) : ProbabilityTheory.iIndepFun m f μ ↔\n (MeasureTheory.Measure.pi fun i => MeasureTheory.Measure.map (f i) μ) = MeasureTheory.Measure.map (fun ω i => f i ω) μ"} +{"name":"ProbabilityTheory.iIndepFun.inv","declaration":"theorem ProbabilityTheory.iIndepFun.inv {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (i : ι) [Inv (α i)] [MeasurableInv (α i)] [DecidableEq ι] (h : ProbabilityTheory.iIndepFun n f μ) : ProbabilityTheory.iIndepFun n (Function.update f i (f i)⁻¹) μ"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.Independence.Conditional.jsonl b/PFR-declarations/PFR.Mathlib.Probability.Independence.Conditional.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..9cb4acb93b25995ef8c6383d3ac717fc6a63b279 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.Independence.Conditional.jsonl @@ -0,0 +1,9 @@ +{"name":"ProbabilityTheory.CondIndepFun.comp_right","declaration":"/-- Composing independent functions with a measurable embedding of conull range gives independent\nfunctions. -/\ntheorem ProbabilityTheory.CondIndepFun.comp_right {Ω : Type u_1} {Ω' : Type u_2} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [MeasurableSpace Ω] [MeasurableSpace Ω'] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure Ω} {f : Ω → α} {g : Ω → β} {h : Ω → γ} [MeasurableSingletonClass γ] {i : Ω' → Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : Ω) ∂μ, a ∈ Set.range i) (hf : Measurable f) (hg : Measurable g) (hh : Measurable h) (hfg : ProbabilityTheory.CondIndepFun f g h μ) : ProbabilityTheory.CondIndepFun (f ∘ i) (g ∘ i) (h ∘ i) (MeasureTheory.Measure.comap i μ)"} +{"name":"ProbabilityTheory.IndepFun.identDistrib_cond","declaration":"/-- If `A` is independent from `B`, then conditioning on an event given by `B` does not change\nthe distribution of `A`. -/\ntheorem ProbabilityTheory.IndepFun.identDistrib_cond {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n {B : Ω → β} [inst : MeasureTheory.IsProbabilityMeasure μ],\n ProbabilityTheory.IndepFun A B μ →\n ∀ {s : Set β},\n MeasurableSet s →\n Measurable A →\n Measurable B → ↑↑μ (B ⁻¹' s) ≠ 0 → ProbabilityTheory.IdentDistrib A A μ (ProbabilityTheory.cond μ (B ⁻¹' s))"} +{"name":"ProbabilityTheory.condIndep_copies","declaration":"/-- For $X, Y$ random variables, there exist conditionally independent trials $X_1, X_2, Y'$. -/\ntheorem ProbabilityTheory.condIndep_copies {Ω : Type u_1} {α : Type u} {β : Type u} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] (X : Ω → α) (Y : Ω → β) (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : ∃ Ω' mΩ' X₁ X₂ Y' ν,\n MeasureTheory.IsProbabilityMeasure ν ∧\n Measurable X₁ ∧\n Measurable X₂ ∧\n Measurable Y' ∧\n ProbabilityTheory.CondIndepFun X₁ X₂ Y' ν ∧\n ProbabilityTheory.IdentDistrib (⟨X₁, Y'⟩) (⟨X, Y⟩) ν μ ∧\n ProbabilityTheory.IdentDistrib (⟨X₂, Y'⟩) (⟨X, Y⟩) ν μ"} +{"name":"ProbabilityTheory.IndepFun.cond_right","declaration":"/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `B ∈ t` of positive probability. -/\ntheorem ProbabilityTheory.IndepFun.cond_right {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n {B : Ω → β},\n ProbabilityTheory.IndepFun A B μ →\n ∀ {t : Set β}, MeasurableSet t → Measurable B → ProbabilityTheory.IndepFun A B (ProbabilityTheory.cond μ (B ⁻¹' t))"} +{"name":"ProbabilityTheory.IndepFun.cond","declaration":"/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `A ∈ s ∩ B ∈ t` of positive probability. -/\ntheorem ProbabilityTheory.IndepFun.cond {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n {B : Ω → β},\n ProbabilityTheory.IndepFun A B μ →\n ∀ {s : Set α} {t : Set β},\n MeasurableSet s →\n MeasurableSet t →\n Measurable A → Measurable B → ProbabilityTheory.IndepFun A B (ProbabilityTheory.cond μ (A ⁻¹' s ∩ B ⁻¹' t))"} +{"name":"ProbabilityTheory.condIndep_copies'","declaration":"/-- For $X, Y$ random variables, there exist conditionally independent trials $X_1, X_2, Y'$. -/\ntheorem ProbabilityTheory.condIndep_copies' {Ω : Type u_1} {α : Type u} {β : Type u} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] (X : Ω → α) (Y : Ω → β) (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (p : α → β → Prop) (hp : Measurable (Function.uncurry p)) (hp' : ∀ᵐ (ω : Ω) ∂μ, p (X ω) (Y ω)) : ∃ Ω' mΩ' X₁ X₂ Y' ν,\n MeasureTheory.IsProbabilityMeasure ν ∧\n Measurable X₁ ∧\n Measurable X₂ ∧\n Measurable Y' ∧\n ProbabilityTheory.CondIndepFun X₁ X₂ Y' ν ∧\n ProbabilityTheory.IdentDistrib (⟨X₁, Y'⟩) (⟨X, Y⟩) ν μ ∧\n ProbabilityTheory.IdentDistrib (⟨X₂, Y'⟩) (⟨X, Y⟩) ν μ ∧\n (∀ (ω : Ω'), p (X₁ ω) (Y' ω)) ∧ ∀ (ω : Ω'), p (X₂ ω) (Y' ω)"} +{"name":"ProbabilityTheory.condIndepFun_iff","declaration":"theorem ProbabilityTheory.condIndepFun_iff {Ω : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure Ω} {f : Ω → α} {g : Ω → β} {h : Ω → γ} : ProbabilityTheory.CondIndepFun f g h μ ↔\n ∀ᵐ (z : γ) ∂MeasureTheory.Measure.map h μ, ProbabilityTheory.IndepFun f g (ProbabilityTheory.cond μ (h ⁻¹' {z}))"} +{"name":"ProbabilityTheory.IndepFun.cond_left","declaration":"/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `A ∈ s` of positive probability. -/\ntheorem ProbabilityTheory.IndepFun.cond_left {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n {B : Ω → β},\n ProbabilityTheory.IndepFun A B μ →\n ∀ {s : Set α}, MeasurableSet s → Measurable A → ProbabilityTheory.IndepFun A B (ProbabilityTheory.cond μ (A ⁻¹' s))"} +{"name":"ProbabilityTheory.CondIndepFun","declaration":"/-- The assertion that `f` and `g` are conditionally independent relative to `h`. -/\ndef ProbabilityTheory.CondIndepFun {Ω : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (f : Ω → α) (g : Ω → β) (h : Ω → γ) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : Prop"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.Independence.FourVariables.jsonl b/PFR-declarations/PFR.Mathlib.Probability.Independence.FourVariables.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..77f5d9f71fe85bcce17952eeb99e8e91bf03e4b7 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.Independence.FourVariables.jsonl @@ -0,0 +1,19 @@ +{"name":"ProbabilityTheory.iIndepFun.reindex_four_abdc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_abdc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, Z₄, Z₃] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_badc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_badc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₁, Z₄, Z₃] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_bdca","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bdca {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₄, Z₃, Z₁] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_adcb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_adcb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₄, Z₃, Z₂] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_dabc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dabc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₁, Z₂, Z₃] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_bcad","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bcad {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₃, Z₁, Z₄] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_bcda","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bcda {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₃, Z₄, Z₁] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_dbca","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dbca {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₂, Z₃, Z₁] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.κ","declaration":"def ProbabilityTheory.iIndepFun.κ : Fin 3 → Type"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_acdb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_acdb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₃, Z₄, Z₂] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_bacd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bacd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₁, Z₃, Z₄] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_acbd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_acbd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₃, Z₂, Z₄] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_cadb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_cadb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₃, Z₁, Z₄, Z₂] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_cbad","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_cbad {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₃, Z₂, Z₁, Z₄] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_abcd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_abcd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_bdac","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bdac {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₄, Z₁, Z₃] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_adbc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_adbc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₄, Z₂, Z₃] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.apply_two_last","declaration":"/-- If `(Z₁, Z₂, Z₃, Z₄)` are independent, so are `(Z₁, Z₂, φ Z₃ Z₄)` for any measurable `φ`. -/\ntheorem ProbabilityTheory.iIndepFun.apply_two_last {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) {phi : G → G → G} (hphi : Measurable (Function.uncurry phi)) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, fun ω => phi (Z₃ ω) (Z₄ ω)] MeasureTheory.volume"} +{"name":"ProbabilityTheory.iIndepFun.reindex_four_dbac","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dbac {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₂, Z₁, Z₃] MeasureTheory.volume"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.Independence.Kernel.jsonl b/PFR-declarations/PFR.Mathlib.Probability.Independence.Kernel.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6c612b1be10929c65b6a6e984fae52cec013b306 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.Independence.Kernel.jsonl @@ -0,0 +1 @@ +{"name":"ProbabilityTheory.kernel.IndepFun.ae_eq'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.kernel.IndepFun.ae_eq' {α : Type u_6} {Ω : Type u_5} {β : Type u_1} {β' : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ↥(ProbabilityTheory.kernel α Ω)} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f : Ω → β} {f' : Ω → β} {g : Ω → β'} {g' : Ω → β'} (hfg : ProbabilityTheory.kernel.IndepFun f g κ μ) (hf : ∀ᵐ (a : α) ∂μ, f =ᶠ[MeasureTheory.Measure.ae (κ a)] f') (hg : ∀ᵐ (a : α) ∂μ, g =ᶠ[MeasureTheory.Measure.ae (κ a)] g') : ProbabilityTheory.kernel.IndepFun f' g' κ μ"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.Kernel.Composition.jsonl b/PFR-declarations/PFR.Mathlib.Probability.Kernel.Composition.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..72903778268c864fbae6821b4473baa421e7cc20 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.Kernel.Composition.jsonl @@ -0,0 +1,21 @@ +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.reverse","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.reverse {γ : Type u_3} {S : Type u_7} {T : Type u_8} {β : Type u_9} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [Countable β] [MeasurableSpace β] [MeasurableSingletonClass β] [Countable γ] [MeasurableSpace γ] [MeasurableSingletonClass γ] {κ : ↥(ProbabilityTheory.kernel T (S × β × γ))} {μ : MeasureTheory.Measure T} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.reverse κ) μ"} +{"name":"ProbabilityTheory.kernel.reverse_reverse","declaration":"theorem ProbabilityTheory.kernel.reverse_reverse {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.reverse (ProbabilityTheory.kernel.reverse κ) = κ"} +{"name":"ProbabilityTheory.kernel.map_swapRight","declaration":"theorem ProbabilityTheory.kernel.map_swapRight {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ))) {f : γ × β → δ} (hf : Measurable f),\n ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.swapRight κ) f hf =\n ProbabilityTheory.kernel.map κ (f ∘ Prod.swap) ⋯"} +{"name":"ProbabilityTheory.kernel.instIsMarkovKernelProdInstMeasurableSpaceDeleteRight","declaration":"instance ProbabilityTheory.kernel.instIsMarkovKernelProdInstMeasurableSpaceDeleteRight {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ)))\n [inst_2 : ProbabilityTheory.IsMarkovKernel κ],\n ProbabilityTheory.IsMarkovKernel (ProbabilityTheory.kernel.deleteRight κ)"} +{"name":"ProbabilityTheory.kernel.deleteMiddle","declaration":"/-- Given a kernel taking values in a product of three spaces, forget the middle one. -/\ndef ProbabilityTheory.kernel.deleteMiddle {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : {x : MeasurableSpace α} →\n {x_1 : MeasurableSpace δ} →\n {β : Type u_9} →\n [inst : MeasurableSpace β] →\n [inst_1 : MeasurableSpace γ] → ↥(ProbabilityTheory.kernel α (β × γ × δ)) → ↥(ProbabilityTheory.kernel α (β × δ))"} +{"name":"ProbabilityTheory.kernel.swapRight_snd_reverse","declaration":"theorem ProbabilityTheory.kernel.swapRight_snd_reverse {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.swapRight (ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.reverse κ)) =\n ProbabilityTheory.kernel.deleteRight κ"} +{"name":"ProbabilityTheory.kernel.snd_deleteMiddle","declaration":"theorem ProbabilityTheory.kernel.snd_deleteMiddle {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.deleteMiddle κ) =\n ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.snd κ)"} +{"name":"ProbabilityTheory.kernel.deleteRight","declaration":"/-- Given a kernel taking values in a product of three spaces, forget the last variable. -/\ndef ProbabilityTheory.kernel.deleteRight {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : {x : MeasurableSpace α} →\n {x_1 : MeasurableSpace δ} →\n {β : Type u_9} →\n [inst : MeasurableSpace β] →\n [inst_1 : MeasurableSpace γ] → ↥(ProbabilityTheory.kernel α (β × γ × δ)) → ↥(ProbabilityTheory.kernel α (β × γ))"} +{"name":"ProbabilityTheory.kernel.reverse","declaration":"/-- Given a kernel taking values in a product of three spaces, reverse the order of the spaces. -/\ndef ProbabilityTheory.kernel.reverse {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : {x : MeasurableSpace α} →\n {x_1 : MeasurableSpace δ} →\n {β : Type u_9} →\n [inst : MeasurableSpace β] →\n [inst_1 : MeasurableSpace γ] →\n ↥(ProbabilityTheory.kernel α (β × γ × δ)) → ↥(ProbabilityTheory.kernel α (δ × γ × β))"} +{"name":"ProbabilityTheory.kernel.deleteMiddle_compProd","declaration":"theorem ProbabilityTheory.kernel.deleteMiddle_compProd {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (ξ : ↥(ProbabilityTheory.kernel α β)) [inst_2 : ProbabilityTheory.IsSFiniteKernel ξ]\n (κ : ↥(ProbabilityTheory.kernel (α × β) (γ × δ))) [inst_3 : ProbabilityTheory.IsSFiniteKernel κ],\n ProbabilityTheory.kernel.deleteMiddle (ProbabilityTheory.kernel.compProd ξ κ) =\n ProbabilityTheory.kernel.compProd ξ (ProbabilityTheory.kernel.snd κ)"} +{"name":"ProbabilityTheory.kernel.instIsMarkovKernelProdInstMeasurableSpaceDeleteMiddle","declaration":"instance ProbabilityTheory.kernel.instIsMarkovKernelProdInstMeasurableSpaceDeleteMiddle {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ)))\n [inst_2 : ProbabilityTheory.IsMarkovKernel κ],\n ProbabilityTheory.IsMarkovKernel (ProbabilityTheory.kernel.deleteMiddle κ)"} +{"name":"ProbabilityTheory.kernel.swapRight_deleteMiddle_reverse","declaration":"theorem ProbabilityTheory.kernel.swapRight_deleteMiddle_reverse {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.swapRight (ProbabilityTheory.kernel.deleteMiddle (ProbabilityTheory.kernel.reverse κ)) =\n ProbabilityTheory.kernel.deleteMiddle κ"} +{"name":"ProbabilityTheory.kernel.swapRight_deleteRight_reverse","declaration":"theorem ProbabilityTheory.kernel.swapRight_deleteRight_reverse {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.swapRight (ProbabilityTheory.kernel.deleteRight (ProbabilityTheory.kernel.reverse κ)) =\n ProbabilityTheory.kernel.snd κ"} +{"name":"ProbabilityTheory.kernel.fst_deleteRight","declaration":"theorem ProbabilityTheory.kernel.fst_deleteRight {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.fst (ProbabilityTheory.kernel.deleteRight κ) = ProbabilityTheory.kernel.fst κ"} +{"name":"ProbabilityTheory.kernel.map_map","declaration":"theorem ProbabilityTheory.kernel.map_map {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α β)) {f : β → γ} (hf : Measurable f) {g : γ → δ}\n (hg : Measurable g),\n ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.map κ f hf) g hg = ProbabilityTheory.kernel.map κ (g ∘ f) ⋯"} +{"name":"ProbabilityTheory.kernel.fst_deleteMiddle","declaration":"theorem ProbabilityTheory.kernel.fst_deleteMiddle {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.fst (ProbabilityTheory.kernel.deleteMiddle κ) = ProbabilityTheory.kernel.fst κ"} +{"name":"ProbabilityTheory.kernel.instIsMarkovKernelProdProdInstMeasurableSpaceInstMeasurableSpaceReverse","declaration":"instance ProbabilityTheory.kernel.instIsMarkovKernelProdProdInstMeasurableSpaceInstMeasurableSpaceReverse {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ)))\n [inst_2 : ProbabilityTheory.IsMarkovKernel κ], ProbabilityTheory.IsMarkovKernel (ProbabilityTheory.kernel.reverse κ)"} +{"name":"ProbabilityTheory.kernel.deleteRight_map_prod","declaration":"theorem ProbabilityTheory.kernel.deleteRight_map_prod {α : Type u_1} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {x_2 : MeasurableSpace ε} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α β)) {f : β → γ} {g : β → δ} {g' : β → ε}\n (hf : Measurable f) (hg : Measurable g) (hg' : Measurable g'),\n ProbabilityTheory.kernel.deleteRight (ProbabilityTheory.kernel.map κ (fun b => (f b, g b, g' b)) ⋯) =\n ProbabilityTheory.kernel.map κ (fun b => (f b, g b)) ⋯"} +{"name":"ProbabilityTheory.kernel.snd_deleteRight","declaration":"theorem ProbabilityTheory.kernel.snd_deleteRight {α : Type u_1} {γ : Type u_3} {δ : Type u_4} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α (β × γ × δ))),\n ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.deleteRight κ) =\n ProbabilityTheory.kernel.fst (ProbabilityTheory.kernel.snd κ)"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_reverse","declaration":"/-- Reversing preserves finite kernel support -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_reverse {α : Type u_1} {γ : Type u_3} : ∀ {x : MeasurableSpace α} {T : Type u_8} {β : Type u_9} [inst : MeasurableSpace T] [inst_1 : Countable β]\n [inst_2 : MeasurableSpace β] [inst_3 : MeasurableSingletonClass β] [inst_4 : Countable γ] [inst_5 : MeasurableSpace γ]\n [inst_6 : MeasurableSingletonClass γ] [inst_7 : Countable α] [inst_8 : MeasurableSingletonClass α]\n {κ : ↥(ProbabilityTheory.kernel T (α × β × γ))},\n ProbabilityTheory.kernel.FiniteKernelSupport κ →\n ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.reverse κ)"} +{"name":"ProbabilityTheory.kernel.deleteMiddle_map_prod","declaration":"theorem ProbabilityTheory.kernel.deleteMiddle_map_prod {α : Type u_1} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace δ} {x_2 : MeasurableSpace ε} {β : Type u_9} [inst : MeasurableSpace β]\n [inst_1 : MeasurableSpace γ] (κ : ↥(ProbabilityTheory.kernel α β)) {f : β → γ} {g : β → δ} {g' : β → ε}\n (hf : Measurable f) (hg : Measurable g) (hg' : Measurable g'),\n ProbabilityTheory.kernel.deleteMiddle (ProbabilityTheory.kernel.map κ (fun b => (f b, g b, g' b)) ⋯) =\n ProbabilityTheory.kernel.map κ (fun b => (f b, g' b)) ⋯"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.Kernel.Disintegration.jsonl b/PFR-declarations/PFR.Mathlib.Probability.Kernel.Disintegration.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ea3c3b286151d7a1488a165644d7711651c70465 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.Kernel.Disintegration.jsonl @@ -0,0 +1,59 @@ +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.fst","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.fst {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.fst κ) μ"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.prodMkRight","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.prodMkRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (ν : MeasureTheory.Measure U) [MeasureTheory.SFinite ν] : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.prodMkRight U κ)\n (MeasureTheory.Measure.prod μ ν)"} +{"name":"MeasureTheory.Measure.ae_of_ae_compProd","declaration":"theorem MeasureTheory.Measure.ae_of_ae_compProd {α : Type u_5} {β : Type u_6} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] {κ : ↥(ProbabilityTheory.kernel α β)} [ProbabilityTheory.IsSFiniteKernel κ] {p : α × β → Prop} (hp : ∀ᵐ (x : α × β) ∂MeasureTheory.Measure.compProd μ κ, p x) : ∀ᵐ (a : α) ∂μ, ∀ᵐ (b : β) ∂κ a, p (a, b)"} +{"name":"ProbabilityTheory.kernel.condKernel_map_prod_mk_left","declaration":"theorem ProbabilityTheory.kernel.condKernel_map_prod_mk_left {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {V : Type u_5} [Nonempty V] [MeasurableSpace V] [DiscreteMeasurableSpace V] [Countable V] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsFiniteMeasure μ] (f : S × U → V) : ⇑(ProbabilityTheory.kernel.condKernel\n (ProbabilityTheory.kernel.map κ (fun p => (p.1, f p))\n ⋯)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ))]\n ⇑(ProbabilityTheory.kernel.snd\n (ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.condKernel κ)\n (ProbabilityTheory.kernel.deterministic (fun x => f (x.1.2, x.2)) ⋯)))"} +{"name":"ProbabilityTheory.kernel.condKernel_compProd_ae_eq","declaration":"theorem ProbabilityTheory.kernel.condKernel_compProd_ae_eq {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsFiniteKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.kernel.condKernel\n (ProbabilityTheory.kernel.compProd κ η)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.compProd μ κ)]\n ⇑η"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_prodMkLeft","declaration":"/-- prodMkLeft preserves finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_prodMkLeft {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.prodMkLeft U κ)"} +{"name":"ProbabilityTheory.condKernel_condDistrib_ae_eq","declaration":"theorem ProbabilityTheory.condKernel_condDistrib_ae_eq {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.kernel.condKernel\n (ProbabilityTheory.condDistrib (fun a => (X a, Y a)) Z\n μ)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ)]\n ⇑(ProbabilityTheory.condDistrib Y (fun ω => (Z ω, X ω)) μ)"} +{"name":"ProbabilityTheory.kernel.aefiniteKernelSupport_iff","declaration":"theorem ProbabilityTheory.kernel.aefiniteKernelSupport_iff {S : Type u_2} {T : Type u_3} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {κ : ↥(ProbabilityTheory.kernel T S)} {μ : MeasureTheory.Measure T} : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ ↔\n ∃ κ', ProbabilityTheory.kernel.FiniteKernelSupport κ' ∧ ⇑κ' =ᶠ[MeasureTheory.Measure.ae μ] ⇑κ"} +{"name":"ProbabilityTheory.kernel.aefiniteKernelSupport_of_cond","declaration":"/-- Conditioning a kernel preserves finite kernel support. -/\ntheorem ProbabilityTheory.kernel.aefiniteKernelSupport_of_cond {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [hU : Nonempty U] (μ : MeasureTheory.Measure T) [MeasureTheory.IsFiniteMeasure μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) [ProbabilityTheory.IsFiniteKernel κ] : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.condKernel κ)\n (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ))"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_fst","declaration":"/-- Projecting a kernel to first coordinate preserves finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_fst {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.fst κ)"} +{"name":"ProbabilityTheory.kernel.condKernel_apply'","declaration":"theorem ProbabilityTheory.kernel.condKernel_apply' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [MeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsFiniteKernel κ] (x : T × S) (hx : ↑↑(κ x.1) (Prod.fst ⁻¹' {x.2}) ≠ 0) {s : Set U} (hs : MeasurableSet s) : ↑↑((ProbabilityTheory.kernel.condKernel κ) x) s = (↑↑(κ x.1) (Prod.fst ⁻¹' {x.2}))⁻¹ * ↑↑(κ x.1) ({x.2} ×ˢ s)"} +{"name":"MeasureTheory.Measure.compProd_apply_singleton","declaration":"theorem MeasureTheory.Measure.compProd_apply_singleton {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure T) [MeasureTheory.SFinite μ] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsSFiniteKernel κ] (t : T) (s : S) : ↑↑(MeasureTheory.Measure.compProd μ κ) {(t, s)} = ↑↑(κ t) {s} * ↑↑μ {t}"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.compProd","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.SFinite μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η (MeasureTheory.Measure.compProd μ κ)) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.compProd κ η) μ"} +{"name":"ProbabilityTheory.condDistrib_snd_ae_eq","declaration":"theorem ProbabilityTheory.condDistrib_snd_ae_eq {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.kernel.snd\n (ProbabilityTheory.condDistrib (fun a => (X a, Y a)) Z\n μ)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map Z μ)]\n ⇑(ProbabilityTheory.condDistrib Y Z μ)"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.snd","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.snd {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.snd κ) μ"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.finiteKernelSupport_mk","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.finiteKernelSupport_mk {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} [hS : Nonempty S] {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.AEFiniteKernelSupport.mk hκ)"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.map","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.map {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {μ : MeasureTheory.Measure T} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) {f : S → U} (hf : Measurable f) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.map κ f hf) μ"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_prodMkRight","declaration":"/-- prodMkRight preserves finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_prodMkRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.prodMkRight U κ)"} +{"name":"ProbabilityTheory.condDistrib_fst_ae_eq","declaration":"theorem ProbabilityTheory.condDistrib_fst_ae_eq {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.kernel.fst\n (ProbabilityTheory.condDistrib (fun a => (X a, Y a)) Z\n μ)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map Z μ)]\n ⇑(ProbabilityTheory.condDistrib X Z μ)"} +{"name":"instStandardBorelSpace_discreteMeasurableSpace","declaration":"instance instStandardBorelSpace_discreteMeasurableSpace {α : Type u_1} [MeasurableSpace α] [DiscreteMeasurableSpace α] [Countable α] : StandardBorelSpace α"} +{"name":"ProbabilityTheory.condDistrib_snd_of_ne_zero","declaration":"theorem ProbabilityTheory.condDistrib_snd_of_ne_zero {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (u : U) (hu : ↑↑μ (Z ⁻¹' {u}) ≠ 0) : (ProbabilityTheory.kernel.snd (ProbabilityTheory.condDistrib (fun a => (X a, Y a)) Z μ)) u =\n (ProbabilityTheory.condDistrib Y Z μ) u"} +{"name":"ProbabilityTheory.kernel.condKernel_compProd_apply","declaration":"theorem ProbabilityTheory.kernel.condKernel_compProd_apply {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [MeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsFiniteKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel η] (x : T × S) (hx : ↑↑(κ x.1) {x.2} ≠ 0) : (ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.compProd κ η)) x = η x"} +{"name":"ProbabilityTheory.condDistrib_unit_right","declaration":"theorem ProbabilityTheory.condDistrib_unit_right {Ω : Type u_1} {S : Type u_2} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] {X : Ω → S} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : ProbabilityTheory.condDistrib X (fun x => ()) μ = ProbabilityTheory.kernel.const Unit (MeasureTheory.Measure.map X μ)"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.ae_eq_mk","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.ae_eq_mk {S : Type u_2} {T : Type u_3} [Nonempty S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ⇑κ =ᶠ[MeasureTheory.Measure.ae μ] ⇑(ProbabilityTheory.kernel.AEFiniteKernelSupport.mk hκ)"} +{"name":"ProbabilityTheory.swap_condDistrib_ae_eq","declaration":"theorem ProbabilityTheory.swap_condDistrib_ae_eq {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.kernel.comap (ProbabilityTheory.condDistrib Y (fun a => (X a, Z a)) μ) Prod.swap\n ⋯) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ)]\n ⇑(ProbabilityTheory.condDistrib Y (fun ω => (Z ω, X ω)) μ)"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_deterministic","declaration":"/-- Deterministic kernels have finite kernel support.-/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_deterministic {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] [MeasurableSingletonClass U] (f : T × S → U) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.deterministic f ⋯)"} +{"name":"ProbabilityTheory.condDistrib_comp","declaration":"theorem ProbabilityTheory.condDistrib_comp {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (f : S → U) : ⇑(ProbabilityTheory.condDistrib (f ∘ X) Y μ) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map Y μ)]\n ⇑(ProbabilityTheory.kernel.map (ProbabilityTheory.condDistrib X Y μ) f ⋯)"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.comap_equiv","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.comap_equiv {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {μ : MeasureTheory.Measure T} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (f : U ≃ᵐ T) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.comap κ ⇑f ⋯)\n (MeasureTheory.Measure.comap (⇑f) μ)"} +{"name":"ProbabilityTheory.condDistrib_fst_of_ne_zero","declaration":"theorem ProbabilityTheory.condDistrib_fst_of_ne_zero {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (u : U) (hu : ↑↑μ (Z ⁻¹' {u}) ≠ 0) : (ProbabilityTheory.kernel.fst (ProbabilityTheory.condDistrib (fun a => (X a, Y a)) Z μ)) u =\n (ProbabilityTheory.condDistrib X Z μ) u"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.prod","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.prod {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.prod κ η) μ"} +{"name":"ProbabilityTheory.condDistrib_apply","declaration":"theorem ProbabilityTheory.condDistrib_apply {Ω : Type u_1} {S : Type u_2} {T : Type u_3} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (x : T) (hYx : ↑↑μ (Y ⁻¹' {x}) ≠ 0) : (ProbabilityTheory.condDistrib X Y μ) x = MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {x}))"} +{"name":"ProbabilityTheory.kernel.compProd_congr","declaration":"theorem ProbabilityTheory.kernel.compProd_congr {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {μ : MeasureTheory.Measure T} [MeasureTheory.SFinite μ] {κ : ↥(ProbabilityTheory.kernel T S)} {κ' : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel κ'] {η : ↥(ProbabilityTheory.kernel (T × S) U)} {η' : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.IsMarkovKernel η'] (hκ : ⇑κ =ᶠ[MeasureTheory.Measure.ae μ] ⇑κ') (hη : ⇑η =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.compProd μ κ)] ⇑η') : ⇑(ProbabilityTheory.kernel.compProd κ η) =ᶠ[MeasureTheory.Measure.ae μ] ⇑(ProbabilityTheory.kernel.compProd κ' η')"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_finite_range","declaration":"/-- Finite range implies finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_finite_range {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [Fintype S] (κ : ↥(ProbabilityTheory.kernel T S)) : ProbabilityTheory.kernel.FiniteKernelSupport κ"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.prodMkLeft","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.prodMkLeft {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (ν : MeasureTheory.Measure U) [MeasureTheory.SFinite μ] : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.prodMkLeft U κ)\n (MeasureTheory.Measure.prod ν μ)"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_snd","declaration":"/-- Projecting a kernel to second coordinate preserves finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_snd {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.snd κ)"} +{"name":"ProbabilityTheory.kernel.condKernel_compProd_apply'","declaration":"theorem ProbabilityTheory.kernel.condKernel_compProd_apply' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [MeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsFiniteKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel η] (x : T × S) (hx : ↑↑(κ x.1) {x.2} ≠ 0) {s : Set U} (hs : MeasurableSet s) : ↑↑((ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.compProd κ η)) x) s = ↑↑(η x) s"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_map","declaration":"/-- maps preserve finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_map {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) {f : S → U} (hf : Measurable f) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.map κ f hf)"} +{"name":"ProbabilityTheory.kernel.FiniteKernelSupport.aefiniteKernelSupport","declaration":"theorem ProbabilityTheory.kernel.FiniteKernelSupport.aefiniteKernelSupport {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (μ : MeasureTheory.Measure T) : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ"} +{"name":"ProbabilityTheory.condDistrib_eq_prod_of_indepFun","declaration":"theorem ProbabilityTheory.condDistrib_eq_prod_of_indepFun {Ω : Type u_1} {S : Type u_2} {T : Type u_3} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {V : Type u_5} [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {W : Ω → V} (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun (fun ω => (X ω, Z ω)) (fun ω => (Y ω, W ω)) μ) : ⇑(ProbabilityTheory.condDistrib (fun ω => (X ω, Y ω)) (fun ω => (Z ω, W ω))\n μ) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map (fun ω => (Z ω, W ω)) μ)]\n ⇑(ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight V (ProbabilityTheory.condDistrib X Z μ))\n (ProbabilityTheory.kernel.prodMkLeft U (ProbabilityTheory.condDistrib Y W μ)))"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_swapRight","declaration":"/-- Swapping a kernel right preserves finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_swapRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.swapRight κ)"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.isMarkovKernel_mk","declaration":"instance ProbabilityTheory.kernel.AEFiniteKernelSupport.isMarkovKernel_mk {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} [hS : Nonempty S] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.IsMarkovKernel (ProbabilityTheory.kernel.AEFiniteKernelSupport.mk hκ)"} +{"name":"ProbabilityTheory.condDistrib_const_unit","declaration":"theorem ProbabilityTheory.condDistrib_const_unit {Ω : Type u_1} {S : Type u_2} {T : Type u_3} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.kernel.condKernel\n (ProbabilityTheory.kernel.const Unit\n (MeasureTheory.Measure.map (fun ω => (X ω, Y ω))\n μ))) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map (fun ω => ((), X ω)) μ)]\n ⇑(ProbabilityTheory.kernel.prodMkLeft Unit (ProbabilityTheory.condDistrib Y X μ))"} +{"name":"ProbabilityTheory.condDistrib_apply'","declaration":"theorem ProbabilityTheory.condDistrib_apply' {Ω : Type u_1} {S : Type u_2} {T : Type u_3} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (x : T) (hYx : ↑↑μ (Y ⁻¹' {x}) ≠ 0) {s : Set S} (hs : MeasurableSet s) : ↑↑((ProbabilityTheory.condDistrib X Y μ) x) s = (↑↑μ (Y ⁻¹' {x}))⁻¹ * ↑↑μ (Y ⁻¹' {x} ∩ X ⁻¹' s)"} +{"name":"ProbabilityTheory.kernel.condKernel_apply","declaration":"theorem ProbabilityTheory.kernel.condKernel_apply {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [MeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsFiniteKernel κ] (x : T × S) (hx : ↑↑(κ x.1) (Prod.fst ⁻¹' {x.2}) ≠ 0) : (ProbabilityTheory.kernel.condKernel κ) x = (MeasureTheory.Measure.condKernel (κ x.1)) x.2"} +{"name":"ProbabilityTheory.map_compProd_condDistrib","declaration":"theorem ProbabilityTheory.map_compProd_condDistrib {Ω : Type u_1} {S : Type u_2} {U : Type u_4} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {X : Ω → S} {Z : Ω → U} (hX : Measurable X) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : MeasureTheory.Measure.compProd (MeasureTheory.Measure.map Z μ) (ProbabilityTheory.condDistrib X Z μ) =\n MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.swapRight","declaration":"theorem ProbabilityTheory.kernel.AEFiniteKernelSupport.swapRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.kernel.swapRight κ) μ"} +{"name":"ProbabilityTheory.condDistrib_ae_eq","declaration":"theorem ProbabilityTheory.condDistrib_ae_eq {Ω : Type u_1} {S : Type u_2} {T : Type u_3} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.condDistrib X Y μ) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.map Y μ)] fun x =>\n MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {x}))"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_prod","declaration":"/-- Products preserve finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_prod {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.prod κ η)"} +{"name":"ProbabilityTheory.kernel.FiniteKernelSupport","declaration":"/-- The analogue of FiniteSupport for probability kernels. -/\ndef ProbabilityTheory.kernel.FiniteKernelSupport {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) : Prop"} +{"name":"MeasureTheory.Measure.ae_of_compProd_eq_zero","declaration":"theorem MeasureTheory.Measure.ae_of_compProd_eq_zero {α : Type u_5} {β : Type u_6} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] {κ : ↥(ProbabilityTheory.kernel α β)} [ProbabilityTheory.IsSFiniteKernel κ] {s : Set (α × β)} (hs : ↑↑(MeasureTheory.Measure.compProd μ κ) s = 0) : ∀ᵐ (a : α) ∂μ, ↑↑(κ a) (Prod.mk a ⁻¹' s) = 0"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport","declaration":"def ProbabilityTheory.kernel.AEFiniteKernelSupport {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) : Prop"} +{"name":"MeasureTheory.lintegral_piecewise","declaration":"theorem MeasureTheory.lintegral_piecewise {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → ENNReal) (g : α → ENNReal) [(j : α) → Decidable (j ∈ s)] : ∫⁻ (a : α), Set.piecewise s f g a ∂μ = ∫⁻ (a : α) in s, f a ∂μ + ∫⁻ (a : α) in sᶜ, g a ∂μ"} +{"name":"ProbabilityTheory.kernel.disintegration","declaration":"theorem ProbabilityTheory.kernel.disintegration {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [MeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsFiniteKernel κ] : κ = ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.fst κ) (ProbabilityTheory.kernel.condKernel κ)"} +{"name":"ProbabilityTheory.kernel.condKernel_prod_ae_eq","declaration":"theorem ProbabilityTheory.kernel.condKernel_prod_ae_eq {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {μ : MeasureTheory.Measure T} (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsFiniteKernel κ] (η : ↥(ProbabilityTheory.kernel T U)) [ProbabilityTheory.IsMarkovKernel η] [MeasureTheory.IsFiniteMeasure μ] : ⇑(ProbabilityTheory.kernel.condKernel\n (ProbabilityTheory.kernel.prod κ η)) =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.compProd μ κ)]\n ⇑(ProbabilityTheory.kernel.prodMkRight S η)"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_comap","declaration":"/-- comaps preserve finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_comap {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) {f : U → T} (hf : Measurable f) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.comap κ f hf)"} +{"name":"ProbabilityTheory.kernel.local_support_of_finiteKernelSupport","declaration":"/-- Finite kernel support locally implies uniform finite kernel support. -/\ntheorem ProbabilityTheory.kernel.local_support_of_finiteKernelSupport {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] {κ : ↥(ProbabilityTheory.kernel T S)} (h : ProbabilityTheory.kernel.FiniteKernelSupport κ) (A : Finset T) : ∃ B, ∀ t ∈ A, ↑↑(κ t) (↑B)ᶜ = 0"} +{"name":"ProbabilityTheory.kernel.ae_eq_condKernel_of_compProd_eq","declaration":"theorem ProbabilityTheory.kernel.ae_eq_condKernel_of_compProd_eq {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [DiscreteMeasurableSpace S] [Countable T] [MeasurableSpace T] [DiscreteMeasurableSpace T] [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U] {μ : MeasureTheory.Measure T} (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsFiniteKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel η] [MeasureTheory.IsFiniteMeasure μ] (h : ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.fst κ) η = κ) : ⇑η =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ))]\n ⇑(ProbabilityTheory.kernel.condKernel κ)"} +{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_compProd","declaration":"/-- Composition-product preserves finite kernel support -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.compProd κ η)"} +{"name":"ProbabilityTheory.kernel.AEFiniteKernelSupport.mk","declaration":"def ProbabilityTheory.kernel.AEFiniteKernelSupport.mk {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} [hS : Nonempty S] {κ : ↥(ProbabilityTheory.kernel T S)} (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ↥(ProbabilityTheory.kernel T S)"} diff --git a/PFR-declarations/PFR.Mathlib.Probability.Kernel.MeasureCompProd.jsonl b/PFR-declarations/PFR.Mathlib.Probability.Kernel.MeasureCompProd.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..acf8293b5e4f482841e70a3cc287f68aee25c092 --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.Probability.Kernel.MeasureCompProd.jsonl @@ -0,0 +1,5 @@ +{"name":"MeasureTheory.Measure.comap_swap","declaration":"theorem MeasureTheory.Measure.comap_swap {α : Type u_5} {β : Type u_6} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure (α × β)) : MeasureTheory.Measure.comap Prod.swap μ = MeasureTheory.Measure.map Prod.swap μ"} +{"name":"ProbabilityTheory.kernel.map_prod_swap","declaration":"theorem ProbabilityTheory.kernel.map_prod_swap {α : Type u_5} {β : Type u_6} {γ : Type u_7} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (κ : ↥(ProbabilityTheory.kernel α β)) (η : ↥(ProbabilityTheory.kernel α γ)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] : ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) Prod.swap ⋯ = ProbabilityTheory.kernel.prod η κ"} +{"name":"ProbabilityTheory.kernel.compProd_deterministic_apply","declaration":"theorem ProbabilityTheory.kernel.compProd_deterministic_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSingletonClass γ] (κ : ↥(ProbabilityTheory.kernel α β)) [ProbabilityTheory.IsSFiniteKernel κ] {f : α × β → γ} (hf : Measurable f) (x : α) {s : Set (β × γ)} (hs : MeasurableSet s) : ↑↑((ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.deterministic f hf)) x) s =\n ↑↑(κ x) {b | (b, f (x, b)) ∈ s}"} +{"name":"ProbabilityTheory.kernel.comap_prod_swap","declaration":"theorem ProbabilityTheory.kernel.comap_prod_swap {δ : Type u_4} {mδ : MeasurableSpace δ} {α : Type u_5} {β : Type u_6} {γ : Type u_7} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (κ : ↥(ProbabilityTheory.kernel α β)) (η : ↥(ProbabilityTheory.kernel γ δ)) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] : ProbabilityTheory.kernel.comap\n (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight α η) (ProbabilityTheory.kernel.prodMkLeft γ κ))\n Prod.swap ⋯ =\n ProbabilityTheory.kernel.map\n (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight γ κ) (ProbabilityTheory.kernel.prodMkLeft α η))\n Prod.swap ⋯"} +{"name":"ProbabilityTheory.kernel.compProd_preimage_fst","declaration":"theorem ProbabilityTheory.kernel.compProd_preimage_fst {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (κ : ↥(ProbabilityTheory.kernel α β)) (η : ↥(ProbabilityTheory.kernel (α × β) γ)) [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsMarkovKernel η] {x : α} {s : Set β} (hs : MeasurableSet s) : ↑↑((ProbabilityTheory.kernel.compProd κ η) x) (Prod.fst ⁻¹' s) = ↑↑(κ x) s"} diff --git a/PFR-declarations/PFR.Mathlib.SetTheory.Cardinal.Finite.jsonl b/PFR-declarations/PFR.Mathlib.SetTheory.Cardinal.Finite.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..85e67c2179b2ed3e490734c1f6bff2cd6e4b806e --- /dev/null +++ b/PFR-declarations/PFR.Mathlib.SetTheory.Cardinal.Finite.jsonl @@ -0,0 +1 @@ +{"name":"Set.Nonempty.card_pos","declaration":"theorem Set.Nonempty.card_pos {α : Type u_1} {s : Set α} (hs : Set.Nonempty s) (hs' : Set.Finite s) : 0 < Nat.card ↑s"} diff --git a/PFR-declarations/PFR.SecondEstimate.jsonl b/PFR-declarations/PFR.SecondEstimate.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cc85ea672a8485128e97f55dc68b52d2c2780980 --- /dev/null +++ b/PFR-declarations/PFR.SecondEstimate.jsonl @@ -0,0 +1,3 @@ +{"name":"second_estimate","declaration":"/-- $$ I_2 \\leq 2 \\eta k + \\frac{2 \\eta (2 \\eta k - I_1)}{1 - \\eta}.$$ -/\ntheorem second_estimate {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : I[X₁ + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] ≤\n 2 * p.η * d[X₁ # X₂] + 2 * p.η * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂']) / (1 - p.η)"} +{"name":"rdist_of_sums_ge'","declaration":"/-- $$ d[X_1+\\tilde X_1; X_2+\\tilde X_2] \\geq k - \\frac{\\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$$\n-/\ntheorem rdist_of_sums_ge' {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : d[X₁ + X₁' # X₂ + X₂'] ≥ d[X₁ # X₂] - p.η * (d[X₁ # X₁] + d[X₂ # X₂]) / 2"} +{"name":"second_estimate_aux","declaration":"theorem second_estimate_aux {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : d[X₁ # X₁] + d[X₂ # X₂] ≤\n 2 * (d[X₁ # X₂] + (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂']) / (1 - p.η))"} diff --git a/PFR-declarations/PFR.Tactic.Finiteness.Attr.jsonl b/PFR-declarations/PFR.Tactic.Finiteness.Attr.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/PFR-declarations/PFR.Tactic.Finiteness.jsonl b/PFR-declarations/PFR.Tactic.Finiteness.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7f162a58055b38e9b264b6034510f64e448df59c --- /dev/null +++ b/PFR-declarations/PFR.Tactic.Finiteness.jsonl @@ -0,0 +1,5 @@ +{"name":"ENNReal.ofNat_ne_top","declaration":"theorem ENNReal.ofNat_ne_top (n : ℕ) [Nat.AtLeastTwo n] : OfNat.ofNat n ≠ ⊤"} +{"name":"ENNReal.add_ne_top'","declaration":"theorem ENNReal.add_ne_top' {a : ENNReal} {b : ENNReal} (ha : a ≠ ⊤) (hb : b ≠ ⊤) : a + b ≠ ⊤"} +{"name":"PositivityForAesop","declaration":"/-- A version of the positivity tactic for use by `aesop`. -/\ndef PositivityForAesop : Lean.Elab.Tactic.TacticM Unit"} +{"name":"finiteness","declaration":"/-- Tactic to solve goals of the form `*** < ∞` and (equivalently) `*** ≠ ∞` in the extended\nnonnegative reals (`ℝ≥0∞`). -/\ndef finiteness : Lean.ParserDescr"} +{"name":"ENNReal.inv_ne_top'","declaration":"theorem ENNReal.inv_ne_top' {a : ENNReal} (h : a ≠ 0) : a⁻¹ ≠ ⊤"} diff --git a/PFR-declarations/PFR.Tactic.RPowSimp.jsonl b/PFR-declarations/PFR.Tactic.RPowSimp.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..4918d1c136972171a65c9ae925886484052568b9 --- /dev/null +++ b/PFR-declarations/PFR.Tactic.RPowSimp.jsonl @@ -0,0 +1,55 @@ +{"name":"Mathlib.Tactic.RPowRing.elabRPowRingConv","declaration":"/-- Elaborator for the `rpow_ring` tactic. -/\ndef Mathlib.Tactic.RPowRing.elabRPowRingConv : Lean.Elab.Tactic.Tactic"} +{"name":"Mathlib.Tactic.RPowRing.instInhabitedSigmaQuotedConstMkStr1NilLevelExProd","declaration":"instance Mathlib.Tactic.RPowRing.instInhabitedSigmaQuotedConstMkStr1NilLevelExProd : Inhabited ((e : Q(ℝ)) × Mathlib.Tactic.RPowRing.ExProd e)"} +{"name":"Mathlib.Tactic.RPowRing.Context.ctx","declaration":"/-- A basically empty simp context, passed to the `simp` traversal in `RPowRing.rewrite`. -/\ndef Mathlib.Tactic.RPowRing.Context.ctx (self : Mathlib.Tactic.RPowRing.Context) : Lean.Meta.Simp.Context"} +{"name":"Mathlib.Tactic.RPowRing.atom_pow_pf'","declaration":"theorem Mathlib.Tactic.RPowRing.atom_pow_pf' {a : ℝ} {a' : ℝ} (p : a = a') : a = a ^ 1 * 1"} +{"name":"Mathlib.Tactic.RPowRing.M.run","declaration":"/-- Runs a tactic in the `RingNF.M` monad, given initial data:\n\n* `s`: a reference to the mutable state of `ring`, for persisting across calls.\n This ensures that atom ordering is used consistently.\n* `cfg`: the configuration options\n* `x`: the tactic to run\n-/\ndef Mathlib.Tactic.RPowRing.M.run {α : Type} (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RPowRing.Config) (x : Mathlib.Tactic.RPowRing.M α) : Lean.MetaM α"} +{"name":"Mathlib.Tactic.RPowRing.pow_congr","declaration":"theorem Mathlib.Tactic.RPowRing.pow_congr {a : ℝ} {a' : ℝ} {b : ℝ} {c : ℝ} : a = a' → a' ^ b = c → a ^ b = c"} +{"name":"Mathlib.Tactic.RPowRing.Result","declaration":"def Mathlib.Tactic.RPowRing.Result (E : Q(ℝ) → Type) (e : Q(ℝ)) : Type"} +{"name":"Mathlib.Tactic.RPowRing.ExProd.one","declaration":"/-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.\nIf `value` is not an integer, then `hyp` should be a proof of `(value.den : α) ≠ 0`. -/\nctor Mathlib.Tactic.RPowRing.ExProd.one : Mathlib.Tactic.RPowRing.ExProd q(1)"} +{"name":"Mathlib.Tactic.RPowRing.fix_cast₁","declaration":"theorem Mathlib.Tactic.RPowRing.fix_cast₁ : ↑(Int.ofNat 1) = 1"} +{"name":"Mathlib.Tactic.RPowRing.delab_ofNat","declaration":"def Mathlib.Tactic.RPowRing.delab_ofNat : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"Mathlib.Tactic.RPowRing.ExBase.pow","declaration":"/-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`\nand `e` is an `ExBase` representing a monomial expression in `ℕ` (it is a monomial instead of\na polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/\nctor Mathlib.Tactic.RPowRing.ExBase.pow (id : ℕ) (x : Q(ℝ)) (h : Q(0 < «$x»)) (e : Q(ℝ)) : Mathlib.Tactic.RPowRing.ExBase q(«$x» ^ «$e»)"} +{"name":"Mathlib.Tactic.RPowRing.evalPow","declaration":"def Mathlib.Tactic.RPowRing.evalPow {a : Q(ℝ)} (va : Mathlib.Tactic.RPowRing.ExProd a) (e : Q(ℝ)) : Option (Q(0 < «$a») × Mathlib.Tactic.RPowRing.Result Mathlib.Tactic.RPowRing.ExProd q(«$a» ^ «$e»))"} +{"name":"Mathlib.Tactic.RPowRing.atom_pf","declaration":"theorem Mathlib.Tactic.RPowRing.atom_pf (a : ℝ) : a = a * 1"} +{"name":"Mathlib.Tactic.RPowRing.Context","declaration":"/-- The read-only state of the `RPowRing` monad. -/\nstructure Mathlib.Tactic.RPowRing.Context : Type"} +{"name":"Mathlib.Tactic.RPowRing.eval","declaration":"opaque Mathlib.Tactic.RPowRing.eval (e : Q(ℝ)) : Mathlib.Tactic.AtomM (Mathlib.Tactic.RPowRing.Result Mathlib.Tactic.RPowRing.ExProd e)"} +{"name":"Mathlib.Tactic.RPowRing.npow_congr","declaration":"theorem Mathlib.Tactic.RPowRing.npow_congr {a : ℝ} {a' : ℝ} {c : ℝ} {b : ℕ} : a = a' → a' ^ ↑b = c → Monoid.npow b a = c"} +{"name":"Mathlib.Tactic.RPowRing.evalMul","declaration":"/-- Multiplies two monomials `va, vb` together to get a normalized result monomial.\n\n* `x * y = (x * y)` (for `x`, `y` coefficients)\n* `x * (b₁ * b₂) = b₁ * (b₂ * x)` (for `x` coefficient)\n* `(a₁ * a₂) * y = a₁ * (a₂ * y)` (for `y` coefficient)\n* `(x ^ ea * a₂) * (x ^ eb * b₂) = x ^ (ea + eb) * (a₂ * b₂)`\n (if `ea` and `eb` are identical except coefficient)\n* `(a₁ * a₂) * (b₁ * b₂) = a₁ * (a₂ * (b₁ * b₂))` (if `a₁.lt b₁`)\n* `(a₁ * a₂) * (b₁ * b₂) = b₁ * ((a₁ * a₂) * b₂)` (if not `a₁.lt b₁`)\n-/\nopaque Mathlib.Tactic.RPowRing.evalMul {a : Q(ℝ)} {b : Q(ℝ)} (va : Mathlib.Tactic.RPowRing.ExProd a) (vb : Mathlib.Tactic.RPowRing.ExProd b) : Mathlib.Tactic.RPowRing.Result Mathlib.Tactic.RPowRing.ExProd q(«$a» * «$b»)"} +{"name":"Mathlib.Tactic.RPowRing.ExBase.atom","declaration":"/-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.\nIf `value` is not an integer, then `hyp` should be a proof of `(value.den : α) ≠ 0`. -/\nctor Mathlib.Tactic.RPowRing.ExBase.atom (id : ℕ) (x : Q(ℝ)) : Mathlib.Tactic.RPowRing.ExBase x"} +{"name":"Mathlib.Tactic.RPowRing.rpowRingTarget","declaration":"/-- Use `rpow_ring` to rewrite the main goal. -/\ndef Mathlib.Tactic.RPowRing.rpowRingTarget (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RPowRing.Config) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"Mathlib.Tactic.RPowRing.pow_pos","declaration":"theorem Mathlib.Tactic.RPowRing.pow_pos {a : ℝ} {b : ℝ} {e : ℝ} (ha : 0 < a) (hb : 0 < b) : 0 < a ^ e * b"} +{"name":"Real.inv_rpow'","declaration":"theorem Real.inv_rpow' {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x⁻¹ ^ y = x ^ (-y)"} +{"name":"Real.pow_neg","declaration":"theorem Real.pow_neg (a : ℝ) (b : ℝ) (h : 0 ≤ a) : a ^ (-b) = a⁻¹ ^ b"} +{"name":"Mathlib.Tactic.RPowRing.instInhabitedConfig","declaration":"instance Mathlib.Tactic.RPowRing.instInhabitedConfig : Inhabited Mathlib.Tactic.RPowRing.Config"} +{"name":"Mathlib.Tactic.RPowRing.rewrite","declaration":"def Mathlib.Tactic.RPowRing.rewrite (parent : Lean.Expr) (root : optParam Bool true) : Mathlib.Tactic.RPowRing.M Lean.Meta.Simp.Result"} +{"name":"Mathlib.Tactic.RPowRing.pow_pf","declaration":"theorem Mathlib.Tactic.RPowRing.pow_pf {a : ℝ} {b : ℝ} {e₂ : ℝ} {b' : ℝ} {e₁ : ℝ} (ha : 0 < a) (hb : 0 < b) : b ^ e₂ = b' → (a ^ e₁ * b) ^ e₂ = a ^ (e₁ * e₂) * b'"} +{"name":"Mathlib.Tactic.RPowRing.mul_pp_pf_overlap","declaration":"theorem Mathlib.Tactic.RPowRing.mul_pp_pf_overlap {x : ℝ} {a₂ : ℝ} {b₂ : ℝ} {c : ℝ} (ea : ℝ) (eb : ℝ) (h : 0 < x) : a₂ * b₂ = c → x ^ ea * a₂ * (x ^ eb * b₂) = x ^ (ea + eb) * c"} +{"name":"Mathlib.Tactic.RPowRing.M","declaration":"/-- The monad for `RingNF` contains, in addition to the `AtomM` state,\na simp context for the main traversal and a simp function (which has another simp context)\nto simplify normalized polynomials. -/\ndef Mathlib.Tactic.RPowRing.M (α : Type) : Type"} +{"name":"Mathlib.Tactic.RPowRing.instInhabitedSigmaQuotedConstMkStr1NilLevelExBase","declaration":"instance Mathlib.Tactic.RPowRing.instInhabitedSigmaQuotedConstMkStr1NilLevelExBase : Inhabited ((e : Q(ℝ)) × Mathlib.Tactic.RPowRing.ExBase e)"} +{"name":"Mathlib.Tactic.RPowRing.Context.simp","declaration":"/-- A cleanup routine, which simplifies normalized polynomials to a more human-friendly\nformat. -/\ndef Mathlib.Tactic.RPowRing.Context.simp (self : Mathlib.Tactic.RPowRing.Context) : Lean.Meta.Simp.Result → Lean.Meta.SimpM Lean.Meta.Simp.Result"} +{"name":"Mathlib.Tactic.RPowRing.fix_cast₃","declaration":"theorem Mathlib.Tactic.RPowRing.fix_cast₃ {n : ℕ} [Nat.AtLeastTwo n] : ↑n = n"} +{"name":"Mathlib.Tactic.RPowRing.atom_pf'","declaration":"theorem Mathlib.Tactic.RPowRing.atom_pf' {a : ℝ} {a' : ℝ} (p : a = a') : a = a * 1"} +{"name":"Mathlib.Tactic.RPowRing.evalAtom","declaration":"/-- Evaluates an atom, an expression where `ring` can find no additional structure.\n\n* `a = a ^ 1 * 1 + 0`\n-/\ndef Mathlib.Tactic.RPowRing.evalAtom (e : Q(ℝ)) : Mathlib.Tactic.AtomM (Mathlib.Tactic.RPowRing.Result Mathlib.Tactic.RPowRing.ExProd e)"} +{"name":"Real.rpow_inv","declaration":"theorem Real.rpow_inv {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : (x ^ y)⁻¹ = x ^ (-y)"} +{"name":"Mathlib.Tactic.RPowRing.ExBase.id","declaration":"def Mathlib.Tactic.RPowRing.ExBase.id {e : Q(ℝ)} : Mathlib.Tactic.RPowRing.ExBase e → ℕ"} +{"name":"Mathlib.Tactic.RPowRing.tacticRpow_simp__","declaration":"def Mathlib.Tactic.RPowRing.tacticRpow_simp__ : Lean.ParserDescr"} +{"name":"Mathlib.Tactic.RPowRing.mul_pf_right","declaration":"theorem Mathlib.Tactic.RPowRing.mul_pf_right {a : ℝ} {b₂ : ℝ} {c : ℝ} (b₁ : ℝ) : a * b₂ = c → a * (b₁ * b₂) = b₁ * c"} +{"name":"Mathlib.Tactic.RPowRing.ExBase","declaration":"/-- A monomial, which is a product of powers of `ExBase` expressions,\nterminated by a (nonzero) constant coefficient.\n-/\ninductive Mathlib.Tactic.RPowRing.ExBase (e : Q(ℝ)) : Type"} +{"name":"Mathlib.Tactic.RPowRing.elabConfig","declaration":"/-- Function elaborating `RingNF.Config`. -/\ndef Mathlib.Tactic.RPowRing.elabConfig : Lean.Syntax → Lean.Elab.TermElabM Mathlib.Tactic.RPowRing.Config"} +{"name":"Mathlib.Tactic.RPowRing.Context.mk","declaration":"ctor Mathlib.Tactic.RPowRing.Context.mk (ctx : Lean.Meta.Simp.Context) (simp : Lean.Meta.Simp.Result → Lean.Meta.SimpM Lean.Meta.Simp.Result) : Mathlib.Tactic.RPowRing.Context"} +{"name":"Mathlib.Tactic.RPowRing.inv_congr","declaration":"theorem Mathlib.Tactic.RPowRing.inv_congr {a : ℝ} {a' : ℝ} {b : ℝ} : a = a' → a' ^ (-1) = b → a⁻¹ = b"} +{"name":"Mathlib.Tactic.RPowRing.rpowRing","declaration":"/-- Simplification tactic for expressions in the language of commutative (semi)rings,\nwhich rewrites all ring expressions into a normal form.\n* `rpow_ring!` will use a more aggressive reducibility setting to identify atoms.\n* `rpow_ring (config := cfg)` allows for additional configuration:\n * `red`: the reducibility setting (overridden by `!`)\n * `recursive`: if true, `rpow_ring` will also recurse into atoms\n* `rpow_ring` works as both a tactic and a conv tactic.\n In tactic mode, `rpow_ring at h` can be used to rewrite in a hypothesis.\n-/\ndef Mathlib.Tactic.RPowRing.rpowRing : Lean.ParserDescr"} +{"name":"Mathlib.Tactic.RPowRing.tacticRpow_ring!__","declaration":"/-- Simplification tactic for expressions in the language of commutative (semi)rings,\nwhich rewrites all ring expressions into a normal form.\n* `rpow_ring!` will use a more aggressive reducibility setting to identify atoms.\n* `rpow_ring (config := cfg)` allows for additional configuration:\n * `red`: the reducibility setting (overridden by `!`)\n * `recursive`: if true, `rpow_ring` will also recurse into atoms\n* `rpow_ring` works as both a tactic and a conv tactic.\n In tactic mode, `rpow_ring at h` can be used to rewrite in a hypothesis.\n-/\ndef Mathlib.Tactic.RPowRing.tacticRpow_ring!__ : Lean.ParserDescr"} +{"name":"Mathlib.Tactic.RPowRing.Config.mk","declaration":"ctor Mathlib.Tactic.RPowRing.Config.mk (red : Lean.Meta.TransparencyMode) : Mathlib.Tactic.RPowRing.Config"} +{"name":"Mathlib.Tactic.RPowRing.ExProd.mul","declaration":"/-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`\nand `e` is an `ExProd` representing a monomial expression in `ℕ` (it is a monomial instead of\na polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/\nctor Mathlib.Tactic.RPowRing.ExProd.mul {x : Q(ℝ)} {b : Q(ℝ)} : Mathlib.Tactic.RPowRing.ExBase x → Mathlib.Tactic.RPowRing.ExProd b → Mathlib.Tactic.RPowRing.ExProd q(«$x» * «$b»)"} +{"name":"Mathlib.Tactic.RPowRing.ExProd","declaration":"/-- A monomial, which is a product of powers of `ExBase` expressions,\nterminated by a (nonzero) constant coefficient.\n-/\ninductive Mathlib.Tactic.RPowRing.ExProd (e : Q(ℝ)) : Type"} +{"name":"Mathlib.Tactic.RPowRing.Config","declaration":"/-- Configuration for `ring_nf`. -/\nstructure Mathlib.Tactic.RPowRing.Config : Type"} +{"name":"Mathlib.Tactic.RPowRing.rpowRingLocalDecl","declaration":"/-- Use `rpow_ring` to rewrite hypothesis `h`. -/\ndef Mathlib.Tactic.RPowRing.rpowRingLocalDecl (s : IO.Ref Mathlib.Tactic.AtomM.State) (cfg : Mathlib.Tactic.RPowRing.Config) (fvarId : Lean.FVarId) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"Mathlib.Tactic.RPowRing.instReprConfig","declaration":"instance Mathlib.Tactic.RPowRing.instReprConfig : Repr Mathlib.Tactic.RPowRing.Config"} +{"name":"Mathlib.Tactic.RPowRing.Config.red","declaration":"/-- the reducibility setting to use when comparing atoms for defeq -/\ndef Mathlib.Tactic.RPowRing.Config.red (self : Mathlib.Tactic.RPowRing.Config) : Lean.Meta.TransparencyMode"} +{"name":"Mathlib.Tactic.RPowRing.fix_cast₂","declaration":"theorem Mathlib.Tactic.RPowRing.fix_cast₂ {n : ℕ} : ↑(Int.ofNat n) = ↑n"} +{"name":"Mathlib.Tactic.RPowRing.ExBase.toProd","declaration":"/-- Embed an exponent (an `ExBase, ExProd` pair) as an `ExProd` by multiplying by 1. -/\ndef Mathlib.Tactic.RPowRing.ExBase.toProd {a : Q(ℝ)} (va : Mathlib.Tactic.RPowRing.ExBase a) : Mathlib.Tactic.RPowRing.ExProd q(«$a» * 1)"} +{"name":"Mathlib.Tactic.RPowRing.instBEqConfig","declaration":"instance Mathlib.Tactic.RPowRing.instBEqConfig : BEq Mathlib.Tactic.RPowRing.Config"} +{"name":"Mathlib.Tactic.RPowRing.mul_pf_left","declaration":"theorem Mathlib.Tactic.RPowRing.mul_pf_left {a₂ : ℝ} {b : ℝ} {c : ℝ} (a₁ : ℝ) : a₂ * b = c → a₁ * a₂ * b = a₁ * c"} +{"name":"Mathlib.Tactic.RPowRing.rpowRingConv","declaration":"/-- Simplification tactic for expressions in the language of commutative (semi)rings,\nwhich rewrites all ring expressions into a normal form.\n* `rpow_ring!` will use a more aggressive reducibility setting to identify atoms.\n* `rpow_ring (config := cfg)` allows for additional configuration:\n * `red`: the reducibility setting (overridden by `!`)\n * `recursive`: if true, `rpow_ring` will also recurse into atoms\n* `rpow_ring` works as both a tactic and a conv tactic.\n In tactic mode, `rpow_ring at h` can be used to rewrite in a hypothesis.\n-/\ndef Mathlib.Tactic.RPowRing.rpowRingConv : Lean.ParserDescr"} +{"name":"Mathlib.Tactic.RPowRing.atom_pow_pf","declaration":"theorem Mathlib.Tactic.RPowRing.atom_pow_pf (a : ℝ) : a = a ^ 1 * 1"} diff --git a/PFR-declarations/PFR.TauFunctional.jsonl b/PFR-declarations/PFR.TauFunctional.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..bbdc05e73b94910ad4b52d2b5c7c3ade8fa962be --- /dev/null +++ b/PFR-declarations/PFR.TauFunctional.jsonl @@ -0,0 +1,24 @@ +{"name":"distance_ge_of_min","declaration":"/-- Let `X₁` and `X₂` be tau-minimizers associated to `p`, with $d[X_1,X_2]=k$, then\n$$ d[X'_1;X'_2] \\geq\n k - \\eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2] - d[X^0_2;X_2] )$$\nfor any $G$-valued random variables $X'_1,X'_2$.\n-/\ntheorem distance_ge_of_min {Ω : Type u_7} {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω'₁ : Type u_5} {Ω'₂ : Type u_6} [MeasureTheory.MeasureSpace Ω] [hΩ₁ : MeasureTheory.MeasureSpace Ω'₁] [hΩ₂ : MeasureTheory.MeasureSpace Ω'₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X'₁ : Ω'₁ → G} {X'₂ : Ω'₂ → G} (h : tau_minimizes p X₁ X₂) (h1 : Measurable X'₁) (h2 : Measurable X'₂) : d[X₁ # X₂] - p.η * (d[p.X₀₁ # X'₁] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X'₂] - d[p.X₀₂ # X₂]) ≤ d[X'₁ # X'₂]"} +{"name":"«termτ[_#_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termτ[_#_|_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"«termτ[_;_#_;_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termτ[_;_#_;_|_]».delab : Lean.PrettyPrinter.Delaborator.Delab"} +{"name":"distance_ge_of_min'","declaration":"/-- Version of `distance_ge_of_min` with the measures made explicit. -/\ntheorem distance_ge_of_min' {Ω : Type u_9} {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) [MeasureTheory.MeasureSpace Ω] {X₁ : Ω → G} {X₂ : Ω → G} {Ω'₁ : Type u_7} {Ω'₂ : Type u_8} (h : tau_minimizes p X₁ X₂) [MeasurableSpace Ω'₁] [MeasurableSpace Ω'₂] {μ : MeasureTheory.Measure Ω'₁} {μ' : MeasureTheory.Measure Ω'₂} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X'₁ : Ω'₁ → G} {X'₂ : Ω'₂ → G} (h1 : Measurable X'₁) (h2 : Measurable X'₂) : d[X₁ # X₂] - p.η * (d[p.X₀₁ ; MeasureTheory.volume # X'₁ ; μ] - d[p.X₀₁ # X₁]) -\n p.η * (d[p.X₀₂ ; MeasureTheory.volume # X'₂ ; μ'] - d[p.X₀₂ # X₂]) ≤\n d[X'₁ ; μ # X'₂ ; μ']"} +{"name":"refPackage.hmeas1","declaration":"def refPackage.hmeas1 {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (self : refPackage Ω₀₁ Ω₀₂ G) : Measurable self.X₀₁"} +{"name":"«termτ[_;_#_;_|_]»","declaration":"/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `τ[X₁ ; μ₁ # X₂ ; μ₂ | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `ℙ`, we\ncan use `τ[X₁ # X₂ | p]`\n-/\ndef «termτ[_;_#_;_|_]» : Lean.ParserDescr"} +{"name":"condRuzsaDistance_ge_of_min","declaration":"/-- For any $G$-valued random variables $X'_1,X'_2$ and random variables $Z,W$, one can lower\nbound $d[X'_1|Z;X'_2|W]$ by\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] ).$$\n-/\ntheorem condRuzsaDistance_ge_of_min {Ω : Type u_9} {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [Fintype G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω'₁ : Type u_5} {Ω'₂ : Type u_6} [MeasureTheory.MeasureSpace Ω] [hΩ₁ : MeasureTheory.MeasureSpace Ω'₁] [hΩ₂ : MeasureTheory.MeasureSpace Ω'₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X'₁ : Ω'₁ → G} {X'₂ : Ω'₂ → G} {S : Type u_7} {T : Type u_8} [MeasurableSingletonClass G] [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] (h : tau_minimizes p X₁ X₂) (h1 : Measurable X'₁) (h2 : Measurable X'₂) (Z : Ω'₁ → S) (W : Ω'₂ → T) (hZ : Measurable Z) (hW : Measurable W) : d[X₁ # X₂] - p.η * (d[p.X₀₁ # X'₁ | Z] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X'₂ | W] - d[p.X₀₂ # X₂]) ≤\n d[X'₁ | Z # X'₂ | W]"} +{"name":"tau","declaration":"/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `τ[X₁ ; μ₁ # X₂ ; μ₂ | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `ℙ`, we\ncan use `τ[X₁ # X₂ | p]`\n-/\ndef tau {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω₁ : Type u_7} {Ω₂ : Type u_8} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] (X₁ : Ω₁ → G) (X₂ : Ω₂ → G) (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) : ℝ"} +{"name":"refPackage.hη'","declaration":"def refPackage.hη' {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (self : refPackage Ω₀₁ Ω₀₂ G) : 8 * self.η ≤ 1"} +{"name":"refPackage.hmeas2","declaration":"def refPackage.hmeas2 {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (self : refPackage Ω₀₁ Ω₀₂ G) : Measurable self.X₀₂"} +{"name":"«termτ[_#_|_]»","declaration":"/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `τ[X₁ ; μ₁ # X₂ ; μ₂ | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `ℙ`, we\ncan use `τ[X₁ # X₂ | p]`\n-/\ndef «termτ[_#_|_]» : Lean.ParserDescr"} +{"name":"ProbabilityTheory.IdentDistrib.tau_minimizes","declaration":"/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\ntheorem ProbabilityTheory.IdentDistrib.tau_minimizes {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_7} {Ω' : Type u_8} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.MeasureSpace Ω'] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω' → G} {X₂' : Ω' → G} (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) : tau_minimizes p X₁ X₂ ↔ tau_minimizes p X₁' X₂'"} +{"name":"refPackage","declaration":"/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the τ functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe η parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage (Ω₀₁ : Type u_1) (Ω₀₂ : Type u_2) [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (G : Type uG) [MeasurableSpace G] : Type (max (max uG u_1) u_2)"} +{"name":"continuous_tau_restrict_probabilityMeasure","declaration":"theorem continuous_tau_restrict_probabilityMeasure {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [Fintype G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] : Continuous fun μ => τ[id ; ↑μ.1 # id ; ↑μ.2 | p]"} +{"name":"tau_minimizer_exists","declaration":"/-- A pair of random variables minimizing $τ$ exists. -/\ntheorem tau_minimizer_exists {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [Fintype G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) [MeasurableSingletonClass G] : ∃ Ω mΩ X₁ X₂,\n Measurable X₁ ∧ Measurable X₂ ∧ MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧ tau_minimizes p X₁ X₂"} +{"name":"is_tau_min","declaration":"theorem is_tau_min {Ω : Type u_7} {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω'₁ : Type u_5} {Ω'₂ : Type u_6} [MeasureTheory.MeasureSpace Ω] [hΩ₁ : MeasureTheory.MeasureSpace Ω'₁] [hΩ₂ : MeasureTheory.MeasureSpace Ω'₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X'₁ : Ω'₁ → G} {X'₂ : Ω'₂ → G} (h : tau_minimizes p X₁ X₂) (h1 : Measurable X'₁) (h2 : Measurable X'₂) : τ[X₁ # X₂ | p] ≤ τ[X'₁ # X'₂ | p]"} +{"name":"ProbabilityTheory.IdentDistrib.tau_eq","declaration":"/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\ntheorem ProbabilityTheory.IdentDistrib.tau_eq {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω₁ : Type u_3} {Ω₂ : Type u_4} {Ω'₁ : Type u_5} {Ω'₂ : Type u_6} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] [MeasurableSpace Ω'₁] [MeasurableSpace Ω'₂] {μ₁ : MeasureTheory.Measure Ω₁} {μ₂ : MeasureTheory.Measure Ω₂} {μ'₁ : MeasureTheory.Measure Ω'₁} {μ'₂ : MeasureTheory.Measure Ω'₂} {X₁ : Ω₁ → G} {X₂ : Ω₂ → G} {X'₁ : Ω'₁ → G} {X'₂ : Ω'₂ → G} (h₁ : ProbabilityTheory.IdentDistrib X₁ X'₁ μ₁ μ'₁) (h₂ : ProbabilityTheory.IdentDistrib X₂ X'₂ μ₂ μ'₂) : τ[X₁ ; μ₁ # X₂ ; μ₂ | p] = τ[X'₁ ; μ'₁ # X'₂ ; μ'₂ | p]"} +{"name":"refPackage.η","declaration":"def refPackage.η {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (self : refPackage Ω₀₁ Ω₀₂ G) : ℝ"} +{"name":"tau_minimizes","declaration":"/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [AddCommGroup G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_7} [MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) : Prop"} +{"name":"refPackage.mk","declaration":"ctor refPackage.mk {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (X₀₁ : Ω₀₁ → G) (X₀₂ : Ω₀₂ → G) (hmeas1 : Measurable X₀₁) (hmeas2 : Measurable X₀₂) (η : ℝ) (hη : 0 < η) (hη' : 8 * η ≤ 1) : refPackage Ω₀₁ Ω₀₂ G"} +{"name":"tau_min_exists_measure","declaration":"/-- A pair of measures minimizing $\\tau$ exists. -/\ntheorem tau_min_exists_measure {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [Fintype G] [MeasurableSpace G] (p : refPackage Ω₀₁ Ω₀₂ G) [MeasurableSingletonClass G] : ∃ μ,\n MeasureTheory.IsProbabilityMeasure μ.1 ∧\n MeasureTheory.IsProbabilityMeasure μ.2 ∧\n ∀ (ν₁ ν₂ : MeasureTheory.Measure G),\n MeasureTheory.IsProbabilityMeasure ν₁ →\n MeasureTheory.IsProbabilityMeasure ν₂ → τ[id ; μ.1 # id ; μ.2 | p] ≤ τ[id ; ν₁ # id ; ν₂ | p]"} +{"name":"refPackage.hη","declaration":"def refPackage.hη {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (self : refPackage Ω₀₁ Ω₀₂ G) : 0 < self.η"} +{"name":"refPackage.X₀₁","declaration":"/-- The first variable in a package. -/\ndef refPackage.X₀₁ {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (self : refPackage Ω₀₁ Ω₀₂ G) : Ω₀₁ → G"} +{"name":"refPackage.X₀₂","declaration":"/-- The second variable in a package. -/\ndef refPackage.X₀₂ {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {G : Type uG} [MeasurableSpace G] (self : refPackage Ω₀₁ Ω₀₂ G) : Ω₀₂ → G"} diff --git a/PFR-declarations/PFR.WeakPFR.jsonl b/PFR-declarations/PFR.WeakPFR.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ed35cc06cac2653fc7f31ae5b996230a3bdee65f --- /dev/null +++ b/PFR-declarations/PFR.WeakPFR.jsonl @@ -0,0 +1,32 @@ +{"name":"dimension_le_of_coset_cover","declaration":"theorem dimension_le_of_coset_cover {G : Type u_1} [AddCommGroup G] (A : Set G) (S : Submodule ℤ G) (v : G) (hA : ∀ a ∈ A, a - v ∈ S) : dimension A ≤ FiniteDimensional.finrank ℤ ↥S"} +{"name":"torsion_dist_shrinking","declaration":"/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\ntheorem torsion_dist_shrinking {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G] [Countable G] {Ω : Type u} {Ω' : Type u} [MeasurableSpace Ω] [MeasurableSpace Ω'] (X : Ω → G) (Y : Ω' → G) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2] [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H] (hG : AddMonoid.IsTorsionFree G) (φ : G →+ H) : H[⇑φ ∘ X ; μ] ≤ 10 * d[X ; μ # Y ; μ']"} +{"name":"weak_PFR_int","declaration":"/-- Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A-A\\rvert\\leq K\\lvert A\\rvert$.\nThere exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A' \\leq \\frac{40}{\\log 2} \\log K$.-/\ntheorem weak_PFR_int {G : Type u} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Set G} [Finite ↑A] (hnA : Set.Nonempty A) {K : ℝ} (hK : 0 < K) (hA : ↑(Nat.card ↑(A - A)) ≤ K * ↑(Nat.card ↑A)) : ∃ A' ⊆ A, ↑(Nat.card ↑A') ≥ K ^ (-17) * ↑(Nat.card ↑A) ∧ ↑(dimension A') ≤ 40 / Real.log 2 * Real.log K"} +{"name":"IsShift.sub_self_congr","declaration":"theorem IsShift.sub_self_congr {G : Type u_1} [AddCommGroup G] {A : Set G} {B : Set G} : IsShift A B → A - A = B - B"} +{"name":"weak_PFR","declaration":"/-- If $A\\subseteq \\mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\\leq \\log K$ then there exists a non-empty $A'\\subseteq A$ such that\n$\\lvert A'\\rvert\\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A'\\leq \\frac{40}{\\log 2} \\log K$. -/\ntheorem weak_PFR {G : Type u} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Set G} [Finite ↑A] {K : ℝ} (hA : Set.Nonempty A) (hK : 0 < K) (hdist : dᵤ[A # A] ≤ Real.log K) : ∃ A' ⊆ A, ↑(Nat.card ↑A') ≥ K ^ (-17) * ↑(Nat.card ↑A) ∧ ↑(dimension A') ≤ 40 / Real.log 2 * Real.log K"} +{"name":"weak_PFR_asymm","declaration":"/-- If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that\n\\[\\log\\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A'\\rvert\\lvert B'\\rvert}\\leq 34 d[U_A;U_B]\\]\nsuch that $\\max(\\dim A',\\dim B')\\leq \\frac{40}{\\log 2} d[U_A;U_B]$. -/\ntheorem weak_PFR_asymm {G : Type u} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] (hA : Set.Nonempty A) (hB : Set.Nonempty B) : WeakPFRAsymmConclusion A B"} +{"name":"app_ent_PFR","declaration":"theorem app_ent_PFR {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (X : Ω → G) (Y : Ω' → G) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (α : ℝ) (hent : 20 * d[X ; μ # Y ; μ'] < α * (H[X ; μ] + H[Y ; μ'])) (hX : Measurable X) (hY : Measurable Y) : ∃ H,\n Real.log ↑(Nat.card ↥H) < (1 + α) / 2 * (H[X ; μ] + H[Y ; μ']) ∧\n H[⇑(QuotientAddGroup.mk' H) ∘ X ; μ] + H[⇑(QuotientAddGroup.mk' H) ∘ Y ; μ'] < α * (H[X ; μ] + H[Y ; μ'])"} +{"name":"wlog_notInCoset","declaration":"/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\ntheorem wlog_notInCoset {G : Type u_1} [AddCommGroup G] {A : Set G} {B : Set G} (hA : Set.Nonempty A) (hB : Set.Nonempty B) : ∃ G' A' B', IsShift A (Subtype.val '' A') ∧ IsShift B (Subtype.val '' B') ∧ NotInCoset A' B'"} +{"name":"instMeasurableSingletonClassQuotientAddSubgroupToAddGroupInstHasQuotientAddSubgroupMeasurableSpace","declaration":"instance instMeasurableSingletonClassQuotientAddSubgroupToAddGroupInstHasQuotientAddSubgroupMeasurableSpace {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G) : MeasurableSingletonClass (G ⧸ H)"} +{"name":"third_iso","declaration":"/-- A version of the third isomorphism theorem: if G₂ ≤ G and H' is a subgroup of G⧸G₂, then there is a canonical isomorphism between H⧸H' and G⧸N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\ntheorem third_iso {G : Type u} [AddCommGroup G] {G₂ : AddSubgroup G} (H' : AddSubgroup (G ⧸ G₂)) : let H := G ⧸ G₂;\nlet φ := QuotientAddGroup.mk' G₂;\nlet N := AddSubgroup.comap φ H';\n∃ e, ∀ (x : G), e ((QuotientAddGroup.mk' H') (φ x)) = (QuotientAddGroup.mk' N) x"} +{"name":"exists_coset_cover","declaration":"theorem exists_coset_cover {G : Type u_1} [AddCommGroup G] (A : Set G) : ∃ d S v, FiniteDimensional.finrank ℤ ↥S = d ∧ ∀ a ∈ A, a - v ∈ S"} +{"name":"dimension_le_rank","declaration":"theorem dimension_le_rank {G : Type u_1} [AddCommGroup G] [Module.Finite ℤ G] (A : Set G) : dimension A ≤ FiniteDimensional.finrank ℤ G"} +{"name":"torsion_free","declaration":"/-- A free Z-module is torsion-free. Move to Mathlib? -/\ntheorem torsion_free {G : Type u} [AddCommGroup G] [Module.Free ℤ G] : AddMonoid.IsTorsionFree G"} +{"name":"dimension","declaration":"/-- The dimension of the affine span over `ℤ` of a subset of an additive group. -/\ndef dimension {G : Type u_1} [AddCommGroup G] (A : Set G) : ℕ"} +{"name":"app_ent_PFR'","declaration":"/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\ntheorem app_ent_PFR' {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_1} {Ω' : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.MeasureSpace Ω'] (X : Ω → G) (Y : Ω' → G) [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {α : ℝ} (hent : 20 * d[X # Y] < α * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) : ∃ H,\n Real.log ↑(Nat.card ↥H) < (1 + α) / 2 * (H[X] + H[Y]) ∧\n H[⇑(QuotientAddGroup.mk' H) ∘ X] + H[⇑(QuotientAddGroup.mk' H) ∘ Y] < α * (H[X] + H[Y])"} +{"name":"PFR_projection'","declaration":"/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + α) / (2 * (1 - α)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\ntheorem PFR_projection' {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (X : Ω → G) (Y : Ω' → G) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (α : ℝ) (hX : Measurable X) (hY : Measurable Y) (αpos : 0 < α) (αone : α < 1) : ∃ H,\n Real.log ↑(Nat.card ↥H) ≤ (1 + α) / (2 * (1 - α)) * (H[X ; μ] + H[Y ; μ']) ∧\n α * (H[⇑(QuotientAddGroup.mk' H) ∘ X ; μ] + H[⇑(QuotientAddGroup.mk' H) ∘ Y ; μ']) ≤\n 20 * d[⇑(QuotientAddGroup.mk' H) ∘ X ; μ # ⇑(QuotientAddGroup.mk' H) ∘ Y ; μ']"} +{"name":"sum_prob_preimage","declaration":"theorem sum_prob_preimage {G : Type u_1} {H : Type u_2} {X : Finset H} {A : Set G} [Finite ↑A] {φ : ↑A → { x // x ∈ X }} {A_ : H → Set G} (hA : Set.Nonempty A) (hφ : ∀ (x : { x // x ∈ X }), A_ ↑x = Subtype.val '' (φ ⁻¹' {x})) : (Finset.sum X fun x => ↑(Nat.card ↑(A_ x)) / ↑(Nat.card ↑A)) = 1"} +{"name":"Finsupp.mapRange_surjective","declaration":"/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\ntheorem Finsupp.mapRange_surjective {α : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M → N) (hf : f 0 = 0) (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange f hf)"} +{"name":"weak_PFR_quotient_prelim","declaration":"/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\ntheorem weak_PFR_quotient_prelim {G : Type u} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G] : let H := G ⧸ AddMonoidHom.range (zsmulAddGroupHom 2);\nElementaryAddCommGroup H 2 ∧ Finite H ∧ Nat.card H = 2 ^ FiniteDimensional.finrank ℤ G"} +{"name":"WeakPFRAsymmConclusion","declaration":"/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion {G : Type u} [AddCommGroup G] [MeasurableSpace G] (A : Set G) (B : Set G) : Prop"} +{"name":"weak_PFR_asymm_prelim","declaration":"/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} ≤ 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 ≤ \\log |G/N| + 40 d[U_A; U_B].$$\n -/\ntheorem weak_PFR_asymm_prelim {G : Type u} [AddCommGroup G] [Module.Free ℤ G] [Module.Finite ℤ G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] (hnA : Set.Nonempty A) (hnB : Set.Nonempty B) : ∃ N x y Ax By,\n Set.Nonempty Ax ∧\n Set.Nonempty By ∧\n Set.Finite Ax ∧\n Set.Finite By ∧\n Ax = {z | z ∈ A ∧ (QuotientAddGroup.mk' N) z = x} ∧\n By = {z | z ∈ B ∧ (QuotientAddGroup.mk' N) z = y} ∧\n Real.log 2 * ↑(FiniteDimensional.finrank ℤ G) ≤ Real.log ↑(Nat.card (G ⧸ N)) + 40 * dᵤ[A # B] ∧\n Real.log ↑(Nat.card ↑A) + Real.log ↑(Nat.card ↑B) - Real.log ↑(Nat.card ↑Ax) -\n Real.log ↑(Nat.card ↑By) ≤\n 34 * (dᵤ[A # B] - dᵤ[Ax # By])"} +{"name":"four_logs","declaration":"theorem four_logs {a : ℝ} {b : ℝ} {c : ℝ} {d : ℝ} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : Real.log (a * b / (c * d)) = Real.log a + Real.log b - Real.log c - Real.log d"} +{"name":"dimension_of_shift","declaration":"/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\ntheorem dimension_of_shift {G : Type u} [AddCommGroup G] {H : AddSubgroup G} (A : Set ↥H) (x : G) : dimension ((fun a => ↑a + x) '' A) ≤ dimension A"} +{"name":"single","declaration":"theorem single {Ω : Type u} [MeasurableSpace Ω] [DiscreteMeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {A : Set Ω} {z : Ω} (hA : μ.real A = 1) (hz : μ.real {z} > 0) : z ∈ A"} +{"name":"NotInCoset","declaration":"/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset {G : Type u_1} [AddCommGroup G] (A : Set G) (B : Set G) : Prop"} +{"name":"IsShift","declaration":"def IsShift {G : Type u_1} [AddCommGroup G] (A : Set G) (B : Set G) : Prop"} +{"name":"PFR_projection","declaration":"/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\ntheorem PFR_projection {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_1} {Ω' : Type u_2} [MeasurableSpace Ω] [MeasurableSpace Ω'] (X : Ω → G) (Y : Ω' → G) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) : ∃ H,\n Real.log ↑(Nat.card ↥H) ≤ 2 * (H[X ; μ] + H[Y ; μ']) ∧\n H[⇑(QuotientAddGroup.mk' H) ∘ X ; μ] + H[⇑(QuotientAddGroup.mk' H) ∘ Y ; μ'] ≤\n 34 * d[⇑(QuotientAddGroup.mk' H) ∘ X ; μ # ⇑(QuotientAddGroup.mk' H) ∘ Y ; μ']"} +{"name":"conclusion_transfers","declaration":"theorem conclusion_transfers {G : Type u} [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Set G} {B : Set G} (G' : AddSubgroup G) (A' : Set ↥G') (B' : Set ↥G') (hA : IsShift A (Subtype.val '' A')) (hB : IsShift B (Subtype.val '' B')) [Finite ↑A'] [Finite ↑B'] (hA' : Set.Nonempty A') (hB' : Set.Nonempty B') (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B"} +{"name":"single_fibres","declaration":"/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\ntheorem single_fibres {G : Type u} {H : Type u} {Ω : Type u} {Ω' : Type u} [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H] [MeasureTheory.MeasureSpace Ω] [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (φ : G →+ H) {A : Set G} {B : Set G} [Finite ↑A] [Finite ↑B] {UA : Ω → G} {UB : Ω' → G} (hA : Set.Nonempty A) (hB : Set.Nonempty B) (hUA' : Measurable UA) (hUB' : Measurable UB) (hUA : ProbabilityTheory.IsUniform A UA MeasureTheory.volume) (hUB : ProbabilityTheory.IsUniform B UB MeasureTheory.volume) (hUA_mem : ∀ (ω : Ω), UA ω ∈ A) (hUB_mem : ∀ (ω : Ω'), UB ω ∈ B) : ∃ x y Ax By,\n Ax = A ∩ φ.toFun ⁻¹' {x} ∧\n By = B ∩ φ.toFun ⁻¹' {y} ∧\n Set.Nonempty Ax ∧\n Set.Nonempty By ∧\n d[φ.toFun ∘ UA # φ.toFun ∘ UB] *\n Real.log (↑(Nat.card ↑A) * ↑(Nat.card ↑B) / (↑(Nat.card ↑Ax) * ↑(Nat.card ↑By))) ≤\n (H[φ.toFun ∘ UA] + H[φ.toFun ∘ UB]) * (d[UA # UB] - dᵤ[Ax # By])"} +{"name":"IsShift.card_congr","declaration":"theorem IsShift.card_congr {G : Type u_1} [AddCommGroup G] {A : Set G} {B : Set G} : IsShift A B → Nat.card ↑A = Nat.card ↑B"} +{"name":"not_in_coset","declaration":"/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G : Type u} [AddCommGroup G] (A : Set G) (B : Set G) : Prop"} +{"name":"torsion_free_doubling","declaration":"/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\ntheorem torsion_free_doubling {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G] [Countable G] {Ω : Type u} {Ω' : Type u} [MeasurableSpace Ω] [MeasurableSpace Ω'] (X : Ω → G) (Y : Ω' → G) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) : d[X ; μ # Y + Y ; μ'] ≤ 5 * d[X ; μ # Y ; μ']"} diff --git a/PFR-declarations/PFR.jsonl b/PFR-declarations/PFR.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/hep-declarations/HepLean.AnomalyCancellation.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..334f6acf994c52593ab55ce504e819c1a4b470cf --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.Basic.jsonl @@ -0,0 +1,63 @@ +{"name":"ACCSystemCharges.Charges","declaration":"/-- The charges as functions from `Fin χ.numberCharges → ℚ`. -/\ndef ACCSystemCharges.Charges (χ : ACCSystemCharges) : Type"} +{"name":"ACCSystemLinear.LinSols.linearSol","declaration":"/-- The condition that the charge satisfies the linear ACCs. -/\ndef ACCSystemLinear.LinSols.linearSol {χ : ACCSystemLinear} (self : ACCSystemLinear.LinSols χ) (i : Fin χ.numberLinear) : (χ.linearACCs i) self.val = 0"} +{"name":"ACCSystem.Hom.anomalyFree","declaration":"/-- The map between solutions. -/\ndef ACCSystem.Hom.anomalyFree {χ : ACCSystem} {η : ACCSystem} (self : ACCSystem.Hom χ η) : ACCSystem.Sols χ → ACCSystem.Sols η"} +{"name":"ACCSystem.Sols.ext","declaration":"/-- Two solutions are equal if the underlying charges are equal. -/\ntheorem ACCSystem.Sols.ext {χ : ACCSystem} {S : ACCSystem.Sols χ} {T : ACCSystem.Sols χ} (h : S.val = T.val) : S = T"} +{"name":"ACCSystemQuad.quadSolsInclLinSols","declaration":"/-- The inclusion of quadratic solutions into linear solutions. -/\ndef ACCSystemQuad.quadSolsInclLinSols (χ : ACCSystemQuad) : ACCSystemQuad.QuadSols χ →[ℚ] ACCSystemLinear.LinSols χ.toACCSystemLinear"} +{"name":"ACCSystem.Sols.mk","declaration":"ctor ACCSystem.Sols.mk {χ : ACCSystem} (toQuadSols : ACCSystemQuad.QuadSols χ.toACCSystemQuad) (cubicSol : χ.cubicACC toQuadSols.val = 0) : ACCSystem.Sols χ"} +{"name":"ACCSystemQuad.QuadSols.mk","declaration":"ctor ACCSystemQuad.QuadSols.mk {χ : ACCSystemQuad} (toLinSols : ACCSystemLinear.LinSols χ.toACCSystemLinear) (quadSol : ∀ (i : Fin χ.numberQuadratic), (χ.quadraticACCs i) toLinSols.val = 0) : ACCSystemQuad.QuadSols χ"} +{"name":"ACCSystem.solsInclLinSols","declaration":"/-- The inclusion of `Sols` into `LinSols`. -/\ndef ACCSystem.solsInclLinSols (χ : ACCSystem) : ACCSystem.Sols χ →[ℚ] ACCSystemLinear.LinSols χ.toACCSystemLinear"} +{"name":"ACCSystem.Hom.commute","declaration":"/-- The condition that the map commutes with the relevant inclusions. -/\ndef ACCSystem.Hom.commute {χ : ACCSystem} {η : ACCSystem} (self : ACCSystem.Hom χ η) : ⇑self.charges ∘ ⇑(ACCSystem.solsIncl χ) = ⇑(ACCSystem.solsIncl η) ∘ self.anomalyFree"} +{"name":"ACCSystem.IsSolution","declaration":"/-- A charge `S` is a solution if it extends to a solution. -/\ndef ACCSystem.IsSolution (χ : ACCSystem) (S : ACCSystemCharges.Charges χ.toACCSystemCharges) : Prop"} +{"name":"ACCSystemLinear.LinSols.val","declaration":"/-- The underlying charge. -/\ndef ACCSystemLinear.LinSols.val {χ : ACCSystemLinear} (self : ACCSystemLinear.LinSols χ) : ACCSystemCharges.Charges χ.toACCSystemCharges"} +{"name":"ACCSystem.solsMulAction","declaration":"/-- An instance giving the properties and structures to define an action of `ℚ` on `Sols`. -/\ninstance ACCSystem.solsMulAction (χ : ACCSystem) : MulAction ℚ (ACCSystem.Sols χ)"} +{"name":"ACCSystem","declaration":"/-- The type of charges plus the anomaly cancellation conditions. -/\nstructure ACCSystem : Type"} +{"name":"ACCSystemLinear","declaration":"/-- The type of charges plus the linear ACCs. -/\nstructure ACCSystemLinear : Type"} +{"name":"ACCSystem.cubicACC","declaration":"/-- The cubic ACC. -/\ndef ACCSystem.cubicACC (self : ACCSystem) : HomogeneousCubic (ACCSystemCharges.Charges self.toACCSystemCharges)"} +{"name":"ACCSystemLinear.linSolsAddCommMonoid_zero_val","declaration":"theorem ACCSystemLinear.linSolsAddCommMonoid_zero_val (χ : ACCSystemLinear) : 0.val = Zero.zero"} +{"name":"ACCSystemCharges.chargesModule","declaration":"/-- An instance to provide the necessary operations and properties for `charges` to form a module\nover the field `ℚ`.\n-/\ninstance ACCSystemCharges.chargesModule (χ : ACCSystemCharges) : Module ℚ (ACCSystemCharges.Charges χ)"} +{"name":"ACCSystemCharges.chargesModule_smul","declaration":"theorem ACCSystemCharges.chargesModule_smul (χ : ACCSystemCharges) : ∀ (x : ℚ) (x_1 : (i : Fin χ.numberCharges) → (fun i => (fun i => (fun a => ℚ) i) i) i) (i : Fin χ.numberCharges),\n SMul.smul x x_1 i = x * x_1 i"} +{"name":"ACCSystemCharges.chargesAddCommMonoid_nsmul","declaration":"theorem ACCSystemCharges.chargesAddCommMonoid_nsmul (χ : ACCSystemCharges) (n : ℕ) (x : (i : Fin χ.numberCharges) → (fun i => (fun a => ℚ) i) i) (i : Fin χ.numberCharges) : (n • x) i = ↑n * x i"} +{"name":"ACCSystemLinear.numberLinear","declaration":"/-- The number of linear ACCs. -/\ndef ACCSystemLinear.numberLinear (self : ACCSystemLinear) : ℕ"} +{"name":"ACCSystemChargesMk","declaration":"/-- Creates an `ACCSystemCharges` object with the specified number of charges.\n-/\ndef ACCSystemChargesMk (n : ℕ) : ACCSystemCharges"} +{"name":"ACCSystem.Hom","declaration":"/-- The structure of a map between two ACCSystems. -/\nstructure ACCSystem.Hom (χ : ACCSystem) (η : ACCSystem) : Type"} +{"name":"ACCSystemLinear.linSolsAddCommMonoid","declaration":"/-- An instance providing the operations and properties for `LinSols` to form an\nadditive commutative monoid. -/\ninstance ACCSystemLinear.linSolsAddCommMonoid (χ : ACCSystemLinear) : AddCommMonoid (ACCSystemLinear.LinSols χ)"} +{"name":"ACCSystemCharges.mk","declaration":"ctor ACCSystemCharges.mk (numberCharges : ℕ) : ACCSystemCharges"} +{"name":"ACCSystemLinear.LinSols.mk","declaration":"ctor ACCSystemLinear.LinSols.mk {χ : ACCSystemLinear} (val : ACCSystemCharges.Charges χ.toACCSystemCharges) (linearSol : ∀ (i : Fin χ.numberLinear), (χ.linearACCs i) val = 0) : ACCSystemLinear.LinSols χ"} +{"name":"ACCSystemLinear.linearACCs","declaration":"/-- The linear ACCs. -/\ndef ACCSystemLinear.linearACCs (self : ACCSystemLinear) : Fin self.numberLinear → ACCSystemCharges.Charges self.toACCSystemCharges →ₗ[ℚ] ℚ"} +{"name":"ACCSystemLinear.linSolsAddCommGroup","declaration":"/-- An instance providing the operations and properties for `LinSols` to form an\nadditive commutative group. -/\ninstance ACCSystemLinear.linSolsAddCommGroup (χ : ACCSystemLinear) : AddCommGroup (ACCSystemLinear.LinSols χ)"} +{"name":"ACCSystemCharges.numberCharges","declaration":"/-- The number of charges. -/\ndef ACCSystemCharges.numberCharges (self : ACCSystemCharges) : ℕ"} +{"name":"ACCSystem.Sols","declaration":"/-- The type of solutions to the anomaly cancellation conditions. -/\nstructure ACCSystem.Sols (χ : ACCSystem) : Type"} +{"name":"ACCSystemQuad.QuadSols.quadSol","declaration":"/-- The condition that the charge satisfies the quadratic ACCs. -/\ndef ACCSystemQuad.QuadSols.quadSol {χ : ACCSystemQuad} (self : ACCSystemQuad.QuadSols χ) (i : Fin χ.numberQuadratic) : (χ.quadraticACCs i) self.val = 0"} +{"name":"ACCSystemLinear.LinSols","declaration":"/-- The type of solutions to the linear ACCs. -/\nstructure ACCSystemLinear.LinSols (χ : ACCSystemLinear) : Type"} +{"name":"ACCSystemLinear.linSolsIncl","declaration":"/-- The inclusion of `LinSols` into `charges`. -/\ndef ACCSystemLinear.linSolsIncl (χ : ACCSystemLinear) : ACCSystemLinear.LinSols χ →ₗ[ℚ] ACCSystemCharges.Charges χ.toACCSystemCharges"} +{"name":"ACCSystem.Hom.charges","declaration":"/-- The linear map between vector spaces of charges. -/\ndef ACCSystem.Hom.charges {χ : ACCSystem} {η : ACCSystem} (self : ACCSystem.Hom χ η) : ACCSystemCharges.Charges χ.toACCSystemCharges →ₗ[ℚ] ACCSystemCharges.Charges η.toACCSystemCharges"} +{"name":"ACCSystemLinear.Nat.cast_smul_eq_nsmul","declaration":"/-- `nsmul` is equal to any other module structure via a cast. -/\ntheorem ACCSystemLinear.Nat.cast_smul_eq_nsmul (R : Type u_1) {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (n : ℕ) (b : M) : ↑n • b = n • b"} +{"name":"ACCSystemCharges.ChargesAddCommGroup","declaration":"/-- An instance provides the necessary operations and properties for `charges` to form an additive\ncommutative group.\n-/\ninstance ACCSystemCharges.ChargesAddCommGroup (χ : ACCSystemCharges) : AddCommGroup (ACCSystemCharges.Charges χ)"} +{"name":"ACCSystemQuad.quadraticACCs","declaration":"/-- The quadratic ACCs. -/\ndef ACCSystemQuad.quadraticACCs (self : ACCSystemQuad) : Fin self.numberQuadratic → HomogeneousQuadratic (ACCSystemCharges.Charges self.toACCSystemCharges)"} +{"name":"ACCSystem.Hom.mk","declaration":"ctor ACCSystem.Hom.mk {χ : ACCSystem} {η : ACCSystem} (charges : ACCSystemCharges.Charges χ.toACCSystemCharges →ₗ[ℚ] ACCSystemCharges.Charges η.toACCSystemCharges) (anomalyFree : ACCSystem.Sols χ → ACCSystem.Sols η) (commute : ⇑charges ∘ ⇑(ACCSystem.solsIncl χ) = ⇑(ACCSystem.solsIncl η) ∘ anomalyFree) : ACCSystem.Hom χ η"} +{"name":"ACCSystem.Hom.comp","declaration":"/-- The definition of composition between two ACCSystems. -/\ndef ACCSystem.Hom.comp {χ : ACCSystem} {η : ACCSystem} {ε : ACCSystem} (g : ACCSystem.Hom η ε) (f : ACCSystem.Hom χ η) : ACCSystem.Hom χ ε"} +{"name":"ACCSystemQuad.QuadSols","declaration":"/-- The type of solutions to the linear and quadratic ACCs. -/\nstructure ACCSystemQuad.QuadSols (χ : ACCSystemQuad) : Type"} +{"name":"ACCSystemQuad.numberQuadratic","declaration":"/-- The number of quadratic ACCs. -/\ndef ACCSystemQuad.numberQuadratic (self : ACCSystemQuad) : ℕ"} +{"name":"ACCSystemQuad.quadSolsIncl","declaration":"/-- The inclusion of quadratic solutions into all charges. -/\ndef ACCSystemQuad.quadSolsIncl (χ : ACCSystemQuad) : ACCSystemQuad.QuadSols χ →[ℚ] ACCSystemCharges.Charges χ.toACCSystemCharges"} +{"name":"ACCSystemCharges.chargesAddCommMonoid_add","declaration":"theorem ACCSystemCharges.chargesAddCommMonoid_add (χ : ACCSystemCharges) (f : (i : Fin χ.numberCharges) → (fun i => (fun i => (fun i => (fun a => ℚ) i) i) i) i) (g : (i : Fin χ.numberCharges) → (fun i => (fun i => (fun i => (fun a => ℚ) i) i) i) i) (i : Fin χ.numberCharges) : (f + g) i = f i + g i"} +{"name":"ACCSystemCharges.instFiniteRatChargesSemiringChargesAddCommMonoidChargesModule","declaration":"instance ACCSystemCharges.instFiniteRatChargesSemiringChargesAddCommMonoidChargesModule (χ : ACCSystemCharges) : Module.Finite ℚ (ACCSystemCharges.Charges χ)"} +{"name":"ACCSystemQuad","declaration":"/-- The type of charges plus the linear ACCs plus the quadratic ACCs. -/\nstructure ACCSystemQuad : Type"} +{"name":"ACCSystemCharges.chargesAddCommMonoid","declaration":"/-- An instance to provide the necessary operations and properties for `charges` to form an additive\ncommutative monoid.\n-/\ninstance ACCSystemCharges.chargesAddCommMonoid (χ : ACCSystemCharges) : AddCommMonoid (ACCSystemCharges.Charges χ)"} +{"name":"ACCSystem.solsInclQuadSols","declaration":"/-- The inclusion of `Sols` into `QuadSols`. -/\ndef ACCSystem.solsInclQuadSols (χ : ACCSystem) : ACCSystem.Sols χ →[ℚ] ACCSystemQuad.QuadSols χ.toACCSystemQuad"} +{"name":"ACCSystemCharges.chargesAddCommMonoid_zero_num","declaration":"theorem ACCSystemCharges.chargesAddCommMonoid_zero_num (χ : ACCSystemCharges) : ∀ (x : Fin χ.numberCharges), (0 x).num = 0"} +{"name":"ACCSystemLinear.linSolsModule_smul_val","declaration":"theorem ACCSystemLinear.linSolsModule_smul_val (χ : ACCSystemLinear) (a : ℚ) (S : ACCSystemLinear.LinSols χ) : (SMul.smul a S).val = a • S.val"} +{"name":"ACCSystemQuad.linSolsInclQuadSolsZero","declaration":"/-- The inclusion of the linear solutions into the quadratic solutions, where there is\nno quadratic equations (i.e. no U(1)'s in the underlying gauge group). -/\ndef ACCSystemQuad.linSolsInclQuadSolsZero (χ : ACCSystemQuad) (h : χ.numberQuadratic = 0) : ACCSystemLinear.LinSols χ.toACCSystemLinear →[ℚ] ACCSystemQuad.QuadSols χ"} +{"name":"ACCSystemLinear.LinSols.ext","declaration":"/-- Two solutions are equal if the underlying charges are equal. -/\ntheorem ACCSystemLinear.LinSols.ext {χ : ACCSystemLinear} {S : ACCSystemLinear.LinSols χ} {T : ACCSystemLinear.LinSols χ} (h : S.val = T.val) : S = T"} +{"name":"ACCSystem.mk","declaration":"ctor ACCSystem.mk (toACCSystemQuad : ACCSystemQuad) (cubicACC : HomogeneousCubic (ACCSystemCharges.Charges toACCSystemQuad.toACCSystemCharges)) : ACCSystem"} +{"name":"LinearMap.toMulActionHom","declaration":"def LinearMap.toMulActionHom {R : Type u_1} [Semiring R] {M : Type u_2} {M₂ : Type u_3} [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (self : M →ₗ[R] M₂) : M →[R] M₂"} +{"name":"ACCSystemQuad.QuadSols.ext","declaration":"/-- Two `QuadSols` are equal if the underlying charges are equal. -/\ntheorem ACCSystemQuad.QuadSols.ext {χ : ACCSystemQuad} {S : ACCSystemQuad.QuadSols χ} {T : ACCSystemQuad.QuadSols χ} (h : S.val = T.val) : S = T"} +{"name":"ACCSystemLinear.mk","declaration":"ctor ACCSystemLinear.mk (toACCSystemCharges : ACCSystemCharges) (numberLinear : ℕ) (linearACCs : Fin numberLinear → ACCSystemCharges.Charges toACCSystemCharges →ₗ[ℚ] ℚ) : ACCSystemLinear"} +{"name":"ACCSystemQuad.mk","declaration":"ctor ACCSystemQuad.mk (toACCSystemLinear : ACCSystemLinear) (numberQuadratic : ℕ) (quadraticACCs : Fin numberQuadratic → HomogeneousQuadratic (ACCSystemCharges.Charges toACCSystemLinear.toACCSystemCharges)) : ACCSystemQuad"} +{"name":"ACCSystemLinear.linSolsAddCommMonoid_add_val","declaration":"theorem ACCSystemLinear.linSolsAddCommMonoid_add_val (χ : ACCSystemLinear) (S : ACCSystemLinear.LinSols χ) (T : ACCSystemLinear.LinSols χ) : (S + T).val = S.val + T.val"} +{"name":"ACCSystemCharges","declaration":"/-- A system of charges, specified by the number of charges. -/\nstructure ACCSystemCharges : Type"} +{"name":"ACCSystemLinear.linSolsAddCommMonoid_nsmul_val","declaration":"theorem ACCSystemLinear.linSolsAddCommMonoid_nsmul_val (χ : ACCSystemLinear) (n : ℕ) (S : ACCSystemLinear.LinSols χ) : (n • S).val = n • S.val"} +{"name":"ACCSystemCharges.chargesAddCommMonoid_zero_den","declaration":"theorem ACCSystemCharges.chargesAddCommMonoid_zero_den (χ : ACCSystemCharges) : ∀ (x : Fin χ.numberCharges), (0 x).den = 1"} +{"name":"ACCSystemLinear.linSolsModule","declaration":"/-- An instance providing the operations and properties for `LinSols` to form a\nmodule over `ℚ`. -/\ninstance ACCSystemLinear.linSolsModule (χ : ACCSystemLinear) : Module ℚ (ACCSystemLinear.LinSols χ)"} +{"name":"ACCSystemQuad.quadSolsMulAction","declaration":"/-- An instance giving the properties and structures to define an action of `ℚ` on `QuadSols`. -/\ninstance ACCSystemQuad.quadSolsMulAction (χ : ACCSystemQuad) : MulAction ℚ (ACCSystemQuad.QuadSols χ)"} +{"name":"ACCSystem.Sols.cubicSol","declaration":"/-- The condition that the charge satisfies the cubic ACC. -/\ndef ACCSystem.Sols.cubicSol {χ : ACCSystem} (self : ACCSystem.Sols χ) : χ.cubicACC self.val = 0"} +{"name":"ACCSystem.solsIncl","declaration":"/-- The inclusion of `Sols` into `LinSols`. -/\ndef ACCSystem.solsIncl (χ : ACCSystem) : ACCSystem.Sols χ →[ℚ] ACCSystemCharges.Charges χ.toACCSystemCharges"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.GroupActions.jsonl b/hep-declarations/HepLean.AnomalyCancellation.GroupActions.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e5d375a5ed4e208b67b83903b5a3ed214aac82a1 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.GroupActions.jsonl @@ -0,0 +1,20 @@ +{"name":"ACCSystemGroupAction.rep_linSolRep_commute","declaration":"/-- The representation on the charges and anomaly free solutions\ncommutes with the inclusion. -/\ntheorem ACCSystemGroupAction.rep_linSolRep_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) (S : ACCSystemLinear.LinSols χ.toACCSystemLinear) : (ACCSystemLinear.linSolsIncl χ.toACCSystemLinear) (((ACCSystemGroupAction.linSolRep G) g) S) =\n (G.rep g) ((ACCSystemLinear.linSolsIncl χ.toACCSystemLinear) S)"} +{"name":"ACCSystemGroupAction.solAction","declaration":"/-- The group action acting on solutions to the anomaly cancellation conditions. -/\ninstance ACCSystemGroupAction.solAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) : MulAction G.group (ACCSystem.Sols χ)"} +{"name":"ACCSystemGroupAction.instGroupGroup","declaration":"instance ACCSystemGroupAction.instGroupGroup {χ : ACCSystem} (G : ACCSystemGroupAction χ) : Group G.group"} +{"name":"ACCSystemGroupAction.mk","declaration":"ctor ACCSystemGroupAction.mk {χ : ACCSystem} (group : Type) (groupInst : Group group) (rep : Representation ℚ group (ACCSystemCharges.Charges χ.toACCSystemCharges)) (linearInvariant : ∀ (i : Fin χ.numberLinear) (g : group) (S : ACCSystemCharges.Charges χ.toACCSystemCharges),\n (χ.linearACCs i) ((rep g) S) = (χ.linearACCs i) S) (quadInvariant : ∀ (i : Fin χ.numberQuadratic) (g : group) (S : ACCSystemCharges.Charges χ.toACCSystemCharges),\n (χ.quadraticACCs i) ((rep g) S) = (χ.quadraticACCs i) S) (cubicInvariant : ∀ (g : group) (S : ACCSystemCharges.Charges χ.toACCSystemCharges), χ.cubicACC ((rep g) S) = χ.cubicACC S) : ACCSystemGroupAction χ"} +{"name":"ACCSystemGroupAction.linSolRep","declaration":"/-- The representation acting on the vector space of solutions to the linear ACCs. -/\ndef ACCSystemGroupAction.linSolRep {χ : ACCSystem} (G : ACCSystemGroupAction χ) : Representation ℚ G.group (ACCSystemLinear.LinSols χ.toACCSystemLinear)"} +{"name":"ACCSystemGroupAction.quadInvariant","declaration":"/-- The invariance of the quadratic ACCs under the group action. -/\ndef ACCSystemGroupAction.quadInvariant {χ : ACCSystem} (self : ACCSystemGroupAction χ) (i : Fin χ.numberQuadratic) (g : self.group) (S : ACCSystemCharges.Charges χ.toACCSystemCharges) : (χ.quadraticACCs i) ((self.rep g) S) = (χ.quadraticACCs i) S"} +{"name":"ACCSystemGroupAction.cubicInvariant","declaration":"/-- The invariance of the cubic ACC under the group action. -/\ndef ACCSystemGroupAction.cubicInvariant {χ : ACCSystem} (self : ACCSystemGroupAction χ) (g : self.group) (S : ACCSystemCharges.Charges χ.toACCSystemCharges) : χ.cubicACC ((self.rep g) S) = χ.cubicACC S"} +{"name":"ACCSystemGroupAction","declaration":"/-- The type of a group action on a system of charges is defined as a representation on\nthe vector spaces of charges under which the anomaly equations are invariant.\n-/\nstructure ACCSystemGroupAction (χ : ACCSystem) : Type 1"} +{"name":"ACCSystemGroupAction.group","declaration":"/-- The underlying type of the group. -/\ndef ACCSystemGroupAction.group {χ : ACCSystem} (self : ACCSystemGroupAction χ) : Type"} +{"name":"ACCSystemGroupAction.quadSolAction","declaration":"/-- A multiplicative action of `G.group` on `quadSols`. -/\ninstance ACCSystemGroupAction.quadSolAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) : MulAction G.group (ACCSystemQuad.QuadSols χ.toACCSystemQuad)"} +{"name":"ACCSystemGroupAction.linSolRep_solAction_commute","declaration":"theorem ACCSystemGroupAction.linSolRep_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) (S : ACCSystem.Sols χ) : (ACCSystem.solsInclLinSols χ) ((MulAction.toFun G.group (ACCSystem.Sols χ)) S g) =\n ((ACCSystemGroupAction.linSolRep G) g) ((ACCSystem.solsInclLinSols χ) S)"} +{"name":"ACCSystemGroupAction.rep","declaration":"/-- The representation of group acting on the vector space of charges. -/\ndef ACCSystemGroupAction.rep {χ : ACCSystem} (self : ACCSystemGroupAction χ) : Representation ℚ self.group (ACCSystemCharges.Charges χ.toACCSystemCharges)"} +{"name":"ACCSystemGroupAction.linSolRep_quadSolAction_commute","declaration":"theorem ACCSystemGroupAction.linSolRep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) (S : ACCSystemQuad.QuadSols χ.toACCSystemQuad) : (ACCSystemQuad.quadSolsInclLinSols χ.toACCSystemQuad)\n ((MulAction.toFun G.group (ACCSystemQuad.QuadSols χ.toACCSystemQuad)) S g) =\n ((ACCSystemGroupAction.linSolRep G) g) ((ACCSystemQuad.quadSolsInclLinSols χ.toACCSystemQuad) S)"} +{"name":"ACCSystemGroupAction.linSolMap","declaration":"/-- The action of a group element on the vector space of linear solutions. -/\ndef ACCSystemGroupAction.linSolMap {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) : ACCSystemLinear.LinSols χ.toACCSystemLinear →ₗ[ℚ] ACCSystemLinear.LinSols χ.toACCSystemLinear"} +{"name":"ACCSystemGroupAction.quadSolAction_solAction_commute","declaration":"theorem ACCSystemGroupAction.quadSolAction_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) (S : ACCSystem.Sols χ) : (ACCSystem.solsInclQuadSols χ) ((MulAction.toFun G.group (ACCSystem.Sols χ)) S g) =\n (MulAction.toFun G.group (ACCSystemQuad.QuadSols χ.toACCSystemQuad)) ((ACCSystem.solsInclQuadSols χ) S) g"} +{"name":"ACCSystemGroupAction.groupInst","declaration":"/-- An instance given the `group` component the structure of a `Group`. -/\ndef ACCSystemGroupAction.groupInst {χ : ACCSystem} (self : ACCSystemGroupAction χ) : Group self.group"} +{"name":"ACCSystemGroupAction.linearInvariant","declaration":"/-- The invariance of the linear ACCs under the group action. -/\ndef ACCSystemGroupAction.linearInvariant {χ : ACCSystem} (self : ACCSystemGroupAction χ) (i : Fin χ.numberLinear) (g : self.group) (S : ACCSystemCharges.Charges χ.toACCSystemCharges) : (χ.linearACCs i) ((self.rep g) S) = (χ.linearACCs i) S"} +{"name":"ACCSystemGroupAction.rep_solAction_commute","declaration":"theorem ACCSystemGroupAction.rep_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) (S : ACCSystem.Sols χ) : (ACCSystem.solsIncl χ) ((MulAction.toFun G.group (ACCSystem.Sols χ)) S g) = (G.rep g) ((ACCSystem.solsIncl χ) S)"} +{"name":"ACCSystemGroupAction.linSolRep_apply_apply_val","declaration":"theorem ACCSystemGroupAction.linSolRep_apply_apply_val {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) (S : ACCSystemLinear.LinSols χ.toACCSystemLinear) : (((ACCSystemGroupAction.linSolRep G) g) S).val = (G.rep g) S.val"} +{"name":"ACCSystemGroupAction.rep_quadSolAction_commute","declaration":"theorem ACCSystemGroupAction.rep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group) (S : ACCSystemQuad.QuadSols χ.toACCSystemQuad) : (ACCSystemQuad.quadSolsIncl χ.toACCSystemQuad)\n ((MulAction.toFun G.group (ACCSystemQuad.QuadSols χ.toACCSystemQuad)) S g) =\n (G.rep g) ((ACCSystemQuad.quadSolsIncl χ.toACCSystemQuad) S)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.B3.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.B3.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..4480239c8a91f01b25552fe24af51005624948cd --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.B3.jsonl @@ -0,0 +1,4 @@ +{"name":"MSSMACC.B₃AsCharge","declaration":"/-- `B₃` is the charge which is $B-L$ in all families, but with the third\nfamily of the opposite sign. -/\ndef MSSMACC.B₃AsCharge : ACCSystemCharges.Charges MSSMACC.toACCSystemCharges"} +{"name":"MSSMACC.B₃_val","declaration":"theorem MSSMACC.B₃_val : MSSMACC.B₃.val = MSSMACC.B₃AsCharge"} +{"name":"MSSMACC.B₃","declaration":"/-- `B₃` as a solution. -/\ndef MSSMACC.B₃ : ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC.doublePoint_B₃_B₃","declaration":"theorem MSSMACC.doublePoint_B₃_B₃ (R : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : ((MSSMACCs.cubeTriLin MSSMACC.B₃.val) MSSMACC.B₃.val) R.val = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d64d0b99cf12b20742514f433a18e062aed36308 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Basic.jsonl @@ -0,0 +1,72 @@ +{"name":"MSSMCharges.Hu","declaration":"/-- The charge `Hu`. -/\ndef MSSMCharges.Hu : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ℚ"} +{"name":"MSSMCharges.toSMSpecies_toSpecies_inv","declaration":"theorem MSSMCharges.toSMSpecies_toSpecies_inv (i : Fin 6) (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) : (MSSMCharges.toSMSpecies i) (MSSMCharges.toSpecies.symm f) = f.1 i"} +{"name":"MSSMCharges.L","declaration":"/-- The `L` charges as a map `Fin 3 → ℚ`. -/\ndef MSSMCharges.L : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMSpecies"} +{"name":"MSSMCharges.Q","declaration":"/-- The `Q` charges as a map `Fin 3 → ℚ`. -/\ndef MSSMCharges.Q : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMSpecies"} +{"name":"MSSMCharges.toSplitSMPlusH","declaration":"/-- An equivalence between `MSSMCharges.charges` and `(Fin 18 → ℚ) × (Fin 2 → ℚ)`. This\nsplits the charges up into the SM and the additional ones for the MSSM. -/\ndef MSSMCharges.toSplitSMPlusH : ACCSystemCharges.Charges MSSMCharges ≃ (Fin 18 → ℚ) × (Fin 2 → ℚ)"} +{"name":"MSSMCharges.splitSMPlusH_apply","declaration":"theorem MSSMCharges.splitSMPlusH_apply (f : Fin 18 ⊕ Fin 2 → ℚ) : MSSMCharges.splitSMPlusH f = (f ∘ Sum.inl, f ∘ Sum.inr)"} +{"name":"MSSMACCs.accYY_ext","declaration":"/-- Extensionality lemma for `accGrav`. -/\ntheorem MSSMACCs.accYY_ext {S : ACCSystemCharges.Charges MSSMCharges} {T : ACCSystemCharges.Charges MSSMCharges} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) S i) =\n Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) T i) (hd : MSSMCharges.Hd S = MSSMCharges.Hd T) (hu : MSSMCharges.Hu S = MSSMCharges.Hu T) : MSSMACCs.accYY S = MSSMACCs.accYY T"} +{"name":"MSSMACCs.accCube_ext","declaration":"/-- Extensionality lemma for `accCube`. -/\ntheorem MSSMACCs.accCube_ext {S : ACCSystemCharges.Charges MSSMCharges} {T : ACCSystemCharges.Charges MSSMCharges} (h : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => ((fun a => a ^ 3) ∘ (MSSMCharges.toSMSpecies j) S) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ 3) ∘ (MSSMCharges.toSMSpecies j) T) i) (hd : MSSMCharges.Hd S = MSSMCharges.Hd T) (hu : MSSMCharges.Hu S = MSSMCharges.Hu T) : MSSMACCs.accCube S = MSSMACCs.accCube T"} +{"name":"MSSMCharges.Hu_apply","declaration":"theorem MSSMCharges.Hu_apply (S : ACCSystemCharges.Charges MSSMCharges) : MSSMCharges.Hu S = S { val := 19, isLt := MSSMCharges.Hu.proof_1 }"} +{"name":"MSSMCharges.toSpecies_apply","declaration":"theorem MSSMCharges.toSpecies_apply : ∀ (a : ACCSystemCharges.Charges MSSMCharges),\n MSSMCharges.toSpecies a =\n (MSSMCharges.toSpeciesMaps' (MSSMCharges.toSMPlusH a ∘ Sum.inl), MSSMCharges.toSMPlusH a ∘ Sum.inr)"} +{"name":"MSSMACC.AnomalyFreeQuadMk'","declaration":"/-- A `QuadSol` from a `LinSol` satisfying the quadratic ACC. -/\ndef MSSMACC.AnomalyFreeQuadMk' (S : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) (hquad : MSSMACCs.accQuad S.val = 0) : ACCSystemQuad.QuadSols MSSMACC.toACCSystemQuad"} +{"name":"MSSMACCs.accCube","declaration":"/-- The cubic ACC. -/\ndef MSSMACCs.accCube : HomogeneousCubic (ACCSystemCharges.Charges MSSMCharges)"} +{"name":"MSSMCharges.splitSMPlusH","declaration":"/-- An equivalence between `Fin 18 ⊕ Fin 2 → ℚ` and `(Fin 18 → ℚ) × (Fin 2 → ℚ)`. -/\ndef MSSMCharges.splitSMPlusH : (Fin 18 ⊕ Fin 2 → ℚ) ≃ (Fin 18 → ℚ) × (Fin 2 → ℚ)"} +{"name":"MSSMCharges.toSMPlusH_apply","declaration":"theorem MSSMCharges.toSMPlusH_apply : ∀ (a : Fin (18 + 2) → ℚ) (a_1 : Fin 18 ⊕ Fin 2), MSSMCharges.toSMPlusH a a_1 = a (finSumFinEquiv a_1)"} +{"name":"MSSMACC.dot","declaration":"/-- The dot product on the vector space of charges. -/\ndef MSSMACC.dot : BiLinearSymm (ACCSystemCharges.Charges MSSMCharges)"} +{"name":"MSSMACCs.accGrav_ext","declaration":"/-- Extensionality lemma for `accGrav`. -/\ntheorem MSSMACCs.accGrav_ext {S : ACCSystemCharges.Charges MSSMCharges} {T : ACCSystemCharges.Charges MSSMCharges} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) S i) =\n Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) T i) (hd : MSSMCharges.Hd S = MSSMCharges.Hd T) (hu : MSSMCharges.Hu S = MSSMCharges.Hu T) : MSSMACCs.accGrav S = MSSMACCs.accGrav T"} +{"name":"MSSMCharges.splitSMPlusH_symm_apply","declaration":"theorem MSSMCharges.splitSMPlusH_symm_apply (f : (Fin 18 → ℚ) × (Fin 2 → ℚ)) : ∀ (a : Fin 18 ⊕ Fin 2), MSSMCharges.splitSMPlusH.symm f a = Sum.elim f.1 f.2 a"} +{"name":"MSSMCharges.toSpeciesMaps'_symm_apply","declaration":"theorem MSSMCharges.toSpeciesMaps'_symm_apply : ∀ (a : Fin 6 → Fin 3 → ℚ) (a_1 : Fin (6 * 3)),\n MSSMCharges.toSpeciesMaps'.symm a a_1 = a (Fin.divNat a_1) (Fin.modNat a_1)"} +{"name":"MSSMSpecies_numberCharges","declaration":"theorem MSSMSpecies_numberCharges : MSSMSpecies.numberCharges = 3"} +{"name":"MSSMACCs.cubeTriLinToFun_map_smul₁","declaration":"theorem MSSMACCs.cubeTriLinToFun_map_smul₁ (a : ℚ) (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) (R : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.cubeTriLinToFun (a • S, T, R) = a * MSSMACCs.cubeTriLinToFun (S, T, R)"} +{"name":"MSSMCharges.D","declaration":"/-- The `D` charges as a map `Fin 3 → ℚ`. -/\ndef MSSMCharges.D : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMSpecies"} +{"name":"MSSMACC_numberQuadratic","declaration":"theorem MSSMACC_numberQuadratic : MSSMACC.numberQuadratic = 1"} +{"name":"MSSMACC_numberLinear","declaration":"theorem MSSMACC_numberLinear : MSSMACC.numberLinear = 4"} +{"name":"MSSMCharges.toSpeciesMaps'","declaration":"/-- An equivalence between `(Fin 18 → ℚ)` and `(Fin 6 → Fin 3 → ℚ)`. -/\ndef MSSMCharges.toSpeciesMaps' : (Fin 18 → ℚ) ≃ (Fin 6 → Fin 3 → ℚ)"} +{"name":"MSSMCharges.Hd_toSpecies_inv","declaration":"theorem MSSMCharges.Hd_toSpecies_inv (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) : MSSMCharges.Hd (MSSMCharges.toSpecies.symm f) = f.2 0"} +{"name":"MSSMACCs.cubeTriLinToFun_swap2","declaration":"theorem MSSMACCs.cubeTriLinToFun_swap2 (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) (R : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.cubeTriLinToFun (S, T, R) = MSSMACCs.cubeTriLinToFun (S, R, T)"} +{"name":"MSSMCharges.toSplitSMPlusH_apply","declaration":"theorem MSSMCharges.toSplitSMPlusH_apply : ∀ (a : ACCSystemCharges.Charges MSSMCharges),\n MSSMCharges.toSplitSMPlusH a = (MSSMCharges.toSMPlusH a ∘ Sum.inl, MSSMCharges.toSMPlusH a ∘ Sum.inr)"} +{"name":"MSSMACC.AnomalyFreeMk_val","declaration":"theorem MSSMACC.AnomalyFreeMk_val (S : ACCSystemCharges.Charges MSSMACC.toACCSystemCharges) (hg : MSSMACCs.accGrav S = 0) (hsu2 : MSSMACCs.accSU2 S = 0) (hsu3 : MSSMACCs.accSU3 S = 0) (hyy : MSSMACCs.accYY S = 0) (hquad : MSSMACCs.accQuad S = 0) (hcube : MSSMACCs.accCube S = 0) : (MSSMACC.AnomalyFreeMk S hg hsu2 hsu3 hyy hquad hcube).val = S"} +{"name":"MSSMCharges.toSplitSMPlusH_symm_apply","declaration":"theorem MSSMCharges.toSplitSMPlusH_symm_apply : ∀ (a : (Fin 18 → ℚ) × (Fin 2 → ℚ)),\n MSSMCharges.toSplitSMPlusH.symm a = MSSMCharges.toSMPlusH.symm (MSSMCharges.splitSMPlusH.symm a)"} +{"name":"MSSMACC.dot_toFun_apply","declaration":"theorem MSSMACC.dot_toFun_apply (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) : (MSSMACC.dot S) T =\n (Finset.sum Finset.univ fun i =>\n S (Fin.castAdd 2 (finProdFinEquiv (0, i))) * T (Fin.castAdd 2 (finProdFinEquiv (0, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (1, i))) * T (Fin.castAdd 2 (finProdFinEquiv (1, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (2, i))) * T (Fin.castAdd 2 (finProdFinEquiv (2, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (3, i))) * T (Fin.castAdd 2 (finProdFinEquiv (3, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (4, i))) * T (Fin.castAdd 2 (finProdFinEquiv (4, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (5, i))) * T (Fin.castAdd 2 (finProdFinEquiv (5, i)))) +\n S { val := 18, isLt := MSSMCharges.Hd.proof_1 } * T { val := 18, isLt := MSSMCharges.Hd.proof_1 } +\n S { val := 19, isLt := MSSMCharges.Hu.proof_1 } * T { val := 19, isLt := MSSMCharges.Hu.proof_1 }"} +{"name":"MSSMCharges.Hd_apply","declaration":"theorem MSSMCharges.Hd_apply (S : ACCSystemCharges.Charges MSSMCharges) : MSSMCharges.Hd S = S { val := 18, isLt := MSSMCharges.Hd.proof_1 }"} +{"name":"MSSMCharges.U","declaration":"/-- The `U` charges as a map `Fin 3 → ℚ`. -/\ndef MSSMCharges.U : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMSpecies"} +{"name":"MSSMACC","declaration":"/-- The ACCSystem for the MSSM without RHN. -/\ndef MSSMACC : ACCSystem"} +{"name":"MSSMACC.AnomalyFreeMk'","declaration":"/-- A `Sol` from a `LinSol` satisfying the quadratic and cubic ACCs. -/\ndef MSSMACC.AnomalyFreeMk' (S : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) (hquad : MSSMACCs.accQuad S.val = 0) (hcube : MSSMACCs.accCube S.val = 0) : ACCSystem.Sols MSSMACC"} +{"name":"MSSMCharges.toSMSpecies_apply","declaration":"theorem MSSMCharges.toSMSpecies_apply (i : Fin 6) (S : ACCSystemCharges.Charges MSSMCharges) : ∀ (a : Fin 3), (MSSMCharges.toSMSpecies i) S a = S (Fin.castAdd 2 (finProdFinEquiv (i, a)))"} +{"name":"MSSMCharges.toSpecies","declaration":"/-- An equivalence between `MSSMCharges.charges` and `(Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ))`.\nThis splits charges up into the SM and additional fermions, and further splits the SM into\nspecies. -/\ndef MSSMCharges.toSpecies : ACCSystemCharges.Charges MSSMCharges ≃ (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)"} +{"name":"MSSMACCs.cubeTriLinToFun","declaration":"/-- The function underlying the symmetric trilinear form used to define the cubic ACC. -/\ndef MSSMACCs.cubeTriLinToFun (S : ACCSystemCharges.Charges MSSMCharges × ACCSystemCharges.Charges MSSMCharges × ACCSystemCharges.Charges MSSMCharges) : ℚ"} +{"name":"MSSMSpecies","declaration":"/-- The vector spaces of charges of one species of fermions in the MSSM. -/\ndef MSSMSpecies : ACCSystemCharges"} +{"name":"MSSMACCs.cubeTriLinToFun_swap1","declaration":"theorem MSSMACCs.cubeTriLinToFun_swap1 (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) (R : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.cubeTriLinToFun (S, T, R) = MSSMACCs.cubeTriLinToFun (T, S, R)"} +{"name":"MSSMACC_linearACCs","declaration":"theorem MSSMACC_linearACCs (i : Fin 4) : MSSMACC.linearACCs i =\n match i with\n | 0 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n (Finset.sum Finset.univ fun i =>\n 6 * S (Fin.castAdd 2 (finProdFinEquiv (0, i))) + 3 * S (Fin.castAdd 2 (finProdFinEquiv (1, i))) +\n 3 * S (Fin.castAdd 2 (finProdFinEquiv (2, i))) +\n 2 * S (Fin.castAdd 2 (finProdFinEquiv (3, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (4, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (5, i)))) +\n 2 * (S { val := 18, isLt := MSSMCharges.Hd.proof_1 } + S { val := 19, isLt := MSSMCharges.Hu.proof_1 }),\n map_add' := MSSMACCs.accGrav.proof_1 },\n map_smul' := MSSMACCs.accGrav.proof_2 }\n | 1 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n (Finset.sum Finset.univ fun i =>\n 3 * S (Fin.castAdd 2 (finProdFinEquiv (0, i))) + S (Fin.castAdd 2 (finProdFinEquiv (3, i)))) +\n S { val := 18, isLt := MSSMCharges.Hd.proof_1 } +\n S { val := 19, isLt := MSSMCharges.Hu.proof_1 },\n map_add' := MSSMACCs.accSU2.proof_1 },\n map_smul' := MSSMACCs.accSU2.proof_2 }\n | 2 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n 2 * S (Fin.castAdd 2 (finProdFinEquiv (0, i))) + S (Fin.castAdd 2 (finProdFinEquiv (1, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (2, i))),\n map_add' := MSSMACCs.accSU3.proof_1 },\n map_smul' := MSSMACCs.accSU3.proof_2 }\n | 3 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n (Finset.sum Finset.univ fun i =>\n S (Fin.castAdd 2 (finProdFinEquiv (0, i))) + 8 * S (Fin.castAdd 2 (finProdFinEquiv (1, i))) +\n 2 * S (Fin.castAdd 2 (finProdFinEquiv (2, i))) +\n 3 * S (Fin.castAdd 2 (finProdFinEquiv (3, i))) +\n 6 * S (Fin.castAdd 2 (finProdFinEquiv (4, i)))) +\n 3 * (S { val := 18, isLt := MSSMCharges.Hd.proof_1 } + S { val := 19, isLt := MSSMCharges.Hu.proof_1 }),\n map_add' := MSSMACCs.accYY.proof_1 },\n map_smul' := MSSMACCs.accYY.proof_2 }"} +{"name":"MSSMCharges","declaration":"/-- The vector space of charges corresponding to the MSSM fermions. -/\ndef MSSMCharges : ACCSystemCharges"} +{"name":"MSSMACC.AnomalyFreeMk''","declaration":"/-- A `Sol` from a `QuadSol` satisfying the cubic ACCs. -/\ndef MSSMACC.AnomalyFreeMk'' (S : ACCSystemQuad.QuadSols MSSMACC.toACCSystemQuad) (hcube : MSSMACCs.accCube S.val = 0) : ACCSystem.Sols MSSMACC"} +{"name":"MSSMCharges.Hu_toSpecies_inv","declaration":"theorem MSSMCharges.Hu_toSpecies_inv (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) : MSSMCharges.Hu (MSSMCharges.toSpecies.symm f) = f.2 1"} +{"name":"MSSMACC.AnomalyFreeMk","declaration":"/-- A solution from a charge satisfying the ACCs. -/\ndef MSSMACC.AnomalyFreeMk (S : ACCSystemCharges.Charges MSSMACC.toACCSystemCharges) (hg : MSSMACCs.accGrav S = 0) (hsu2 : MSSMACCs.accSU2 S = 0) (hsu3 : MSSMACCs.accSU3 S = 0) (hyy : MSSMACCs.accYY S = 0) (hquad : MSSMACCs.accQuad S = 0) (hcube : MSSMACCs.accCube S = 0) : ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC_cubicACC_toFun","declaration":"theorem MSSMACC_cubicACC_toFun (S : ACCSystemCharges.Charges MSSMCharges) : MSSMACC.cubicACC.toFun S =\n (Finset.sum Finset.univ fun i =>\n 6 *\n (S (Fin.castAdd 2 (finProdFinEquiv (0, i))) * S (Fin.castAdd 2 (finProdFinEquiv (0, i))) *\n S (Fin.castAdd 2 (finProdFinEquiv (0, i)))) +\n 3 *\n (S (Fin.castAdd 2 (finProdFinEquiv (1, i))) * S (Fin.castAdd 2 (finProdFinEquiv (1, i))) *\n S (Fin.castAdd 2 (finProdFinEquiv (1, i)))) +\n 3 *\n (S (Fin.castAdd 2 (finProdFinEquiv (2, i))) * S (Fin.castAdd 2 (finProdFinEquiv (2, i))) *\n S (Fin.castAdd 2 (finProdFinEquiv (2, i)))) +\n 2 *\n (S (Fin.castAdd 2 (finProdFinEquiv (3, i))) * S (Fin.castAdd 2 (finProdFinEquiv (3, i))) *\n S (Fin.castAdd 2 (finProdFinEquiv (3, i)))) +\n S (Fin.castAdd 2 (finProdFinEquiv (4, i))) * S (Fin.castAdd 2 (finProdFinEquiv (4, i))) *\n S (Fin.castAdd 2 (finProdFinEquiv (4, i))) +\n S (Fin.castAdd 2 (finProdFinEquiv (5, i))) * S (Fin.castAdd 2 (finProdFinEquiv (5, i))) *\n S (Fin.castAdd 2 (finProdFinEquiv (5, i)))) +\n (2 * S { val := 18, isLt := MSSMCharges.Hd.proof_1 } * S { val := 18, isLt := MSSMCharges.Hd.proof_1 } *\n S { val := 18, isLt := MSSMCharges.Hd.proof_1 } +\n 2 * S { val := 19, isLt := MSSMCharges.Hu.proof_1 } * S { val := 19, isLt := MSSMCharges.Hu.proof_1 } *\n S { val := 19, isLt := MSSMCharges.Hu.proof_1 })"} +{"name":"MSSMCharges.toSpeciesMaps'_apply","declaration":"theorem MSSMCharges.toSpeciesMaps'_apply : ∀ (a : Fin (6 * 3) → ℚ) (a_1 : Fin 6) (a_2 : Fin 3),\n MSSMCharges.toSpeciesMaps' a a_1 a_2 = a (finProdFinEquiv (a_1, a_2))"} +{"name":"MSSMACCs.accYY","declaration":"/-- The ACC for `Y²`. -/\ndef MSSMACCs.accYY : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ℚ"} +{"name":"MSSMCharges.toSMPlusH","declaration":"/-- An equivalence between `MSSMCharges.charges` and the space of maps\n`(Fin 18 ⊕ Fin 2 → ℚ)`. The first 18 factors corresponds to the SM fermions, while the last two\nare the higgsions. -/\ndef MSSMCharges.toSMPlusH : ACCSystemCharges.Charges MSSMCharges ≃ (Fin 18 ⊕ Fin 2 → ℚ)"} +{"name":"MSSMACC.quadSol","declaration":"theorem MSSMACC.quadSol (S : ACCSystemQuad.QuadSols MSSMACC.toACCSystemQuad) : MSSMACCs.accQuad S.val = 0"} +{"name":"MSSMACCs.accSU3","declaration":"/-- The anomaly cancellation condition for SU(3) anomaly. -/\ndef MSSMACCs.accSU3 : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ℚ"} +{"name":"MSSMACCs.accSU2","declaration":"/-- The anomaly cancellation condition for SU(2) anomaly. -/\ndef MSSMACCs.accSU2 : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ℚ"} +{"name":"MSSMCharges.toSpecies_symm_apply","declaration":"theorem MSSMCharges.toSpecies_symm_apply : ∀ (a : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)),\n MSSMCharges.toSpecies.symm a =\n MSSMCharges.toSMPlusH.symm (MSSMCharges.splitSMPlusH.symm (MSSMCharges.toSpeciesMaps'.symm a.1, a.2))"} +{"name":"MSSMACCs.accSU2_ext","declaration":"/-- Extensionality lemma for `accSU2`. -/\ntheorem MSSMACCs.accSU2_ext {S : ACCSystemCharges.Charges MSSMCharges} {T : ACCSystemCharges.Charges MSSMCharges} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) S i) =\n Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) T i) (hd : MSSMCharges.Hd S = MSSMCharges.Hd T) (hu : MSSMCharges.Hu S = MSSMCharges.Hu T) : MSSMACCs.accSU2 S = MSSMACCs.accSU2 T"} +{"name":"MSSMACC_quadraticACCs","declaration":"theorem MSSMACC_quadraticACCs (i : Fin 1) : MSSMACC.quadraticACCs i =\n match i with\n | 0 => BiLinearSymm.toHomogeneousQuad MSSMACCs.quadBiLin"} +{"name":"MSSMACCs.quadBiLin","declaration":"/-- The symmetric bilinear function used to define the quadratic ACC. -/\ndef MSSMACCs.quadBiLin : BiLinearSymm (ACCSystemCharges.Charges MSSMCharges)"} +{"name":"MSSMCharges.E","declaration":"/-- The `E` charges as a map `Fin 3 → ℚ`. -/\ndef MSSMCharges.E : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMSpecies"} +{"name":"MSSMACCs.accQuad_ext","declaration":"/-- Extensionality lemma for `accQuad`. -/\ntheorem MSSMACCs.accQuad_ext {S : ACCSystemCharges.Charges MSSMCharges} {T : ACCSystemCharges.Charges MSSMCharges} (h : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => ((fun a => a ^ 2) ∘ (MSSMCharges.toSMSpecies j) S) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ 2) ∘ (MSSMCharges.toSMSpecies j) T) i) (hd : MSSMCharges.Hd S = MSSMCharges.Hd T) (hu : MSSMCharges.Hu S = MSSMCharges.Hu T) : MSSMACCs.accQuad S = MSSMACCs.accQuad T"} +{"name":"MSSMACCs.quadBiLin_toFun_apply","declaration":"theorem MSSMACCs.quadBiLin_toFun_apply (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) : (MSSMACCs.quadBiLin S) T =\n (Finset.sum Finset.univ fun x =>\n S (Fin.castAdd 2 (finProdFinEquiv (0, x))) * T (Fin.castAdd 2 (finProdFinEquiv (0, x))) +\n -(2 * (S (Fin.castAdd 2 (finProdFinEquiv (1, x))) * T (Fin.castAdd 2 (finProdFinEquiv (1, x))))) +\n S (Fin.castAdd 2 (finProdFinEquiv (2, x))) * T (Fin.castAdd 2 (finProdFinEquiv (2, x))) +\n -(S (Fin.castAdd 2 (finProdFinEquiv (3, x))) * T (Fin.castAdd 2 (finProdFinEquiv (3, x)))) +\n S (Fin.castAdd 2 (finProdFinEquiv (4, x))) * T (Fin.castAdd 2 (finProdFinEquiv (4, x)))) +\n (-(S { val := 18, isLt := MSSMCharges.Hd.proof_1 } * T { val := 18, isLt := MSSMCharges.Hd.proof_1 }) +\n S { val := 19, isLt := MSSMCharges.Hu.proof_1 } * T { val := 19, isLt := MSSMCharges.Hu.proof_1 })"} +{"name":"MSSMCharges_numberCharges","declaration":"theorem MSSMCharges_numberCharges : MSSMCharges.numberCharges = 20"} +{"name":"MSSMACC_numberCharges","declaration":"theorem MSSMACC_numberCharges : MSSMACC.numberCharges = 20"} +{"name":"MSSMCharges.Hd","declaration":"/-- The charge `Hd`. -/\ndef MSSMCharges.Hd : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ℚ"} +{"name":"MSSMCharges.N","declaration":"/-- The `N` charges as a map `Fin 3 → ℚ`. -/\ndef MSSMCharges.N : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMSpecies"} +{"name":"MSSMACCs.accQuad","declaration":"/-- The quadratic ACC. -/\ndef MSSMACCs.accQuad : HomogeneousQuadratic (ACCSystemCharges.Charges MSSMCharges)"} +{"name":"MSSMACC.AnomalyFreeMk''_val","declaration":"theorem MSSMACC.AnomalyFreeMk''_val (S : ACCSystemQuad.QuadSols MSSMACC.toACCSystemQuad) (hcube : MSSMACCs.accCube S.val = 0) : (MSSMACC.AnomalyFreeMk'' S hcube).val = S.val"} +{"name":"MSSMCharges.charges_eq_toSpecies_eq","declaration":"theorem MSSMCharges.charges_eq_toSpecies_eq (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) : S = T ↔\n (∀ (i : Fin 6), (MSSMCharges.toSMSpecies i) S = (MSSMCharges.toSMSpecies i) T) ∧\n MSSMCharges.Hd S = MSSMCharges.Hd T ∧ MSSMCharges.Hu S = MSSMCharges.Hu T"} +{"name":"MSSMACCs.accGrav","declaration":"/-- The gravitational anomaly equation. -/\ndef MSSMACCs.accGrav : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ℚ"} +{"name":"MSSMACCs.cubeTriLin","declaration":"/-- The symmetric trilinear form used to define the cubic ACC. -/\ndef MSSMACCs.cubeTriLin : TriLinearSymm (ACCSystemCharges.Charges MSSMCharges)"} +{"name":"MSSMACCs.accSU3_ext","declaration":"/-- Extensionality lemma for `accSU3`. -/\ntheorem MSSMACCs.accSU3_ext {S : ACCSystemCharges.Charges MSSMCharges} {T : ACCSystemCharges.Charges MSSMCharges} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) S i) =\n Finset.sum Finset.univ fun i => (MSSMCharges.toSMSpecies j) T i) : MSSMACCs.accSU3 S = MSSMACCs.accSU3 T"} +{"name":"MSSMACCs.cubeTriLin_toFun_apply_apply","declaration":"theorem MSSMACCs.cubeTriLin_toFun_apply_apply (S : ACCSystemCharges.Charges MSSMCharges) (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) : ((MSSMACCs.cubeTriLin S✝) S) T =\n (Finset.sum Finset.univ fun i =>\n 6 *\n (S✝ (Fin.castAdd 2 (finProdFinEquiv (0, i))) * S (Fin.castAdd 2 (finProdFinEquiv (0, i))) *\n T (Fin.castAdd 2 (finProdFinEquiv (0, i)))) +\n 3 *\n (S✝ (Fin.castAdd 2 (finProdFinEquiv (1, i))) * S (Fin.castAdd 2 (finProdFinEquiv (1, i))) *\n T (Fin.castAdd 2 (finProdFinEquiv (1, i)))) +\n 3 *\n (S✝ (Fin.castAdd 2 (finProdFinEquiv (2, i))) * S (Fin.castAdd 2 (finProdFinEquiv (2, i))) *\n T (Fin.castAdd 2 (finProdFinEquiv (2, i)))) +\n 2 *\n (S✝ (Fin.castAdd 2 (finProdFinEquiv (3, i))) * S (Fin.castAdd 2 (finProdFinEquiv (3, i))) *\n T (Fin.castAdd 2 (finProdFinEquiv (3, i)))) +\n S✝ (Fin.castAdd 2 (finProdFinEquiv (4, i))) * S (Fin.castAdd 2 (finProdFinEquiv (4, i))) *\n T (Fin.castAdd 2 (finProdFinEquiv (4, i))) +\n S✝ (Fin.castAdd 2 (finProdFinEquiv (5, i))) * S (Fin.castAdd 2 (finProdFinEquiv (5, i))) *\n T (Fin.castAdd 2 (finProdFinEquiv (5, i)))) +\n (2 * S✝ { val := 18, isLt := MSSMCharges.Hd.proof_1 } * S { val := 18, isLt := MSSMCharges.Hd.proof_1 } *\n T { val := 18, isLt := MSSMCharges.Hd.proof_1 } +\n 2 * S✝ { val := 19, isLt := MSSMCharges.Hu.proof_1 } * S { val := 19, isLt := MSSMCharges.Hu.proof_1 } *\n T { val := 19, isLt := MSSMCharges.Hu.proof_1 })"} +{"name":"MSSMACCs.cubeTriLinToFun_map_add₁","declaration":"theorem MSSMACCs.cubeTriLinToFun_map_add₁ (S : ACCSystemCharges.Charges MSSMCharges) (T : ACCSystemCharges.Charges MSSMCharges) (R : ACCSystemCharges.Charges MSSMCharges) (L : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.cubeTriLinToFun (S + T, R, L) = MSSMACCs.cubeTriLinToFun (S, R, L) + MSSMACCs.cubeTriLinToFun (T, R, L)"} +{"name":"MSSMCharges.toSMSpecies","declaration":"/-- For a given `i ∈ Fin 6` the projection of `MSSMCharges.charges` down to the\ncorresponding SM species of charges. -/\ndef MSSMCharges.toSMSpecies (i : Fin 6) : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMSpecies"} +{"name":"MSSMCharges.toSMPlusH_symm_apply","declaration":"theorem MSSMCharges.toSMPlusH_symm_apply : ∀ (a : Fin 18 ⊕ Fin 2 → ℚ) (a_1 : Fin (18 + 2)), MSSMCharges.toSMPlusH.symm a a_1 = a (finSumFinEquiv.symm a_1)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.HyperCharge.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.HyperCharge.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..fb6446cbfdaeb0ed918f2fb5fd920b7211327736 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.HyperCharge.jsonl @@ -0,0 +1,2 @@ +{"name":"MSSMACC.YAsCharge","declaration":"/-- The hypercharge as an element of `MSSMACC.charges`. -/\ndef MSSMACC.YAsCharge : ACCSystemCharges.Charges MSSMACC.toACCSystemCharges"} +{"name":"MSSMACC.Y","declaration":"/-- The hypercharge as an element of `MSSMACC.Sols`. -/\ndef MSSMACC.Y : ACCSystem.Sols MSSMACC"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.LineY3B3.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.LineY3B3.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a3ea498b2853f752dc13b622af1f8756f20dde92 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.LineY3B3.jsonl @@ -0,0 +1,6 @@ +{"name":"MSSMACC.lineY₃B₃","declaration":"/-- The line through $Y_3$ and $B_3$ as `Sols`. -/\ndef MSSMACC.lineY₃B₃ (a : ℚ) (b : ℚ) : ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC.doublePoint_Y₃_B₃","declaration":"theorem MSSMACC.doublePoint_Y₃_B₃ (R : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : ((MSSMACCs.cubeTriLin MSSMACC.Y₃.val) MSSMACC.B₃.val) R.val = 0"} +{"name":"MSSMACC.lineY₃B₃_doublePoint","declaration":"theorem MSSMACC.lineY₃B₃_doublePoint (R : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) (a : ℚ) (b : ℚ) : ((MSSMACCs.cubeTriLin (MSSMACC.lineY₃B₃ a b).val) (MSSMACC.lineY₃B₃ a b).val) R.val = 0"} +{"name":"MSSMACC.lineY₃B₃Charges","declaration":"/-- The line through $Y_3$ and $B_3$ as `LinSols`. -/\ndef MSSMACC.lineY₃B₃Charges (a : ℚ) (b : ℚ) : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear"} +{"name":"MSSMACC.lineY₃B₃Charges_cubic","declaration":"theorem MSSMACC.lineY₃B₃Charges_cubic (a : ℚ) (b : ℚ) : MSSMACCs.accCube (MSSMACC.lineY₃B₃Charges a b).val = 0"} +{"name":"MSSMACC.lineY₃B₃Charges_quad","declaration":"theorem MSSMACC.lineY₃B₃Charges_quad (a : ℚ) (b : ℚ) : MSSMACCs.accQuad (MSSMACC.lineY₃B₃Charges a b).val = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..668fbdc1ccbc9642c74f312a055365650891d3e7 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.Basic.jsonl @@ -0,0 +1,15 @@ +{"name":"MSSMACC.AnomalyFreePerp.perpB₃","declaration":"def MSSMACC.AnomalyFreePerp.perpB₃ (self : MSSMACC.AnomalyFreePerp) : (MSSMACC.dot MSSMACC.B₃.val) self.val = 0"} +{"name":"MSSMACC.proj_val","declaration":"theorem MSSMACC.proj_val (T : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : (MSSMACC.proj T).val =\n ((MSSMACC.dot MSSMACC.B₃.val) T.val - (MSSMACC.dot MSSMACC.Y₃.val) T.val) • MSSMACC.Y₃.val +\n ((MSSMACC.dot MSSMACC.Y₃.val) T.val - 2 * (MSSMACC.dot MSSMACC.B₃.val) T.val) • MSSMACC.B₃.val +\n (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val • T.val"} +{"name":"MSSMACC.AnomalyFreePerp.mk","declaration":"ctor MSSMACC.AnomalyFreePerp.mk (toLinSols : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) (perpY₃ : (MSSMACC.dot MSSMACC.Y₃.val) toLinSols.val = 0) (perpB₃ : (MSSMACC.dot MSSMACC.B₃.val) toLinSols.val = 0) : MSSMACC.AnomalyFreePerp"} +{"name":"MSSMACC.AnomalyFreePerp.perpY₃","declaration":"def MSSMACC.AnomalyFreePerp.perpY₃ (self : MSSMACC.AnomalyFreePerp) : (MSSMACC.dot MSSMACC.Y₃.val) self.val = 0"} +{"name":"MSSMACC.cube_proj_proj_B₃","declaration":"theorem MSSMACC.cube_proj_proj_B₃ (T : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : ((MSSMACCs.cubeTriLin (MSSMACC.proj T).val) (MSSMACC.proj T).val) MSSMACC.B₃.val =\n (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val ^ 2 * ((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.B₃.val"} +{"name":"MSSMACC.cube_proj_proj_Y₃","declaration":"theorem MSSMACC.cube_proj_proj_Y₃ (T : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : ((MSSMACCs.cubeTriLin (MSSMACC.proj T).val) (MSSMACC.proj T).val) MSSMACC.Y₃.val =\n (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val ^ 2 * ((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.Y₃.val"} +{"name":"MSSMACC.quad_self_proj","declaration":"theorem MSSMACC.quad_self_proj (T : ACCSystem.Sols MSSMACC) : (MSSMACCs.quadBiLin T.val) (MSSMACC.proj T.toLinSols).val =\n ((MSSMACC.dot MSSMACC.B₃.val) T.val - (MSSMACC.dot MSSMACC.Y₃.val) T.val) *\n (MSSMACCs.quadBiLin MSSMACC.Y₃.val) T.val +\n ((MSSMACC.dot MSSMACC.Y₃.val) T.val - 2 * (MSSMACC.dot MSSMACC.B₃.val) T.val) *\n (MSSMACCs.quadBiLin MSSMACC.B₃.val) T.val"} +{"name":"MSSMACC.proj","declaration":"/-- The projection of an object in `MSSMACC.AnomalyFreeLinear` onto the subspace\northgonal to `Y₃` and`B₃`. -/\ndef MSSMACC.proj (T : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : MSSMACC.AnomalyFreePerp"} +{"name":"MSSMACC.Y₃_plus_B₃_plus_proj","declaration":"theorem MSSMACC.Y₃_plus_B₃_plus_proj (T : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) (a : ℚ) (b : ℚ) (c : ℚ) : a • MSSMACC.Y₃.val + b • MSSMACC.B₃.val + c • (MSSMACC.proj T).val =\n (a + c * ((MSSMACC.dot MSSMACC.B₃.val) T.val - (MSSMACC.dot MSSMACC.Y₃.val) T.val)) • MSSMACC.Y₃.val +\n (b + c * ((MSSMACC.dot MSSMACC.Y₃.val) T.val - 2 * (MSSMACC.dot MSSMACC.B₃.val) T.val)) • MSSMACC.B₃.val +\n ((MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val * c) • T.val"} +{"name":"MSSMACC.AnomalyFreePerp","declaration":"/-- The type of linear solutions orthogonal to $Y_3$ and $B_3$. -/\nstructure MSSMACC.AnomalyFreePerp : Type"} +{"name":"MSSMACC.quad_B₃_proj","declaration":"theorem MSSMACC.quad_B₃_proj (T : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : (MSSMACCs.quadBiLin MSSMACC.B₃.val) (MSSMACC.proj T).val =\n (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val * (MSSMACCs.quadBiLin MSSMACC.B₃.val) T.val"} +{"name":"MSSMACC.cube_proj_proj_self","declaration":"theorem MSSMACC.cube_proj_proj_self (T : ACCSystem.Sols MSSMACC) : ((MSSMACCs.cubeTriLin (MSSMACC.proj T.toLinSols).val) (MSSMACC.proj T.toLinSols).val) T.val =\n 2 * (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val *\n (((MSSMACC.dot MSSMACC.B₃.val) T.val - (MSSMACC.dot MSSMACC.Y₃.val) T.val) *\n ((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.Y₃.val +\n ((MSSMACC.dot MSSMACC.Y₃.val) T.val - 2 * (MSSMACC.dot MSSMACC.B₃.val) T.val) *\n ((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.B₃.val)"} +{"name":"MSSMACC.cube_proj","declaration":"theorem MSSMACC.cube_proj (T : ACCSystem.Sols MSSMACC) : ((MSSMACCs.cubeTriLin (MSSMACC.proj T.toLinSols).val) (MSSMACC.proj T.toLinSols).val) (MSSMACC.proj T.toLinSols).val =\n 3 * (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val ^ 2 *\n (((MSSMACC.dot MSSMACC.B₃.val) T.val - (MSSMACC.dot MSSMACC.Y₃.val) T.val) *\n ((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.Y₃.val +\n ((MSSMACC.dot MSSMACC.Y₃.val) T.val - 2 * (MSSMACC.dot MSSMACC.B₃.val) T.val) *\n ((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.B₃.val)"} +{"name":"MSSMACC.quad_Y₃_proj","declaration":"theorem MSSMACC.quad_Y₃_proj (T : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : (MSSMACCs.quadBiLin MSSMACC.Y₃.val) (MSSMACC.proj T).val =\n (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val * (MSSMACCs.quadBiLin MSSMACC.Y₃.val) T.val"} +{"name":"MSSMACC.quad_proj","declaration":"theorem MSSMACC.quad_proj (T : ACCSystem.Sols MSSMACC) : (MSSMACCs.quadBiLin (MSSMACC.proj T.toLinSols).val) (MSSMACC.proj T.toLinSols).val =\n 2 * (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val *\n (((MSSMACC.dot MSSMACC.B₃.val) T.val - (MSSMACC.dot MSSMACC.Y₃.val) T.val) *\n (MSSMACCs.quadBiLin MSSMACC.Y₃.val) T.val +\n ((MSSMACC.dot MSSMACC.Y₃.val) T.val - 2 * (MSSMACC.dot MSSMACC.B₃.val) T.val) *\n (MSSMACCs.quadBiLin MSSMACC.B₃.val) T.val)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.PlaneWithY3B3.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.PlaneWithY3B3.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1c21e7c0f46a8cc4d2985554760f709becf49e53 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.PlaneWithY3B3.jsonl @@ -0,0 +1,25 @@ +{"name":"MSSMACC.α₃_proj","declaration":"theorem MSSMACC.α₃_proj (T : ACCSystem.Sols MSSMACC) : MSSMACC.α₃ (MSSMACC.proj T.toLinSols) =\n 6 * (MSSMACC.dot MSSMACC.Y₃.val) MSSMACC.B₃.val ^ 3 *\n (((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.Y₃.val * (MSSMACCs.quadBiLin MSSMACC.B₃.val) T.val -\n ((MSSMACCs.cubeTriLin T.val) T.val) MSSMACC.B₃.val * (MSSMACCs.quadBiLin MSSMACC.Y₃.val) T.val)"} +{"name":"MSSMACC.lineCube_smul","declaration":"theorem MSSMACC.lineCube_smul (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) (d : ℚ) : MSSMACC.lineCube R (d * a) (d * b) (d * c) = d • MSSMACC.lineCube R a b c"} +{"name":"MSSMACC.lineQuadAFL","declaration":"/-- The line in the plane spanned by `Y₃`, `B₃` and `R` which is in the quadratic,\nas `LinSols`. -/\ndef MSSMACC.lineQuadAFL (R : MSSMACC.AnomalyFreePerp) (c1 : ℚ) (c2 : ℚ) (c3 : ℚ) : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear"} +{"name":"MSSMACC.lineQuad","declaration":"/-- The line in the plane spanned by `Y₃`, `B₃` and `R` which is in the quadratic. -/\ndef MSSMACC.lineQuad (R : MSSMACC.AnomalyFreePerp) (c1 : ℚ) (c2 : ℚ) (c3 : ℚ) : ACCSystemQuad.QuadSols MSSMACC.toACCSystemQuad"} +{"name":"MSSMACC.planeY₃B₃_val","declaration":"theorem MSSMACC.planeY₃B₃_val (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) : (MSSMACC.planeY₃B₃ R a b c).val = a • MSSMACC.Y₃.val + b • MSSMACC.B₃.val + c • R.val"} +{"name":"MSSMACC.α₁_proj","declaration":"theorem MSSMACC.α₁_proj (T : ACCSystem.Sols MSSMACC) : MSSMACC.α₁ (MSSMACC.proj T.toLinSols) =\n -MSSMACC.α₃ (MSSMACC.proj T.toLinSols) * ((MSSMACC.dot MSSMACC.B₃.val) T.val - (MSSMACC.dot MSSMACC.Y₃.val) T.val)"} +{"name":"MSSMACC.α₂_proj","declaration":"theorem MSSMACC.α₂_proj (T : ACCSystem.Sols MSSMACC) : MSSMACC.α₂ (MSSMACC.proj T.toLinSols) =\n -MSSMACC.α₃ (MSSMACC.proj T.toLinSols) * ((MSSMACC.dot MSSMACC.Y₃.val) T.val - 2 * (MSSMACC.dot MSSMACC.B₃.val) T.val)"} +{"name":"MSSMACC.planeY₃B₃_cubic","declaration":"theorem MSSMACC.planeY₃B₃_cubic (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) : MSSMACCs.accCube (MSSMACC.planeY₃B₃ R a b c).val =\n c ^ 2 *\n (3 * a * ((MSSMACCs.cubeTriLin R.val) R.val) MSSMACC.Y₃.val +\n 3 * b * ((MSSMACCs.cubeTriLin R.val) R.val) MSSMACC.B₃.val +\n c * ((MSSMACCs.cubeTriLin R.val) R.val) R.val)"} +{"name":"MSSMACC.planeY₃B₃_val_eq'","declaration":"theorem MSSMACC.planeY₃B₃_val_eq' {a' : ℚ} {b' : ℚ} {c' : ℚ} (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) (hR' : R.val ≠ 0) (h : (MSSMACC.planeY₃B₃ R a b c).val = (MSSMACC.planeY₃B₃ R a' b' c').val) : a = a' ∧ b = b' ∧ c = c'"} +{"name":"MSSMACC.lineCube","declaration":"/-- The line in the plane spanned by `Y₃`, `B₃` and `R` which is in the cubic. -/\ndef MSSMACC.lineCube (R : MSSMACC.AnomalyFreePerp) (a₁ : ℚ) (a₂ : ℚ) (a₃ : ℚ) : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear"} +{"name":"MSSMACC.α₂","declaration":"/-- A helper function to simplify following expressions. -/\ndef MSSMACC.α₂ (T : MSSMACC.AnomalyFreePerp) : ℚ"} +{"name":"MSSMACC.planeY₃B₃_smul","declaration":"theorem MSSMACC.planeY₃B₃_smul (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) (d : ℚ) : MSSMACC.planeY₃B₃ R (d * a) (d * b) (d * c) = d • MSSMACC.planeY₃B₃ R a b c"} +{"name":"MSSMACC.lineQuadAFL_quad","declaration":"theorem MSSMACC.lineQuadAFL_quad (R : MSSMACC.AnomalyFreePerp) (c1 : ℚ) (c2 : ℚ) (c3 : ℚ) : MSSMACCs.accQuad (MSSMACC.lineQuadAFL R c1 c2 c3).val = 0"} +{"name":"MSSMACC.planeY₃B₃_quad","declaration":"theorem MSSMACC.planeY₃B₃_quad (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) : MSSMACCs.accQuad (MSSMACC.planeY₃B₃ R a b c).val =\n c *\n (2 * a * (MSSMACCs.quadBiLin MSSMACC.Y₃.val) R.val + 2 * b * (MSSMACCs.quadBiLin MSSMACC.B₃.val) R.val +\n c * (MSSMACCs.quadBiLin R.val) R.val)"} +{"name":"MSSMACC.α₁_proj_zero","declaration":"theorem MSSMACC.α₁_proj_zero (T : ACCSystem.Sols MSSMACC) (h1 : MSSMACC.α₃ (MSSMACC.proj T.toLinSols) = 0) : MSSMACC.α₁ (MSSMACC.proj T.toLinSols) = 0"} +{"name":"MSSMACC.lineQuad_cube","declaration":"theorem MSSMACC.lineQuad_cube (R : MSSMACC.AnomalyFreePerp) (c₁ : ℚ) (c₂ : ℚ) (c₃ : ℚ) : MSSMACCs.accCube (MSSMACC.lineQuad R c₁ c₂ c₃).val =\n -4 * (c₁ * (MSSMACCs.quadBiLin MSSMACC.B₃.val) R.val - c₂ * (MSSMACCs.quadBiLin MSSMACC.Y₃.val) R.val) ^ 2 *\n (MSSMACC.α₁ R * c₁ + MSSMACC.α₂ R * c₂ + MSSMACC.α₃ R * c₃)"} +{"name":"MSSMACC.lineQuad_smul","declaration":"theorem MSSMACC.lineQuad_smul (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) (d : ℚ) : MSSMACC.lineQuad R (d * a) (d * b) (d * c) = d • MSSMACC.lineQuad R a b c"} +{"name":"MSSMACC.lineCube_quad","declaration":"theorem MSSMACC.lineCube_quad (R : MSSMACC.AnomalyFreePerp) (a₁ : ℚ) (a₂ : ℚ) (a₃ : ℚ) : MSSMACCs.accQuad (MSSMACC.lineCube R a₁ a₂ a₃).val =\n 3 *\n (a₁ * ((MSSMACCs.cubeTriLin R.val) R.val) MSSMACC.B₃.val -\n a₂ * ((MSSMACCs.cubeTriLin R.val) R.val) MSSMACC.Y₃.val) *\n (MSSMACC.α₁ R * a₁ + MSSMACC.α₂ R * a₂ + MSSMACC.α₃ R * a₃)"} +{"name":"MSSMACC.α₂_proj_zero","declaration":"theorem MSSMACC.α₂_proj_zero (T : ACCSystem.Sols MSSMACC) (h1 : MSSMACC.α₃ (MSSMACC.proj T.toLinSols) = 0) : MSSMACC.α₂ (MSSMACC.proj T.toLinSols) = 0"} +{"name":"MSSMACC.α₁","declaration":"/-- A helper function to simplify following expressions. -/\ndef MSSMACC.α₁ (T : MSSMACC.AnomalyFreePerp) : ℚ"} +{"name":"MSSMACC.planeY₃B₃","declaration":"/-- The plane of linear solutions spanned by `Y₃`, `B₃` and `R`, a point orthogonal\nto `Y₃` and `B₃`. -/\ndef MSSMACC.planeY₃B₃ (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear"} +{"name":"MSSMACC.α₃","declaration":"/-- A helper function to simplify following expressions. -/\ndef MSSMACC.α₃ (T : MSSMACC.AnomalyFreePerp) : ℚ"} +{"name":"MSSMACC.lineQuad_val","declaration":"theorem MSSMACC.lineQuad_val (R : MSSMACC.AnomalyFreePerp) (c1 : ℚ) (c2 : ℚ) (c3 : ℚ) : (MSSMACC.lineQuad R c1 c2 c3).val =\n (MSSMACC.planeY₃B₃ R (c2 * (MSSMACCs.quadBiLin R.val) R.val - 2 * c3 * (MSSMACCs.quadBiLin MSSMACC.B₃.val) R.val)\n (2 * c3 * (MSSMACCs.quadBiLin MSSMACC.Y₃.val) R.val - c1 * (MSSMACCs.quadBiLin R.val) R.val)\n (2 * c1 * (MSSMACCs.quadBiLin MSSMACC.B₃.val) R.val - 2 * c2 * (MSSMACCs.quadBiLin MSSMACC.Y₃.val) R.val)).val"} +{"name":"MSSMACC.planeY₃B₃_eq","declaration":"theorem MSSMACC.planeY₃B₃_eq {a' : ℚ} {b' : ℚ} {c' : ℚ} (R : MSSMACC.AnomalyFreePerp) (a : ℚ) (b : ℚ) (c : ℚ) (h : a = a' ∧ b = b' ∧ c = c') : MSSMACC.planeY₃B₃ R a b c = MSSMACC.planeY₃B₃ R a' b' c'"} +{"name":"MSSMACC.lineCube_cube","declaration":"theorem MSSMACC.lineCube_cube (R : MSSMACC.AnomalyFreePerp) (a₁ : ℚ) (a₂ : ℚ) (a₃ : ℚ) : MSSMACCs.accCube (MSSMACC.lineCube R a₁ a₂ a₃).val = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.ToSols.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.ToSols.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ae9cee078c5a62439c97c36361e52488d60c9638 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.ToSols.jsonl @@ -0,0 +1,49 @@ +{"name":"MSSMACC.AnomalyFreePerp.toSol_inQuadCube","declaration":"theorem MSSMACC.AnomalyFreePerp.toSol_inQuadCube (T : MSSMACC.AnomalyFreePerp.InQuadCubeSol) : ∃ X, MSSMACC.AnomalyFreePerp.toSol X = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.lineEqCoeff","declaration":"/-- A rational which appears in `toSolNS` acting on sols, and which being zero is\nequivalent to satisfying `lineEqPropSol`. -/\ndef MSSMACC.AnomalyFreePerp.lineEqCoeff (T : ACCSystem.Sols MSSMACC) : ℚ"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadCubeToSol_proj","declaration":"theorem MSSMACC.AnomalyFreePerp.inQuadCubeToSol_proj (T : MSSMACC.AnomalyFreePerp.InQuadCubeSol) : MSSMACC.AnomalyFreePerp.inQuadCubeToSol (MSSMACC.AnomalyFreePerp.inQuadCubeProj T) = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.inLineEqTo_smul","declaration":"theorem MSSMACC.AnomalyFreePerp.inLineEqTo_smul (R : MSSMACC.AnomalyFreePerp.InLineEq) (c₁ : ℚ) (c₂ : ℚ) (c₃ : ℚ) (d : ℚ) : MSSMACC.AnomalyFreePerp.inLineEqToSol (R, d * c₁, d * c₂, d * c₃) =\n d • MSSMACC.AnomalyFreePerp.inLineEqToSol (R, c₁, c₂, c₃)"} +{"name":"MSSMACC.AnomalyFreePerp.instDecidableInCubeProp","declaration":"instance MSSMACC.AnomalyFreePerp.instDecidableInCubeProp (R : MSSMACC.AnomalyFreePerp) : Decidable (MSSMACC.AnomalyFreePerp.InCubeProp R)"} +{"name":"MSSMACC.AnomalyFreePerp.NotInLineEqSol","declaration":"/-- Those solutions which do not satisfy the condition `lineEqPropSol`. -/\ndef MSSMACC.AnomalyFreePerp.NotInLineEqSol : Type"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadSolProp_iff_proj_inQuadProp","declaration":"/-- The conditions `inQuadSolProp R` and `inQuadProp (proj R.1.1)` are equivalent. This is to be\nexpected since both `R` and `proj R.1.1` define the same plane with `Y₃` and `B₃`. -/\ntheorem MSSMACC.AnomalyFreePerp.inQuadSolProp_iff_proj_inQuadProp (R : ACCSystem.Sols MSSMACC) : MSSMACC.AnomalyFreePerp.InQuadSolProp R ↔ MSSMACC.AnomalyFreePerp.InQuadProp (MSSMACC.proj R.toLinSols)"} +{"name":"MSSMACC.AnomalyFreePerp.InCubeProp","declaration":"/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies\nentirely in the cubic surface. -/\ndef MSSMACC.AnomalyFreePerp.InCubeProp (R : MSSMACC.AnomalyFreePerp) : Prop"} +{"name":"MSSMACC.AnomalyFreePerp.toSolNSQuad_eq_planeY₃B₃_on_α","declaration":"theorem MSSMACC.AnomalyFreePerp.toSolNSQuad_eq_planeY₃B₃_on_α (R : MSSMACC.AnomalyFreePerp) : (MSSMACC.AnomalyFreePerp.toSolNSQuad R).toLinSols = MSSMACC.planeY₃B₃ R (MSSMACC.α₁ R) (MSSMACC.α₂ R) (MSSMACC.α₃ R)"} +{"name":"MSSMACC.AnomalyFreePerp.toSol_surjective","declaration":"theorem MSSMACC.AnomalyFreePerp.toSol_surjective : Function.Surjective MSSMACC.AnomalyFreePerp.toSol"} +{"name":"MSSMACC.AnomalyFreePerp.toSolNS_proj","declaration":"theorem MSSMACC.AnomalyFreePerp.toSolNS_proj (T : MSSMACC.AnomalyFreePerp.NotInLineEqSol) : MSSMACC.AnomalyFreePerp.toSolNS (MSSMACC.AnomalyFreePerp.toSolNSProj ↑T) = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadToSol_proj","declaration":"theorem MSSMACC.AnomalyFreePerp.inQuadToSol_proj (T : MSSMACC.AnomalyFreePerp.InQuadSol) : MSSMACC.AnomalyFreePerp.inQuadToSol (MSSMACC.AnomalyFreePerp.inQuadProj T) = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.inCubeSolProp_iff_cubicCoeff_zero","declaration":"theorem MSSMACC.AnomalyFreePerp.inCubeSolProp_iff_cubicCoeff_zero (T : ACCSystem.Sols MSSMACC) : MSSMACC.AnomalyFreePerp.InCubeSolProp T ↔ MSSMACC.AnomalyFreePerp.cubicCoeff T = 0"} +{"name":"MSSMACC.AnomalyFreePerp.LineEqProp","declaration":"/-- A condition for the quad line in the plane spanned by R, Y₃ and B₃ to sit in the cubic,\nand for the cube line to sit in the quad. -/\ndef MSSMACC.AnomalyFreePerp.LineEqProp (R : MSSMACC.AnomalyFreePerp) : Prop"} +{"name":"MSSMACC.AnomalyFreePerp.linEqPropSol_iff_proj_linEqProp","declaration":"theorem MSSMACC.AnomalyFreePerp.linEqPropSol_iff_proj_linEqProp (R : ACCSystem.Sols MSSMACC) : MSSMACC.AnomalyFreePerp.LineEqPropSol R ↔ MSSMACC.AnomalyFreePerp.LineEqProp (MSSMACC.proj R.toLinSols)"} +{"name":"MSSMACC.AnomalyFreePerp.toSol_inLineEq","declaration":"theorem MSSMACC.AnomalyFreePerp.toSol_inLineEq (T : MSSMACC.AnomalyFreePerp.InLineEqSol) : ∃ X, MSSMACC.AnomalyFreePerp.toSol X = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.inLineEqProj","declaration":"/-- On elements of `inLineEqSol` a right-inverse to `inLineEqSol`. -/\ndef MSSMACC.AnomalyFreePerp.inLineEqProj (T : MSSMACC.AnomalyFreePerp.InLineEqSol) : MSSMACC.AnomalyFreePerp.InLineEq × ℚ × ℚ × ℚ"} +{"name":"MSSMACC.AnomalyFreePerp.instDecidableInQuadProp","declaration":"instance MSSMACC.AnomalyFreePerp.instDecidableInQuadProp (R : MSSMACC.AnomalyFreePerp) : Decidable (MSSMACC.AnomalyFreePerp.InQuadProp R)"} +{"name":"MSSMACC.AnomalyFreePerp.InQuadSolProp","declaration":"/-- A condition which is satisfied if the plane spanned by the solutions `R`, `Y₃` and `B₃`\nlies entirely in the quadratic surface. -/\ndef MSSMACC.AnomalyFreePerp.InQuadSolProp (R : ACCSystem.Sols MSSMACC) : Prop"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadCubeToSol","declaration":"/-- Given a element of `inQuadCube × ℚ × ℚ × ℚ`, a solution to the ACCs. -/\ndef MSSMACC.AnomalyFreePerp.inQuadCubeToSol : MSSMACC.AnomalyFreePerp.InQuadCube × ℚ × ℚ × ℚ → ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC.AnomalyFreePerp.InQuadCubeSol","declaration":"/-- Those solutions which satisfy the conditions `lineEqPropSol`, `inQuadSolProp`\nand `inCubeSolProp`. -/\ndef MSSMACC.AnomalyFreePerp.InQuadCubeSol : Type"} +{"name":"MSSMACC.AnomalyFreePerp.LineEqPropSol","declaration":"/-- A condition on `Sols` which we will show in `linEqPropSol_iff_proj_linEqProp` that is equivalent\nto the condition that the `proj` of the solution satisfies `lineEqProp`. -/\ndef MSSMACC.AnomalyFreePerp.LineEqPropSol (R : ACCSystem.Sols MSSMACC) : Prop"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadToSol","declaration":"/-- Given an element of `inQuad × ℚ × ℚ × ℚ`, a solution to the ACCs. -/\ndef MSSMACC.AnomalyFreePerp.inQuadToSol : MSSMACC.AnomalyFreePerp.InQuad × ℚ × ℚ × ℚ → ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC.AnomalyFreePerp.InQuadCube","declaration":"/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the conditions\n`lineEqProp`, `inQuadProp` and `inCubeProp`. -/\ndef MSSMACC.AnomalyFreePerp.InQuadCube : Type"} +{"name":"MSSMACC.AnomalyFreePerp.lineEqPropSol_iff_lineEqCoeff_zero","declaration":"theorem MSSMACC.AnomalyFreePerp.lineEqPropSol_iff_lineEqCoeff_zero (T : ACCSystem.Sols MSSMACC) : MSSMACC.AnomalyFreePerp.LineEqPropSol T ↔ MSSMACC.AnomalyFreePerp.lineEqCoeff T = 0"} +{"name":"MSSMACC.AnomalyFreePerp.toSolNSQuad","declaration":"/-- Given an `R` perpendicular to `Y₃` and `B₃` a quadratic solution. -/\ndef MSSMACC.AnomalyFreePerp.toSolNSQuad (R : MSSMACC.AnomalyFreePerp) : ACCSystemQuad.QuadSols MSSMACC.toACCSystemQuad"} +{"name":"MSSMACC.AnomalyFreePerp.inLineEqToSol_proj","declaration":"theorem MSSMACC.AnomalyFreePerp.inLineEqToSol_proj (T : MSSMACC.AnomalyFreePerp.InLineEqSol) : MSSMACC.AnomalyFreePerp.inLineEqToSol (MSSMACC.AnomalyFreePerp.inLineEqProj T) = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.instDecidableLineEqProp","declaration":"instance MSSMACC.AnomalyFreePerp.instDecidableLineEqProp (R : MSSMACC.AnomalyFreePerp) : Decidable (MSSMACC.AnomalyFreePerp.LineEqProp R)"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadCubeToSol_smul","declaration":"theorem MSSMACC.AnomalyFreePerp.inQuadCubeToSol_smul (R : MSSMACC.AnomalyFreePerp.InQuadCube) (c₁ : ℚ) (c₂ : ℚ) (c₃ : ℚ) (d : ℚ) : MSSMACC.AnomalyFreePerp.inQuadCubeToSol (R, d * c₁, d * c₂, d * c₃) =\n d • MSSMACC.AnomalyFreePerp.inQuadCubeToSol (R, c₁, c₂, c₃)"} +{"name":"MSSMACC.AnomalyFreePerp.InQuadSol","declaration":"/-- Those solutions which satisfy the condition `lineEqPropSol` and `inQuadSolProp` but\nnot `inCubeSolProp`. -/\ndef MSSMACC.AnomalyFreePerp.InQuadSol : Type"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadToSol_smul","declaration":"theorem MSSMACC.AnomalyFreePerp.inQuadToSol_smul (R : MSSMACC.AnomalyFreePerp.InQuad) (c₁ : ℚ) (c₂ : ℚ) (c₃ : ℚ) (d : ℚ) : MSSMACC.AnomalyFreePerp.inQuadToSol (R, d * c₁, d * c₂, d * c₃) =\n d • MSSMACC.AnomalyFreePerp.inQuadToSol (R, c₁, c₂, c₃)"} +{"name":"MSSMACC.AnomalyFreePerp.toSolNSProj","declaration":"/-- A map from `Sols` to `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` which on elements of\n`notInLineEqSol` will produce a right inverse to `toSolNS`. -/\ndef MSSMACC.AnomalyFreePerp.toSolNSProj (T : ACCSystem.Sols MSSMACC) : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ"} +{"name":"MSSMACC.AnomalyFreePerp.toSol_inQuad","declaration":"theorem MSSMACC.AnomalyFreePerp.toSol_inQuad (T : MSSMACC.AnomalyFreePerp.InQuadSol) : ∃ X, MSSMACC.AnomalyFreePerp.toSol X = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.toSolNSQuad_cube","declaration":"theorem MSSMACC.AnomalyFreePerp.toSolNSQuad_cube (R : MSSMACC.AnomalyFreePerp) : MSSMACCs.accCube (MSSMACC.AnomalyFreePerp.toSolNSQuad R).val = 0"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadCubeProj","declaration":"/-- On elements of `inQuadCubeSol` a right-inverse to `inQuadCubeToSol`. -/\ndef MSSMACC.AnomalyFreePerp.inQuadCubeProj (T : MSSMACC.AnomalyFreePerp.InQuadCubeSol) : MSSMACC.AnomalyFreePerp.InQuadCube × ℚ × ℚ × ℚ"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadSolProp_iff_quadCoeff_zero","declaration":"theorem MSSMACC.AnomalyFreePerp.inQuadSolProp_iff_quadCoeff_zero (T : ACCSystem.Sols MSSMACC) : MSSMACC.AnomalyFreePerp.InQuadSolProp T ↔ MSSMACC.AnomalyFreePerp.quadCoeff T = 0"} +{"name":"MSSMACC.AnomalyFreePerp.InQuadProp","declaration":"/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies\nentirely in the quadratic surface. -/\ndef MSSMACC.AnomalyFreePerp.InQuadProp (R : MSSMACC.AnomalyFreePerp) : Prop"} +{"name":"MSSMACC.AnomalyFreePerp.inQuadProj","declaration":"/-- On elements of `inQuadSol` a right-inverse to `inQuadToSol`. -/\ndef MSSMACC.AnomalyFreePerp.inQuadProj (T : MSSMACC.AnomalyFreePerp.InQuadSol) : MSSMACC.AnomalyFreePerp.InQuad × ℚ × ℚ × ℚ"} +{"name":"MSSMACC.AnomalyFreePerp.toSol","declaration":"/-- A solution from an element of `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ`. We will\nshow that this map is a surjection. -/\ndef MSSMACC.AnomalyFreePerp.toSol : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC.AnomalyFreePerp.InCubeSolProp","declaration":"/-- A condition which is satisfied if the plane spanned by the solutions `R`, `Y₃` and `B₃`\nlies entirely in the cubic surface. -/\ndef MSSMACC.AnomalyFreePerp.InCubeSolProp (R : ACCSystem.Sols MSSMACC) : Prop"} +{"name":"MSSMACC.AnomalyFreePerp.toSol_toSolNSProj","declaration":"theorem MSSMACC.AnomalyFreePerp.toSol_toSolNSProj (T : MSSMACC.AnomalyFreePerp.NotInLineEqSol) : ∃ X, MSSMACC.AnomalyFreePerp.toSol X = ↑T"} +{"name":"MSSMACC.AnomalyFreePerp.inCubeSolProp_iff_proj_inCubeProp","declaration":"theorem MSSMACC.AnomalyFreePerp.inCubeSolProp_iff_proj_inCubeProp (R : ACCSystem.Sols MSSMACC) : MSSMACC.AnomalyFreePerp.InCubeSolProp R ↔ MSSMACC.AnomalyFreePerp.InCubeProp (MSSMACC.proj R.toLinSols)"} +{"name":"MSSMACC.AnomalyFreePerp.cubicCoeff","declaration":"/-- A rational which has two properties. It is zero for a solution `T` if and only if\nthat solution satisfies `inCubeSolProp`. It appears in the definition of `inLineEqProj`. -/\ndef MSSMACC.AnomalyFreePerp.cubicCoeff (T : ACCSystem.Sols MSSMACC) : ℚ"} +{"name":"MSSMACC.AnomalyFreePerp.InQuad","declaration":"/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the conditions\n`lineEqProp` and `inQuadProp`. -/\ndef MSSMACC.AnomalyFreePerp.InQuad : Type"} +{"name":"MSSMACC.AnomalyFreePerp.quadCoeff","declaration":"/-- A rational which has two properties. It is zero for a solution `T` if and only if\nthat solution satisfies `inQuadSolProp`. It appears in the definition of `inQuadProj`. -/\ndef MSSMACC.AnomalyFreePerp.quadCoeff (T : ACCSystem.Sols MSSMACC) : ℚ"} +{"name":"MSSMACC.AnomalyFreePerp.inLineEqToSol","declaration":"/-- A solution to the ACCs, given an element of `inLineEq × ℚ × ℚ × ℚ`. -/\ndef MSSMACC.AnomalyFreePerp.inLineEqToSol : MSSMACC.AnomalyFreePerp.InLineEq × ℚ × ℚ × ℚ → ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC.AnomalyFreePerp.InLineEq","declaration":"/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition\n`lineEqProp`. -/\ndef MSSMACC.AnomalyFreePerp.InLineEq : Type"} +{"name":"MSSMACC.AnomalyFreePerp.toSolNS","declaration":"/-- Given an `R` perpendicular to `Y₃` and `B₃`, an element of `Sols`. This map is\nnot surjective. -/\ndef MSSMACC.AnomalyFreePerp.toSolNS : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → ACCSystem.Sols MSSMACC"} +{"name":"MSSMACC.AnomalyFreePerp.InLineEqSol","declaration":"/-- Those solutions which satisfy the condition `lineEqPropSol` but not `inQuadSolProp`. -/\ndef MSSMACC.AnomalyFreePerp.InLineEqSol : Type"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Permutations.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Permutations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b3b4bc85a29a536ee26d5b158b947ff2bcd1b267 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Permutations.jsonl @@ -0,0 +1,16 @@ +{"name":"MSSM.PermGroup","declaration":"/-- The group of family permutations is `S₃⁶`-/\ndef MSSM.PermGroup : Type"} +{"name":"MSSM.accSU2_invariant","declaration":"theorem MSSM.accSU2_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.accSU2 ((MSSM.repCharges f) S) = MSSMACCs.accSU2 S"} +{"name":"MSSM.accSU3_invariant","declaration":"theorem MSSM.accSU3_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.accSU3 ((MSSM.repCharges f) S) = MSSMACCs.accSU3 S"} +{"name":"MSSM.accCube_invariant","declaration":"theorem MSSM.accCube_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.accCube ((MSSM.repCharges f) S) = MSSMACCs.accCube S"} +{"name":"MSSM.Hu_invariant","declaration":"theorem MSSM.Hu_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMCharges.Hu ((MSSM.repCharges f) S) = MSSMCharges.Hu S"} +{"name":"MSSM.accQuad_invariant","declaration":"theorem MSSM.accQuad_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.accQuad ((MSSM.repCharges f) S) = MSSMACCs.accQuad S"} +{"name":"MSSM.accGrav_invariant","declaration":"theorem MSSM.accGrav_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.accGrav ((MSSM.repCharges f) S) = MSSMACCs.accGrav S"} +{"name":"MSSM.chargeMap","declaration":"/-- The image of an element of `permGroup` under the representation on charges. -/\ndef MSSM.chargeMap (f : MSSM.PermGroup) : ACCSystemCharges.Charges MSSMCharges →ₗ[ℚ] ACCSystemCharges.Charges MSSMCharges"} +{"name":"MSSM.repCharges_toSMSpecies","declaration":"theorem MSSM.repCharges_toSMSpecies (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) (j : Fin 6) : (MSSMCharges.toSMSpecies j) ((MSSM.repCharges f) S) = (MSSMCharges.toSMSpecies j) S ∘ ⇑(f⁻¹ j)"} +{"name":"MSSM.instGroupPermGroup","declaration":"instance MSSM.instGroupPermGroup : Group MSSM.PermGroup"} +{"name":"MSSM.chargeMap_apply","declaration":"theorem MSSM.chargeMap_apply (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : (MSSM.chargeMap f) S =\n MSSMCharges.toSMPlusH.symm\n (MSSMCharges.splitSMPlusH.symm\n (MSSMCharges.toSpeciesMaps'.symm fun i => (MSSMCharges.toSMSpecies i) S ∘ ⇑(f i),\n MSSMCharges.toSMPlusH S ∘ Sum.inr))"} +{"name":"MSSM.toSpecies_sum_invariant","declaration":"theorem MSSM.toSpecies_sum_invariant (m : ℕ) (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) (j : Fin 6) : (Finset.sum Finset.univ fun i => ((fun a => a ^ m) ∘ (MSSMCharges.toSMSpecies j) ((MSSM.repCharges f) S)) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ m) ∘ (MSSMCharges.toSMSpecies j) S) i"} +{"name":"MSSM.repCharges","declaration":"/-- The representation of `permGroup` acting on the vector space of charges. -/\ndef MSSM.repCharges : Representation ℚ MSSM.PermGroup (ACCSystemCharges.Charges MSSMCharges)"} +{"name":"MSSM.accYY_invariant","declaration":"theorem MSSM.accYY_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMACCs.accYY ((MSSM.repCharges f) S) = MSSMACCs.accYY S"} +{"name":"MSSM.Hd_invariant","declaration":"theorem MSSM.Hd_invariant (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) : MSSMCharges.Hd ((MSSM.repCharges f) S) = MSSMCharges.Hd S"} +{"name":"MSSM.chargeMap_toSpecies","declaration":"theorem MSSM.chargeMap_toSpecies (f : MSSM.PermGroup) (S : ACCSystemCharges.Charges MSSMCharges) (j : Fin 6) : (MSSMCharges.toSMSpecies j) ((MSSM.chargeMap f) S) = (MSSMCharges.toSMSpecies j) S ∘ ⇑(f j)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Y3.jsonl b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Y3.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c3cfe6145638c1a78605cdd312b4e1b164fbef4d --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.MSSMNu.Y3.jsonl @@ -0,0 +1,4 @@ +{"name":"MSSMACC.Y₃_val","declaration":"theorem MSSMACC.Y₃_val : MSSMACC.Y₃.val = MSSMACC.Y₃AsCharge"} +{"name":"MSSMACC.doublePoint_Y₃_Y₃","declaration":"theorem MSSMACC.doublePoint_Y₃_Y₃ (R : ACCSystemLinear.LinSols MSSMACC.toACCSystemLinear) : ((MSSMACCs.cubeTriLin MSSMACC.Y₃.val) MSSMACC.Y₃.val) R.val = 0"} +{"name":"MSSMACC.Y₃AsCharge","declaration":"/-- $Y_3$ is the charge which is hypercharge in all families, but with the third\nfamily of the opposite sign. -/\ndef MSSMACC.Y₃AsCharge : ACCSystemCharges.Charges MSSMACC.toACCSystemCharges"} +{"name":"MSSMACC.Y₃","declaration":"/-- $Y_3$ as a solution. -/\ndef MSSMACC.Y₃ : ACCSystem.Sols MSSMACC"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cdfcd7f6dab980055bb130df19d13d9a00610ca0 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Basic.jsonl @@ -0,0 +1,21 @@ +{"name":"PureU1.accCubeTriLinSymm_toFun_apply_apply","declaration":"theorem PureU1.accCubeTriLinSymm_toFun_apply_apply {n : ℕ} (S : ACCSystemCharges.Charges (PureU1.PureU1Charges n)) (S : ACCSystemCharges.Charges (PureU1.PureU1Charges n)) (T : ACCSystemCharges.Charges (PureU1.PureU1Charges n)) : ((PureU1.accCubeTriLinSymm S✝) S) T = Finset.sum Finset.univ fun i => S✝ i * S i * T i"} +{"name":"PureU1_quadraticACCs","declaration":"theorem PureU1_quadraticACCs (n : ℕ) : ∀ (a : Fin 0), (PureU1 n).quadraticACCs a = Fin.elim0 a"} +{"name":"PureU1.PureU1Charges_numberCharges","declaration":"theorem PureU1.PureU1Charges_numberCharges (n : ℕ) : (PureU1.PureU1Charges n).numberCharges = n"} +{"name":"pureU1_cube","declaration":"theorem pureU1_cube {n : ℕ} (S : ACCSystem.Sols (PureU1 (Nat.succ n))) : (Finset.sum Finset.univ fun i => S.val i ^ 3) = 0"} +{"name":"pureU1_anomalyFree_ext","declaration":"theorem pureU1_anomalyFree_ext {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear} {T : ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear} (h : ∀ (i : Fin n), S.val (Fin.castSucc i) = T.val (Fin.castSucc i)) : S = T"} +{"name":"pureU1_linear","declaration":"theorem pureU1_linear {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear) : (Finset.sum Finset.univ fun i => S.val i) = 0"} +{"name":"PureU1_linearACCs","declaration":"theorem PureU1_linearACCs (n : ℕ) (i : Fin 1) : (PureU1 n).linearACCs i =\n match i with\n | 0 => PureU1.accGrav n"} +{"name":"PureU1_numberLinear","declaration":"theorem PureU1_numberLinear (n : ℕ) : (PureU1 n).numberLinear = 1"} +{"name":"PureU1.sum_of_anomaly_free_linear","declaration":"theorem PureU1.sum_of_anomaly_free_linear {k : ℕ} {n : ℕ} (f : Fin k → ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear) (j : Fin n) : (Finset.sum Finset.univ fun i => f i).val j = Finset.sum Finset.univ fun i => (f i).val j"} +{"name":"PureU1.accGrav","declaration":"/-- The gravitational anomaly. -/\ndef PureU1.accGrav (n : ℕ) : ACCSystemCharges.Charges (PureU1.PureU1Charges n) →ₗ[ℚ] ℚ"} +{"name":"PureU1.accCube_explicit","declaration":"theorem PureU1.accCube_explicit (n : ℕ) (S : ACCSystemCharges.Charges (PureU1.PureU1Charges n)) : (PureU1.accCube n) S = Finset.sum Finset.univ fun i => S i ^ 3"} +{"name":"PureU1_cubicACC_toFun","declaration":"theorem PureU1_cubicACC_toFun (n : ℕ) (S : ACCSystemCharges.Charges (PureU1.PureU1Charges n)) : (PureU1 n).cubicACC.toFun S = Finset.sum Finset.univ fun i => S i * S i * S i"} +{"name":"PureU1.accCube","declaration":"/-- The cubic anomaly equation. -/\ndef PureU1.accCube (n : ℕ) : HomogeneousCubic (ACCSystemCharges.Charges (PureU1.PureU1Charges n))"} +{"name":"PureU1_numberCharges","declaration":"theorem PureU1_numberCharges (n : ℕ) : (PureU1 n).numberCharges = n"} +{"name":"PureU1.PureU1Charges","declaration":"/-- The vector space of charges. -/\ndef PureU1.PureU1Charges (n : ℕ) : ACCSystemCharges"} +{"name":"PureU1","declaration":"/-- The ACC system for a pure $U(1)$ gauge theory with $n$ fermions. -/\ndef PureU1 (n : ℕ) : ACCSystem"} +{"name":"PureU1_numberQuadratic","declaration":"theorem PureU1_numberQuadratic (n : ℕ) : (PureU1 n).numberQuadratic = 0"} +{"name":"pureU1EqCharges","declaration":"/-- An equivalence of vector spaces of charges when the number of fermions is equal. -/\ndef pureU1EqCharges {n : ℕ} {m : ℕ} (h : n = m) : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges ≃ₗ[ℚ] ACCSystemCharges.Charges (PureU1 m).toACCSystemCharges"} +{"name":"pureU1_last","declaration":"theorem pureU1_last {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear) : S.val (Fin.last n) = -Finset.sum Finset.univ fun i => S.val (Fin.castSucc i)"} +{"name":"PureU1.accCubeTriLinSymm","declaration":"/-- The symmetric trilinear form used to define the cubic anomaly. -/\ndef PureU1.accCubeTriLinSymm {n : ℕ} : TriLinearSymm (ACCSystemCharges.Charges (PureU1.PureU1Charges n))"} +{"name":"PureU1.sum_of_charges","declaration":"theorem PureU1.sum_of_charges {k : ℕ} {n : ℕ} (f : Fin k → ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) (j : Fin n) : Finset.sum Finset.univ (fun i => f i) j = Finset.sum Finset.univ fun i => f i j"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.BasisLinear.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.BasisLinear.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..81057714cf0a14fb0eb731d0004f16e57e024dfd --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.BasisLinear.jsonl @@ -0,0 +1,10 @@ +{"name":"PureU1.BasisLinear.asLinSols","declaration":"/-- The basis elements as `LinSols`. -/\ndef PureU1.BasisLinear.asLinSols {n : ℕ} (j : Fin n) : ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.BasisLinear.asCharges_eq_castSucc","declaration":"theorem PureU1.BasisLinear.asCharges_eq_castSucc {n : ℕ} (j : Fin n) : PureU1.BasisLinear.asCharges j (Fin.castSucc j) = 1"} +{"name":"PureU1.BasisLinear.sum_of_vectors","declaration":"theorem PureU1.BasisLinear.sum_of_vectors {k : ℕ} {n : ℕ} (f : Fin k → ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear) (j : Fin n) : (Finset.sum Finset.univ fun i => f i).val j = Finset.sum Finset.univ fun i => (f i).val j"} +{"name":"PureU1.BasisLinear.asLinSols_val","declaration":"theorem PureU1.BasisLinear.asLinSols_val {n : ℕ} (j : Fin n) : (PureU1.BasisLinear.asLinSols j).val = PureU1.BasisLinear.asCharges j"} +{"name":"PureU1.BasisLinear.coordinateMap","declaration":"/-- The coordinate map for the basis. -/\ndef PureU1.BasisLinear.coordinateMap {n : ℕ} : ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear ≃ₗ[ℚ] Fin n →₀ ℚ"} +{"name":"PureU1.BasisLinear.finrank_AnomalyFreeLinear","declaration":"theorem PureU1.BasisLinear.finrank_AnomalyFreeLinear {n : ℕ} : FiniteDimensional.finrank ℚ (ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear) = n"} +{"name":"PureU1.BasisLinear.asBasis","declaration":"/-- The basis of `LinSols`. -/\ndef PureU1.BasisLinear.asBasis {n : ℕ} : Basis (Fin n) ℚ (ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear)"} +{"name":"PureU1.BasisLinear.asCharges","declaration":"/-- The basis elements as charges, defined to have a `1` in the `j`th position and a `-1` in the\nlast position. -/\ndef PureU1.BasisLinear.asCharges {n : ℕ} (j : Fin n) : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges"} +{"name":"PureU1.BasisLinear.instFiniteRatLinSolsToACCSystemLinearToACCSystemQuadPureU1SuccSemiringLinSolsAddCommMonoidLinSolsModule","declaration":"instance PureU1.BasisLinear.instFiniteRatLinSolsToACCSystemLinearToACCSystemQuadPureU1SuccSemiringLinSolsAddCommMonoidLinSolsModule {n : ℕ} : Module.Finite ℚ (ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear)"} +{"name":"PureU1.BasisLinear.asCharges_ne_castSucc","declaration":"theorem PureU1.BasisLinear.asCharges_ne_castSucc {n : ℕ} {k : Fin n} {j : Fin n} (h : k ≠ j) : PureU1.BasisLinear.asCharges k (Fin.castSucc j) = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.ConstAbs.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.ConstAbs.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b61538a2469291b31cfe9e767e98b6b7dc84926c --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.ConstAbs.jsonl @@ -0,0 +1,32 @@ +{"name":"PureU1.ConstAbsSorted.AFL_odd_noBoundary","declaration":"theorem PureU1.ConstAbsSorted.AFL_odd_noBoundary {n : ℕ} {A : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} (h : PureU1.ConstAbsSorted A.val) (hA : A.val 0 ≠ 0) : ¬PureU1.ConstAbsSorted.HasBoundary A.val"} +{"name":"PureU1.ConstAbsSorted.AFL_odd","declaration":"theorem PureU1.ConstAbsSorted.AFL_odd {n : ℕ} (A : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear) (h : PureU1.ConstAbsSorted A.val) : A = 0"} +{"name":"PureU1.ConstAbsSorted.HasBoundary","declaration":"/-- A `S ∈ charges` has a boundary if there exists a `k ∈ Fin n` which is a boundary. -/\ndef PureU1.ConstAbsSorted.HasBoundary {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges) : Prop"} +{"name":"PureU1.ConstAbs.boundary_value_odd","declaration":"theorem PureU1.ConstAbs.boundary_value_odd {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear) (hs : PureU1.ConstAbs S.val) : S = 0"} +{"name":"PureU1.ConstAbsSorted.AFL_even_Boundary","declaration":"theorem PureU1.ConstAbsSorted.AFL_even_Boundary {n : ℕ} {A : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (h : PureU1.ConstAbsSorted A.val) (hA : A.val 0 ≠ 0) {k : Fin (2 * n + 1)} (hk : PureU1.ConstAbsSorted.Boundary A.val k) : ↑k = n"} +{"name":"PureU1.ConstAbsSorted.boundary_castSucc","declaration":"theorem PureU1.ConstAbsSorted.boundary_castSucc {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) {k : Fin n} (hk : PureU1.ConstAbsSorted.Boundary S k) : S (Fin.castSucc k) = S 0"} +{"name":"PureU1.ConstAbsSorted.not_hasBoundary_grav","declaration":"theorem PureU1.ConstAbsSorted.not_hasBoundary_grav {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) (hnot : ¬PureU1.ConstAbsSorted.HasBoundary S) : (PureU1.accGrav (Nat.succ n)) S = ↑(Nat.succ n) * S 0"} +{"name":"PureU1.ConstAbs","declaration":"/-- The condition on a charge assignment `S` to have constant absolute value among charges. -/\ndef PureU1.ConstAbs {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : Prop"} +{"name":"PureU1.ConstAbsSorted.lt_eq","declaration":"theorem PureU1.ConstAbsSorted.lt_eq {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) {k : Fin (Nat.succ n)} {i : Fin (Nat.succ n)} (hk : S k ≤ 0) (hik : i ≤ k) : S i = S k"} +{"name":"PureU1.ConstAbsSorted.AFL_even_below","declaration":"theorem PureU1.ConstAbsSorted.AFL_even_below {n : ℕ} (A : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) (h : PureU1.ConstAbsSorted A.val) (i : Fin (Nat.succ n)) : A.val (Fin.cast ⋯ (Fin.castAdd (Nat.succ n) i)) = A.val 0"} +{"name":"PureU1.constAbs_sort","declaration":"theorem PureU1.constAbs_sort {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges} (CA : PureU1.ConstAbs S) : PureU1.ConstAbs (PureU1.sort S)"} +{"name":"PureU1.ConstAbsSorted.boundary_succ","declaration":"theorem PureU1.ConstAbsSorted.boundary_succ {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) {k : Fin n} (hk : PureU1.ConstAbsSorted.Boundary S k) : S (Fin.succ k) = -S 0"} +{"name":"PureU1.ConstAbsSorted.gt_eq","declaration":"theorem PureU1.ConstAbsSorted.gt_eq {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) {k : Fin (Nat.succ n)} {i : Fin (Nat.succ n)} (hk : 0 ≤ S k) (hik : k ≤ i) : S i = S k"} +{"name":"PureU1.ConstAbsProp","declaration":"/-- The condition for two rationals to have the same square (equivalent to same abs). -/\ndef PureU1.ConstAbsProp : ℚ × ℚ → Prop"} +{"name":"PureU1.ConstAbsSorted.not_hasBoundary_zero_le","declaration":"theorem PureU1.ConstAbsSorted.not_hasBoundary_zero_le {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) (hnot : ¬PureU1.ConstAbsSorted.HasBoundary S) (h0 : S 0 < 0) (i : Fin (PureU1 (Nat.succ n)).numberCharges) : S 0 = S i"} +{"name":"PureU1.ConstAbsSorted.zero_gt","declaration":"theorem PureU1.ConstAbsSorted.zero_gt {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) (h0 : 0 ≤ S 0) (i : Fin (Nat.succ n)) : S 0 = S i"} +{"name":"PureU1.ConstAbsSorted.AFL_even_above'","declaration":"theorem PureU1.ConstAbsSorted.AFL_even_above' {n : ℕ} {A : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (h : PureU1.ConstAbsSorted A.val) (hA : A.val 0 ≠ 0) (i : Fin (Nat.succ n)) : A.val (Fin.cast ⋯ (Fin.natAdd (Nat.succ n) i)) = -A.val 0"} +{"name":"PureU1.ConstAbsSorted.boundary_split","declaration":"theorem PureU1.ConstAbsSorted.boundary_split {n : ℕ} (k : Fin n) : ↑(Fin.succ k) + (Nat.succ n - ↑(Fin.succ k)) = Nat.succ n"} +{"name":"PureU1.ConstAbsSorted.boundary_accGrav''","declaration":"theorem PureU1.ConstAbsSorted.boundary_accGrav'' {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) (k : Fin n) (hk : PureU1.ConstAbsSorted.Boundary S k) : (PureU1.accGrav (Nat.succ n)) S = (2 * ↑↑k + 1 - ↑n) * S 0"} +{"name":"PureU1.ConstAbsSorted","declaration":"/-- The condition for a set of charges to be `sorted`, and have `constAbs`-/\ndef PureU1.ConstAbsSorted {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : Prop"} +{"name":"PureU1.ConstAbsSorted.val_le_zero","declaration":"theorem PureU1.ConstAbsSorted.val_le_zero {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) {i : Fin (Nat.succ n)} (hi : S i ≤ 0) : S i = S 0"} +{"name":"PureU1.ConstAbsSorted.not_hasBoundry_zero","declaration":"theorem PureU1.ConstAbsSorted.not_hasBoundry_zero {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) (hnot : ¬PureU1.ConstAbsSorted.HasBoundary S) (i : Fin (Nat.succ n)) : S 0 = S i"} +{"name":"PureU1.constAbs_perm","declaration":"theorem PureU1.constAbs_perm {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) (M : (PureU1.FamilyPermutations n).group) : PureU1.ConstAbs (((PureU1.FamilyPermutations n).rep M) S) ↔ PureU1.ConstAbs S"} +{"name":"PureU1.ConstAbs.boundary_value_even","declaration":"theorem PureU1.ConstAbs.boundary_value_even {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) (hs : PureU1.ConstAbs S.val) : PureU1.VectorLikeEven S.val"} +{"name":"PureU1.ConstAbsSorted.AFL_odd_zero","declaration":"theorem PureU1.ConstAbsSorted.AFL_odd_zero {n : ℕ} {A : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} (h : PureU1.ConstAbsSorted A.val) : A.val 0 = 0"} +{"name":"PureU1.ConstAbsSorted.AFL_hasBoundary","declaration":"theorem PureU1.ConstAbsSorted.AFL_hasBoundary {n : ℕ} {A : ACCSystemLinear.LinSols (PureU1 (Nat.succ n)).toACCSystemLinear} (hA : PureU1.ConstAbsSorted A.val) (h : A.val 0 ≠ 0) : PureU1.ConstAbsSorted.HasBoundary A.val"} +{"name":"PureU1.ConstAbsSorted.AFL_even_above","declaration":"theorem PureU1.ConstAbsSorted.AFL_even_above {n : ℕ} (A : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) (h : PureU1.ConstAbsSorted A.val) (i : Fin (Nat.succ n)) : A.val (Fin.cast ⋯ (Fin.natAdd (Nat.succ n) i)) = -A.val 0"} +{"name":"PureU1.ConstAbsSorted.opposite_signs_eq_neg","declaration":"theorem PureU1.ConstAbsSorted.opposite_signs_eq_neg {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) {i : Fin (Nat.succ n)} {j : Fin (Nat.succ n)} (hi : S i ≤ 0) (hj : 0 ≤ S j) : S i = -S j"} +{"name":"PureU1.ConstAbsSorted.AFL_even_below'","declaration":"theorem PureU1.ConstAbsSorted.AFL_even_below' {n : ℕ} {A : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (h : PureU1.ConstAbsSorted A.val) (hA : A.val 0 ≠ 0) (i : Fin (Nat.succ n)) : A.val (Fin.cast ⋯ (Fin.castAdd (Nat.succ n) i)) = A.val 0"} +{"name":"PureU1.ConstAbsSorted.is_zero","declaration":"theorem PureU1.ConstAbsSorted.is_zero {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (hS : PureU1.ConstAbsSorted S) (h0 : S 0 = 0) : S = 0"} +{"name":"PureU1.ConstAbsSorted.Boundary","declaration":"/-- A boundary of `S : (PureU1 n.succ).charges` (assumed sorted, constAbs and non-zero)\nis defined as a element of `k ∈ Fin n` such that `S k.castSucc` and `S k.succ` are different signs.\n-/\ndef PureU1.ConstAbsSorted.Boundary {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges) (k : Fin n) : Prop"} +{"name":"PureU1.ConstAbsSorted.boundary_accGrav'","declaration":"theorem PureU1.ConstAbsSorted.boundary_accGrav' {n : ℕ} {S : ACCSystemCharges.Charges (PureU1 (Nat.succ n)).toACCSystemCharges} (k : Fin n) : (PureU1.accGrav (Nat.succ n)) S = Finset.sum Finset.univ fun i => S (Fin.cast ⋯ i)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.BasisLinear.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.BasisLinear.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..dc211fbe37735a31c09d93230b18f6a113459c92 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.BasisLinear.jsonl @@ -0,0 +1,85 @@ +{"name":"PureU1.VectorLikeEvenPlane.P_δ₁","declaration":"theorem PureU1.VectorLikeEvenPlane.P_δ₁ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (j : Fin (Nat.succ n)) : PureU1.VectorLikeEvenPlane.P f (PureU1.VectorLikeEvenPlane.δ₁ j) = f j"} +{"name":"PureU1.VectorLikeEvenPlane.basis_δ₂_eq_minus_δ₁","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_δ₂_eq_minus_δ₁ {n : ℕ} (j : Fin (Nat.succ n)) (i : Fin (Nat.succ n)) : PureU1.VectorLikeEvenPlane.basisAsCharges j (PureU1.VectorLikeEvenPlane.δ₂ i) =\n -PureU1.VectorLikeEvenPlane.basisAsCharges j (PureU1.VectorLikeEvenPlane.δ₁ i)"} +{"name":"PureU1.VectorLikeEvenPlane.basis_on_δ₂_self","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_on_δ₂_self {n : ℕ} (j : Fin (Nat.succ n)) : PureU1.VectorLikeEvenPlane.basisAsCharges j (PureU1.VectorLikeEvenPlane.δ₂ j) = -1"} +{"name":"PureU1.VectorLikeEvenPlane.P_δ₂","declaration":"theorem PureU1.VectorLikeEvenPlane.P_δ₂ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (j : Fin (Nat.succ n)) : PureU1.VectorLikeEvenPlane.P f (PureU1.VectorLikeEvenPlane.δ₂ j) = -f j"} +{"name":"PureU1.VectorLikeEvenPlane.Pa_δ!₂","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa_δ!₂ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeEvenPlane.Pa f g (PureU1.VectorLikeEvenPlane.δ!₂ j) = -f (Fin.castSucc j) - g j"} +{"name":"PureU1.VectorLikeEvenPlane.P'","declaration":"/-- A point in the span of the first part of the basis. -/\ndef PureU1.VectorLikeEvenPlane.P' {n : ℕ} (f : Fin (Nat.succ n) → ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeEvenPlane.basis_on_δ₂_other","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_on_δ₂_other {n : ℕ} {k : Fin (Nat.succ n)} {j : Fin (Nat.succ n)} (h : k ≠ j) : PureU1.VectorLikeEvenPlane.basisAsCharges k (PureU1.VectorLikeEvenPlane.δ₂ j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.δ₂","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeEvenPlane.δ₂ {n : ℕ} (j : Fin (Nat.succ n)) : Fin (2 * Nat.succ n)"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_linearACC","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_linearACC {n : ℕ} (j : Fin n) : (PureU1.accGrav (2 * Nat.succ n)) (PureU1.VectorLikeEvenPlane.basis!AsCharges j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.Pa_zero","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa_zero {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin n → ℚ) (h : PureU1.VectorLikeEvenPlane.Pa f g = 0) (i : Fin (Nat.succ n)) : f i = 0"} +{"name":"PureU1.VectorLikeEvenPlane.δ₁","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeEvenPlane.δ₁ {n : ℕ} (j : Fin (Nat.succ n)) : Fin (2 * Nat.succ n)"} +{"name":"PureU1.VectorLikeEvenPlane.P'_val","declaration":"theorem PureU1.VectorLikeEvenPlane.P'_val {n : ℕ} (f : Fin (Nat.succ n) → ℚ) : (PureU1.VectorLikeEvenPlane.P' f).val = PureU1.VectorLikeEvenPlane.P f"} +{"name":"PureU1.VectorLikeEvenPlane.P_linearACC","declaration":"theorem PureU1.VectorLikeEvenPlane.P_linearACC {n : ℕ} (f : Fin (Nat.succ n) → ℚ) : (PureU1.accGrav (2 * Nat.succ n)) (PureU1.VectorLikeEvenPlane.P f) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.Pa'_eq","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa'_eq {n : ℕ} (f : Fin (Nat.succ n) ⊕ Fin n → ℚ) (f' : Fin (Nat.succ n) ⊕ Fin n → ℚ) : PureU1.VectorLikeEvenPlane.Pa' f = PureU1.VectorLikeEvenPlane.Pa' f' ↔ f = f'"} +{"name":"PureU1.VectorLikeEvenPlane.basis_val","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_val {n : ℕ} (j : Fin (Nat.succ n)) : (PureU1.VectorLikeEvenPlane.basis j).val = PureU1.VectorLikeEvenPlane.basisAsCharges j"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_on_δ!₁_self","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_on_δ!₁_self {n : ℕ} (j : Fin n) : PureU1.VectorLikeEvenPlane.basis!AsCharges j (PureU1.VectorLikeEvenPlane.δ!₁ j) = 1"} +{"name":"PureU1.VectorLikeEvenPlane.basis!AsCharges","declaration":"/-- The second part of the basis as charges. -/\ndef PureU1.VectorLikeEvenPlane.basis!AsCharges {n : ℕ} (j : Fin n) : ACCSystemCharges.Charges (PureU1 (2 * Nat.succ n)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeEvenPlane.Pa_δ!₃","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa_δ!₃ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.Pa f g PureU1.VectorLikeEvenPlane.δ!₃ = f 0"} +{"name":"PureU1.VectorLikeEvenPlane.sum_δ₁_δ₂","declaration":"theorem PureU1.VectorLikeEvenPlane.sum_δ₁_δ₂ {n : ℕ} (S : Fin (2 * Nat.succ n) → ℚ) : (Finset.sum Finset.univ fun i => S i) =\n Finset.sum Finset.univ fun i => (S ∘ PureU1.VectorLikeEvenPlane.δ₁) i + (S ∘ PureU1.VectorLikeEvenPlane.δ₂) i"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₄_δ₂Last","declaration":"theorem PureU1.VectorLikeEvenPlane.δ!₄_δ₂Last {n : ℕ} : PureU1.VectorLikeEvenPlane.δ!₄ = PureU1.VectorLikeEvenPlane.δ₂ (Fin.last n)"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_linear_independent","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_linear_independent {n : ℕ} : LinearIndependent ℚ PureU1.VectorLikeEvenPlane.basis!"} +{"name":"PureU1.VectorLikeEvenPlane.P!_zero","declaration":"theorem PureU1.VectorLikeEvenPlane.P!_zero {n : ℕ} (f : Fin n → ℚ) (h : PureU1.VectorLikeEvenPlane.P! f = 0) (i : Fin n) : f i = 0"} +{"name":"PureU1.VectorLikeEvenPlane.join","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeEvenPlane.join {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) : Fin (Nat.succ n) ⊕ Fin n → ℚ"} +{"name":"PureU1.VectorLikeEvenPlane.basis_accCube","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_accCube {n : ℕ} (j : Fin (Nat.succ n)) : (PureU1.accCube (2 * Nat.succ n)) (PureU1.VectorLikeEvenPlane.basisAsCharges j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₂_δ₂","declaration":"theorem PureU1.VectorLikeEvenPlane.δ!₂_δ₂ {n : ℕ} (j : Fin n) : PureU1.VectorLikeEvenPlane.δ!₂ j = PureU1.VectorLikeEvenPlane.δ₂ (Fin.castSucc j)"} +{"name":"PureU1.VectorLikeEvenPlane.P!'_val","declaration":"theorem PureU1.VectorLikeEvenPlane.P!'_val {n : ℕ} (f : Fin n → ℚ) : (PureU1.VectorLikeEvenPlane.P!' f).val = PureU1.VectorLikeEvenPlane.P! f"} +{"name":"PureU1.VectorLikeEvenPlane.P_P_P!_accCube","declaration":"theorem PureU1.VectorLikeEvenPlane.P_P_P!_accCube {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (j : Fin n) : ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.basis!AsCharges j) =\n g (Fin.succ j) ^ 2 - g (Fin.castSucc j) ^ 2"} +{"name":"PureU1.VectorLikeEvenPlane.Pa","declaration":"/-- A point in the span of the basis as a charge. -/\ndef PureU1.VectorLikeEvenPlane.Pa {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin n → ℚ) : ACCSystemCharges.Charges (PureU1 (2 * Nat.succ n)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeEvenPlane.P!","declaration":"/-- A point in the span of the second part of the basis as a charge. -/\ndef PureU1.VectorLikeEvenPlane.P! {n : ℕ} (f : Fin n → ℚ) : ACCSystemCharges.Charges (PureU1 (2 * Nat.succ n)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeEvenPlane.vectorLikeEven_in_span","declaration":"theorem PureU1.VectorLikeEvenPlane.vectorLikeEven_in_span {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) (hS : PureU1.VectorLikeEven S.val) : ∃ M,\n ((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * Nat.succ n))) M) S ∈\n Submodule.span ℚ (Set.range PureU1.VectorLikeEvenPlane.basis)"} +{"name":"PureU1.VectorLikeEvenPlane.P!_in_span","declaration":"theorem PureU1.VectorLikeEvenPlane.P!_in_span {n : ℕ} (f : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.P! f ∈ Submodule.span ℚ (Set.range PureU1.VectorLikeEvenPlane.basis!AsCharges)"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_on_δ!₂_other","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_on_δ!₂_other {n : ℕ} {k : Fin n} {j : Fin n} (h : k ≠ j) : PureU1.VectorLikeEvenPlane.basis!AsCharges k (PureU1.VectorLikeEvenPlane.δ!₂ j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.P_zero","declaration":"theorem PureU1.VectorLikeEvenPlane.P_zero {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (h : PureU1.VectorLikeEvenPlane.P f = 0) (i : Fin (Nat.succ n)) : f i = 0"} +{"name":"PureU1.VectorLikeEvenPlane.P!'","declaration":"/-- A point in the span of the second part of the basis. -/\ndef PureU1.VectorLikeEvenPlane.P!' {n : ℕ} (f : Fin n → ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeEvenPlane.basis_on_δ₁_self","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_on_δ₁_self {n : ℕ} (j : Fin (Nat.succ n)) : PureU1.VectorLikeEvenPlane.basisAsCharges j (PureU1.VectorLikeEvenPlane.δ₁ j) = 1"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₃_δ₁0","declaration":"theorem PureU1.VectorLikeEvenPlane.δ!₃_δ₁0 {n : ℕ} : PureU1.VectorLikeEvenPlane.δ!₃ = PureU1.VectorLikeEvenPlane.δ₁ 0"} +{"name":"PureU1.VectorLikeEvenPlane.P_accCube","declaration":"theorem PureU1.VectorLikeEvenPlane.P_accCube {n : ℕ} (f : Fin (Nat.succ n) → ℚ) : (PureU1.accCube (2 * Nat.succ n)) (PureU1.VectorLikeEvenPlane.P f) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.Pa_zero!","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa_zero! {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin n → ℚ) (h : PureU1.VectorLikeEvenPlane.Pa f g = 0) (i : Fin n) : g i = 0"} +{"name":"PureU1.VectorLikeEvenPlane.sum_δ!₁_δ!₂","declaration":"theorem PureU1.VectorLikeEvenPlane.sum_δ!₁_δ!₂ {n : ℕ} (S : Fin (2 * Nat.succ n) → ℚ) : (Finset.sum Finset.univ fun i => S i) =\n S PureU1.VectorLikeEvenPlane.δ!₃ + S PureU1.VectorLikeEvenPlane.δ!₄ +\n Finset.sum Finset.univ fun i => (S ∘ PureU1.VectorLikeEvenPlane.δ!₁) i + (S ∘ PureU1.VectorLikeEvenPlane.δ!₂) i"} +{"name":"PureU1.VectorLikeEvenPlane.sum_δ₁_δ₂'","declaration":"theorem PureU1.VectorLikeEvenPlane.sum_δ₁_δ₂' {n : ℕ} (S : Fin (2 * Nat.succ n) → ℚ) : (Finset.sum Finset.univ fun i => S i) =\n Finset.sum Finset.univ fun i => (S ∘ PureU1.VectorLikeEvenPlane.δ₁) i + (S ∘ PureU1.VectorLikeEvenPlane.δ₂) i"} +{"name":"PureU1.VectorLikeEvenPlane.P","declaration":"/-- A point in the span of the first part of the basis as a charge. -/\ndef PureU1.VectorLikeEvenPlane.P {n : ℕ} (f : Fin (Nat.succ n) → ℚ) : ACCSystemCharges.Charges (PureU1 (2 * Nat.succ n)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeEvenPlane.basisAsCharges","declaration":"/-- The first part of the basis as charges. -/\ndef PureU1.VectorLikeEvenPlane.basisAsCharges {n : ℕ} (j : Fin (Nat.succ n)) : ACCSystemCharges.Charges (PureU1 (2 * Nat.succ n)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeEvenPlane.span_basis_swap!","declaration":"theorem PureU1.VectorLikeEvenPlane.span_basis_swap! {n : ℕ} {S' : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (j : Fin n) (hS : ((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * Nat.succ n)))\n (PureU1.pairSwap (PureU1.VectorLikeEvenPlane.δ!₁ j) (PureU1.VectorLikeEvenPlane.δ!₂ j)))\n S =\n S') (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (h : S.val = PureU1.VectorLikeEvenPlane.P g + PureU1.VectorLikeEvenPlane.P! f) : ∃ g' f',\n S'.val = PureU1.VectorLikeEvenPlane.P g' + PureU1.VectorLikeEvenPlane.P! f' ∧\n PureU1.VectorLikeEvenPlane.P! f' =\n PureU1.VectorLikeEvenPlane.P! f +\n (S.val (PureU1.VectorLikeEvenPlane.δ!₂ j) - S.val (PureU1.VectorLikeEvenPlane.δ!₁ j)) •\n PureU1.VectorLikeEvenPlane.basis!AsCharges j ∧\n g' = g"} +{"name":"PureU1.VectorLikeEvenPlane.Pa_eq","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa_eq {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (g' : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (f' : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.Pa g f = PureU1.VectorLikeEvenPlane.Pa g' f' ↔ g = g' ∧ f = f'"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_accCube","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_accCube {n : ℕ} (j : Fin n) : (PureU1.accCube (2 * Nat.succ n)) (PureU1.VectorLikeEvenPlane.basis!AsCharges j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.P!_δ!₁","declaration":"theorem PureU1.VectorLikeEvenPlane.P!_δ!₁ {n : ℕ} (f : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeEvenPlane.P! f (PureU1.VectorLikeEvenPlane.δ!₁ j) = f j"} +{"name":"PureU1.VectorLikeEvenPlane.P_δ₁_δ₂","declaration":"theorem PureU1.VectorLikeEvenPlane.P_δ₁_δ₂ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) : PureU1.VectorLikeEvenPlane.P f ∘ PureU1.VectorLikeEvenPlane.δ₂ =\n -PureU1.VectorLikeEvenPlane.P f ∘ PureU1.VectorLikeEvenPlane.δ₁"} +{"name":"PureU1.VectorLikeEvenPlane.Pa'_P'_P!'","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa'_P'_P!' {n : ℕ} (f : Fin (Nat.succ n) ⊕ Fin n → ℚ) : PureU1.VectorLikeEvenPlane.Pa' f =\n PureU1.VectorLikeEvenPlane.P' (f ∘ Sum.inl) + PureU1.VectorLikeEvenPlane.P!' (f ∘ Sum.inr)"} +{"name":"PureU1.VectorLikeEvenPlane.P!_δ!₂","declaration":"theorem PureU1.VectorLikeEvenPlane.P!_δ!₂ {n : ℕ} (f : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeEvenPlane.P! f (PureU1.VectorLikeEvenPlane.δ!₂ j) = -f j"} +{"name":"PureU1.VectorLikeEvenPlane.smul_basis!AsCharges_in_span","declaration":"theorem PureU1.VectorLikeEvenPlane.smul_basis!AsCharges_in_span {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) (j : Fin n) : (S.val (PureU1.VectorLikeEvenPlane.δ!₂ j) - S.val (PureU1.VectorLikeEvenPlane.δ!₁ j)) •\n PureU1.VectorLikeEvenPlane.basis!AsCharges j ∈\n Submodule.span ℚ (Set.range PureU1.VectorLikeEvenPlane.basis!AsCharges)"} +{"name":"PureU1.VectorLikeEvenPlane.join_Pa","declaration":"theorem PureU1.VectorLikeEvenPlane.join_Pa {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (g' : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (f' : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.Pa' (PureU1.VectorLikeEvenPlane.join g f) =\n PureU1.VectorLikeEvenPlane.Pa' (PureU1.VectorLikeEvenPlane.join g' f') ↔\n PureU1.VectorLikeEvenPlane.Pa g f = PureU1.VectorLikeEvenPlane.Pa g' f'"} +{"name":"PureU1.VectorLikeEvenPlane.swap!_as_add","declaration":"/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/\ntheorem PureU1.VectorLikeEvenPlane.swap!_as_add {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} {S' : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (j : Fin n) (hS : ((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * Nat.succ n)))\n (PureU1.pairSwap (PureU1.VectorLikeEvenPlane.δ!₁ j) (PureU1.VectorLikeEvenPlane.δ!₂ j)))\n S =\n S') : S'.val =\n S.val +\n (S.val (PureU1.VectorLikeEvenPlane.δ!₂ j) - S.val (PureU1.VectorLikeEvenPlane.δ!₁ j)) •\n PureU1.VectorLikeEvenPlane.basis!AsCharges j"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₁","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeEvenPlane.δ!₁ {n : ℕ} (j : Fin n) : Fin (2 * Nat.succ n)"} +{"name":"PureU1.VectorLikeEvenPlane.Pa'","declaration":"/-- A point in the span of the whole basis. -/\ndef PureU1.VectorLikeEvenPlane.Pa' {n : ℕ} (f : Fin (Nat.succ n) ⊕ Fin n → ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₁_δ₁","declaration":"theorem PureU1.VectorLikeEvenPlane.δ!₁_δ₁ {n : ℕ} (j : Fin n) : PureU1.VectorLikeEvenPlane.δ!₁ j = PureU1.VectorLikeEvenPlane.δ₁ (Fin.succ j)"} +{"name":"PureU1.VectorLikeEvenPlane.basisa_card","declaration":"theorem PureU1.VectorLikeEvenPlane.basisa_card {n : ℕ} : Fintype.card (Fin (Nat.succ n) ⊕ Fin n) =\n FiniteDimensional.finrank ℚ (ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear)"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_on_δ!₁_other","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_on_δ!₁_other {n : ℕ} {k : Fin n} {j : Fin n} (h : k ≠ j) : PureU1.VectorLikeEvenPlane.basis!AsCharges k (PureU1.VectorLikeEvenPlane.δ!₁ j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.P!_accCube","declaration":"theorem PureU1.VectorLikeEvenPlane.P!_accCube {n : ℕ} (f : Fin n → ℚ) : (PureU1.accCube (2 * Nat.succ n)) (PureU1.VectorLikeEvenPlane.P! f) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.P!_δ!₄","declaration":"theorem PureU1.VectorLikeEvenPlane.P!_δ!₄ {n : ℕ} (f : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.P! f PureU1.VectorLikeEvenPlane.δ!₄ = 0"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_val","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_val {n : ℕ} (j : Fin n) : (PureU1.VectorLikeEvenPlane.basis! j).val = PureU1.VectorLikeEvenPlane.basis!AsCharges j"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_on_δ!₂_self","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_on_δ!₂_self {n : ℕ} (j : Fin n) : PureU1.VectorLikeEvenPlane.basis!AsCharges j (PureU1.VectorLikeEvenPlane.δ!₂ j) = -1"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₄","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeEvenPlane.δ!₄ {n : ℕ} : Fin (2 * Nat.succ n)"} +{"name":"PureU1.VectorLikeEvenPlane.basis_on_δ₁_other","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_on_δ₁_other {n : ℕ} {k : Fin (Nat.succ n)} {j : Fin (Nat.succ n)} (h : k ≠ j) : PureU1.VectorLikeEvenPlane.basisAsCharges k (PureU1.VectorLikeEvenPlane.δ₁ j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.basisaAsBasis","declaration":"/-- The basis formed out of our `basisa` vectors. -/\ndef PureU1.VectorLikeEvenPlane.basisaAsBasis {n : ℕ} : Basis (Fin (Nat.succ n) ⊕ Fin n) ℚ (ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear)"} +{"name":"PureU1.VectorLikeEvenPlane.basisa_linear_independent","declaration":"theorem PureU1.VectorLikeEvenPlane.basisa_linear_independent {n : ℕ} : LinearIndependent ℚ PureU1.VectorLikeEvenPlane.basisa"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_on_other","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_on_other {n : ℕ} {k : Fin n} {j : Fin (2 * Nat.succ n)} (h1 : j ≠ PureU1.VectorLikeEvenPlane.δ!₁ k) (h2 : j ≠ PureU1.VectorLikeEvenPlane.δ!₂ k) : PureU1.VectorLikeEvenPlane.basis!AsCharges k j = 0"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₃","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeEvenPlane.δ!₃ {n : ℕ} : Fin (2 * Nat.succ n)"} +{"name":"PureU1.VectorLikeEvenPlane.P!_δ!₃","declaration":"theorem PureU1.VectorLikeEvenPlane.P!_δ!₃ {n : ℕ} (f : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.P! f PureU1.VectorLikeEvenPlane.δ!₃ = 0"} +{"name":"PureU1.VectorLikeEvenPlane.Pa_δ!₄","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa_δ!₄ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.Pa f g PureU1.VectorLikeEvenPlane.δ!₄ = -f (Fin.last n)"} +{"name":"PureU1.VectorLikeEvenPlane.basis","declaration":"/-- The first part of the basis as `LinSols`. -/\ndef PureU1.VectorLikeEvenPlane.basis {n : ℕ} (j : Fin (Nat.succ n)) : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_on_δ!₄","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_on_δ!₄ {n : ℕ} (j : Fin n) : PureU1.VectorLikeEvenPlane.basis!AsCharges j PureU1.VectorLikeEvenPlane.δ!₄ = 0"} +{"name":"PureU1.VectorLikeEvenPlane.n_cond₂","declaration":"theorem PureU1.VectorLikeEvenPlane.n_cond₂ (n : ℕ) : 1 + (n + n + 1) = 2 * Nat.succ n"} +{"name":"PureU1.VectorLikeEvenPlane.P_P!_P!_accCube","declaration":"theorem PureU1.VectorLikeEvenPlane.P_P!_P!_accCube {n : ℕ} (g : Fin n → ℚ) (j : Fin (Nat.succ n)) : ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P! g)) (PureU1.VectorLikeEvenPlane.P! g))\n (PureU1.VectorLikeEvenPlane.basisAsCharges j) =\n PureU1.VectorLikeEvenPlane.P! g (PureU1.VectorLikeEvenPlane.δ₁ j) ^ 2 -\n PureU1.VectorLikeEvenPlane.P! g (PureU1.VectorLikeEvenPlane.δ₂ j) ^ 2"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_on_δ!₃","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_on_δ!₃ {n : ℕ} (j : Fin n) : PureU1.VectorLikeEvenPlane.basis!AsCharges j PureU1.VectorLikeEvenPlane.δ!₃ = 0"} +{"name":"PureU1.VectorLikeEvenPlane.basisa","declaration":"/-- The whole basis as `LinSols`. -/\ndef PureU1.VectorLikeEvenPlane.basisa {n : ℕ} : Fin (Nat.succ n) ⊕ Fin n → ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeEvenPlane.Pa_δ!₁","declaration":"theorem PureU1.VectorLikeEvenPlane.Pa_δ!₁ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeEvenPlane.Pa f g (PureU1.VectorLikeEvenPlane.δ!₁ j) = f (Fin.succ j) + g j"} +{"name":"PureU1.VectorLikeEvenPlane.basis!_δ!₂_eq_minus_δ!₁","declaration":"theorem PureU1.VectorLikeEvenPlane.basis!_δ!₂_eq_minus_δ!₁ {n : ℕ} (j : Fin n) (i : Fin n) : PureU1.VectorLikeEvenPlane.basis!AsCharges j (PureU1.VectorLikeEvenPlane.δ!₂ i) =\n -PureU1.VectorLikeEvenPlane.basis!AsCharges j (PureU1.VectorLikeEvenPlane.δ!₁ i)"} +{"name":"PureU1.VectorLikeEvenPlane.span_basis","declaration":"theorem PureU1.VectorLikeEvenPlane.span_basis {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) : ∃ g f, S.val = PureU1.VectorLikeEvenPlane.P g + PureU1.VectorLikeEvenPlane.P! f"} +{"name":"PureU1.VectorLikeEvenPlane.δ!₂","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeEvenPlane.δ!₂ {n : ℕ} (j : Fin n) : Fin (2 * Nat.succ n)"} +{"name":"PureU1.VectorLikeEvenPlane.basis!","declaration":"/-- The second part of the basis as `LinSols`. -/\ndef PureU1.VectorLikeEvenPlane.basis! {n : ℕ} (j : Fin n) : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeEvenPlane.basis_linear_independent","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_linear_independent {n : ℕ} : LinearIndependent ℚ PureU1.VectorLikeEvenPlane.basis"} +{"name":"PureU1.VectorLikeEvenPlane.join_ext","declaration":"theorem PureU1.VectorLikeEvenPlane.join_ext {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (g' : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (f' : Fin n → ℚ) : PureU1.VectorLikeEvenPlane.join g f = PureU1.VectorLikeEvenPlane.join g' f' ↔ g = g' ∧ f = f'"} +{"name":"PureU1.VectorLikeEvenPlane.ext_δ","declaration":"theorem PureU1.VectorLikeEvenPlane.ext_δ {n : ℕ} (S : Fin (2 * Nat.succ n) → ℚ) (T : Fin (2 * Nat.succ n) → ℚ) (h1 : ∀ (i : Fin (Nat.succ n)), S (PureU1.VectorLikeEvenPlane.δ₁ i) = T (PureU1.VectorLikeEvenPlane.δ₁ i)) (h2 : ∀ (i : Fin (Nat.succ n)), S (PureU1.VectorLikeEvenPlane.δ₂ i) = T (PureU1.VectorLikeEvenPlane.δ₂ i)) : S = T"} +{"name":"PureU1.VectorLikeEvenPlane.basis_linearACC","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_linearACC {n : ℕ} (j : Fin (Nat.succ n)) : (PureU1.accGrav (2 * Nat.succ n)) (PureU1.VectorLikeEvenPlane.basisAsCharges j) = 0"} +{"name":"PureU1.VectorLikeEvenPlane.basis_on_other","declaration":"theorem PureU1.VectorLikeEvenPlane.basis_on_other {n : ℕ} {k : Fin (Nat.succ n)} {j : Fin (2 * Nat.succ n)} (h1 : j ≠ PureU1.VectorLikeEvenPlane.δ₁ k) (h2 : j ≠ PureU1.VectorLikeEvenPlane.δ₂ k) : PureU1.VectorLikeEvenPlane.basisAsCharges k j = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.LineInCubic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.LineInCubic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6c23925f0a5991f292ab1a8f89b9280aace40231 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.LineInCubic.jsonl @@ -0,0 +1,13 @@ +{"name":"PureU1.Even.line_in_cubic_P_P_P!","declaration":"/-- This lemma states that for a given `S` of type `(PureU1 (2 * n.succ)).AnomalyFreeLinear` and\na proof `h` that the line through `S` lies on a cubic curve,\nfor any functions `g : Fin n.succ → ℚ` and `f : Fin n → ℚ`, if `S.val = P g + P! f`,\nthen `accCubeTriLinSymm.toFun (P g, P g, P! f) = 0`.\n-/\ntheorem PureU1.Even.line_in_cubic_P_P_P! {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (h : PureU1.Even.LineInCubic S) (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) : S.val = PureU1.VectorLikeEvenPlane.P g + PureU1.VectorLikeEvenPlane.P! f →\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.P! f) =\n 0"} +{"name":"PureU1.Even.lineInCubicPerm_swap","declaration":"theorem PureU1.Even.lineInCubicPerm_swap {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (LIC : PureU1.Even.LineInCubicPerm S) (j : Fin n) (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) : S.val = PureU1.VectorLikeEvenPlane.Pa g f →\n (S.val (PureU1.VectorLikeEvenPlane.δ!₂ j) - S.val (PureU1.VectorLikeEvenPlane.δ!₁ j)) *\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.basis!AsCharges j) =\n 0"} +{"name":"PureU1.Even.lineInCubicPerm_self","declaration":"/-- If `lineInCubicPerm S` then `lineInCubic S`. -/\ntheorem PureU1.Even.lineInCubicPerm_self {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (hS : PureU1.Even.LineInCubicPerm S) : PureU1.Even.LineInCubic S"} +{"name":"PureU1.Even.LineInCubic","declaration":"/-- A property on `LinSols`, satisfied if every point on the line between the two planes\nin the basis through that point is in the cubic. -/\ndef PureU1.Even.LineInCubic {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) : Prop"} +{"name":"PureU1.Even.lineInCubicPerm_permute","declaration":"/-- If `lineInCubicPerm S` then `lineInCubicPerm (M S)` for all permutations `M`. -/\ntheorem PureU1.Even.lineInCubicPerm_permute {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (hS : PureU1.Even.LineInCubicPerm S) (M' : (PureU1.FamilyPermutations (2 * Nat.succ n)).group) : PureU1.Even.LineInCubicPerm (((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * Nat.succ n))) M') S)"} +{"name":"PureU1.Even.lineInCubicPerm_constAbs","declaration":"theorem PureU1.Even.lineInCubicPerm_constAbs {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n)))} (LIC : PureU1.Even.LineInCubicPerm S.toLinSols) : PureU1.ConstAbs S.val"} +{"name":"PureU1.Even.lineInCubicPerm_in_plane","declaration":"theorem PureU1.Even.lineInCubicPerm_in_plane {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n)))) (LIC : PureU1.Even.LineInCubicPerm S.toLinSols) : ∃ M,\n ((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * Nat.succ (Nat.succ n)))) M) S.toLinSols ∈\n Submodule.span ℚ (Set.range PureU1.VectorLikeEvenPlane.basis)"} +{"name":"PureU1.Even.lineInCubicPerm_last_cond","declaration":"theorem PureU1.Even.lineInCubicPerm_last_cond {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n))).toACCSystemLinear} (LIC : PureU1.Even.LineInCubicPerm S) : PureU1.LineInPlaneProp\n (S.val (PureU1.VectorLikeEvenPlane.δ!₂ (Fin.last n)), S.val (PureU1.VectorLikeEvenPlane.δ!₁ (Fin.last n)),\n S.val PureU1.VectorLikeEvenPlane.δ!₄)"} +{"name":"PureU1.Even.lineInCubicPerm_vectorLike","declaration":"theorem PureU1.Even.lineInCubicPerm_vectorLike {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n)))} (LIC : PureU1.Even.LineInCubicPerm S.toLinSols) : PureU1.VectorLikeEven S.val"} +{"name":"PureU1.Even.lineInCubicPerm_last_perm","declaration":"theorem PureU1.Even.lineInCubicPerm_last_perm {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n))).toACCSystemLinear} (LIC : PureU1.Even.LineInCubicPerm S) : PureU1.LineInPlaneCond S"} +{"name":"PureU1.Even.LineInCubicPerm","declaration":"/-- A `LinSol` satisfies `LineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/\ndef PureU1.Even.LineInCubicPerm {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear) : Prop"} +{"name":"PureU1.Even.P_P_P!_accCube'","declaration":"theorem PureU1.Even.P_P_P!_accCube' {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n))).toACCSystemLinear} (f : Fin (Nat.succ (Nat.succ n)) → ℚ) (g : Fin (Nat.succ n) → ℚ) (hS : S.val = PureU1.VectorLikeEvenPlane.Pa f g) : ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P f)) (PureU1.VectorLikeEvenPlane.P f))\n (PureU1.VectorLikeEvenPlane.basis!AsCharges (Fin.last n)) =\n -(S.val (PureU1.VectorLikeEvenPlane.δ!₂ (Fin.last n)) + S.val (PureU1.VectorLikeEvenPlane.δ!₁ (Fin.last n))) *\n (2 * S.val PureU1.VectorLikeEvenPlane.δ!₄ + S.val (PureU1.VectorLikeEvenPlane.δ!₂ (Fin.last n)) +\n S.val (PureU1.VectorLikeEvenPlane.δ!₁ (Fin.last n)))"} +{"name":"PureU1.Even.lineInCubic_expand","declaration":"theorem PureU1.Even.lineInCubic_expand {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear} (h : PureU1.Even.LineInCubic S) (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) : S.val = PureU1.VectorLikeEvenPlane.Pa g f →\n ∀ (a b : ℚ),\n 3 * a * b *\n (a *\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.P! f) +\n b *\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P! f)) (PureU1.VectorLikeEvenPlane.P! f))\n (PureU1.VectorLikeEvenPlane.P g)) =\n 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.Parameterization.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.Parameterization.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..701d297e050f30a4b1f073d7f00d1943120bcdad --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Even.Parameterization.jsonl @@ -0,0 +1,14 @@ +{"name":"PureU1.Even.generic_case","declaration":"theorem PureU1.Even.generic_case {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))} (h : PureU1.Even.GenericCase S) : ∃ g f a, S = PureU1.Even.parameterization g f a"} +{"name":"PureU1.Even.parameterization","declaration":"/-- The construction of a `Sol` from a `Fin n.succ → ℚ`, a `Fin n → ℚ` and a `ℚ`. -/\ndef PureU1.Even.parameterization {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (a : ℚ) : ACCSystem.Sols (PureU1 (2 * Nat.succ n))"} +{"name":"PureU1.Even.parameterizationCharge_cube","declaration":"theorem PureU1.Even.parameterizationCharge_cube {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (a : ℚ) : (PureU1.accCube (2 * Nat.succ n)) (PureU1.Even.parameterizationAsLinear g f a).val = 0"} +{"name":"PureU1.Even.anomalyFree_param","declaration":"theorem PureU1.Even.anomalyFree_param {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))} (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (hS : S.val = PureU1.VectorLikeEvenPlane.P g + PureU1.VectorLikeEvenPlane.P! f) : ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.P! f) =\n -((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P! f)) (PureU1.VectorLikeEvenPlane.P! f))\n (PureU1.VectorLikeEvenPlane.P g)"} +{"name":"PureU1.Even.specialCase_exists","declaration":"theorem PureU1.Even.specialCase_exists {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))) (hs : ∃ g f,\n S.val = PureU1.VectorLikeEvenPlane.P g + PureU1.VectorLikeEvenPlane.P! f ∧\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.P! f) =\n 0) : PureU1.Even.SpecialCase S"} +{"name":"PureU1.Even.parameterizationAsLinear_val","declaration":"theorem PureU1.Even.parameterizationAsLinear_val {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (a : ℚ) : (PureU1.Even.parameterizationAsLinear g f a).val =\n a •\n (((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P! f)) (PureU1.VectorLikeEvenPlane.P! f))\n (PureU1.VectorLikeEvenPlane.P g) •\n PureU1.VectorLikeEvenPlane.P g +\n -((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.P! f) •\n PureU1.VectorLikeEvenPlane.P! f)"} +{"name":"PureU1.Even.special_case_lineInCubic","declaration":"theorem PureU1.Even.special_case_lineInCubic {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))} (h : PureU1.Even.SpecialCase S) : PureU1.Even.LineInCubic S.toLinSols"} +{"name":"PureU1.Even.GenericCase","declaration":"/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.\nIn this case our parameterization above will be able to recover this point. -/\ndef PureU1.Even.GenericCase {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))) : Prop"} +{"name":"PureU1.Even.special_case_lineInCubic_perm","declaration":"theorem PureU1.Even.special_case_lineInCubic_perm {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))} (h : ∀ (M : (PureU1.FamilyPermutations (2 * Nat.succ n)).group),\n PureU1.Even.SpecialCase\n ((MulAction.toFun (PureU1.FamilyPermutations (2 * Nat.succ n)).group (ACCSystem.Sols (PureU1 (2 * Nat.succ n)))) S\n M)) : PureU1.Even.LineInCubicPerm S.toLinSols"} +{"name":"PureU1.Even.parameterizationAsLinear","declaration":"/-- Given coefficients `g` of a point in `P` and `f` of a point in `P!`, and a rational, we get a\nrational `a ∈ ℚ`, we get a\npoint in `(PureU1 (2 * n.succ)).AnomalyFreeLinear`, which we will later show extends to an anomaly\nfree point. -/\ndef PureU1.Even.parameterizationAsLinear {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (f : Fin n → ℚ) (a : ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n)).toACCSystemLinear"} +{"name":"PureU1.Even.SpecialCase","declaration":"/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) = 0`. -/\ndef PureU1.Even.SpecialCase {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))) : Prop"} +{"name":"PureU1.Even.generic_or_special","declaration":"theorem PureU1.Even.generic_or_special {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))) : PureU1.Even.GenericCase S ∨ PureU1.Even.SpecialCase S"} +{"name":"PureU1.Even.special_case","declaration":"theorem PureU1.Even.special_case {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n)))} (h : ∀ (M : (PureU1.FamilyPermutations (2 * Nat.succ (Nat.succ n))).group),\n PureU1.Even.SpecialCase\n ((MulAction.toFun (PureU1.FamilyPermutations (2 * Nat.succ (Nat.succ n))).group\n (ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n)))))\n S M)) : ∃ M,\n ((MulAction.toFun (PureU1.FamilyPermutations (2 * Nat.succ (Nat.succ n))).group\n (ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n)))))\n S M).toLinSols ∈\n Submodule.span ℚ (Set.range PureU1.VectorLikeEvenPlane.basis)"} +{"name":"PureU1.Even.genericCase_exists","declaration":"theorem PureU1.Even.genericCase_exists {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n))) (hs : ∃ g f,\n S.val = PureU1.VectorLikeEvenPlane.P g + PureU1.VectorLikeEvenPlane.P! f ∧\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeEvenPlane.P g)) (PureU1.VectorLikeEvenPlane.P g))\n (PureU1.VectorLikeEvenPlane.P! f) ≠\n 0) : PureU1.Even.GenericCase S"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.LineInPlaneCond.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LineInPlaneCond.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d282153d4ebb34fca992a96cb71646a143f0712a --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LineInPlaneCond.jsonl @@ -0,0 +1,11 @@ +{"name":"PureU1.LineInPlaneProp","declaration":"/-- The proposition on three rationals to satisfy the `linInPlane` condition. -/\ndef PureU1.LineInPlaneProp : ℚ × ℚ × ℚ → Prop"} +{"name":"PureU1.lineInPlaneCond_perm","declaration":"theorem PureU1.lineInPlaneCond_perm {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear} (hS : PureU1.LineInPlaneCond S) (M : (PureU1.FamilyPermutations n).group) : PureU1.LineInPlaneCond (((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations n)) M) S)"} +{"name":"PureU1.linesInPlane_four","declaration":"theorem PureU1.linesInPlane_four (S : ACCSystem.Sols (PureU1 4)) (hS : PureU1.LineInPlaneCond S.toLinSols) : PureU1.ConstAbsProp (S.val 0, S.val 1)"} +{"name":"PureU1.linesInPlane_constAbs_four","declaration":"theorem PureU1.linesInPlane_constAbs_four (S : ACCSystem.Sols (PureU1 4)) (hS : PureU1.LineInPlaneCond S.toLinSols) : PureU1.ConstAbs S.val"} +{"name":"PureU1.linesInPlane_eq_sq","declaration":"theorem PureU1.linesInPlane_eq_sq {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))).toACCSystemLinear} (hS : PureU1.LineInPlaneCond S) (i : Fin (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (j : Fin (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) : i ≠ j → PureU1.ConstAbsProp (S.val i, S.val j)"} +{"name":"PureU1.linesInPlane_eq_sq_four","declaration":"theorem PureU1.linesInPlane_eq_sq_four {S : ACCSystem.Sols (PureU1 4)} (hS : PureU1.LineInPlaneCond S.toLinSols) (i : Fin 4) (j : Fin 4) : i ≠ j → PureU1.ConstAbsProp (S.val i, S.val j)"} +{"name":"PureU1.LineInPlaneCond","declaration":"/-- The proposition on a `LinSol` to satisfy the `linInPlane` condition. -/\ndef PureU1.LineInPlaneCond {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear) : Prop"} +{"name":"PureU1.lineInPlaneCond_eq_last'","declaration":"theorem PureU1.lineInPlaneCond_eq_last' {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (Nat.succ (Nat.succ n))).toACCSystemLinear} (hS : PureU1.LineInPlaneCond S) (h : ¬S.val (Fin.castSucc (Fin.last n)) ^ 2 = S.val (Fin.succ (Fin.last n)) ^ 2) : (2 - ↑n) * S.val (Fin.last (n + 1)) = -(2 - ↑n) * S.val (Fin.castSucc (Fin.last n))"} +{"name":"PureU1.linesInPlane_constAbs_AF","declaration":"theorem PureU1.linesInPlane_constAbs_AF {n : ℕ} (S : ACCSystem.Sols (PureU1 (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (hS : PureU1.LineInPlaneCond S.toLinSols) : PureU1.ConstAbs S.val"} +{"name":"PureU1.linesInPlane_constAbs","declaration":"theorem PureU1.linesInPlane_constAbs {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))).toACCSystemLinear} (hS : PureU1.LineInPlaneCond S) : PureU1.ConstAbs S.val"} +{"name":"PureU1.lineInPlaneCond_eq_last","declaration":"theorem PureU1.lineInPlaneCond_eq_last {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (Nat.succ (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))).toACCSystemLinear} (hS : PureU1.LineInPlaneCond S) : PureU1.ConstAbsProp\n (S.val (Fin.castSucc (Fin.last (Nat.succ (Nat.succ (Nat.succ n))))),\n S.val (Fin.succ (Fin.last (Nat.succ (Nat.succ (Nat.succ n))))))"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.One.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.One.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d12fa308c8400b603dbb30bab9381eb44983e95d --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.One.jsonl @@ -0,0 +1 @@ +{"name":"PureU1.One.solEqZero","declaration":"theorem PureU1.One.solEqZero (S : ACCSystemLinear.LinSols (PureU1 1).toACCSystemLinear) : S = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.Three.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.Three.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..4f7850ca0a6b55fb277eb6e75c5a3343882594a7 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.Three.jsonl @@ -0,0 +1,5 @@ +{"name":"PureU1.Three.three_sol_zero","declaration":"theorem PureU1.Three.three_sol_zero (S : ACCSystem.Sols (PureU1 3)) : S.val 0 = 0 ∨ S.val 1 = 0 ∨ S.val 2 = 0"} +{"name":"PureU1.Three.cube_for_linSol'","declaration":"theorem PureU1.Three.cube_for_linSol' (S : ACCSystemLinear.LinSols (PureU1 3).toACCSystemLinear) : 3 * S.val 0 * S.val 1 * S.val 2 = 0 ↔ (PureU1 3).cubicACC S.val = 0"} +{"name":"PureU1.Three.cube_for_linSol","declaration":"theorem PureU1.Three.cube_for_linSol (S : ACCSystemLinear.LinSols (PureU1 3).toACCSystemLinear) : S.val 0 = 0 ∨ S.val 1 = 0 ∨ S.val 2 = 0 ↔ (PureU1 3).cubicACC S.val = 0"} +{"name":"PureU1.Three.solOfLinear","declaration":"/-- Given a `LinSol` with a charge equal to zero a `Sol`. -/\ndef PureU1.Three.solOfLinear (S : ACCSystemLinear.LinSols (PureU1 3).toACCSystemLinear) (hS : S.val 0 = 0 ∨ S.val 1 = 0 ∨ S.val 2 = 0) : ACCSystem.Sols (PureU1 3)"} +{"name":"PureU1.Three.solOfLinear_surjects","declaration":"theorem PureU1.Three.solOfLinear_surjects (S : ACCSystem.Sols (PureU1 3)) : ∃ T, ∃ (hT : T.val 0 = 0 ∨ T.val 1 = 0 ∨ T.val 2 = 0), PureU1.Three.solOfLinear T hT = S"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.Two.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.Two.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d24e1c9b2f65590c51e48e77ea7e954af69539ef --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.LowDim.Two.jsonl @@ -0,0 +1 @@ +{"name":"PureU1.Two.equiv","declaration":"/-- An equivalence between `LinSols` and `Sols`. -/\ndef PureU1.Two.equiv : ACCSystemLinear.LinSols (PureU1 2).toACCSystemLinear ≃ ACCSystem.Sols (PureU1 2)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.BasisLinear.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.BasisLinear.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..08e4640aa1b61e02dcc18b2d84cd814ba424975c --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.BasisLinear.jsonl @@ -0,0 +1,87 @@ +{"name":"PureU1.VectorLikeOddPlane.δa₃","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δa₃ {n : ℕ} : Fin (2 * Nat.succ n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.basis_on_δ₃","declaration":"theorem PureU1.VectorLikeOddPlane.basis_on_δ₃ {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.basisAsCharges j PureU1.VectorLikeOddPlane.δ₃ = 0"} +{"name":"PureU1.VectorLikeOddPlane.basis!AsCharges","declaration":"/-- The second part of the basis as charge assignments. -/\ndef PureU1.VectorLikeOddPlane.basis!AsCharges {n : ℕ} (j : Fin n) : ACCSystemCharges.Charges (PureU1 (2 * n + 1)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeOddPlane.basisa","declaration":"/-- The whole basis as `LinSols`. -/\ndef PureU1.VectorLikeOddPlane.basisa {n : ℕ} : Fin n ⊕ Fin n → ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeOddPlane.P_zero","declaration":"theorem PureU1.VectorLikeOddPlane.P_zero {n : ℕ} (f : Fin n → ℚ) (h : PureU1.VectorLikeOddPlane.P f = 0) (i : Fin n) : f i = 0"} +{"name":"PureU1.VectorLikeOddPlane.δa₄_δ!₂","declaration":"theorem PureU1.VectorLikeOddPlane.δa₄_δ!₂ {n : ℕ} (j : Fin (Nat.succ n)) : PureU1.VectorLikeOddPlane.δa₄ j = PureU1.VectorLikeOddPlane.δ!₂ j"} +{"name":"PureU1.VectorLikeOddPlane.sum_δ!","declaration":"theorem PureU1.VectorLikeOddPlane.sum_δ! {n : ℕ} (S : Fin (2 * n + 1) → ℚ) : (Finset.sum Finset.univ fun i => S i) =\n S PureU1.VectorLikeOddPlane.δ!₃ +\n Finset.sum Finset.univ fun i => (S ∘ PureU1.VectorLikeOddPlane.δ!₁) i + (S ∘ PureU1.VectorLikeOddPlane.δ!₂) i"} +{"name":"PureU1.VectorLikeOddPlane.Pa","declaration":"/-- A point in the span of the basis as a charge. -/\ndef PureU1.VectorLikeOddPlane.Pa {n : ℕ} (f : Fin n → ℚ) (g : Fin n → ℚ) : ACCSystemCharges.Charges (PureU1 (2 * n + 1)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeOddPlane.δa₁_δ₁","declaration":"theorem PureU1.VectorLikeOddPlane.δa₁_δ₁ {n : ℕ} : PureU1.VectorLikeOddPlane.δa₁ = PureU1.VectorLikeOddPlane.δ₁ 0"} +{"name":"PureU1.VectorLikeOddPlane.basis_val","declaration":"theorem PureU1.VectorLikeOddPlane.basis_val {n : ℕ} (j : Fin n) : (PureU1.VectorLikeOddPlane.basis j).val = PureU1.VectorLikeOddPlane.basisAsCharges j"} +{"name":"PureU1.VectorLikeOddPlane.basis!_val","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_val {n : ℕ} (j : Fin n) : (PureU1.VectorLikeOddPlane.basis! j).val = PureU1.VectorLikeOddPlane.basis!AsCharges j"} +{"name":"PureU1.VectorLikeOddPlane.basis_on_δ₂_self","declaration":"theorem PureU1.VectorLikeOddPlane.basis_on_δ₂_self {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.basisAsCharges j (PureU1.VectorLikeOddPlane.δ₂ j) = -1"} +{"name":"PureU1.VectorLikeOddPlane.Pa'","declaration":"/-- A point in the span of the whole basis. -/\ndef PureU1.VectorLikeOddPlane.Pa' {n : ℕ} (f : Fin n ⊕ Fin n → ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeOddPlane.split_adda","declaration":"theorem PureU1.VectorLikeOddPlane.split_adda (n : ℕ) : 1 + n + 1 + Nat.succ n = 2 * Nat.succ n + 1"} +{"name":"PureU1.VectorLikeOddPlane.P_δ₂","declaration":"theorem PureU1.VectorLikeOddPlane.P_δ₂ {n : ℕ} (f : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeOddPlane.P f (PureU1.VectorLikeOddPlane.δ₂ j) = -f j"} +{"name":"PureU1.VectorLikeOddPlane.basis!_linear_independent","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_linear_independent {n : ℕ} : LinearIndependent ℚ PureU1.VectorLikeOddPlane.basis!"} +{"name":"PureU1.VectorLikeOddPlane.Pa_δa₃","declaration":"theorem PureU1.VectorLikeOddPlane.Pa_δa₃ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin (Nat.succ n) → ℚ) : PureU1.VectorLikeOddPlane.Pa f g PureU1.VectorLikeOddPlane.δa₃ = g (Fin.last n)"} +{"name":"PureU1.VectorLikeOddPlane.join","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.join {n : ℕ} (g : Fin n → ℚ) (f : Fin n → ℚ) : Fin n ⊕ Fin n → ℚ"} +{"name":"PureU1.VectorLikeOddPlane.P!_zero","declaration":"theorem PureU1.VectorLikeOddPlane.P!_zero {n : ℕ} (f : Fin n → ℚ) (h : PureU1.VectorLikeOddPlane.P! f = 0) (i : Fin n) : f i = 0"} +{"name":"PureU1.VectorLikeOddPlane.basis!_on_δ!₁_self","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_on_δ!₁_self {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.basis!AsCharges j (PureU1.VectorLikeOddPlane.δ!₁ j) = 1"} +{"name":"PureU1.VectorLikeOddPlane.basis_linearACC","declaration":"theorem PureU1.VectorLikeOddPlane.basis_linearACC {n : ℕ} (j : Fin n) : (PureU1.accGrav (2 * n + 1)) (PureU1.VectorLikeOddPlane.basisAsCharges j) = 0"} +{"name":"PureU1.VectorLikeOddPlane.δa₂_δ!₁","declaration":"theorem PureU1.VectorLikeOddPlane.δa₂_δ!₁ {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.δa₂ j = PureU1.VectorLikeOddPlane.δ!₁ (Fin.castSucc j)"} +{"name":"PureU1.VectorLikeOddPlane.Pa_eq","declaration":"theorem PureU1.VectorLikeOddPlane.Pa_eq {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (g' : Fin (Nat.succ n) → ℚ) (f : Fin (Nat.succ n) → ℚ) (f' : Fin (Nat.succ n) → ℚ) : PureU1.VectorLikeOddPlane.Pa g f = PureU1.VectorLikeOddPlane.Pa g' f' ↔ g = g' ∧ f = f'"} +{"name":"PureU1.VectorLikeOddPlane.sum_δ","declaration":"theorem PureU1.VectorLikeOddPlane.sum_δ {n : ℕ} (S : Fin (2 * n + 1) → ℚ) : (Finset.sum Finset.univ fun i => S i) =\n S PureU1.VectorLikeOddPlane.δ₃ +\n Finset.sum Finset.univ fun i => (S ∘ PureU1.VectorLikeOddPlane.δ₁) i + (S ∘ PureU1.VectorLikeOddPlane.δ₂) i"} +{"name":"PureU1.VectorLikeOddPlane.δ₂_δ!₂","declaration":"theorem PureU1.VectorLikeOddPlane.δ₂_δ!₂ {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.δ₂ j = PureU1.VectorLikeOddPlane.δ!₂ j"} +{"name":"PureU1.VectorLikeOddPlane.P_accCube","declaration":"theorem PureU1.VectorLikeOddPlane.P_accCube {n : ℕ} (f : Fin n → ℚ) : (PureU1.accCube (2 * n + 1)) (PureU1.VectorLikeOddPlane.P f) = 0"} +{"name":"PureU1.VectorLikeOddPlane.join_ext","declaration":"theorem PureU1.VectorLikeOddPlane.join_ext {n : ℕ} (g : Fin n → ℚ) (g' : Fin n → ℚ) (f : Fin n → ℚ) (f' : Fin n → ℚ) : PureU1.VectorLikeOddPlane.join g f = PureU1.VectorLikeOddPlane.join g' f' ↔ g = g' ∧ f = f'"} +{"name":"PureU1.VectorLikeOddPlane.P!_linearACC","declaration":"theorem PureU1.VectorLikeOddPlane.P!_linearACC {n : ℕ} (f : Fin n → ℚ) : (PureU1.accGrav (2 * n + 1)) (PureU1.VectorLikeOddPlane.P! f) = 0"} +{"name":"PureU1.VectorLikeOddPlane.P!_δ!₂","declaration":"theorem PureU1.VectorLikeOddPlane.P!_δ!₂ {n : ℕ} (f : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeOddPlane.P! f (PureU1.VectorLikeOddPlane.δ!₂ j) = -f j"} +{"name":"PureU1.VectorLikeOddPlane.δa₁","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δa₁ {n : ℕ} : Fin (2 * Nat.succ n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.basis!","declaration":"/-- The second part of the basis as `LinSols`. -/\ndef PureU1.VectorLikeOddPlane.basis! {n : ℕ} (j : Fin n) : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeOddPlane.span_basis","declaration":"theorem PureU1.VectorLikeOddPlane.span_basis {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n + 1)).toACCSystemLinear) : ∃ g f, S.val = PureU1.VectorLikeOddPlane.P g + PureU1.VectorLikeOddPlane.P! f"} +{"name":"PureU1.VectorLikeOddPlane.basisa_card","declaration":"theorem PureU1.VectorLikeOddPlane.basisa_card {n : ℕ} : Fintype.card (Fin (Nat.succ n) ⊕ Fin (Nat.succ n)) =\n FiniteDimensional.finrank ℚ (ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n + 1)).toACCSystemLinear)"} +{"name":"PureU1.VectorLikeOddPlane.δ!₃","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δ!₃ {n : ℕ} : Fin (2 * n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.P_P_P!_accCube","declaration":"theorem PureU1.VectorLikeOddPlane.P_P_P!_accCube {n : ℕ} (g : Fin n → ℚ) (j : Fin n) : ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.basis!AsCharges j) =\n PureU1.VectorLikeOddPlane.P g (PureU1.VectorLikeOddPlane.δ!₁ j) ^ 2 - g j ^ 2"} +{"name":"PureU1.VectorLikeOddPlane.P!_δ!₁","declaration":"theorem PureU1.VectorLikeOddPlane.P!_δ!₁ {n : ℕ} (f : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeOddPlane.P! f (PureU1.VectorLikeOddPlane.δ!₁ j) = f j"} +{"name":"PureU1.VectorLikeOddPlane.δ!₁","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δ!₁ {n : ℕ} (j : Fin n) : Fin (2 * n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.basis_δ₂_eq_minus_δ₁","declaration":"theorem PureU1.VectorLikeOddPlane.basis_δ₂_eq_minus_δ₁ {n : ℕ} (j : Fin n) (i : Fin n) : PureU1.VectorLikeOddPlane.basisAsCharges j (PureU1.VectorLikeOddPlane.δ₂ i) =\n -PureU1.VectorLikeOddPlane.basisAsCharges j (PureU1.VectorLikeOddPlane.δ₁ i)"} +{"name":"PureU1.VectorLikeOddPlane.Pa'_P'_P!'","declaration":"theorem PureU1.VectorLikeOddPlane.Pa'_P'_P!' {n : ℕ} (f : Fin n ⊕ Fin n → ℚ) : PureU1.VectorLikeOddPlane.Pa' f =\n PureU1.VectorLikeOddPlane.P' (f ∘ Sum.inl) + PureU1.VectorLikeOddPlane.P!' (f ∘ Sum.inr)"} +{"name":"PureU1.VectorLikeOddPlane.join_Pa","declaration":"theorem PureU1.VectorLikeOddPlane.join_Pa {n : ℕ} (g : Fin (Nat.succ n) → ℚ) (g' : Fin (Nat.succ n) → ℚ) (f : Fin (Nat.succ n) → ℚ) (f' : Fin (Nat.succ n) → ℚ) : PureU1.VectorLikeOddPlane.Pa' (PureU1.VectorLikeOddPlane.join g f) =\n PureU1.VectorLikeOddPlane.Pa' (PureU1.VectorLikeOddPlane.join g' f') ↔\n PureU1.VectorLikeOddPlane.Pa g f = PureU1.VectorLikeOddPlane.Pa g' f'"} +{"name":"PureU1.VectorLikeOddPlane.δa₁_δ!₃","declaration":"theorem PureU1.VectorLikeOddPlane.δa₁_δ!₃ {n : ℕ} : PureU1.VectorLikeOddPlane.δa₁ = PureU1.VectorLikeOddPlane.δ!₃"} +{"name":"PureU1.VectorLikeOddPlane.Pa_zero!","declaration":"theorem PureU1.VectorLikeOddPlane.Pa_zero! {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin (Nat.succ n) → ℚ) (h : PureU1.VectorLikeOddPlane.Pa f g = 0) (i : Fin (Nat.succ n)) : g i = 0"} +{"name":"PureU1.VectorLikeOddPlane.Pa_δa₄","declaration":"theorem PureU1.VectorLikeOddPlane.Pa_δa₄ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin (Nat.succ n) → ℚ) (j : Fin (Nat.succ n)) : PureU1.VectorLikeOddPlane.Pa f g (PureU1.VectorLikeOddPlane.δa₄ j) = -f j - g j"} +{"name":"PureU1.VectorLikeOddPlane.δa₄_δ₂","declaration":"theorem PureU1.VectorLikeOddPlane.δa₄_δ₂ {n : ℕ} (j : Fin (Nat.succ n)) : PureU1.VectorLikeOddPlane.δa₄ j = PureU1.VectorLikeOddPlane.δ₂ j"} +{"name":"PureU1.VectorLikeOddPlane.swap!_as_add","declaration":"/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/\ntheorem PureU1.VectorLikeOddPlane.swap!_as_add {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} {S' : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} (j : Fin n) (hS : ((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * n + 1)))\n (PureU1.pairSwap (PureU1.VectorLikeOddPlane.δ!₁ j) (PureU1.VectorLikeOddPlane.δ!₂ j)))\n S =\n S') : S'.val =\n S.val +\n (S.val (PureU1.VectorLikeOddPlane.δ!₂ j) - S.val (PureU1.VectorLikeOddPlane.δ!₁ j)) •\n PureU1.VectorLikeOddPlane.basis!AsCharges j"} +{"name":"PureU1.VectorLikeOddPlane.δ₃","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δ₃ {n : ℕ} : Fin (2 * n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.δa₃_δ₃","declaration":"theorem PureU1.VectorLikeOddPlane.δa₃_δ₃ {n : ℕ} : PureU1.VectorLikeOddPlane.δa₃ = PureU1.VectorLikeOddPlane.δ₃"} +{"name":"PureU1.VectorLikeOddPlane.basis!_on_other","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_on_other {n : ℕ} {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ PureU1.VectorLikeOddPlane.δ!₁ k) (h2 : j ≠ PureU1.VectorLikeOddPlane.δ!₂ k) : PureU1.VectorLikeOddPlane.basis!AsCharges k j = 0"} +{"name":"PureU1.VectorLikeOddPlane.P!'","declaration":"/-- A point in the span of the second part of the basis. -/\ndef PureU1.VectorLikeOddPlane.P!' {n : ℕ} (f : Fin n → ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeOddPlane.basisa_linear_independent","declaration":"theorem PureU1.VectorLikeOddPlane.basisa_linear_independent {n : ℕ} : LinearIndependent ℚ PureU1.VectorLikeOddPlane.basisa"} +{"name":"PureU1.VectorLikeOddPlane.δa₂_δ₁","declaration":"theorem PureU1.VectorLikeOddPlane.δa₂_δ₁ {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.δa₂ j = PureU1.VectorLikeOddPlane.δ₁ (Fin.succ j)"} +{"name":"PureU1.VectorLikeOddPlane.P","declaration":"/-- A point in the span of the first part of the basis as a charge. -/\ndef PureU1.VectorLikeOddPlane.P {n : ℕ} (f : Fin n → ℚ) : ACCSystemCharges.Charges (PureU1 (2 * n + 1)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeOddPlane.basis_on_δ₁_other","declaration":"theorem PureU1.VectorLikeOddPlane.basis_on_δ₁_other {n : ℕ} {k : Fin n} {j : Fin n} (h : k ≠ j) : PureU1.VectorLikeOddPlane.basisAsCharges k (PureU1.VectorLikeOddPlane.δ₁ j) = 0"} +{"name":"PureU1.VectorLikeOddPlane.δ₂","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δ₂ {n : ℕ} (j : Fin n) : Fin (2 * n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.P_linearACC","declaration":"theorem PureU1.VectorLikeOddPlane.P_linearACC {n : ℕ} (f : Fin n → ℚ) : (PureU1.accGrav (2 * n + 1)) (PureU1.VectorLikeOddPlane.P f) = 0"} +{"name":"PureU1.VectorLikeOddPlane.δ₁","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δ₁ {n : ℕ} (j : Fin n) : Fin (2 * n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.basis!_on_δ!₃","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_on_δ!₃ {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.basis!AsCharges j PureU1.VectorLikeOddPlane.δ!₃ = 0"} +{"name":"PureU1.VectorLikeOddPlane.δa₂","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δa₂ {n : ℕ} (j : Fin n) : Fin (2 * Nat.succ n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.δa₄","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δa₄ {n : ℕ} (j : Fin (Nat.succ n)) : Fin (2 * Nat.succ n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.δa₃_δ!₁","declaration":"theorem PureU1.VectorLikeOddPlane.δa₃_δ!₁ {n : ℕ} : PureU1.VectorLikeOddPlane.δa₃ = PureU1.VectorLikeOddPlane.δ!₁ (Fin.last n)"} +{"name":"PureU1.VectorLikeOddPlane.P!","declaration":"/-- A point in the span of the second part of the basis as a charge. -/\ndef PureU1.VectorLikeOddPlane.P! {n : ℕ} (f : Fin n → ℚ) : ACCSystemCharges.Charges (PureU1 (2 * n + 1)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeOddPlane.P_δ₁","declaration":"theorem PureU1.VectorLikeOddPlane.P_δ₁ {n : ℕ} (f : Fin n → ℚ) (j : Fin n) : PureU1.VectorLikeOddPlane.P f (PureU1.VectorLikeOddPlane.δ₁ j) = f j"} +{"name":"PureU1.VectorLikeOddPlane.Pa_δa₂","declaration":"theorem PureU1.VectorLikeOddPlane.Pa_δa₂ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin (Nat.succ n) → ℚ) (j : Fin n) : PureU1.VectorLikeOddPlane.Pa f g (PureU1.VectorLikeOddPlane.δa₂ j) = f (Fin.succ j) + g (Fin.castSucc j)"} +{"name":"PureU1.VectorLikeOddPlane.split_odd!","declaration":"theorem PureU1.VectorLikeOddPlane.split_odd! (n : ℕ) : 1 + n + n = 2 * n + 1"} +{"name":"PureU1.VectorLikeOddPlane.basisAsCharges","declaration":"/-- The first part of the basis as charge assignments. -/\ndef PureU1.VectorLikeOddPlane.basisAsCharges {n : ℕ} (j : Fin n) : ACCSystemCharges.Charges (PureU1 (2 * n + 1)).toACCSystemCharges"} +{"name":"PureU1.VectorLikeOddPlane.P!'_val","declaration":"theorem PureU1.VectorLikeOddPlane.P!'_val {n : ℕ} (f : Fin n → ℚ) : (PureU1.VectorLikeOddPlane.P!' f).val = PureU1.VectorLikeOddPlane.P! f"} +{"name":"PureU1.VectorLikeOddPlane.basis!_on_δ!₁_other","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_on_δ!₁_other {n : ℕ} {k : Fin n} {j : Fin n} (h : k ≠ j) : PureU1.VectorLikeOddPlane.basis!AsCharges k (PureU1.VectorLikeOddPlane.δ!₁ j) = 0"} +{"name":"PureU1.VectorLikeOddPlane.Pa_δa₁","declaration":"theorem PureU1.VectorLikeOddPlane.Pa_δa₁ {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin (Nat.succ n) → ℚ) : PureU1.VectorLikeOddPlane.Pa f g PureU1.VectorLikeOddPlane.δa₁ = f 0"} +{"name":"PureU1.VectorLikeOddPlane.P!_accCube","declaration":"theorem PureU1.VectorLikeOddPlane.P!_accCube {n : ℕ} (f : Fin n → ℚ) : (PureU1.accCube (2 * n + 1)) (PureU1.VectorLikeOddPlane.P! f) = 0"} +{"name":"PureU1.VectorLikeOddPlane.Pa'_eq","declaration":"theorem PureU1.VectorLikeOddPlane.Pa'_eq {n : ℕ} (f : Fin (Nat.succ n) ⊕ Fin (Nat.succ n) → ℚ) (f' : Fin (Nat.succ n) ⊕ Fin (Nat.succ n) → ℚ) : PureU1.VectorLikeOddPlane.Pa' f = PureU1.VectorLikeOddPlane.Pa' f' ↔ f = f'"} +{"name":"PureU1.VectorLikeOddPlane.basis!_δ!₂_eq_minus_δ!₁","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_δ!₂_eq_minus_δ!₁ {n : ℕ} (j : Fin n) (i : Fin n) : PureU1.VectorLikeOddPlane.basis!AsCharges j (PureU1.VectorLikeOddPlane.δ!₂ i) =\n -PureU1.VectorLikeOddPlane.basis!AsCharges j (PureU1.VectorLikeOddPlane.δ!₁ i)"} +{"name":"PureU1.VectorLikeOddPlane.P'","declaration":"/-- A point in the span of the first part of the basis. -/\ndef PureU1.VectorLikeOddPlane.P' {n : ℕ} (f : Fin n → ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeOddPlane.basis!_on_δ!₂_other","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_on_δ!₂_other {n : ℕ} {k : Fin n} {j : Fin n} (h : k ≠ j) : PureU1.VectorLikeOddPlane.basis!AsCharges k (PureU1.VectorLikeOddPlane.δ!₂ j) = 0"} +{"name":"PureU1.VectorLikeOddPlane.basis!_on_δ!₂_self","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_on_δ!₂_self {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.basis!AsCharges j (PureU1.VectorLikeOddPlane.δ!₂ j) = -1"} +{"name":"PureU1.VectorLikeOddPlane.basis_on_other","declaration":"theorem PureU1.VectorLikeOddPlane.basis_on_other {n : ℕ} {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ PureU1.VectorLikeOddPlane.δ₁ k) (h2 : j ≠ PureU1.VectorLikeOddPlane.δ₂ k) : PureU1.VectorLikeOddPlane.basisAsCharges k j = 0"} +{"name":"PureU1.VectorLikeOddPlane.P'_val","declaration":"theorem PureU1.VectorLikeOddPlane.P'_val {n : ℕ} (f : Fin n → ℚ) : (PureU1.VectorLikeOddPlane.P' f).val = PureU1.VectorLikeOddPlane.P f"} +{"name":"PureU1.VectorLikeOddPlane.basisaAsBasis","declaration":"/-- The basis formed out of our basisa vectors. -/\ndef PureU1.VectorLikeOddPlane.basisaAsBasis {n : ℕ} : Basis (Fin (Nat.succ n) ⊕ Fin (Nat.succ n)) ℚ (ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n + 1)).toACCSystemLinear)"} +{"name":"PureU1.VectorLikeOddPlane.P_δ₃","declaration":"theorem PureU1.VectorLikeOddPlane.P_δ₃ {n : ℕ} (f : Fin n → ℚ) : PureU1.VectorLikeOddPlane.P f PureU1.VectorLikeOddPlane.δ₃ = 0"} +{"name":"PureU1.VectorLikeOddPlane.span_basis_swap!","declaration":"theorem PureU1.VectorLikeOddPlane.span_basis_swap! {n : ℕ} {S' : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n + 1)).toACCSystemLinear} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n + 1)).toACCSystemLinear} (j : Fin (Nat.succ n)) (hS : ((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * Nat.succ n + 1)))\n (PureU1.pairSwap (PureU1.VectorLikeOddPlane.δ!₁ j) (PureU1.VectorLikeOddPlane.δ!₂ j)))\n S =\n S') (g : Fin (Nat.succ n) → ℚ) (f : Fin (Nat.succ n) → ℚ) (hS1 : S.val = PureU1.VectorLikeOddPlane.P g + PureU1.VectorLikeOddPlane.P! f) : ∃ g' f',\n S'.val = PureU1.VectorLikeOddPlane.P g' + PureU1.VectorLikeOddPlane.P! f' ∧\n PureU1.VectorLikeOddPlane.P! f' =\n PureU1.VectorLikeOddPlane.P! f +\n (S.val (PureU1.VectorLikeOddPlane.δ!₂ j) - S.val (PureU1.VectorLikeOddPlane.δ!₁ j)) •\n PureU1.VectorLikeOddPlane.basis!AsCharges j ∧\n g' = g"} +{"name":"PureU1.VectorLikeOddPlane.basis_on_δ₂_other","declaration":"theorem PureU1.VectorLikeOddPlane.basis_on_δ₂_other {n : ℕ} {k : Fin n} {j : Fin n} (h : k ≠ j) : PureU1.VectorLikeOddPlane.basisAsCharges k (PureU1.VectorLikeOddPlane.δ₂ j) = 0"} +{"name":"PureU1.VectorLikeOddPlane.basis!_linearACC","declaration":"theorem PureU1.VectorLikeOddPlane.basis!_linearACC {n : ℕ} (j : Fin n) : (PureU1.accGrav (2 * n + 1)) (PureU1.VectorLikeOddPlane.basis!AsCharges j) = 0"} +{"name":"PureU1.VectorLikeOddPlane.basis_on_δ₁_self","declaration":"theorem PureU1.VectorLikeOddPlane.basis_on_δ₁_self {n : ℕ} (j : Fin n) : PureU1.VectorLikeOddPlane.basisAsCharges j (PureU1.VectorLikeOddPlane.δ₁ j) = 1"} +{"name":"PureU1.VectorLikeOddPlane.basis","declaration":"/-- The first part of the basis as `LinSols`. -/\ndef PureU1.VectorLikeOddPlane.basis {n : ℕ} (j : Fin n) : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear"} +{"name":"PureU1.VectorLikeOddPlane.Pa_zero","declaration":"theorem PureU1.VectorLikeOddPlane.Pa_zero {n : ℕ} (f : Fin (Nat.succ n) → ℚ) (g : Fin (Nat.succ n) → ℚ) (h : PureU1.VectorLikeOddPlane.Pa f g = 0) (i : Fin (Nat.succ n)) : f i = 0"} +{"name":"PureU1.VectorLikeOddPlane.basis_linear_independent","declaration":"theorem PureU1.VectorLikeOddPlane.basis_linear_independent {n : ℕ} : LinearIndependent ℚ PureU1.VectorLikeOddPlane.basis"} +{"name":"PureU1.VectorLikeOddPlane.δ!₂","declaration":"/-- A helper function for what follows. -/\ndef PureU1.VectorLikeOddPlane.δ!₂ {n : ℕ} (j : Fin n) : Fin (2 * n + 1)"} +{"name":"PureU1.VectorLikeOddPlane.P!_δ!₃","declaration":"theorem PureU1.VectorLikeOddPlane.P!_δ!₃ {n : ℕ} (f : Fin n → ℚ) : PureU1.VectorLikeOddPlane.P! f PureU1.VectorLikeOddPlane.δ!₃ = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.LineInCubic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.LineInCubic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..01b2bcf2a44cdacb724299548f797c0adce3c6e1 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.LineInCubic.jsonl @@ -0,0 +1,12 @@ +{"name":"PureU1.Odd.lineInCubicPerm_constAbs","declaration":"theorem PureU1.Odd.lineInCubicPerm_constAbs {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n) + 1)).toACCSystemLinear} (LIC : PureU1.Odd.LineInCubicPerm S) : PureU1.ConstAbs S.val"} +{"name":"PureU1.Odd.lineInCubicPerm_last_perm","declaration":"theorem PureU1.Odd.lineInCubicPerm_last_perm {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n) + 1)).toACCSystemLinear} (LIC : PureU1.Odd.LineInCubicPerm S) : PureU1.LineInPlaneCond S"} +{"name":"PureU1.Odd.P_P_P!_accCube'","declaration":"theorem PureU1.Odd.P_P_P!_accCube' {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n) + 1)).toACCSystemLinear} (f : Fin (Nat.succ (Nat.succ n)) → ℚ) (g : Fin (Nat.succ (Nat.succ n)) → ℚ) (hS : S.val = PureU1.VectorLikeOddPlane.Pa f g) : ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P f)) (PureU1.VectorLikeOddPlane.P f))\n (PureU1.VectorLikeOddPlane.basis!AsCharges 0) =\n (S.val (PureU1.VectorLikeOddPlane.δ!₁ 0) + S.val (PureU1.VectorLikeOddPlane.δ!₂ 0)) *\n (2 * S.val PureU1.VectorLikeOddPlane.δ!₃ + S.val (PureU1.VectorLikeOddPlane.δ!₁ 0) +\n S.val (PureU1.VectorLikeOddPlane.δ!₂ 0))"} +{"name":"PureU1.Odd.lineInCubicPerm_permute","declaration":"/-- If `lineInCubicPerm S`, then `lineInCubicPerm (M S)` for all permutations `M`. -/\ntheorem PureU1.Odd.lineInCubicPerm_permute {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} (hS : PureU1.Odd.LineInCubicPerm S) (M' : (PureU1.FamilyPermutations (2 * n + 1)).group) : PureU1.Odd.LineInCubicPerm (((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations (2 * n + 1))) M') S)"} +{"name":"PureU1.Odd.line_in_cubic_P_P_P!","declaration":"theorem PureU1.Odd.line_in_cubic_P_P_P! {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} (h : PureU1.Odd.LineInCubic S) (g : Fin n → ℚ) (f : Fin n → ℚ) : S.val = PureU1.VectorLikeOddPlane.P g + PureU1.VectorLikeOddPlane.P! f →\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.P! f) =\n 0"} +{"name":"PureU1.Odd.lineInCubicPerm_last_cond","declaration":"theorem PureU1.Odd.lineInCubicPerm_last_cond {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n) + 1)).toACCSystemLinear} (LIC : PureU1.Odd.LineInCubicPerm S) : PureU1.LineInPlaneProp\n (S.val (PureU1.VectorLikeOddPlane.δ!₂ 0), S.val (PureU1.VectorLikeOddPlane.δ!₁ 0),\n S.val PureU1.VectorLikeOddPlane.δ!₃)"} +{"name":"PureU1.Odd.lineInCubicPerm_zero","declaration":"theorem PureU1.Odd.lineInCubicPerm_zero {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ (Nat.succ n) + 1)).toACCSystemLinear} (LIC : PureU1.Odd.LineInCubicPerm S) : S = 0"} +{"name":"PureU1.Odd.lineInCubic_expand","declaration":"theorem PureU1.Odd.lineInCubic_expand {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} (h : PureU1.Odd.LineInCubic S) (g : Fin n → ℚ) (f : Fin n → ℚ) : S.val = PureU1.VectorLikeOddPlane.P g + PureU1.VectorLikeOddPlane.P! f →\n ∀ (a b : ℚ),\n 3 * a * b *\n (a *\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.P! f) +\n b *\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P! f)) (PureU1.VectorLikeOddPlane.P! f))\n (PureU1.VectorLikeOddPlane.P g)) =\n 0"} +{"name":"PureU1.Odd.LineInCubic","declaration":"/-- A property on `LinSols`, satisfied if every point on the line between the two planes\nin the basis through that point is in the cubic. -/\ndef PureU1.Odd.LineInCubic {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear) : Prop"} +{"name":"PureU1.Odd.LineInCubicPerm","declaration":"/-- A `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/\ndef PureU1.Odd.LineInCubicPerm {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear) : Prop"} +{"name":"PureU1.Odd.lineInCubicPerm_swap","declaration":"theorem PureU1.Odd.lineInCubicPerm_swap {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * Nat.succ n + 1)).toACCSystemLinear} (LIC : PureU1.Odd.LineInCubicPerm S) (j : Fin (Nat.succ n)) (g : Fin (Nat.succ n) → ℚ) (f : Fin (Nat.succ n) → ℚ) : S.val = PureU1.VectorLikeOddPlane.Pa g f →\n (S.val (PureU1.VectorLikeOddPlane.δ!₂ j) - S.val (PureU1.VectorLikeOddPlane.δ!₁ j)) *\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.basis!AsCharges j) =\n 0"} +{"name":"PureU1.Odd.lineInCubicPerm_self","declaration":"/-- If `lineInCubicPerm S`, then `lineInCubic S`. -/\ntheorem PureU1.Odd.lineInCubicPerm_self {n : ℕ} {S : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear} (hS : PureU1.Odd.LineInCubicPerm S) : PureU1.Odd.LineInCubic S"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.Parameterization.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.Parameterization.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..dfe941235a2f66dcd12f3b0c716da93be643df25 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Odd.Parameterization.jsonl @@ -0,0 +1,14 @@ +{"name":"PureU1.Odd.parameterizationAsLinear_val","declaration":"theorem PureU1.Odd.parameterizationAsLinear_val {n : ℕ} (g : Fin n → ℚ) (f : Fin n → ℚ) (a : ℚ) : (PureU1.Odd.parameterizationAsLinear g f a).val =\n a •\n (((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P! f)) (PureU1.VectorLikeOddPlane.P! f))\n (PureU1.VectorLikeOddPlane.P g) •\n PureU1.VectorLikeOddPlane.P g +\n -((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.P! f) •\n PureU1.VectorLikeOddPlane.P! f)"} +{"name":"PureU1.Odd.generic_case","declaration":"theorem PureU1.Odd.generic_case {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))} (h : PureU1.Odd.GenericCase S) : ∃ g f a, S = PureU1.Odd.parameterization g f a"} +{"name":"PureU1.Odd.genericCase_exists","declaration":"theorem PureU1.Odd.genericCase_exists {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))) (hs : ∃ g f,\n S.val = PureU1.VectorLikeOddPlane.P g + PureU1.VectorLikeOddPlane.P! f ∧\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.P! f) ≠\n 0) : PureU1.Odd.GenericCase S"} +{"name":"PureU1.Odd.generic_or_special","declaration":"theorem PureU1.Odd.generic_or_special {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))) : PureU1.Odd.GenericCase S ∨ PureU1.Odd.SpecialCase S"} +{"name":"PureU1.Odd.parameterizationAsLinear","declaration":"/-- Given a `g f : Fin n → ℚ` and a `a : ℚ` we form a linear solution. We will later\nshow that this can be extended to a complete solution. -/\ndef PureU1.Odd.parameterizationAsLinear {n : ℕ} (g : Fin n → ℚ) (f : Fin n → ℚ) (a : ℚ) : ACCSystemLinear.LinSols (PureU1 (2 * n + 1)).toACCSystemLinear"} +{"name":"PureU1.Odd.specialCase_exists","declaration":"theorem PureU1.Odd.specialCase_exists {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))) (hs : ∃ g f,\n S.val = PureU1.VectorLikeOddPlane.P g + PureU1.VectorLikeOddPlane.P! f ∧\n ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.P! f) =\n 0) : PureU1.Odd.SpecialCase S"} +{"name":"PureU1.Odd.anomalyFree_param","declaration":"theorem PureU1.Odd.anomalyFree_param {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * n + 1))} (g : Fin n → ℚ) (f : Fin n → ℚ) (hS : S.val = PureU1.VectorLikeOddPlane.P g + PureU1.VectorLikeOddPlane.P! f) : ((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P g)) (PureU1.VectorLikeOddPlane.P g))\n (PureU1.VectorLikeOddPlane.P! f) =\n -((PureU1.accCubeTriLinSymm (PureU1.VectorLikeOddPlane.P! f)) (PureU1.VectorLikeOddPlane.P! f))\n (PureU1.VectorLikeOddPlane.P g)"} +{"name":"PureU1.Odd.GenericCase","declaration":"/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.\nIn this case our parameterization above will be able to recover this point. -/\ndef PureU1.Odd.GenericCase {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))) : Prop"} +{"name":"PureU1.Odd.parameterization","declaration":"/-- Given a `g f : Fin n → ℚ` and a `a : ℚ` we form a solution. -/\ndef PureU1.Odd.parameterization {n : ℕ} (g : Fin n → ℚ) (f : Fin n → ℚ) (a : ℚ) : ACCSystem.Sols (PureU1 (2 * n + 1))"} +{"name":"PureU1.Odd.special_case_lineInCubic_perm","declaration":"theorem PureU1.Odd.special_case_lineInCubic_perm {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))} (h : ∀ (M : (PureU1.FamilyPermutations (2 * Nat.succ n + 1)).group),\n PureU1.Odd.SpecialCase\n ((MulAction.toFun (PureU1.FamilyPermutations (2 * Nat.succ n + 1)).group\n (ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))))\n S M)) : PureU1.Odd.LineInCubicPerm S.toLinSols"} +{"name":"PureU1.Odd.parameterizationCharge_cube","declaration":"theorem PureU1.Odd.parameterizationCharge_cube {n : ℕ} (g : Fin n → ℚ) (f : Fin n → ℚ) (a : ℚ) : (PureU1.accCube (2 * n + 1)) (PureU1.Odd.parameterizationAsLinear g f a).val = 0"} +{"name":"PureU1.Odd.special_case","declaration":"theorem PureU1.Odd.special_case {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n) + 1))} (h : ∀ (M : (PureU1.FamilyPermutations (2 * Nat.succ (Nat.succ n) + 1)).group),\n PureU1.Odd.SpecialCase\n ((MulAction.toFun (PureU1.FamilyPermutations (2 * Nat.succ (Nat.succ n) + 1)).group\n (ACCSystem.Sols (PureU1 (2 * Nat.succ (Nat.succ n) + 1))))\n S M)) : S.toLinSols = 0"} +{"name":"PureU1.Odd.SpecialCase","declaration":"/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.\nIn this case we will show that S is zero if it is true for all permutations. -/\ndef PureU1.Odd.SpecialCase {n : ℕ} (S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))) : Prop"} +{"name":"PureU1.Odd.special_case_lineInCubic","declaration":"theorem PureU1.Odd.special_case_lineInCubic {n : ℕ} {S : ACCSystem.Sols (PureU1 (2 * Nat.succ n + 1))} (h : PureU1.Odd.SpecialCase S) : PureU1.Odd.LineInCubic S.toLinSols"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Permutations.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Permutations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b0c422542951aea92e86610a4a5250a1ed375171 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Permutations.jsonl @@ -0,0 +1,40 @@ +{"name":"PureU1.pairSwap_snd","declaration":"theorem PureU1.pairSwap_snd {n : ℕ} (i : Fin n) (j : Fin n) : (PureU1.pairSwap i j).toFun j = i"} +{"name":"PureU1.permOfInjection","declaration":"/-- A permutation of fermions which takes one ordered subset into another. -/\ndef PureU1.permOfInjection {m : ℕ} {n : ℕ} (f : Fin m ↪ Fin n) (g : Fin m ↪ Fin n) : (PureU1.FamilyPermutations n).group"} +{"name":"PureU1.pairSwap_inv_snd","declaration":"theorem PureU1.pairSwap_inv_snd {n : ℕ} (i : Fin n) (j : Fin n) : (PureU1.pairSwap i j).invFun j = i"} +{"name":"PureU1.FamilyPermutations","declaration":"/-- The permutations acting on the ACC system. -/\ndef PureU1.FamilyPermutations (n : ℕ) : ACCSystemGroupAction (PureU1 n)"} +{"name":"PureU1.permCharges","declaration":"/-- The representation of `permGroup` acting on the vector space of charges. -/\ndef PureU1.permCharges {n : ℕ} : Representation ℚ (PureU1.PermGroup n) (ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges)"} +{"name":"PureU1.permTwoInj_fst_apply","declaration":"theorem PureU1.permTwoInj_fst_apply {n : ℕ} {i : Fin n} {j : Fin n} (hij : i ≠ j) : (Function.Embedding.toEquivRange (PureU1.permTwoInj hij)).symm { val := i, property := ⋯ } = 0"} +{"name":"PureU1.FamilyPermutations_anomalyFreeLinear_apply","declaration":"theorem PureU1.FamilyPermutations_anomalyFreeLinear_apply {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear) (i : Fin n) (f : (PureU1.FamilyPermutations n).group) : (((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations n)) f) S).val i = S.val (f.invFun i)"} +{"name":"PureU1.permTwoInj_snd_apply","declaration":"theorem PureU1.permTwoInj_snd_apply {n : ℕ} {i : Fin n} {j : Fin n} (hij : i ≠ j) : (Function.Embedding.toEquivRange (PureU1.permTwoInj hij)).symm { val := j, property := ⋯ } = 1"} +{"name":"PureU1.instGroupPermGroup","declaration":"instance PureU1.instGroupPermGroup {n : ℕ} : Group (PureU1.PermGroup n)"} +{"name":"PureU1.pairSwap_inv_other","declaration":"theorem PureU1.pairSwap_inv_other {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hik : i ≠ k) (hjk : j ≠ k) : (PureU1.pairSwap i j).invFun k = k"} +{"name":"PureU1.permTwo_fst","declaration":"theorem PureU1.permTwo_fst {n : ℕ} {i : Fin n} {j : Fin n} {i' : Fin n} {j' : Fin n} (hij : i ≠ j) (hij' : i' ≠ j') : (PureU1.permTwo hij hij').toFun i' = i"} +{"name":"PureU1.permThreeInj_fst","declaration":"theorem PureU1.permThreeInj_fst {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) : i ∈ Set.range ⇑(PureU1.permThreeInj hij hjk hik)"} +{"name":"PureU1.FamilyPermutations_charges_apply","declaration":"theorem PureU1.FamilyPermutations_charges_apply {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) (i : Fin n) (f : (PureU1.FamilyPermutations n).group) : ((PureU1.FamilyPermutations n).rep f) S i = S (f.invFun i)"} +{"name":"PureU1.permTwo_snd","declaration":"theorem PureU1.permTwo_snd {n : ℕ} {i : Fin n} {j : Fin n} {i' : Fin n} {j' : Fin n} (hij : i ≠ j) (hij' : i' ≠ j') : (PureU1.permTwo hij hij').toFun j' = j"} +{"name":"PureU1.permThree_snd","declaration":"theorem PureU1.permThree_snd {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} {i' : Fin n} {j' : Fin n} {k' : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) (hij' : i' ≠ j') (hjk' : j' ≠ k') (hik' : i' ≠ k') : (PureU1.permThree hij hjk hik hij' hjk' hik').toFun j' = j"} +{"name":"PureU1.permThree_fst","declaration":"theorem PureU1.permThree_fst {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} {i' : Fin n} {j' : Fin n} {k' : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) (hij' : i' ≠ j') (hjk' : j' ≠ k') (hik' : i' ≠ k') : (PureU1.permThree hij hjk hik hij' hjk' hik').toFun i' = i"} +{"name":"PureU1.permThreeInj_snd_apply","declaration":"theorem PureU1.permThreeInj_snd_apply {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) : (Function.Embedding.toEquivRange (PureU1.permThreeInj hij hjk hik)).symm { val := j, property := ⋯ } = 1"} +{"name":"PureU1.permThree_thd","declaration":"theorem PureU1.permThree_thd {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} {i' : Fin n} {j' : Fin n} {k' : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) (hij' : i' ≠ j') (hjk' : j' ≠ k') (hik' : i' ≠ k') : (PureU1.permThree hij hjk hik hij' hjk' hik').toFun k' = k"} +{"name":"PureU1.pairSwap_self_inv","declaration":"theorem PureU1.pairSwap_self_inv {n : ℕ} (i : Fin n) (j : Fin n) : (PureU1.pairSwap i j)⁻¹ = PureU1.pairSwap i j"} +{"name":"PureU1.Prop_two","declaration":"theorem PureU1.Prop_two {n : ℕ} (P : ℚ × ℚ → Prop) {S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear} {a : Fin n} {b : Fin n} (hab : a ≠ b) (h : ∀ (f : (PureU1.FamilyPermutations n).group),\n P\n ((((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations n)) f) S).val a,\n (((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations n)) f) S).val b)) (i : Fin n) (j : Fin n) : i ≠ j → P (S.val i, S.val j)"} +{"name":"PureU1.permThreeInj_thd_apply","declaration":"theorem PureU1.permThreeInj_thd_apply {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) : (Function.Embedding.toEquivRange (PureU1.permThreeInj hij hjk hik)).symm { val := k, property := ⋯ } = 2"} +{"name":"PureU1.Prop_three","declaration":"theorem PureU1.Prop_three {n : ℕ} (P : ℚ × ℚ × ℚ → Prop) {S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear} {a : Fin n} {b : Fin n} {c : Fin n} (hab : a ≠ b) (hac : a ≠ c) (hbc : b ≠ c) (h : ∀ (f : (PureU1.FamilyPermutations n).group),\n P\n ((((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations n)) f) S).val a,\n (((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations n)) f) S).val b,\n (((ACCSystemGroupAction.linSolRep (PureU1.FamilyPermutations n)) f) S).val c)) (i : Fin n) (j : Fin n) (k : Fin n) : i ≠ j → j ≠ k → i ≠ k → P (S.val i, S.val j, S.val k)"} +{"name":"PureU1.pairSwap_other","declaration":"theorem PureU1.pairSwap_other {n : ℕ} (i : Fin n) (j : Fin n) (k : Fin n) (hik : i ≠ k) (hjk : j ≠ k) : (PureU1.pairSwap i j).toFun k = k"} +{"name":"PureU1.PermGroup","declaration":"/-- The permutation group of the n-fermions. -/\ndef PureU1.PermGroup (n : ℕ) : Type"} +{"name":"PureU1.accCube_invariant","declaration":"theorem PureU1.accCube_invariant {n : ℕ} (f : PureU1.PermGroup n) (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : (PureU1.accCube n) ((PureU1.permCharges f) S) = (PureU1.accCube n) S"} +{"name":"PureU1.permThreeInj_thd","declaration":"theorem PureU1.permThreeInj_thd {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) : k ∈ Set.range ⇑(PureU1.permThreeInj hij hjk hik)"} +{"name":"PureU1.accGrav_invariant","declaration":"theorem PureU1.accGrav_invariant {n : ℕ} (f : PureU1.PermGroup n) (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : (PureU1.accGrav n) ((PureU1.permCharges f) S) = (PureU1.accGrav n) S"} +{"name":"PureU1.chargeMap_apply","declaration":"theorem PureU1.chargeMap_apply {n : ℕ} (f : PureU1.PermGroup n) (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : ∀ (a : Fin (PureU1 n).numberCharges), (PureU1.chargeMap f) S a = S (f a)"} +{"name":"PureU1.permTwoInj_fst","declaration":"theorem PureU1.permTwoInj_fst {n : ℕ} {i : Fin n} {j : Fin n} (hij : i ≠ j) : i ∈ Set.range ⇑(PureU1.permTwoInj hij)"} +{"name":"PureU1.permTwoInj","declaration":"/-- Given two distinct elements, an embedding of `Fin 2` into `Fin n`. -/\ndef PureU1.permTwoInj {n : ℕ} {i : Fin n} {j : Fin n} (hij : i ≠ j) : Fin 2 ↪ Fin n"} +{"name":"PureU1.permTwo","declaration":"/-- A permutation which swaps `i` with `i'` and `j` with `j'`. -/\ndef PureU1.permTwo {n : ℕ} {i : Fin n} {j : Fin n} {i' : Fin n} {j' : Fin n} (hij : i ≠ j) (hij' : i' ≠ j') : (PureU1.FamilyPermutations n).group"} +{"name":"PureU1.permThree","declaration":"/-- A permutation which swaps three distinct elements with another three. -/\ndef PureU1.permThree {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} {i' : Fin n} {j' : Fin n} {k' : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) (hij' : i' ≠ j') (hjk' : j' ≠ k') (hik' : i' ≠ k') : (PureU1.FamilyPermutations n).group"} +{"name":"PureU1.pairSwap_fst","declaration":"theorem PureU1.pairSwap_fst {n : ℕ} (i : Fin n) (j : Fin n) : (PureU1.pairSwap i j).toFun i = j"} +{"name":"PureU1.permTwoInj_snd","declaration":"theorem PureU1.permTwoInj_snd {n : ℕ} {i : Fin n} {j : Fin n} (hij : i ≠ j) : j ∈ Set.range ⇑(PureU1.permTwoInj hij)"} +{"name":"PureU1.pairSwap","declaration":"/-- The permutation which swaps i and j. -/\ndef PureU1.pairSwap {n : ℕ} (i : Fin n) (j : Fin n) : (PureU1.FamilyPermutations n).group"} +{"name":"PureU1.permThreeInj","declaration":"/-- Given three distinct elements an embedding of `Fin 3` into `Fin n`. -/\ndef PureU1.permThreeInj {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) : Fin 3 ↪ Fin n"} +{"name":"PureU1.chargeMap","declaration":"/-- The image of an element of `permGroup` under the representation on charges. -/\ndef PureU1.chargeMap {n : ℕ} (f : PureU1.PermGroup n) : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges →ₗ[ℚ] ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges"} +{"name":"PureU1.pairSwap_inv_fst","declaration":"theorem PureU1.pairSwap_inv_fst {n : ℕ} (i : Fin n) (j : Fin n) : (PureU1.pairSwap i j).invFun i = j"} +{"name":"PureU1.permThreeInj_fst_apply","declaration":"theorem PureU1.permThreeInj_fst_apply {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) : (Function.Embedding.toEquivRange (PureU1.permThreeInj hij hjk hik)).symm { val := i, property := ⋯ } = 0"} +{"name":"PureU1.permThreeInj_snd","declaration":"theorem PureU1.permThreeInj_snd {n : ℕ} {i : Fin n} {j : Fin n} {k : Fin n} (hij : i ≠ j) (hjk : j ≠ k) (hik : i ≠ k) : j ∈ Set.range ⇑(PureU1.permThreeInj hij hjk hik)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.Sorts.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Sorts.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1b4e44e13a85a97450f648c49445af083610a24a --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.Sorts.jsonl @@ -0,0 +1,12 @@ +{"name":"PureU1.sort_perm","declaration":"theorem PureU1.sort_perm {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) (M : (PureU1.FamilyPermutations n).group) : PureU1.sort (((PureU1.FamilyPermutations n).rep M) S) = PureU1.sort S"} +{"name":"PureU1.sort_apply","declaration":"theorem PureU1.sort_apply {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) (j : Fin n) : PureU1.sort S j = S ((Tuple.sort S) j)"} +{"name":"AddSemiconjBy.eq_zero_iff","declaration":"theorem AddSemiconjBy.eq_zero_iff {G : Type u_1} [AddGroup G] (a : G) {x : G} {y : G} (h : AddSemiconjBy a x y) : x = 0 ↔ y = 0"} +{"name":"PureU1.sort_sorted","declaration":"theorem PureU1.sort_sorted {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : PureU1.Sorted (PureU1.sort S)"} +{"name":"PureU1.sort","declaration":"/-- Given a charge assignment `S`, the corresponding sorted charge assignment. -/\ndef PureU1.sort {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges"} +{"name":"PureU1.sortAFL","declaration":"/-- The sort function acting on `LinSols`. -/\ndef PureU1.sortAFL {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear) : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear"} +{"name":"PureU1.Sorted","declaration":"/-- A charge is sorted if for all `i ≤ j`, then `S i ≤ S j`. -/\ndef PureU1.Sorted {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : Prop"} +{"name":"PureU1.sortAFL_zero","declaration":"theorem PureU1.sortAFL_zero {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear) (hS : PureU1.sortAFL S = 0) : S = 0"} +{"name":"PureU1.sort_projection","declaration":"theorem PureU1.sort_projection {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) : PureU1.sort (PureU1.sort S) = PureU1.sort S"} +{"name":"SemiconjBy.eq_one_iff","declaration":"theorem SemiconjBy.eq_one_iff {G : Type u_1} [Group G] (a : G) {x : G} {y : G} (h : SemiconjBy a x y) : x = 1 ↔ y = 1"} +{"name":"PureU1.sort_zero","declaration":"theorem PureU1.sort_zero {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 n).toACCSystemCharges) (hS : PureU1.sort S = 0) : S = 0"} +{"name":"PureU1.sortAFL_val","declaration":"theorem PureU1.sortAFL_val {n : ℕ} (S : ACCSystemLinear.LinSols (PureU1 n).toACCSystemLinear) : (PureU1.sortAFL S).val = PureU1.sort S.val"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.PureU1.VectorLike.jsonl b/hep-declarations/HepLean.AnomalyCancellation.PureU1.VectorLike.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..03ce5c24a88680a5a881e9ef109ccc18e2eba378 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.PureU1.VectorLike.jsonl @@ -0,0 +1,3 @@ +{"name":"PureU1.VectorLikeEven","declaration":"/-- A charge configuration for n even is vector like if when sorted the `i`th element\nis equal to the negative of the `n + i`th element. -/\ndef PureU1.VectorLikeEven {n : ℕ} (S : ACCSystemCharges.Charges (PureU1 (2 * n)).toACCSystemCharges) : Prop"} +{"name":"PureU1.split_equal","declaration":"/-- Given a natural number `n`, this lemma proves that `n + n` is equal to `2 * n`.\n-/\ntheorem PureU1.split_equal (n : ℕ) : n + n = 2 * n"} +{"name":"PureU1.split_odd","declaration":"theorem PureU1.split_odd (n : ℕ) : n + 1 + n = 2 * n + 1"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SM.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SM.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f5d673655a9585b6a20a5177cdd4f153a9b130c3 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SM.Basic.jsonl @@ -0,0 +1,32 @@ +{"name":"SMCharges.L","declaration":"/-- The `L` charges as a map `Fin n → ℚ`. -/\ndef SMCharges.L {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SMACCs.quadBiLin_toFun_apply","declaration":"theorem SMACCs.quadBiLin_toFun_apply {n : ℕ} (S : ACCSystemCharges.Charges (SMCharges n)) (T : ACCSystemCharges.Charges (SMCharges n)) : (SMACCs.quadBiLin S) T =\n Finset.sum Finset.univ fun x =>\n S (finProdFinEquiv (0, x)) * T (finProdFinEquiv (0, x)) +\n -(2 * (S (finProdFinEquiv (1, x)) * T (finProdFinEquiv (1, x)))) +\n S (finProdFinEquiv (2, x)) * T (finProdFinEquiv (2, x)) +\n -(S (finProdFinEquiv (3, x)) * T (finProdFinEquiv (3, x))) +\n S (finProdFinEquiv (4, x)) * T (finProdFinEquiv (4, x))"} +{"name":"SMSpecies_numberCharges","declaration":"theorem SMSpecies_numberCharges (n : ℕ) : (SMSpecies n).numberCharges = n"} +{"name":"SMCharges.E","declaration":"/-- The `E` charges as a map `Fin n → ℚ`. -/\ndef SMCharges.E {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SMCharges","declaration":"/-- Associate to each (including RHN) SM fermion a set of charges-/\ndef SMCharges (n : ℕ) : ACCSystemCharges"} +{"name":"SMCharges.Q","declaration":"/-- The `Q` charges as a map `Fin n → ℚ`. -/\ndef SMCharges.Q {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SMCharges.charges_eq_toSpecies_eq","declaration":"theorem SMCharges.charges_eq_toSpecies_eq {n : ℕ} (S : ACCSystemCharges.Charges (SMCharges n)) (T : ACCSystemCharges.Charges (SMCharges n)) : S = T ↔ ∀ (i : Fin 5), (SMCharges.toSpecies i) S = (SMCharges.toSpecies i) T"} +{"name":"SMACCs.accCube_ext","declaration":"/-- Extensionality lemma for `accCube`. -/\ntheorem SMACCs.accCube_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMCharges n)} {T : ACCSystemCharges.Charges (SMCharges n)} (h : ∀ (j : Fin 5),\n (Finset.sum Finset.univ fun i => ((fun a => a ^ 3) ∘ (SMCharges.toSpecies j) S) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ 3) ∘ (SMCharges.toSpecies j) T) i) : SMACCs.accCube S = SMACCs.accCube T"} +{"name":"SMACCs.cubeTriLin_toFun_apply_apply","declaration":"theorem SMACCs.cubeTriLin_toFun_apply_apply {n : ℕ} (S : ACCSystemCharges.Charges (SMCharges n)) (S : ACCSystemCharges.Charges (SMCharges n)) (T : ACCSystemCharges.Charges (SMCharges n)) : ((SMACCs.cubeTriLin S✝) S) T =\n Finset.sum Finset.univ fun i =>\n 6 * (S✝ (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i)) * T (finProdFinEquiv (0, i))) +\n 3 * (S✝ (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i)) * T (finProdFinEquiv (1, i))) +\n 3 * (S✝ (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i)) * T (finProdFinEquiv (2, i))) +\n 2 * (S✝ (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i)) * T (finProdFinEquiv (3, i))) +\n S✝ (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) * T (finProdFinEquiv (4, i))"} +{"name":"SMACCs.accGrav_ext","declaration":"/-- Extensionality lemma for `accGrav`. -/\ntheorem SMACCs.accGrav_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMCharges n)} {T : ACCSystemCharges.Charges (SMCharges n)} (hj : ∀ (j : Fin 5),\n (Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) T i) : SMACCs.accGrav S = SMACCs.accGrav T"} +{"name":"SMACCs.accYY","declaration":"/-- The `Y²` anomaly equation. -/\ndef SMACCs.accYY {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMACCs.accSU3","declaration":"/-- The `SU(3)` anomaly equations. -/\ndef SMACCs.accSU3 {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMCharges.toSpecies","declaration":"/-- For a given `i ∈ Fin 5`, the projection of a charge onto that species. -/\ndef SMCharges.toSpecies {n : ℕ} (i : Fin 5) : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SMCharges_numberCharges","declaration":"theorem SMCharges_numberCharges (n : ℕ) : (SMCharges n).numberCharges = 5 * n"} +{"name":"SMCharges.D","declaration":"/-- The `D` charges as a map `Fin n → ℚ`. -/\ndef SMCharges.D {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SMACCs.accYY_ext","declaration":"/-- Extensionality lemma for `accYY`. -/\ntheorem SMACCs.accYY_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMCharges n)} {T : ACCSystemCharges.Charges (SMCharges n)} (hj : ∀ (j : Fin 5),\n (Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) T i) : SMACCs.accYY S = SMACCs.accYY T"} +{"name":"SMCharges.toSpeciesEquiv_symm_apply","declaration":"theorem SMCharges.toSpeciesEquiv_symm_apply {n : ℕ} : ∀ (a : Fin 5 → Fin n → ℚ) (a_1 : Fin (5 * n)), SMCharges.toSpeciesEquiv.symm a a_1 = a (Fin.divNat a_1) (Fin.modNat a_1)"} +{"name":"SMACCs.accSU2_ext","declaration":"/-- Extensionality lemma for `accSU2`. -/\ntheorem SMACCs.accSU2_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMCharges n)} {T : ACCSystemCharges.Charges (SMCharges n)} (hj : ∀ (j : Fin 5),\n (Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) T i) : SMACCs.accSU2 S = SMACCs.accSU2 T"} +{"name":"SMACCs.cubeTriLin","declaration":"/-- The trilinear function defining the cubic. -/\ndef SMACCs.cubeTriLin {n : ℕ} : TriLinearSymm (ACCSystemCharges.Charges (SMCharges n))"} +{"name":"SMACCs.accGrav","declaration":"/-- The gravitational anomaly equation. -/\ndef SMACCs.accGrav {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMACCs.quadBiLin","declaration":"/-- The quadratic bilinear map. -/\ndef SMACCs.quadBiLin {n : ℕ} : BiLinearSymm (ACCSystemCharges.Charges (SMCharges n))"} +{"name":"SMACCs.accQuad_ext","declaration":"/-- Extensionality lemma for `accQuad`. -/\ntheorem SMACCs.accQuad_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMCharges n)} {T : ACCSystemCharges.Charges (SMCharges n)} (h : ∀ (j : Fin 5),\n (Finset.sum Finset.univ fun i => ((fun a => a ^ 2) ∘ (SMCharges.toSpecies j) S) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ 2) ∘ (SMCharges.toSpecies j) T) i) : SMACCs.accQuad S = SMACCs.accQuad T"} +{"name":"SMACCs.accQuad","declaration":"/-- The quadratic anomaly cancellation condition. -/\ndef SMACCs.accQuad {n : ℕ} : HomogeneousQuadratic (ACCSystemCharges.Charges (SMCharges n))"} +{"name":"SMACCs.accSU3_ext","declaration":"/-- Extensionality lemma for `accSU3`. -/\ntheorem SMACCs.accSU3_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMCharges n)} {T : ACCSystemCharges.Charges (SMCharges n)} (hj : ∀ (j : Fin 5),\n (Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMCharges.toSpecies j) T i) : SMACCs.accSU3 S = SMACCs.accSU3 T"} +{"name":"SMCharges.toSpeciesEquiv","declaration":"/-- An equivalence between the set `(SMCharges n).charges` and the set\n`(Fin 5 → Fin n → ℚ)`. -/\ndef SMCharges.toSpeciesEquiv {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) ≃ (Fin 5 → Fin n → ℚ)"} +{"name":"SMCharges.U","declaration":"/-- The `U` charges as a map `Fin n → ℚ`. -/\ndef SMCharges.U {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SMSpecies","declaration":"/-- The vector space associated with a single species of fermions. -/\ndef SMSpecies (n : ℕ) : ACCSystemCharges"} +{"name":"SMCharges.toSpeciesEquiv_apply","declaration":"theorem SMCharges.toSpeciesEquiv_apply {n : ℕ} : ∀ (a : Fin (5 * n) → ℚ) (a_1 : Fin 5) (a_2 : Fin n), SMCharges.toSpeciesEquiv a a_1 a_2 = a (finProdFinEquiv (a_1, a_2))"} +{"name":"SMACCs.accSU2","declaration":"/-- The `SU(2)` anomaly equation. -/\ndef SMACCs.accSU2 {n : ℕ} : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMACCs.accCube","declaration":"/-- The cubic acc. -/\ndef SMACCs.accCube {n : ℕ} : HomogeneousCubic (ACCSystemCharges.Charges (SMCharges n))"} +{"name":"SMCharges.toSpecies_apply","declaration":"theorem SMCharges.toSpecies_apply {n : ℕ} (i : Fin 5) (S : ACCSystemCharges.Charges (SMCharges n)) : ∀ (a : Fin (SMSpecies n).numberCharges), (SMCharges.toSpecies i) S a = S (finProdFinEquiv (i, a))"} +{"name":"SMCharges.toSMSpecies_toSpecies_inv","declaration":"theorem SMCharges.toSMSpecies_toSpecies_inv {n : ℕ} (i : Fin 5) (f : Fin 5 → Fin n → ℚ) : (SMCharges.toSpecies i) (SMCharges.toSpeciesEquiv.symm f) = f i"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SM.FamilyMaps.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SM.FamilyMaps.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f239d7f26b367388ed9eb9f03d3418dfa5d0f684 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SM.FamilyMaps.jsonl @@ -0,0 +1,11 @@ +{"name":"SM.chargesMapOfSpeciesMap","declaration":"/-- Given a map of for a generic species, the corresponding map for charges. -/\ndef SM.chargesMapOfSpeciesMap {n : ℕ} {m : ℕ} (f : ACCSystemCharges.Charges (SMSpecies n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies m)) : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMCharges m)"} +{"name":"SM.familyEmbedding","declaration":"/-- The embedding of the `m`-family charges onto the `n`-family charges, with all\nother charges zero. -/\ndef SM.familyEmbedding (m : ℕ) (n : ℕ) : ACCSystemCharges.Charges (SMCharges m) →ₗ[ℚ] ACCSystemCharges.Charges (SMCharges n)"} +{"name":"SM.chargesMapOfSpeciesMap_apply","declaration":"theorem SM.chargesMapOfSpeciesMap_apply {n : ℕ} {m : ℕ} (f : ACCSystemCharges.Charges (SMSpecies n) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies m)) (S : ACCSystemCharges.Charges (SMCharges n)) : (SM.chargesMapOfSpeciesMap f) S = SMCharges.toSpeciesEquiv.symm fun i => f ((SMCharges.toSpecies i) S)"} +{"name":"SM.speciesFamilyUniversial_apply","declaration":"theorem SM.speciesFamilyUniversial_apply (n : ℕ) (S : ACCSystemCharges.Charges (SMSpecies 1)) : ∀ (x : Fin (SMSpecies n).numberCharges), (SM.speciesFamilyUniversial n) S x = S 0"} +{"name":"SM.speciesFamilyUniversial","declaration":"/-- For species, the embedding of the `1`-family charges into the `n`-family charges in\na universal manor. -/\ndef SM.speciesFamilyUniversial (n : ℕ) : ACCSystemCharges.Charges (SMSpecies 1) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SM.speciesFamilyProj","declaration":"/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/\ndef SM.speciesFamilyProj {m : ℕ} {n : ℕ} (h : n ≤ m) : ACCSystemCharges.Charges (SMSpecies m) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} +{"name":"SM.speciesEmbed_apply","declaration":"theorem SM.speciesEmbed_apply (m : ℕ) (n : ℕ) (S : ACCSystemCharges.Charges (SMSpecies m)) (i : Fin (SMSpecies n).numberCharges) : (SM.speciesEmbed m n) S i = if hi : ↑i < m then S { val := ↑i, isLt := hi } else 0"} +{"name":"SM.speciesFamilyProj_apply","declaration":"theorem SM.speciesFamilyProj_apply {m : ℕ} {n : ℕ} (h : n ≤ m) (S : ACCSystemCharges.Charges (SMSpecies m)) : ∀ (a : Fin (SMSpecies n).numberCharges), (SM.speciesFamilyProj h) S a = S (Fin.castLE h a)"} +{"name":"SM.familyProjection","declaration":"/-- The projection of the `m`-family charges onto the first `n`-family charges. -/\ndef SM.familyProjection {m : ℕ} {n : ℕ} (h : n ≤ m) : ACCSystemCharges.Charges (SMCharges m) →ₗ[ℚ] ACCSystemCharges.Charges (SMCharges n)"} +{"name":"SM.familyUniversal","declaration":"/-- The embedding of the `1`-family charges into the `n`-family charges in\na universal manor. -/\ndef SM.familyUniversal (n : ℕ) : ACCSystemCharges.Charges (SMCharges 1) →ₗ[ℚ] ACCSystemCharges.Charges (SMCharges n)"} +{"name":"SM.speciesEmbed","declaration":"/-- For species, the embedding of the `m`-family charges onto the `n`-family charges, with all\nother charges zero. -/\ndef SM.speciesEmbed (m : ℕ) (n : ℕ) : ACCSystemCharges.Charges (SMSpecies m) →ₗ[ℚ] ACCSystemCharges.Charges (SMSpecies n)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e2e39e18c5e36ef37c30a7ed12f7da086245cea2 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.Basic.jsonl @@ -0,0 +1,16 @@ +{"name":"SM.SMNoGrav_linearACCs","declaration":"theorem SM.SMNoGrav_linearACCs (n : ℕ) (i : Fin 2) : (SM.SMNoGrav n).linearACCs i =\n match i with\n | 0 =>\n {\n toAddHom :=\n { toFun := fun S => Finset.sum Finset.univ fun i => 3 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (3, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }\n | 1 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n 2 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (1, i)) + S (finProdFinEquiv (2, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }"} +{"name":"SM.SMNoGrav_numberQuadratic","declaration":"theorem SM.SMNoGrav_numberQuadratic (n : ℕ) : (SM.SMNoGrav n).numberQuadratic = 0"} +{"name":"SM.SMNoGrav.SU3Sol","declaration":"theorem SM.SMNoGrav.SU3Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SM.SMNoGrav n).toACCSystemLinear) : SMACCs.accSU3 S.val = 0"} +{"name":"SM.SMNoGrav.linearToAF","declaration":"/-- An element of `AnomalyFreeLinear` which satisfies the quadratic and cubic ACCs\ngives us a element of `AnomalyFree`. -/\ndef SM.SMNoGrav.linearToAF {n : ℕ} (S : ACCSystemLinear.LinSols (SM.SMNoGrav n).toACCSystemLinear) (hc : SMACCs.accCube S.val = 0) : ACCSystem.Sols (SM.SMNoGrav n)"} +{"name":"SM.SMNoGrav_quadraticACCs","declaration":"theorem SM.SMNoGrav_quadraticACCs (n : ℕ) (i : Fin 0) : (SM.SMNoGrav n).quadraticACCs i = Fin.elim0 i"} +{"name":"SM.SMNoGrav.chargeToLinear","declaration":"/-- An element of `charges` which satisfies the linear ACCs\ngives us a element of `AnomalyFreeLinear`. -/\ndef SM.SMNoGrav.chargeToLinear {n : ℕ} (S : ACCSystemCharges.Charges (SM.SMNoGrav n).toACCSystemCharges) (hSU2 : SMACCs.accSU2 S = 0) (hSU3 : SMACCs.accSU3 S = 0) : ACCSystemLinear.LinSols (SM.SMNoGrav n).toACCSystemLinear"} +{"name":"SM.SMNoGrav_numberLinear","declaration":"theorem SM.SMNoGrav_numberLinear (n : ℕ) : (SM.SMNoGrav n).numberLinear = 2"} +{"name":"SM.SMNoGrav.linearToQuad","declaration":"/-- An element of `AnomalyFreeLinear` which satisfies the quadratic ACCs\ngives us a element of `AnomalyFreeQuad`. -/\ndef SM.SMNoGrav.linearToQuad {n : ℕ} (S : ACCSystemLinear.LinSols (SM.SMNoGrav n).toACCSystemLinear) : ACCSystemQuad.QuadSols (SM.SMNoGrav n).toACCSystemQuad"} +{"name":"SM.SMNoGrav.chargeToQuad","declaration":"/-- An element of `charges` which satisfies the linear and quadratic ACCs\ngives us a element of `AnomalyFreeQuad`. -/\ndef SM.SMNoGrav.chargeToQuad {n : ℕ} (S : ACCSystemCharges.Charges (SM.SMNoGrav n).toACCSystemCharges) (hSU2 : SMACCs.accSU2 S = 0) (hSU3 : SMACCs.accSU3 S = 0) : ACCSystemQuad.QuadSols (SM.SMNoGrav n).toACCSystemQuad"} +{"name":"SM.SMNoGrav.SU2Sol","declaration":"theorem SM.SMNoGrav.SU2Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SM.SMNoGrav n).toACCSystemLinear) : SMACCs.accSU2 S.val = 0"} +{"name":"SM.SMNoGrav.chargeToAF","declaration":"/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs\ngives us a element of `AnomalyFree`. -/\ndef SM.SMNoGrav.chargeToAF {n : ℕ} (S : ACCSystemCharges.Charges (SM.SMNoGrav n).toACCSystemCharges) (hSU2 : SMACCs.accSU2 S = 0) (hSU3 : SMACCs.accSU3 S = 0) (hc : SMACCs.accCube S = 0) : ACCSystem.Sols (SM.SMNoGrav n)"} +{"name":"SM.SMNoGrav_numberCharges","declaration":"theorem SM.SMNoGrav_numberCharges (n : ℕ) : (SM.SMNoGrav n).numberCharges = 5 * n"} +{"name":"SM.SMNoGrav.quadToAF","declaration":"/-- An element of `AnomalyFreeQuad` which satisfies the quadratic ACCs\ngives us a element of `AnomalyFree`. -/\ndef SM.SMNoGrav.quadToAF {n : ℕ} (S : ACCSystemQuad.QuadSols (SM.SMNoGrav n).toACCSystemQuad) (hc : SMACCs.accCube S.val = 0) : ACCSystem.Sols (SM.SMNoGrav n)"} +{"name":"SM.SMNoGrav","declaration":"/-- The ACC system for the standard model without RHN and without the gravitational ACC. -/\ndef SM.SMNoGrav (n : ℕ) : ACCSystem"} +{"name":"SM.SMNoGrav.cubeSol","declaration":"theorem SM.SMNoGrav.cubeSol {n : ℕ} (S : ACCSystem.Sols (SM.SMNoGrav n)) : SMACCs.accCube S.val = 0"} +{"name":"SM.SMNoGrav_cubicACC_toFun","declaration":"theorem SM.SMNoGrav_cubicACC_toFun (n : ℕ) (S : ACCSystemCharges.Charges (SMCharges n)) : (SM.SMNoGrav n).cubicACC.toFun S =\n Finset.sum Finset.univ fun i =>\n 6 * (S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i))) +\n 3 * (S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i))) +\n 3 * (S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i))) +\n 2 * (S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i))) +\n S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i))"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.One.Lemmas.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.One.Lemmas.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b88d62dee641336253d222d4ec416064a6b407f0 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.One.Lemmas.jsonl @@ -0,0 +1,4 @@ +{"name":"SM.SMNoGrav.One.accGrav_Q_neq_zero","declaration":"theorem SM.SMNoGrav.One.accGrav_Q_neq_zero {S : ACCSystem.Sols (SM.SMNoGrav 1)} (hQ : SMCharges.Q S.val 0 ≠ 0) (FLTThree : FermatLastTheoremWith ℚ 3) : SMACCs.accGrav S.val = 0"} +{"name":"SM.SMNoGrav.One.accGravSatisfied","declaration":"/-- Any solution to the ACCs without gravity satisfies the gravitational ACC. -/\ntheorem SM.SMNoGrav.One.accGravSatisfied {S : ACCSystem.Sols (SM.SMNoGrav 1)} (FLTThree : FermatLastTheoremWith ℚ 3) : SMACCs.accGrav S.val = 0"} +{"name":"SM.SMNoGrav.One.E_zero_iff_Q_zero","declaration":"theorem SM.SMNoGrav.One.E_zero_iff_Q_zero {S : ACCSystem.Sols (SM.SMNoGrav 1)} : SMCharges.Q S.val 0 = 0 ↔ SMCharges.E S.val 0 = 0"} +{"name":"SM.SMNoGrav.One.accGrav_Q_zero","declaration":"theorem SM.SMNoGrav.One.accGrav_Q_zero {S : ACCSystem.Sols (SM.SMNoGrav 1)} (hQ : SMCharges.Q S.val 0 = 0) : SMACCs.accGrav S.val = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.One.LinearParameterization.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.One.LinearParameterization.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c2003a38a49b219c3acc115d0ebf409e1e18ffd2 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SM.NoGrav.One.LinearParameterization.jsonl @@ -0,0 +1,47 @@ +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero","declaration":"/-- A map from `linearParameters` to `linearParametersQENeqZero` in the special case when\n`Q'` and `E'` of the linear parameters are non-zero. -/\ndef SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero (S : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }) : SM.SMNoGrav.One.linearParametersQENeqZero"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.hvw","declaration":"def SM.SMNoGrav.One.linearParametersQENeqZero.hvw (self : SM.SMNoGrav.One.linearParametersQENeqZero) : self.v + self.w ≠ 0"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.grav_of_cubic","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.grav_of_cubic (S : SM.SMNoGrav.One.linearParametersQENeqZero) (h : SMACCs.accCube (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0) (FLTThree : FermatLastTheoremWith ℚ 3) : SMACCs.accGrav (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0"} +{"name":"SM.SMNoGrav.One.linearParameters.asCharges","declaration":"/-- The map from the linear parameters to elements of `(SMNoGrav 1).charges`. -/\ndef SM.SMNoGrav.One.linearParameters.asCharges (S : SM.SMNoGrav.One.linearParameters) : ACCSystemCharges.Charges (SM.SMNoGrav 1).toACCSystemCharges"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_symm_apply_w","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_symm_apply_w (S : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }) : (SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters.symm S).w = ((↑S).Y - 3 * (↑S).Q') / (↑S).E'"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_apply_coe_Y","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_apply_coe_Y (S : SM.SMNoGrav.One.linearParametersQENeqZero) : (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters S)).Y = 3 * S.x * (S.v - S.w) / (S.v + S.w)"} +{"name":"SM.SMNoGrav.One.linearParameters.E'","declaration":"/-- The parameter `E'`. -/\ndef SM.SMNoGrav.One.linearParameters.E' (self : SM.SMNoGrav.One.linearParameters) : ℚ"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters_coe_E'","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters_coe_E' (S : SM.SMNoGrav.One.linearParametersQENeqZero) : (↑(SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters S)).E' = -(6 * S.x) / (S.v + S.w)"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.ext","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.ext {S : SM.SMNoGrav.One.linearParametersQENeqZero} {T : SM.SMNoGrav.One.linearParametersQENeqZero} (hx : S.x = T.x) (hv : S.v = T.v) (hw : S.w = T.w) : S = T"} +{"name":"SM.SMNoGrav.One.linearParameters.cubic_zero_E'_zero","declaration":"theorem SM.SMNoGrav.One.linearParameters.cubic_zero_E'_zero (S : SM.SMNoGrav.One.linearParameters) (hc : SMACCs.accCube (SM.SMNoGrav.One.linearParameters.asCharges S) = 0) (h : S.E' = 0) : S.Q' = 0"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters","declaration":"/-- A bijection between the type `linearParametersQENeqZero` and linear parameters\nwith `Q'` and `E'` non-zero. -/\ndef SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters : SM.SMNoGrav.One.linearParametersQENeqZero ≃ { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_symm_apply_v","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_symm_apply_v (S : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }) : (SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters.symm S).v = (-(↑S).Y + -(3 * (↑S).Q')) / (↑S).E'"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.mk","declaration":"ctor SM.SMNoGrav.One.linearParametersQENeqZero.mk (x : ℚ) (v : ℚ) (w : ℚ) (hx : x ≠ 0) (hvw : v + w ≠ 0) : SM.SMNoGrav.One.linearParametersQENeqZero"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.hx","declaration":"def SM.SMNoGrav.One.linearParametersQENeqZero.hx (self : SM.SMNoGrav.One.linearParametersQENeqZero) : self.x ≠ 0"} +{"name":"SM.SMNoGrav.One.linearParameters.bijection","declaration":"/-- The bijection between the type of linear parameters and `(SMNoGrav 1).LinSols`. -/\ndef SM.SMNoGrav.One.linearParameters.bijection : SM.SMNoGrav.One.linearParameters ≃ ACCSystemLinear.LinSols (SM.SMNoGrav 1).toACCSystemLinear"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters_coe_Y","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters_coe_Y (S : SM.SMNoGrav.One.linearParametersQENeqZero) : (↑(SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters S)).Y = 3 * S.x * (S.v - S.w) / (S.v + S.w)"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_symm_apply_x","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_symm_apply_x (S : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }) : (SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters.symm S).x = (↑S).Q'"} +{"name":"SM.SMNoGrav.One.linearParameters.mk","declaration":"ctor SM.SMNoGrav.One.linearParameters.mk (Q' : ℚ) (Y : ℚ) (E' : ℚ) : SM.SMNoGrav.One.linearParameters"} +{"name":"SM.SMNoGrav.One.linearParameters.grav","declaration":"theorem SM.SMNoGrav.One.linearParameters.grav (S : SM.SMNoGrav.One.linearParameters) : SMACCs.accGrav (SM.SMNoGrav.One.linearParameters.asCharges S) = 0 ↔ S.E' = 6 * S.Q'"} +{"name":"SM.SMNoGrav.One.linearParameters.Y","declaration":"/-- The parameter `Y`. -/\ndef SM.SMNoGrav.One.linearParameters.Y (self : SM.SMNoGrav.One.linearParameters) : ℚ"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero_v","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero_v (S : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }) : (SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero S).v = (-(↑S).Y + -(3 * (↑S).Q')) / (↑S).E'"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters_coe_Q'","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters_coe_Q' (S : SM.SMNoGrav.One.linearParametersQENeqZero) : (↑(SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters S)).Q' = S.x"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.cubic_v_or_w_zero","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.cubic_v_or_w_zero (S : SM.SMNoGrav.One.linearParametersQENeqZero) (h : SMACCs.accCube (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0) (FLTThree : FermatLastTheoremWith ℚ 3) : S.v = 0 ∨ S.w = 0"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_apply_coe_E'","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_apply_coe_E' (S : SM.SMNoGrav.One.linearParametersQENeqZero) : (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters S)).E' = -(6 * S.x) / (S.v + S.w)"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.v","declaration":"/-- The parameter `v`. -/\ndef SM.SMNoGrav.One.linearParametersQENeqZero.v (self : SM.SMNoGrav.One.linearParametersQENeqZero) : ℚ"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.cube_w_zero","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.cube_w_zero (S : SM.SMNoGrav.One.linearParametersQENeqZero) (h : SMACCs.accCube (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0) (hw : S.w = 0) : S.v = -1"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.cube_w_v","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.cube_w_v (S : SM.SMNoGrav.One.linearParametersQENeqZero) (h : SMACCs.accCube (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0) (FLTThree : FermatLastTheoremWith ℚ 3) : S.v = -1 ∧ S.w = 0 ∨ S.v = 0 ∧ S.w = -1"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero","declaration":"/-- The parameters for solutions to the linear ACCs with the condition that Q and E are\nnon-zero. -/\nstructure SM.SMNoGrav.One.linearParametersQENeqZero : Type"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.cubic_v_zero","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.cubic_v_zero (S : SM.SMNoGrav.One.linearParametersQENeqZero) (h : SMACCs.accCube (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0) (hv : S.v = 0) : S.w = -1"} +{"name":"SM.SMNoGrav.One.linearParameters.Q'","declaration":"/-- The parameter `Q'`. -/\ndef SM.SMNoGrav.One.linearParameters.Q' (self : SM.SMNoGrav.One.linearParameters) : ℚ"} +{"name":"SM.SMNoGrav.One.linearParameters.cubic_zero_Q'_zero","declaration":"theorem SM.SMNoGrav.One.linearParameters.cubic_zero_Q'_zero (S : SM.SMNoGrav.One.linearParameters) (hc : SMACCs.accCube (SM.SMNoGrav.One.linearParameters.asCharges S) = 0) (h : S.Q' = 0) : S.E' = 0"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters","declaration":"/-- A map from `linearParametersQENeqZero` to `linearParameters`. -/\ndef SM.SMNoGrav.One.linearParametersQENeqZero.toLinearParameters (S : SM.SMNoGrav.One.linearParametersQENeqZero) : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero_x","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero_x (S : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }) : (SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero S).x = (↑S).Q'"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.cubic","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.cubic (S : SM.SMNoGrav.One.linearParametersQENeqZero) : SMACCs.accCube (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0 ↔ S.v ^ 3 + S.w ^ 3 = -1"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.w","declaration":"/-- The parameter `w`. -/\ndef SM.SMNoGrav.One.linearParametersQENeqZero.w (self : SM.SMNoGrav.One.linearParametersQENeqZero) : ℚ"} +{"name":"SM.SMNoGrav.One.linearParameters.ext","declaration":"theorem SM.SMNoGrav.One.linearParameters.ext {S : SM.SMNoGrav.One.linearParameters} {T : SM.SMNoGrav.One.linearParameters} (hQ : S.Q' = T.Q') (hY : S.Y = T.Y) (hE : S.E' = T.E') : S = T"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_apply_coe_Q'","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters_apply_coe_Q' (S : SM.SMNoGrav.One.linearParametersQENeqZero) : (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijectionLinearParameters S)).Q' = S.x"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.grav","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.grav (S : SM.SMNoGrav.One.linearParametersQENeqZero) : SMACCs.accGrav (↑(SM.SMNoGrav.One.linearParametersQENeqZero.bijection S)).val = 0 ↔ S.v + S.w = -1"} +{"name":"SM.SMNoGrav.One.linearParameters.speciesVal","declaration":"theorem SM.SMNoGrav.One.linearParameters.speciesVal {i : Fin 5} (S : SM.SMNoGrav.One.linearParameters) : (SMCharges.toSpecies i) (SM.SMNoGrav.One.linearParameters.asCharges S) 0 =\n SM.SMNoGrav.One.linearParameters.asCharges S i"} +{"name":"SM.SMNoGrav.One.linearParameters.asLinear_val","declaration":"theorem SM.SMNoGrav.One.linearParameters.asLinear_val (S : SM.SMNoGrav.One.linearParameters) : (SM.SMNoGrav.One.linearParameters.asLinear S).val = SM.SMNoGrav.One.linearParameters.asCharges S"} +{"name":"SM.SMNoGrav.One.linearParameters","declaration":"/-- The parameters for a linear parameterization to the solution of the linear ACCs. -/\nstructure SM.SMNoGrav.One.linearParameters : Type"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero_w","declaration":"theorem SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero_w (S : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 }) : (SM.SMNoGrav.One.linearParametersQENeqZero.tolinearParametersQNeqZero S).w = ((↑S).Y - 3 * (↑S).Q') / (↑S).E'"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.bijection","declaration":"/-- The bijection between `linearParametersQENeqZero` and `LinSols` with `Q` and `E` non-zero. -/\ndef SM.SMNoGrav.One.linearParametersQENeqZero.bijection : SM.SMNoGrav.One.linearParametersQENeqZero ≃ { S // SMCharges.Q S.val 0 ≠ 0 ∧ SMCharges.E S.val 0 ≠ 0 }"} +{"name":"SM.SMNoGrav.One.linearParametersQENeqZero.x","declaration":"/-- The parameter `x`. -/\ndef SM.SMNoGrav.One.linearParametersQENeqZero.x (self : SM.SMNoGrav.One.linearParametersQENeqZero) : ℚ"} +{"name":"SM.SMNoGrav.One.linearParameters.bijectionQEZero","declaration":"/-- The bijection between the linear parameters and `(SMNoGrav 1).LinSols` in the special\ncase when Q and E are both not zero. -/\ndef SM.SMNoGrav.One.linearParameters.bijectionQEZero : { S // S.Q' ≠ 0 ∧ S.E' ≠ 0 } ≃ { S // SMCharges.Q S.val 0 ≠ 0 ∧ SMCharges.E S.val 0 ≠ 0 }"} +{"name":"SM.SMNoGrav.One.linearParameters.asLinear","declaration":"/-- The map from the linear paramaters to elements of `(SMNoGrav 1).LinSols`. -/\ndef SM.SMNoGrav.One.linearParameters.asLinear (S : SM.SMNoGrav.One.linearParameters) : ACCSystemLinear.LinSols (SM.SMNoGrav 1).toACCSystemLinear"} +{"name":"SM.SMNoGrav.One.linearParameters.cubic","declaration":"theorem SM.SMNoGrav.One.linearParameters.cubic (S : SM.SMNoGrav.One.linearParameters) : SMACCs.accCube (SM.SMNoGrav.One.linearParameters.asCharges S) = -54 * S.Q' ^ 3 - 18 * S.Q' * S.Y ^ 2 + S.E' ^ 3"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SM.Permutations.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SM.Permutations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..771fea0926bced021e7f75cda8ed98ffdf20b93b --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SM.Permutations.jsonl @@ -0,0 +1,13 @@ +{"name":"SM.toSpecies_sum_invariant","declaration":"theorem SM.toSpecies_sum_invariant {n : ℕ} (m : ℕ) (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) (j : Fin 5) : (Finset.sum Finset.univ fun i => ((fun a => a ^ m) ∘ (SMCharges.toSpecies j) ((SM.repCharges f) S)) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ m) ∘ (SMCharges.toSpecies j) S) i"} +{"name":"SM.accSU2_invariant","declaration":"theorem SM.accSU2_invariant {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) : SMACCs.accSU2 ((SM.repCharges f) S) = SMACCs.accSU2 S"} +{"name":"SM.accQuad_invariant","declaration":"theorem SM.accQuad_invariant {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) : SMACCs.accQuad ((SM.repCharges f) S) = SMACCs.accQuad S"} +{"name":"SM.accCube_invariant","declaration":"theorem SM.accCube_invariant {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) : SMACCs.accCube ((SM.repCharges f) S) = SMACCs.accCube S"} +{"name":"SM.chargeMap_apply","declaration":"theorem SM.chargeMap_apply {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) : (SM.chargeMap f) S = SMCharges.toSpeciesEquiv.symm fun i => (SMCharges.toSpecies i) S ∘ ⇑(f i)"} +{"name":"SM.PermGroup","declaration":"/-- The group of `Sₙ` permutations for each species. -/\ndef SM.PermGroup (n : ℕ) : Type"} +{"name":"SM.accGrav_invariant","declaration":"theorem SM.accGrav_invariant {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) : SMACCs.accGrav ((SM.repCharges f) S) = SMACCs.accGrav S"} +{"name":"SM.chargeMap","declaration":"/-- The image of an element of `permGroup n` under the representation on charges. -/\ndef SM.chargeMap {n : ℕ} (f : SM.PermGroup n) : ACCSystemCharges.Charges (SMCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMCharges n)"} +{"name":"SM.repCharges_toSpecies","declaration":"theorem SM.repCharges_toSpecies {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) (j : Fin 5) : (SMCharges.toSpecies j) ((SM.repCharges f) S) = (SMCharges.toSpecies j) S ∘ ⇑(f⁻¹ j)"} +{"name":"SM.accYY_invariant","declaration":"theorem SM.accYY_invariant {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) : SMACCs.accYY ((SM.repCharges f) S) = SMACCs.accYY S"} +{"name":"SM.instGroupPermGroup","declaration":"instance SM.instGroupPermGroup {n : ℕ} : Group (SM.PermGroup n)"} +{"name":"SM.accSU3_invariant","declaration":"theorem SM.accSU3_invariant {n : ℕ} (f : SM.PermGroup n) (S : ACCSystemCharges.Charges (SMCharges n)) : SMACCs.accSU3 ((SM.repCharges f) S) = SMACCs.accSU3 S"} +{"name":"SM.repCharges","declaration":"/-- The representation of `(permGroup n)` acting on the vector space of charges. -/\ndef SM.repCharges {n : ℕ} : Representation ℚ (SM.PermGroup n) (ACCSystemCharges.Charges (SMCharges n))"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..219deebc8f97209e86888865a2ac850f6b9a5fc9 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Basic.jsonl @@ -0,0 +1,42 @@ +{"name":"SMνCharges.toSpecies","declaration":"/-- Given an `i ∈ Fin 6`, the projection of charges onto a given species. -/\ndef SMνCharges.toSpecies {n : ℕ} (i : Fin 6) : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMνACCs.quadBiLin","declaration":"/-- The quadratic bilinear map. -/\ndef SMνACCs.quadBiLin {n : ℕ} : BiLinearSymm (ACCSystemCharges.Charges (SMνCharges n))"} +{"name":"SMνCharges.toSpeciesEquiv_apply","declaration":"theorem SMνCharges.toSpeciesEquiv_apply {n : ℕ} : ∀ (a : Fin (6 * n) → ℚ) (a_1 : Fin 6) (a_2 : Fin n),\n SMνCharges.toSpeciesEquiv a a_1 a_2 = a (finProdFinEquiv (a_1, a_2))"} +{"name":"SMνACCs.accCube_decomp","declaration":"theorem SMνACCs.accCube_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accCube S =\n (((((6 * Finset.sum Finset.univ fun i => SMνCharges.Q S i ^ 3) +\n 3 * Finset.sum Finset.univ fun i => SMνCharges.U S i ^ 3) +\n 3 * Finset.sum Finset.univ fun i => SMνCharges.D S i ^ 3) +\n 2 * Finset.sum Finset.univ fun i => SMνCharges.L S i ^ 3) +\n Finset.sum Finset.univ fun i => SMνCharges.E S i ^ 3) +\n Finset.sum Finset.univ fun i => SMνCharges.N S i ^ 3"} +{"name":"SMνACCs.quadBiLin_decomp","declaration":"theorem SMνACCs.quadBiLin_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) (T : ACCSystemCharges.Charges (SMνCharges n)) : (SMνACCs.quadBiLin S) T =\n ((((Finset.sum Finset.univ fun i => SMνCharges.Q S i * SMνCharges.Q T i) -\n 2 * Finset.sum Finset.univ fun i => SMνCharges.U S i * SMνCharges.U T i) +\n Finset.sum Finset.univ fun i => SMνCharges.D S i * SMνCharges.D T i) -\n Finset.sum Finset.univ fun i => SMνCharges.L S i * SMνCharges.L T i) +\n Finset.sum Finset.univ fun i => SMνCharges.E S i * SMνCharges.E T i"} +{"name":"SMνACCs.accSU3_decomp","declaration":"theorem SMνACCs.accSU3_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accSU3 S =\n ((2 * Finset.sum Finset.univ fun i => SMνCharges.Q S i) + Finset.sum Finset.univ fun i => SMνCharges.U S i) +\n Finset.sum Finset.univ fun i => SMνCharges.D S i"} +{"name":"SMνSpecies","declaration":"/-- The vector spaces of charges of one species of fermions in the SM. -/\ndef SMνSpecies (n : ℕ) : ACCSystemCharges"} +{"name":"SMνCharges.N","declaration":"/-- The `N` charges as a map `Fin n → ℚ`. -/\ndef SMνCharges.N {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMνCharges.toSMSpecies_toSpecies_inv","declaration":"theorem SMνCharges.toSMSpecies_toSpecies_inv {n : ℕ} (i : Fin 6) (f : Fin 6 → Fin n → ℚ) : (SMνCharges.toSpecies i) (SMνCharges.toSpeciesEquiv.symm f) = f i"} +{"name":"SMνACCs.accSU3_ext","declaration":"/-- Extensionality lemma for `accSU3`. -/\ntheorem SMνACCs.accSU3_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMνCharges n)} {T : ACCSystemCharges.Charges (SMνCharges n)} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) T i) : SMνACCs.accSU3 S = SMνACCs.accSU3 T"} +{"name":"SMνCharges.toSpecies_one","declaration":"theorem SMνCharges.toSpecies_one (S : ACCSystemCharges.Charges (SMνCharges 1)) (j : Fin 6) : (SMνCharges.toSpecies j) S { val := 0, isLt := ⋯ } = S j"} +{"name":"SMνACCs.cubeTriLin_decomp","declaration":"theorem SMνACCs.cubeTriLin_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) (T : ACCSystemCharges.Charges (SMνCharges n)) (R : ACCSystemCharges.Charges (SMνCharges n)) : ((SMνACCs.cubeTriLin S) T) R =\n (((((6 * Finset.sum Finset.univ fun i => SMνCharges.Q S i * SMνCharges.Q T i * SMνCharges.Q R i) +\n 3 * Finset.sum Finset.univ fun i => SMνCharges.U S i * SMνCharges.U T i * SMνCharges.U R i) +\n 3 * Finset.sum Finset.univ fun i => SMνCharges.D S i * SMνCharges.D T i * SMνCharges.D R i) +\n 2 * Finset.sum Finset.univ fun i => SMνCharges.L S i * SMνCharges.L T i * SMνCharges.L R i) +\n Finset.sum Finset.univ fun i => SMνCharges.E S i * SMνCharges.E T i * SMνCharges.E R i) +\n Finset.sum Finset.univ fun i => SMνCharges.N S i * SMνCharges.N T i * SMνCharges.N R i"} +{"name":"SMνCharges.U","declaration":"/-- The `U` charges as a map `Fin n → ℚ`. -/\ndef SMνCharges.U {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMνACCs.accCube_ext","declaration":"/-- Extensionality lemma for `accCube`. -/\ntheorem SMνACCs.accCube_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMνCharges n)} {T : ACCSystemCharges.Charges (SMνCharges n)} (h : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => ((fun a => a ^ 3) ∘ (SMνCharges.toSpecies j) S) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ 3) ∘ (SMνCharges.toSpecies j) T) i) : SMνACCs.accCube S = SMνACCs.accCube T"} +{"name":"SMνCharges","declaration":"/-- The vector space of charges corresponding to the SM fermions with RHN. -/\ndef SMνCharges (n : ℕ) : ACCSystemCharges"} +{"name":"SMνACCs.accQuad_decomp","declaration":"theorem SMνACCs.accQuad_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accQuad S =\n ((((Finset.sum Finset.univ fun i => SMνCharges.Q S i ^ 2) -\n 2 * Finset.sum Finset.univ fun i => SMνCharges.U S i ^ 2) +\n Finset.sum Finset.univ fun i => SMνCharges.D S i ^ 2) -\n Finset.sum Finset.univ fun i => SMνCharges.L S i ^ 2) +\n Finset.sum Finset.univ fun i => SMνCharges.E S i ^ 2"} +{"name":"SMνACCs.quadBiLin_toFun_apply","declaration":"theorem SMνACCs.quadBiLin_toFun_apply {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) (T : ACCSystemCharges.Charges (SMνCharges n)) : (SMνACCs.quadBiLin S) T =\n Finset.sum Finset.univ fun x =>\n S (finProdFinEquiv (0, x)) * T (finProdFinEquiv (0, x)) +\n -(2 * (S (finProdFinEquiv (1, x)) * T (finProdFinEquiv (1, x)))) +\n S (finProdFinEquiv (2, x)) * T (finProdFinEquiv (2, x)) +\n -(S (finProdFinEquiv (3, x)) * T (finProdFinEquiv (3, x))) +\n S (finProdFinEquiv (4, x)) * T (finProdFinEquiv (4, x))"} +{"name":"SMνACCs.accGrav","declaration":"/-- The gravitational anomaly equation. -/\ndef SMνACCs.accGrav {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMνACCs.accQuad","declaration":"/-- The quadratic anomaly cancellation condition. -/\ndef SMνACCs.accQuad {n : ℕ} : HomogeneousQuadratic (ACCSystemCharges.Charges (SMνCharges n))"} +{"name":"SMνCharges.toSpeciesEquiv","declaration":"/-- An equivalence between `(SMνCharges n).charges` and `(Fin 6 → Fin n → ℚ)`\nsplitting the charges into species. -/\ndef SMνCharges.toSpeciesEquiv {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) ≃ (Fin 6 → Fin n → ℚ)"} +{"name":"SMνACCs.accQuad_ext","declaration":"/-- Extensionality lemma for `accQuad`. -/\ntheorem SMνACCs.accQuad_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMνCharges n)} {T : ACCSystemCharges.Charges (SMνCharges n)} (h : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => ((fun a => a ^ 2) ∘ (SMνCharges.toSpecies j) S) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ 2) ∘ (SMνCharges.toSpecies j) T) i) : SMνACCs.accQuad S = SMνACCs.accQuad T"} +{"name":"SMνACCs.accGrav_ext","declaration":"/-- Extensionality lemma for `accGrav`. -/\ntheorem SMνACCs.accGrav_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMνCharges n)} {T : ACCSystemCharges.Charges (SMνCharges n)} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) T i) : SMνACCs.accGrav S = SMνACCs.accGrav T"} +{"name":"SMνACCs.accSU3","declaration":"/-- The `SU(3)` anomaly equations. -/\ndef SMνACCs.accSU3 {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMνACCs.accSU2","declaration":"/-- The `SU(2)` anomaly equation. -/\ndef SMνACCs.accSU2 {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMνCharges.charges_eq_toSpecies_eq","declaration":"theorem SMνCharges.charges_eq_toSpecies_eq {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) (T : ACCSystemCharges.Charges (SMνCharges n)) : S = T ↔ ∀ (i : Fin 6), (SMνCharges.toSpecies i) S = (SMνCharges.toSpecies i) T"} +{"name":"SMνCharges.E","declaration":"/-- The `E` charges as a map `Fin n → ℚ`. -/\ndef SMνCharges.E {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMνACCs.accCube","declaration":"/-- The cubic ACC. -/\ndef SMνACCs.accCube {n : ℕ} : HomogeneousCubic (ACCSystemCharges.Charges (SMνCharges n))"} +{"name":"SMνCharges_numberCharges","declaration":"theorem SMνCharges_numberCharges (n : ℕ) : (SMνCharges n).numberCharges = 6 * n"} +{"name":"SMνCharges.Q","declaration":"/-- The `Q` charges as a map `Fin n → ℚ`. -/\ndef SMνCharges.Q {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMνCharges.D","declaration":"/-- The `D` charges as a map `Fin n → ℚ`. -/\ndef SMνCharges.D {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMνACCs.accYY","declaration":"/-- The `Y²` anomaly equation. -/\ndef SMνACCs.accYY {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ℚ"} +{"name":"SMνCharges.toSpeciesEquiv_symm_apply","declaration":"theorem SMνCharges.toSpeciesEquiv_symm_apply {n : ℕ} : ∀ (a : Fin 6 → Fin n → ℚ) (a_1 : Fin (6 * n)),\n SMνCharges.toSpeciesEquiv.symm a a_1 = a (Fin.divNat a_1) (Fin.modNat a_1)"} +{"name":"SMνACCs.accSU2_decomp","declaration":"theorem SMνACCs.accSU2_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accSU2 S =\n (3 * Finset.sum Finset.univ fun i => SMνCharges.Q S i) + Finset.sum Finset.univ fun i => SMνCharges.L S i"} +{"name":"SMνACCs.cubeTriLin_toFun_apply_apply","declaration":"theorem SMνACCs.cubeTriLin_toFun_apply_apply {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) (S : ACCSystemCharges.Charges (SMνCharges n)) (T : ACCSystemCharges.Charges (SMνCharges n)) : ((SMνACCs.cubeTriLin S✝) S) T =\n Finset.sum Finset.univ fun i =>\n 6 * (S✝ (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i)) * T (finProdFinEquiv (0, i))) +\n 3 * (S✝ (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i)) * T (finProdFinEquiv (1, i))) +\n 3 * (S✝ (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i)) * T (finProdFinEquiv (2, i))) +\n 2 * (S✝ (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i)) * T (finProdFinEquiv (3, i))) +\n S✝ (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) * T (finProdFinEquiv (4, i)) +\n S✝ (finProdFinEquiv (5, i)) * S (finProdFinEquiv (5, i)) * T (finProdFinEquiv (5, i))"} +{"name":"SMνACCs.accYY_ext","declaration":"/-- Extensionality lemma for `accYY`. -/\ntheorem SMνACCs.accYY_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMνCharges n)} {T : ACCSystemCharges.Charges (SMνCharges n)} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) T i) : SMνACCs.accYY S = SMνACCs.accYY T"} +{"name":"SMνSpecies_numberCharges","declaration":"theorem SMνSpecies_numberCharges (n : ℕ) : (SMνSpecies n).numberCharges = n"} +{"name":"SMνACCs.cubeTriLin","declaration":"/-- The symmetric trilinear form used to define the cubic acc. -/\ndef SMνACCs.cubeTriLin {n : ℕ} : TriLinearSymm (ACCSystemCharges.Charges (SMνCharges n))"} +{"name":"SMνACCs.accGrav_decomp","declaration":"theorem SMνACCs.accGrav_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accGrav S =\n (((((6 * Finset.sum Finset.univ fun i => SMνCharges.Q S i) + 3 * Finset.sum Finset.univ fun i => SMνCharges.U S i) +\n 3 * Finset.sum Finset.univ fun i => SMνCharges.D S i) +\n 2 * Finset.sum Finset.univ fun i => SMνCharges.L S i) +\n Finset.sum Finset.univ fun i => SMνCharges.E S i) +\n Finset.sum Finset.univ fun i => SMνCharges.N S i"} +{"name":"SMνACCs.accSU2_ext","declaration":"/-- Extensionality lemma for `accSU2`. -/\ntheorem SMνACCs.accSU2_ext {n : ℕ} {S : ACCSystemCharges.Charges (SMνCharges n)} {T : ACCSystemCharges.Charges (SMνCharges n)} (hj : ∀ (j : Fin 6),\n (Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) S i) =\n Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) T i) : SMνACCs.accSU2 S = SMνACCs.accSU2 T"} +{"name":"SMνCharges.toSpecies_apply","declaration":"theorem SMνCharges.toSpecies_apply {n : ℕ} (i : Fin 6) (S : ACCSystemCharges.Charges (SMνCharges n)) : ∀ (a : Fin (SMνSpecies n).numberCharges), (SMνCharges.toSpecies i) S a = S (finProdFinEquiv (i, a))"} +{"name":"SMνCharges.L","declaration":"/-- The `L` charges as a map `Fin n → ℚ`. -/\ndef SMνCharges.L {n : ℕ} : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMνACCs.accYY_decomp","declaration":"theorem SMνACCs.accYY_decomp {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accYY S =\n ((((Finset.sum Finset.univ fun i => SMνCharges.Q S i) + 8 * Finset.sum Finset.univ fun i => SMνCharges.U S i) +\n 2 * Finset.sum Finset.univ fun i => SMνCharges.D S i) +\n 3 * Finset.sum Finset.univ fun i => SMνCharges.L S i) +\n 6 * Finset.sum Finset.univ fun i => SMνCharges.E S i"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.FamilyMaps.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.FamilyMaps.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6d8dac134b6bcf2da1df4dde881793283a5a1ded --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.FamilyMaps.jsonl @@ -0,0 +1,26 @@ +{"name":"SMRHN.sum_familyUniversal_one","declaration":"theorem SMRHN.sum_familyUniversal_one {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) (j : Fin 6) : (Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) ((SMRHN.familyUniversal n) S) i) =\n ↑n * (SMνCharges.toSpecies j) S { val := 0, isLt := ⋯ }"} +{"name":"SMRHN.toSpecies_familyUniversal","declaration":"theorem SMRHN.toSpecies_familyUniversal {n : ℕ} (j : Fin 6) (S : ACCSystemCharges.Charges (SMνCharges 1)) (i : Fin n) : (SMνCharges.toSpecies j) ((SMRHN.familyUniversal n) S) i = (SMνCharges.toSpecies j) S { val := 0, isLt := ⋯ }"} +{"name":"SMRHN.sum_familyUniversal_three","declaration":"theorem SMRHN.sum_familyUniversal_three {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) (T : ACCSystemCharges.Charges (SMνCharges n)) (L : ACCSystemCharges.Charges (SMνCharges n)) (j : Fin 6) : (Finset.sum Finset.univ fun i =>\n (SMνCharges.toSpecies j) ((SMRHN.familyUniversal n) S) i * (SMνCharges.toSpecies j) T i *\n (SMνCharges.toSpecies j) L i) =\n (SMνCharges.toSpecies j) S { val := 0, isLt := ⋯ } *\n Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) T i * (SMνCharges.toSpecies j) L i"} +{"name":"SMRHN.speciesEmbed","declaration":"/-- For species, the embedding of the `m`-family charges onto the `n`-family charges, with all\nother charges zero. -/\ndef SMRHN.speciesEmbed (m : ℕ) (n : ℕ) : ACCSystemCharges.Charges (SMνSpecies m) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMRHN.familyUniversal_quadBiLin","declaration":"theorem SMRHN.familyUniversal_quadBiLin {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) (T : ACCSystemCharges.Charges (SMνCharges n)) : (SMνACCs.quadBiLin ((SMRHN.familyUniversal n) S)) T =\n ((((S 0 * Finset.sum Finset.univ fun i => SMνCharges.Q T i) -\n 2 * S 1 * Finset.sum Finset.univ fun i => SMνCharges.U T i) +\n S 2 * Finset.sum Finset.univ fun i => SMνCharges.D T i) -\n S 3 * Finset.sum Finset.univ fun i => SMνCharges.L T i) +\n S 4 * Finset.sum Finset.univ fun i => SMνCharges.E T i"} +{"name":"SMRHN.familyUniversal_accSU2","declaration":"theorem SMRHN.familyUniversal_accSU2 {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) : SMνACCs.accSU2 ((SMRHN.familyUniversal n) S) = ↑n * SMνACCs.accSU2 S"} +{"name":"SMRHN.familyProjection","declaration":"/-- The projection of the `m`-family charges onto the first `n`-family charges. -/\ndef SMRHN.familyProjection {m : ℕ} {n : ℕ} (h : n ≤ m) : ACCSystemCharges.Charges (SMνCharges m) →ₗ[ℚ] ACCSystemCharges.Charges (SMνCharges n)"} +{"name":"SMRHN.familyEmbedding","declaration":"/-- The embedding of the `m`-family charges onto the `n`-family charges, with all\nother charges zero. -/\ndef SMRHN.familyEmbedding (m : ℕ) (n : ℕ) : ACCSystemCharges.Charges (SMνCharges m) →ₗ[ℚ] ACCSystemCharges.Charges (SMνCharges n)"} +{"name":"SMRHN.speciesFamilyProj_apply","declaration":"theorem SMRHN.speciesFamilyProj_apply {m : ℕ} {n : ℕ} (h : n ≤ m) (S : ACCSystemCharges.Charges (SMνSpecies m)) : ∀ (a : Fin (SMνSpecies n).numberCharges), (SMRHN.speciesFamilyProj h) S a = S (Fin.castLE h a)"} +{"name":"SMRHN.chargesMapOfSpeciesMap_toSpecies","declaration":"theorem SMRHN.chargesMapOfSpeciesMap_toSpecies {n : ℕ} {m : ℕ} (f : ACCSystemCharges.Charges (SMνSpecies n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies m)) (S : ACCSystemCharges.Charges (SMνCharges n)) (j : Fin 6) : (SMνCharges.toSpecies j) ((SMRHN.chargesMapOfSpeciesMap f) S) = (f ∘ₗ SMνCharges.toSpecies j) S"} +{"name":"SMRHN.speciesFamilyProj","declaration":"/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/\ndef SMRHN.speciesFamilyProj {m : ℕ} {n : ℕ} (h : n ≤ m) : ACCSystemCharges.Charges (SMνSpecies m) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMRHN.familyUniversal","declaration":"/-- The embedding of the `1`-family charges into the `n`-family charges in\na universal manor. -/\ndef SMRHN.familyUniversal (n : ℕ) : ACCSystemCharges.Charges (SMνCharges 1) →ₗ[ℚ] ACCSystemCharges.Charges (SMνCharges n)"} +{"name":"SMRHN.chargesMapOfSpeciesMap","declaration":"/-- Given a map of for a generic species, the corresponding map for charges. -/\ndef SMRHN.chargesMapOfSpeciesMap {n : ℕ} {m : ℕ} (f : ACCSystemCharges.Charges (SMνSpecies n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies m)) : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνCharges m)"} +{"name":"SMRHN.speciesEmbed_apply","declaration":"theorem SMRHN.speciesEmbed_apply (m : ℕ) (n : ℕ) (S : ACCSystemCharges.Charges (SMνSpecies m)) (i : Fin (SMνSpecies n).numberCharges) : (SMRHN.speciesEmbed m n) S i = if hi : ↑i < m then S { val := ↑i, isLt := hi } else 0"} +{"name":"SMRHN.familyUniversal_accCube","declaration":"theorem SMRHN.familyUniversal_accCube {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) : SMνACCs.accCube ((SMRHN.familyUniversal n) S) = ↑n * SMνACCs.accCube S"} +{"name":"SMRHN.familyUniversal_accGrav","declaration":"theorem SMRHN.familyUniversal_accGrav {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) : SMνACCs.accGrav ((SMRHN.familyUniversal n) S) = ↑n * SMνACCs.accGrav S"} +{"name":"SMRHN.chargesMapOfSpeciesMap_apply","declaration":"theorem SMRHN.chargesMapOfSpeciesMap_apply {n : ℕ} {m : ℕ} (f : ACCSystemCharges.Charges (SMνSpecies n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies m)) (S : ACCSystemCharges.Charges (SMνCharges n)) : (SMRHN.chargesMapOfSpeciesMap f) S = SMνCharges.toSpeciesEquiv.symm fun i => f ((SMνCharges.toSpecies i) S)"} +{"name":"SMRHN.speciesFamilyUniversial","declaration":"/-- For species, the embedding of the `1`-family charges into the `n`-family charges in\na universal manor. -/\ndef SMRHN.speciesFamilyUniversial (n : ℕ) : ACCSystemCharges.Charges (SMνSpecies 1) →ₗ[ℚ] ACCSystemCharges.Charges (SMνSpecies n)"} +{"name":"SMRHN.speciesFamilyUniversial_apply","declaration":"theorem SMRHN.speciesFamilyUniversial_apply (n : ℕ) (S : ACCSystemCharges.Charges (SMνSpecies 1)) : ∀ (x : Fin (SMνSpecies n).numberCharges), (SMRHN.speciesFamilyUniversial n) S x = S 0"} +{"name":"SMRHN.sum_familyUniversal","declaration":"theorem SMRHN.sum_familyUniversal {n : ℕ} (m : ℕ) (S : ACCSystemCharges.Charges (SMνCharges 1)) (j : Fin 6) : (Finset.sum Finset.univ fun i => ((fun a => a ^ m) ∘ (SMνCharges.toSpecies j) ((SMRHN.familyUniversal n) S)) i) =\n ↑n * (SMνCharges.toSpecies j) S { val := 0, isLt := ⋯ } ^ m"} +{"name":"SMRHN.sum_familyUniversal_two","declaration":"theorem SMRHN.sum_familyUniversal_two {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) (T : ACCSystemCharges.Charges (SMνCharges n)) (j : Fin 6) : (Finset.sum Finset.univ fun i =>\n (SMνCharges.toSpecies j) ((SMRHN.familyUniversal n) S) i * (SMνCharges.toSpecies j) T i) =\n (SMνCharges.toSpecies j) S { val := 0, isLt := ⋯ } * Finset.sum Finset.univ fun i => (SMνCharges.toSpecies j) T i"} +{"name":"SMRHN.familyUniversal_accQuad","declaration":"theorem SMRHN.familyUniversal_accQuad {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) : SMνACCs.accQuad ((SMRHN.familyUniversal n) S) = ↑n * SMνACCs.accQuad S"} +{"name":"SMRHN.familyUniversal_accSU3","declaration":"theorem SMRHN.familyUniversal_accSU3 {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) : SMνACCs.accSU3 ((SMRHN.familyUniversal n) S) = ↑n * SMνACCs.accSU3 S"} +{"name":"SMRHN.familyUniversal_cubeTriLin'","declaration":"theorem SMRHN.familyUniversal_cubeTriLin' {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) (T : ACCSystemCharges.Charges (SMνCharges 1)) (R : ACCSystemCharges.Charges (SMνCharges n)) : ((SMνACCs.cubeTriLin ((SMRHN.familyUniversal n) S)) ((SMRHN.familyUniversal n) T)) R =\n (((((6 * S 0 * T 0 * Finset.sum Finset.univ fun i => SMνCharges.Q R i) +\n 3 * S 1 * T 1 * Finset.sum Finset.univ fun i => SMνCharges.U R i) +\n 3 * S 2 * T 2 * Finset.sum Finset.univ fun i => SMνCharges.D R i) +\n 2 * S 3 * T 3 * Finset.sum Finset.univ fun i => SMνCharges.L R i) +\n S 4 * T 4 * Finset.sum Finset.univ fun i => SMνCharges.E R i) +\n S 5 * T 5 * Finset.sum Finset.univ fun i => SMνCharges.N R i"} +{"name":"SMRHN.familyUniversal_accYY","declaration":"theorem SMRHN.familyUniversal_accYY {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) : SMνACCs.accYY ((SMRHN.familyUniversal n) S) = ↑n * SMνACCs.accYY S"} +{"name":"SMRHN.familyUniversal_cubeTriLin","declaration":"theorem SMRHN.familyUniversal_cubeTriLin {n : ℕ} (S : ACCSystemCharges.Charges (SMνCharges 1)) (T : ACCSystemCharges.Charges (SMνCharges n)) (R : ACCSystemCharges.Charges (SMνCharges n)) : ((SMνACCs.cubeTriLin ((SMRHN.familyUniversal n) S)) T) R =\n (((((6 * S 0 * Finset.sum Finset.univ fun i => SMνCharges.Q T i * SMνCharges.Q R i) +\n 3 * S 1 * Finset.sum Finset.univ fun i => SMνCharges.U T i * SMνCharges.U R i) +\n 3 * S 2 * Finset.sum Finset.univ fun i => SMνCharges.D T i * SMνCharges.D R i) +\n 2 * S 3 * Finset.sum Finset.univ fun i => SMνCharges.L T i * SMνCharges.L R i) +\n S 4 * Finset.sum Finset.univ fun i => SMνCharges.E T i * SMνCharges.E R i) +\n S 5 * Finset.sum Finset.univ fun i => SMνCharges.N T i * SMνCharges.N R i"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.NoGrav.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.NoGrav.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7c764bef04205a7cefc054735480d38fccc78d30 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.NoGrav.Basic.jsonl @@ -0,0 +1,17 @@ +{"name":"SMRHN.SMNoGrav_cubicACC_toFun","declaration":"theorem SMRHN.SMNoGrav_cubicACC_toFun (n : ℕ) (S : ACCSystemCharges.Charges (SMνCharges n)) : (SMRHN.SMNoGrav n).cubicACC.toFun S =\n Finset.sum Finset.univ fun i =>\n 6 * (S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i))) +\n 3 * (S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i))) +\n 3 * (S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i))) +\n 2 * (S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i))) +\n S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) +\n S (finProdFinEquiv (5, i)) * S (finProdFinEquiv (5, i)) * S (finProdFinEquiv (5, i))"} +{"name":"SMRHN.SMNoGrav_quadraticACCs","declaration":"theorem SMRHN.SMNoGrav_quadraticACCs (n : ℕ) (i : Fin 0) : (SMRHN.SMNoGrav n).quadraticACCs i = Fin.elim0 i"} +{"name":"SMRHN.SMNoGrav_linearACCs","declaration":"theorem SMRHN.SMNoGrav_linearACCs (n : ℕ) (i : Fin 2) : (SMRHN.SMNoGrav n).linearACCs i =\n match i with\n | 0 =>\n {\n toAddHom :=\n { toFun := fun S => Finset.sum Finset.univ fun i => 3 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (3, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }\n | 1 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n 2 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (1, i)) + S (finProdFinEquiv (2, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }"} +{"name":"SMRHN.SMNoGrav","declaration":"/-- The ACC system for the SM plus RHN with no gravitational anomaly. -/\ndef SMRHN.SMNoGrav (n : ℕ) : ACCSystem"} +{"name":"SMRHN.SMNoGrav.chargeToLinear","declaration":"/-- An element of `charges` which satisfies the linear ACCs\ngives us a element of `LinSols`. -/\ndef SMRHN.SMNoGrav.chargeToLinear {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.SMNoGrav n).toACCSystemCharges) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) : ACCSystemLinear.LinSols (SMRHN.SMNoGrav n).toACCSystemLinear"} +{"name":"SMRHN.SMNoGrav.linearToAF","declaration":"/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.SMNoGrav.linearToAF {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SMNoGrav n).toACCSystemLinear) (hc : SMνACCs.accCube S.val = 0) : ACCSystem.Sols (SMRHN.SMNoGrav n)"} +{"name":"SMRHN.SMNoGrav_numberQuadratic","declaration":"theorem SMRHN.SMNoGrav_numberQuadratic (n : ℕ) : (SMRHN.SMNoGrav n).numberQuadratic = 0"} +{"name":"SMRHN.SMNoGrav.SU2Sol","declaration":"theorem SMRHN.SMNoGrav.SU2Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SMNoGrav n).toACCSystemLinear) : SMνACCs.accSU2 S.val = 0"} +{"name":"SMRHN.SMNoGrav.quadToAF","declaration":"/-- An element of `QuadSols` which satisfies the quadratic ACCs\ngives us a element of `LinSols`. -/\ndef SMRHN.SMNoGrav.quadToAF {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.SMNoGrav n).toACCSystemQuad) (hc : SMνACCs.accCube S.val = 0) : ACCSystem.Sols (SMRHN.SMNoGrav n)"} +{"name":"SMRHN.SMNoGrav.chargeToQuad","declaration":"/-- An element of `charges` which satisfies the linear and quadratic ACCs\ngives us a element of `QuadSols`. -/\ndef SMRHN.SMNoGrav.chargeToQuad {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.SMNoGrav n).toACCSystemCharges) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) : ACCSystemQuad.QuadSols (SMRHN.SMNoGrav n).toACCSystemQuad"} +{"name":"SMRHN.SMNoGrav_numberCharges","declaration":"theorem SMRHN.SMNoGrav_numberCharges (n : ℕ) : (SMRHN.SMNoGrav n).numberCharges = 6 * n"} +{"name":"SMRHN.SMNoGrav.linearToQuad","declaration":"/-- An element of `LinSols` which satisfies the quadratic ACCs\ngives us a element of `QuadSols`. -/\ndef SMRHN.SMNoGrav.linearToQuad {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SMNoGrav n).toACCSystemLinear) : ACCSystemQuad.QuadSols (SMRHN.SMNoGrav n).toACCSystemQuad"} +{"name":"SMRHN.SMNoGrav.SU3Sol","declaration":"theorem SMRHN.SMNoGrav.SU3Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SMNoGrav n).toACCSystemLinear) : SMνACCs.accSU3 S.val = 0"} +{"name":"SMRHN.SMNoGrav.chargeToAF","declaration":"/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.SMNoGrav.chargeToAF {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.SMNoGrav n).toACCSystemCharges) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) (hc : SMνACCs.accCube S = 0) : ACCSystem.Sols (SMRHN.SMNoGrav n)"} +{"name":"SMRHN.SMNoGrav.perm","declaration":"/-- The permutations acting on the ACC system corresponding to the SM with RHN,\nand no gravitational anomaly. -/\ndef SMRHN.SMNoGrav.perm (n : ℕ) : ACCSystemGroupAction (SMRHN.SMNoGrav n)"} +{"name":"SMRHN.SMNoGrav.cubeSol","declaration":"theorem SMRHN.SMNoGrav.cubeSol {n : ℕ} (S : ACCSystem.Sols (SMRHN.SMNoGrav n)) : SMνACCs.accCube S.val = 0"} +{"name":"SMRHN.SMNoGrav_numberLinear","declaration":"theorem SMRHN.SMNoGrav_numberLinear (n : ℕ) : (SMRHN.SMNoGrav n).numberLinear = 2"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b702d99430cbcd5963236563332f92ccb3d41525 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.Basic.jsonl @@ -0,0 +1,18 @@ +{"name":"SMRHN.SM.quadToAF","declaration":"/-- An element of `QuadSols` which satisfies the quadratic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.SM.quadToAF {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.SM n).toACCSystemQuad) (hc : SMνACCs.accCube S.val = 0) : ACCSystem.Sols (SMRHN.SM n)"} +{"name":"SMRHN.SM_numberCharges","declaration":"theorem SMRHN.SM_numberCharges (n : ℕ) : (SMRHN.SM n).numberCharges = 6 * n"} +{"name":"SMRHN.SM.linearToQuad","declaration":"/-- An element of `LinSols` which satisfies the quadratic ACCs\ngives us a element of `QuadSols`. -/\ndef SMRHN.SM.linearToQuad {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SM n).toACCSystemLinear) : ACCSystemQuad.QuadSols (SMRHN.SM n).toACCSystemQuad"} +{"name":"SMRHN.SM_numberQuadratic","declaration":"theorem SMRHN.SM_numberQuadratic (n : ℕ) : (SMRHN.SM n).numberQuadratic = 0"} +{"name":"SMRHN.SM_linearACCs","declaration":"theorem SMRHN.SM_linearACCs (n : ℕ) (i : Fin 3) : (SMRHN.SM n).linearACCs i =\n match i with\n | 0 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n 6 * S (finProdFinEquiv (0, i)) + 3 * S (finProdFinEquiv (1, i)) + 3 * S (finProdFinEquiv (2, i)) +\n 2 * S (finProdFinEquiv (3, i)) +\n S (finProdFinEquiv (4, i)) +\n S (finProdFinEquiv (5, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }\n | 1 =>\n {\n toAddHom :=\n { toFun := fun S => Finset.sum Finset.univ fun i => 3 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (3, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }\n | 2 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n 2 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (1, i)) + S (finProdFinEquiv (2, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }"} +{"name":"SMRHN.SM.cubeSol","declaration":"theorem SMRHN.SM.cubeSol {n : ℕ} (S : ACCSystem.Sols (SMRHN.SM n)) : SMνACCs.accCube S.val = 0"} +{"name":"SMRHN.SM_numberLinear","declaration":"theorem SMRHN.SM_numberLinear (n : ℕ) : (SMRHN.SM n).numberLinear = 3"} +{"name":"SMRHN.SM.gravSol","declaration":"theorem SMRHN.SM.gravSol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SM n).toACCSystemLinear) : SMνACCs.accGrav S.val = 0"} +{"name":"SMRHN.SM.SU3Sol","declaration":"theorem SMRHN.SM.SU3Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SM n).toACCSystemLinear) : SMνACCs.accSU3 S.val = 0"} +{"name":"SMRHN.SM.perm","declaration":"/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/\ndef SMRHN.SM.perm (n : ℕ) : ACCSystemGroupAction (SMRHN.SM n)"} +{"name":"SMRHN.SM.chargeToLinear","declaration":"/-- An element of `charges` which satisfies the linear ACCs\ngives us a element of `LinSols`. -/\ndef SMRHN.SM.chargeToLinear {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.SM n).toACCSystemCharges) (hGrav : SMνACCs.accGrav S = 0) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) : ACCSystemLinear.LinSols (SMRHN.SM n).toACCSystemLinear"} +{"name":"SMRHN.SM_quadraticACCs","declaration":"theorem SMRHN.SM_quadraticACCs (n : ℕ) (i : Fin 0) : (SMRHN.SM n).quadraticACCs i = Fin.elim0 i"} +{"name":"SMRHN.SM.SU2Sol","declaration":"theorem SMRHN.SM.SU2Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SM n).toACCSystemLinear) : SMνACCs.accSU2 S.val = 0"} +{"name":"SMRHN.SM.chargeToAF","declaration":"/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.SM.chargeToAF {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.SM n).toACCSystemCharges) (hGrav : SMνACCs.accGrav S = 0) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) (hc : SMνACCs.accCube S = 0) : ACCSystem.Sols (SMRHN.SM n)"} +{"name":"SMRHN.SM_cubicACC_toFun","declaration":"theorem SMRHN.SM_cubicACC_toFun (n : ℕ) (S : ACCSystemCharges.Charges (SMνCharges n)) : (SMRHN.SM n).cubicACC.toFun S =\n Finset.sum Finset.univ fun i =>\n 6 * (S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i))) +\n 3 * (S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i))) +\n 3 * (S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i))) +\n 2 * (S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i))) +\n S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) +\n S (finProdFinEquiv (5, i)) * S (finProdFinEquiv (5, i)) * S (finProdFinEquiv (5, i))"} +{"name":"SMRHN.SM.linearToAF","declaration":"/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.SM.linearToAF {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.SM n).toACCSystemLinear) (hc : SMνACCs.accCube S.val = 0) : ACCSystem.Sols (SMRHN.SM n)"} +{"name":"SMRHN.SM.chargeToQuad","declaration":"/-- An element of `charges` which satisfies the linear and quadratic ACCs\ngives us a element of `QuadSols`. -/\ndef SMRHN.SM.chargeToQuad {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.SM n).toACCSystemCharges) (hGrav : SMνACCs.accGrav S = 0) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) : ACCSystemQuad.QuadSols (SMRHN.SM n).toACCSystemQuad"} +{"name":"SMRHN.SM","declaration":"/-- The ACC system for the SM plus RHN. -/\ndef SMRHN.SM (n : ℕ) : ACCSystem"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.DimSevenPlane.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.DimSevenPlane.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..8e570e173e04ed7ada1f813197406954969128e2 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.DimSevenPlane.jsonl @@ -0,0 +1,36 @@ +{"name":"SMRHN.SM.PlaneSeven.Bi_Bj_ne_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.Bi_Bj_ne_cubic {i : Fin 7} {j : Fin 7} (h : i ≠ j) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B i)) (SMRHN.SM.PlaneSeven.B j)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₁_B₁_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₁_B₁_Bi_cubic {i : Fin 7} : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 1)) (SMRHN.SM.PlaneSeven.B 1)) (SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.basis_linear_independent","declaration":"theorem SMRHN.SM.PlaneSeven.basis_linear_independent : LinearIndependent ℚ SMRHN.SM.PlaneSeven.B"} +{"name":"SMRHN.SM.PlaneSeven.B₆_B₆_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₆_B₆_Bi_cubic {i : Fin 7} : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 6)) (SMRHN.SM.PlaneSeven.B 6)) (SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₄_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 4)) (SMRHN.SM.PlaneSeven.B i)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₅_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 5)) (SMRHN.SM.PlaneSeven.B i)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.Bi_Bi_Bj_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.Bi_Bi_Bj_cubic (i : Fin 7) (j : Fin 7) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B i)) (SMRHN.SM.PlaneSeven.B i)) (SMRHN.SM.PlaneSeven.B j) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₃_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₃_cubic (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) (T : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin SMRHN.SM.PlaneSeven.B₃) S) T = 2 * (S 9 * T 9 - S 10 * T 10)"} +{"name":"SMRHN.SM.PlaneSeven.B₁","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.SM.PlaneSeven.B₁ : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} +{"name":"SMRHN.SM.PlaneSeven.B₂_B₂_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₂_B₂_Bi_cubic {i : Fin 7} : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 2)) (SMRHN.SM.PlaneSeven.B 2)) (SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.Bi_Bj_Bk_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.Bi_Bj_Bk_cubic (i : Fin 7) (j : Fin 7) (k : Fin 7) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B i)) (SMRHN.SM.PlaneSeven.B j)) (SMRHN.SM.PlaneSeven.B k) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₀","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.SM.PlaneSeven.B₀ : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} +{"name":"SMRHN.SM.PlaneSeven.B₆_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 6)) (SMRHN.SM.PlaneSeven.B i)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₂","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.SM.PlaneSeven.B₂ : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} +{"name":"SMRHN.SM.PlaneSeven.B_in_accCube","declaration":"theorem SMRHN.SM.PlaneSeven.B_in_accCube (f : Fin 7 → ℚ) : SMνACCs.accCube (Finset.sum Finset.univ fun i => f i • SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B_sum_is_sol","declaration":"theorem SMRHN.SM.PlaneSeven.B_sum_is_sol (f : Fin 7 → ℚ) : ACCSystem.IsSolution (SMRHN.SM 3) (Finset.sum Finset.univ fun i => f i • SMRHN.SM.PlaneSeven.B i)"} +{"name":"SMRHN.SM.PlaneSeven.B₅_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₅_cubic (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) (T : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin SMRHN.SM.PlaneSeven.B₅) S) T = S 15 * T 15 - S 16 * T 16"} +{"name":"SMRHN.SM.PlaneSeven.B₄_B₄_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₄_B₄_Bi_cubic {i : Fin 7} : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 4)) (SMRHN.SM.PlaneSeven.B 4)) (SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₀_B₀_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₀_B₀_Bi_cubic {i : Fin 7} : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 0)) (SMRHN.SM.PlaneSeven.B 0)) (SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₃_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 3)) (SMRHN.SM.PlaneSeven.B i)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₆_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₆_cubic (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) (T : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin SMRHN.SM.PlaneSeven.B₆) S) T = 3 * (S 5 * T 5 - S 8 * T 8)"} +{"name":"SMRHN.SM.PlaneSeven.B₀_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₀_cubic (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) (T : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin SMRHN.SM.PlaneSeven.B₀) S) T = 6 * (S 0 * T 0 - S 1 * T 1)"} +{"name":"SMRHN.SM.PlaneSeven.B₁_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 1)) (SMRHN.SM.PlaneSeven.B i)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₅_B₅_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₅_B₅_Bi_cubic {i : Fin 7} : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 5)) (SMRHN.SM.PlaneSeven.B 5)) (SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₃","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.SM.PlaneSeven.B₃ : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} +{"name":"SMRHN.SM.PlaneSeven.B₀_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 0)) (SMRHN.SM.PlaneSeven.B i)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.B","declaration":"/-- The charge assignments forming a basis of the plane. -/\ndef SMRHN.SM.PlaneSeven.B : Fin 7 → ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} +{"name":"SMRHN.SM.PlaneSeven.B₂_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₂_cubic (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) (T : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin SMRHN.SM.PlaneSeven.B₂) S) T = 3 * (S 6 * T 6 - S 7 * T 7)"} +{"name":"SMRHN.SM.PlaneSeven.B₃_B₃_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₃_B₃_Bi_cubic {i : Fin 7} : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 3)) (SMRHN.SM.PlaneSeven.B 3)) (SMRHN.SM.PlaneSeven.B i) = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₂_Bi_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.SM.PlaneSeven.B 2)) (SMRHN.SM.PlaneSeven.B i)) S = 0"} +{"name":"SMRHN.SM.PlaneSeven.B₄_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₄_cubic (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) (T : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin SMRHN.SM.PlaneSeven.B₄) S) T = S 12 * T 12 - S 13 * T 13"} +{"name":"SMRHN.SM.seven_dim_plane_exists","declaration":"theorem SMRHN.SM.seven_dim_plane_exists : ∃ B,\n LinearIndependent ℚ B ∧\n ∀ (f : Fin 7 → ℚ), ACCSystem.IsSolution (SMRHN.SM 3) (Finset.sum Finset.univ fun i => f i • B i)"} +{"name":"SMRHN.SM.PlaneSeven.B₄","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.SM.PlaneSeven.B₄ : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} +{"name":"SMRHN.SM.PlaneSeven.B₁_cubic","declaration":"theorem SMRHN.SM.PlaneSeven.B₁_cubic (S : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) (T : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges) : ((SMνACCs.cubeTriLin SMRHN.SM.PlaneSeven.B₁) S) T = 3 * (S 3 * T 3 - S 4 * T 4)"} +{"name":"SMRHN.SM.PlaneSeven.B₅","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.SM.PlaneSeven.B₅ : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} +{"name":"SMRHN.SM.PlaneSeven.B₆","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.SM.PlaneSeven.B₆ : ACCSystemCharges.Charges (SMRHN.SM 3).toACCSystemCharges"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.FamilyMaps.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.FamilyMaps.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a618dfce451c1b8e739f911b4db1530ee97d2772 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Ordinary.FamilyMaps.jsonl @@ -0,0 +1,3 @@ +{"name":"SMRHN.SM.familyUniversalQuad","declaration":"/-- The family universal maps on `QuadSols`. -/\ndef SMRHN.SM.familyUniversalQuad (n : ℕ) : ACCSystemQuad.QuadSols (SMRHN.SM 1).toACCSystemQuad → ACCSystemQuad.QuadSols (SMRHN.SM n).toACCSystemQuad"} +{"name":"SMRHN.SM.familyUniversalLinear","declaration":"/-- The family universal maps on `LinSols`. -/\ndef SMRHN.SM.familyUniversalLinear (n : ℕ) : ACCSystemLinear.LinSols (SMRHN.SM 1).toACCSystemLinear →ₗ[ℚ] ACCSystemLinear.LinSols (SMRHN.SM n).toACCSystemLinear"} +{"name":"SMRHN.SM.familyUniversalAF","declaration":"/-- The family universal maps on `Sols`. -/\ndef SMRHN.SM.familyUniversalAF (n : ℕ) : ACCSystem.Sols (SMRHN.SM 1) → ACCSystem.Sols (SMRHN.SM n)"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.Permutations.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Permutations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d19107fbc2c319c1a50e005db9ce20b861187122 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.Permutations.jsonl @@ -0,0 +1,13 @@ +{"name":"SMRHN.accSU3_invariant","declaration":"theorem SMRHN.accSU3_invariant {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accSU3 ((SMRHN.repCharges f) S) = SMνACCs.accSU3 S"} +{"name":"SMRHN.PermGroup","declaration":"/-- The group of `Sₙ` permutations for each species. -/\ndef SMRHN.PermGroup (n : ℕ) : Type"} +{"name":"SMRHN.repCharges_toSpecies","declaration":"theorem SMRHN.repCharges_toSpecies {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) (j : Fin 6) : (SMνCharges.toSpecies j) ((SMRHN.repCharges f) S) = (SMνCharges.toSpecies j) S ∘ ⇑(f⁻¹ j)"} +{"name":"SMRHN.accSU2_invariant","declaration":"theorem SMRHN.accSU2_invariant {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accSU2 ((SMRHN.repCharges f) S) = SMνACCs.accSU2 S"} +{"name":"SMRHN.accGrav_invariant","declaration":"theorem SMRHN.accGrav_invariant {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accGrav ((SMRHN.repCharges f) S) = SMνACCs.accGrav S"} +{"name":"SMRHN.instGroupPermGroup","declaration":"instance SMRHN.instGroupPermGroup {n : ℕ} : Group (SMRHN.PermGroup n)"} +{"name":"SMRHN.chargeMap","declaration":"/-- The image of an element of `permGroup n` under the representation on charges. -/\ndef SMRHN.chargeMap {n : ℕ} (f : SMRHN.PermGroup n) : ACCSystemCharges.Charges (SMνCharges n) →ₗ[ℚ] ACCSystemCharges.Charges (SMνCharges n)"} +{"name":"SMRHN.toSpecies_sum_invariant","declaration":"theorem SMRHN.toSpecies_sum_invariant {n : ℕ} (m : ℕ) (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) (j : Fin 6) : (Finset.sum Finset.univ fun i => ((fun a => a ^ m) ∘ (SMνCharges.toSpecies j) ((SMRHN.repCharges f) S)) i) =\n Finset.sum Finset.univ fun i => ((fun a => a ^ m) ∘ (SMνCharges.toSpecies j) S) i"} +{"name":"SMRHN.chargeMap_apply","declaration":"theorem SMRHN.chargeMap_apply {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) : (SMRHN.chargeMap f) S = SMνCharges.toSpeciesEquiv.symm fun i => (SMνCharges.toSpecies i) S ∘ ⇑(f i)"} +{"name":"SMRHN.repCharges","declaration":"/-- The representation of `(permGroup n)` acting on the vector space of charges. -/\ndef SMRHN.repCharges {n : ℕ} : Representation ℚ (SMRHN.PermGroup n) (ACCSystemCharges.Charges (SMνCharges n))"} +{"name":"SMRHN.accQuad_invariant","declaration":"theorem SMRHN.accQuad_invariant {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accQuad ((SMRHN.repCharges f) S) = SMνACCs.accQuad S"} +{"name":"SMRHN.accCube_invariant","declaration":"theorem SMRHN.accCube_invariant {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accCube ((SMRHN.repCharges f) S) = SMνACCs.accCube S"} +{"name":"SMRHN.accYY_invariant","declaration":"theorem SMRHN.accYY_invariant {n : ℕ} (f : SMRHN.PermGroup n) (S : ACCSystemCharges.Charges (SMνCharges n)) : SMνACCs.accYY ((SMRHN.repCharges f) S) = SMνACCs.accYY S"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.BMinusL.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.BMinusL.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d4c814a6fd39deb054e9fd673759c6ce2933384a --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.BMinusL.jsonl @@ -0,0 +1,13 @@ +{"name":"SMRHN.PlusU1.BL.on_cubeTriLin","declaration":"theorem SMRHN.PlusU1.BL.on_cubeTriLin {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.PlusU1.BL n).val) (SMRHN.PlusU1.BL n).val) S = 9 * SMνACCs.accGrav S - 24 * SMνACCs.accSU3 S"} +{"name":"SMRHN.PlusU1.BL.addQuad_zero","declaration":"theorem SMRHN.PlusU1.BL.addQuad_zero {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) : SMRHN.PlusU1.BL.addQuad S a 0 = a • S"} +{"name":"SMRHN.PlusU1.BL₁","declaration":"/-- $B - L$ in the 1-family case. -/\ndef SMRHN.PlusU1.BL₁ : ACCSystem.Sols (SMRHN.PlusU1 1)"} +{"name":"SMRHN.PlusU1.BL₁_val","declaration":"theorem SMRHN.PlusU1.BL₁_val (i : Fin (SMRHN.PlusU1 1).numberCharges) : SMRHN.PlusU1.BL₁.val i =\n match i with\n | 0 => 1\n | 1 => -1\n | 2 => -1\n | 3 => -3\n | 4 => 3\n | 5 => 3"} +{"name":"SMRHN.PlusU1.BL.add_quad","declaration":"theorem SMRHN.PlusU1.BL.add_quad {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) (b : ℚ) : SMνACCs.accQuad (a • S.val + b • (SMRHN.PlusU1.BL n).val) = 0"} +{"name":"SMRHN.PlusU1.BL.on_quadBiLin_AFL","declaration":"theorem SMRHN.PlusU1.BL.on_quadBiLin_AFL {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : (SMνACCs.quadBiLin (SMRHN.PlusU1.BL n).val) S.val = 0"} +{"name":"SMRHN.PlusU1.BL_val","declaration":"theorem SMRHN.PlusU1.BL_val (n : ℕ) : (SMRHN.PlusU1.BL n).val = (SMRHN.familyUniversal n) SMRHN.PlusU1.BL₁.val"} +{"name":"SMRHN.PlusU1.BL","declaration":"/-- $B - L$ in the $n$-family case. -/\ndef SMRHN.PlusU1.BL (n : ℕ) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.BL.add_AFL_cube","declaration":"theorem SMRHN.PlusU1.BL.add_AFL_cube {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (a : ℚ) (b : ℚ) : SMνACCs.accCube (a • S.val + b • (SMRHN.PlusU1.BL n).val) =\n a ^ 2 * (a * SMνACCs.accCube S.val + 3 * b * ((SMνACCs.cubeTriLin S.val) S.val) (SMRHN.PlusU1.BL n).val)"} +{"name":"SMRHN.PlusU1.BL.add_AFL_quad","declaration":"theorem SMRHN.PlusU1.BL.add_AFL_quad {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (a : ℚ) (b : ℚ) : SMνACCs.accQuad (a • S.val + b • (SMRHN.PlusU1.BL n).val) = a ^ 2 * SMνACCs.accQuad S.val"} +{"name":"SMRHN.PlusU1.BL.addQuad","declaration":"/-- The `QuadSol` obtained by adding $B-L$ to a `QuadSol`. -/\ndef SMRHN.PlusU1.BL.addQuad {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) (b : ℚ) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} +{"name":"SMRHN.PlusU1.BL.on_quadBiLin","declaration":"theorem SMRHN.PlusU1.BL.on_quadBiLin {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) : (SMνACCs.quadBiLin (SMRHN.PlusU1.BL n).val) S =\n 1 / 2 * SMνACCs.accYY S + 3 / 2 * SMνACCs.accSU2 S - 2 * SMνACCs.accSU3 S"} +{"name":"SMRHN.PlusU1.BL.on_cubeTriLin_AFL","declaration":"theorem SMRHN.PlusU1.BL.on_cubeTriLin_AFL {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : ((SMνACCs.cubeTriLin (SMRHN.PlusU1.BL n).val) (SMRHN.PlusU1.BL n).val) S.val = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.Basic.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..8d2350f8f4f3ea07e578068e620bd385b3d8c538 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.Basic.jsonl @@ -0,0 +1,20 @@ +{"name":"SMRHN.PlusU1.chargeToAF","declaration":"/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.PlusU1.chargeToAF {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) (hGrav : SMνACCs.accGrav S = 0) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) (hYY : SMνACCs.accYY S = 0) (hQ : SMνACCs.accQuad S = 0) (hc : SMνACCs.accCube S = 0) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.chargeToQuad","declaration":"/-- An element of `charges` which satisfies the linear and quadratic ACCs\ngives us a element of `QuadSols`. -/\ndef SMRHN.PlusU1.chargeToQuad {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) (hGrav : SMνACCs.accGrav S = 0) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) (hYY : SMνACCs.accYY S = 0) (hQ : SMνACCs.accQuad S = 0) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} +{"name":"SMRHN.PlusU1_numberQuadratic","declaration":"theorem SMRHN.PlusU1_numberQuadratic (n : ℕ) : (SMRHN.PlusU1 n).numberQuadratic = 1"} +{"name":"SMRHN.PlusU1_cubicACC_toFun","declaration":"theorem SMRHN.PlusU1_cubicACC_toFun (n : ℕ) (S : ACCSystemCharges.Charges (SMνCharges n)) : (SMRHN.PlusU1 n).cubicACC.toFun S =\n Finset.sum Finset.univ fun i =>\n 6 * (S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i)) * S (finProdFinEquiv (0, i))) +\n 3 * (S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i)) * S (finProdFinEquiv (1, i))) +\n 3 * (S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i)) * S (finProdFinEquiv (2, i))) +\n 2 * (S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i)) * S (finProdFinEquiv (3, i))) +\n S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) * S (finProdFinEquiv (4, i)) +\n S (finProdFinEquiv (5, i)) * S (finProdFinEquiv (5, i)) * S (finProdFinEquiv (5, i))"} +{"name":"SMRHN.PlusU1_linearACCs","declaration":"theorem SMRHN.PlusU1_linearACCs (n : ℕ) (i : Fin 4) : (SMRHN.PlusU1 n).linearACCs i =\n match i with\n | 0 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n 6 * S (finProdFinEquiv (0, i)) + 3 * S (finProdFinEquiv (1, i)) + 3 * S (finProdFinEquiv (2, i)) +\n 2 * S (finProdFinEquiv (3, i)) +\n S (finProdFinEquiv (4, i)) +\n S (finProdFinEquiv (5, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }\n | 1 =>\n {\n toAddHom :=\n { toFun := fun S => Finset.sum Finset.univ fun i => 3 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (3, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }\n | 2 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n 2 * S (finProdFinEquiv (0, i)) + S (finProdFinEquiv (1, i)) + S (finProdFinEquiv (2, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }\n | 3 =>\n {\n toAddHom :=\n {\n toFun := fun S =>\n Finset.sum Finset.univ fun i =>\n S (finProdFinEquiv (0, i)) + 8 * S (finProdFinEquiv (1, i)) + 2 * S (finProdFinEquiv (2, i)) +\n 3 * S (finProdFinEquiv (3, i)) +\n 6 * S (finProdFinEquiv (4, i)),\n map_add' := ⋯ },\n map_smul' := ⋯ }"} +{"name":"SMRHN.PlusU1.gravSol","declaration":"theorem SMRHN.PlusU1.gravSol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : SMνACCs.accGrav S.val = 0"} +{"name":"SMRHN.PlusU1.perm","declaration":"/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/\ndef SMRHN.PlusU1.perm (n : ℕ) : ACCSystemGroupAction (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.quadToAF","declaration":"/-- An element of `QuadSols` which satisfies the quadratic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.PlusU1.quadToAF {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (hc : SMνACCs.accCube S.val = 0) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.YYsol","declaration":"theorem SMRHN.PlusU1.YYsol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : SMνACCs.accYY S.val = 0"} +{"name":"SMRHN.PlusU1_quadraticACCs","declaration":"theorem SMRHN.PlusU1_quadraticACCs (n : ℕ) (i : Fin 1) : (SMRHN.PlusU1 n).quadraticACCs i =\n match i with\n | 0 => BiLinearSymm.toHomogeneousQuad SMνACCs.quadBiLin"} +{"name":"SMRHN.PlusU1.linearToAF","declaration":"/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs\ngives us a element of `Sols`. -/\ndef SMRHN.PlusU1.linearToAF {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (hQ : SMνACCs.accQuad S.val = 0) (hc : SMνACCs.accCube S.val = 0) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.cubeSol","declaration":"theorem SMRHN.PlusU1.cubeSol {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) : SMνACCs.accCube S.val = 0"} +{"name":"SMRHN.PlusU1.SU2Sol","declaration":"theorem SMRHN.PlusU1.SU2Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : SMνACCs.accSU2 S.val = 0"} +{"name":"SMRHN.PlusU1.linearToQuad","declaration":"/-- An element of `LinSols` which satisfies the quadratic ACCs\ngives us a element of `AnomalyFreeQuad`. -/\ndef SMRHN.PlusU1.linearToQuad {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (hQ : SMνACCs.accQuad S.val = 0) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} +{"name":"SMRHN.PlusU1.chargeToLinear","declaration":"/-- An element of `charges` which satisfies the linear ACCs\ngives us a element of `LinSols`. -/\ndef SMRHN.PlusU1.chargeToLinear {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) (hGrav : SMνACCs.accGrav S = 0) (hSU2 : SMνACCs.accSU2 S = 0) (hSU3 : SMνACCs.accSU3 S = 0) (hYY : SMνACCs.accYY S = 0) : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear"} +{"name":"SMRHN.PlusU1_numberCharges","declaration":"theorem SMRHN.PlusU1_numberCharges (n : ℕ) : (SMRHN.PlusU1 n).numberCharges = 6 * n"} +{"name":"SMRHN.PlusU1_numberLinear","declaration":"theorem SMRHN.PlusU1_numberLinear (n : ℕ) : (SMRHN.PlusU1 n).numberLinear = 4"} +{"name":"SMRHN.PlusU1","declaration":"/-- The ACC system for the SM plus RHN with an additional U1. -/\ndef SMRHN.PlusU1 (n : ℕ) : ACCSystem"} +{"name":"SMRHN.PlusU1.SU3Sol","declaration":"theorem SMRHN.PlusU1.SU3Sol {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : SMνACCs.accSU3 S.val = 0"} +{"name":"SMRHN.PlusU1.quadSol","declaration":"theorem SMRHN.PlusU1.quadSol {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : SMνACCs.accQuad S.val = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.BoundPlaneDim.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.BoundPlaneDim.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..4df356fa26ca9d3eb59ad13d92f5c0e0066de5ae --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.BoundPlaneDim.jsonl @@ -0,0 +1,3 @@ +{"name":"SMRHN.PlusU1.ExistsPlane","declaration":"/-- A proposition which is true if for a given `n`, a plane of charges of dimension `n` exists\nin which each point is a solution. -/\ndef SMRHN.PlusU1.ExistsPlane (n : ℕ) : Prop"} +{"name":"SMRHN.PlusU1.plane_exists_dim_le_7","declaration":"theorem SMRHN.PlusU1.plane_exists_dim_le_7 {n : ℕ} (hn : SMRHN.PlusU1.ExistsPlane n) : n ≤ 7"} +{"name":"SMRHN.PlusU1.exists_plane_exists_basis","declaration":"theorem SMRHN.PlusU1.exists_plane_exists_basis {n : ℕ} (hE : SMRHN.PlusU1.ExistsPlane n) : ∃ B, LinearIndependent ℚ B"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ca151a3c38eb30678c56bac4a813ce25add3425b --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps.jsonl @@ -0,0 +1,3 @@ +{"name":"SMRHN.PlusU1.familyUniversalAF","declaration":"/-- The family universal maps on `Sols`. -/\ndef SMRHN.PlusU1.familyUniversalAF (n : ℕ) : ACCSystem.Sols (SMRHN.PlusU1 1) → ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.familyUniversalLinear","declaration":"/-- The family universal maps on `LinSols`. -/\ndef SMRHN.PlusU1.familyUniversalLinear (n : ℕ) : ACCSystemLinear.LinSols (SMRHN.PlusU1 1).toACCSystemLinear →ₗ[ℚ]\n ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear"} +{"name":"SMRHN.PlusU1.familyUniversalQuad","declaration":"/-- The family universal maps on `QuadSols`. -/\ndef SMRHN.PlusU1.familyUniversalQuad (n : ℕ) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 1).toACCSystemQuad → ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.HyperCharge.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.HyperCharge.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..94225f6a6436cbda6288231ad9ef7201ac1a44f6 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.HyperCharge.jsonl @@ -0,0 +1,18 @@ +{"name":"SMRHN.PlusU1.Y.on_quadBiLin_AFL","declaration":"theorem SMRHN.PlusU1.Y.on_quadBiLin_AFL {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : (SMνACCs.quadBiLin (SMRHN.PlusU1.Y n).val) S.val = 0"} +{"name":"SMRHN.PlusU1.Y.addQuad_zero","declaration":"theorem SMRHN.PlusU1.Y.addQuad_zero {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) : SMRHN.PlusU1.Y.addQuad S a 0 = a • S"} +{"name":"SMRHN.PlusU1.Y_val","declaration":"theorem SMRHN.PlusU1.Y_val (n : ℕ) : (SMRHN.PlusU1.Y n).val = (SMRHN.familyUniversal n) SMRHN.PlusU1.Y₁.val"} +{"name":"SMRHN.PlusU1.Y","declaration":"/-- The hypercharge for `n` family. -/\ndef SMRHN.PlusU1.Y (n : ℕ) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.Y.on_quadBiLin","declaration":"theorem SMRHN.PlusU1.Y.on_quadBiLin {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) : (SMνACCs.quadBiLin (SMRHN.PlusU1.Y n).val) S = SMνACCs.accYY S"} +{"name":"SMRHN.PlusU1.Y.addCube","declaration":"/-- The `Sol` obtained by adding hypercharge to a `Sol`. -/\ndef SMRHN.PlusU1.Y.addCube {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (a : ℚ) (b : ℚ) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.Y.addQuad","declaration":"/-- The `QuadSol` obtained by adding hypercharge to a `QuadSol`. -/\ndef SMRHN.PlusU1.Y.addQuad {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) (b : ℚ) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} +{"name":"SMRHN.PlusU1.Y.on_cubeTriLin","declaration":"theorem SMRHN.PlusU1.Y.on_cubeTriLin {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.PlusU1.Y n).val) (SMRHN.PlusU1.Y n).val) S = 6 * SMνACCs.accYY S"} +{"name":"SMRHN.PlusU1.Y₁","declaration":"/-- The hypercharge for 1 family. -/\ndef SMRHN.PlusU1.Y₁ : ACCSystem.Sols (SMRHN.PlusU1 1)"} +{"name":"SMRHN.PlusU1.Y.on_cubeTriLin'","declaration":"theorem SMRHN.PlusU1.Y.on_cubeTriLin' {n : ℕ} (S : ACCSystemCharges.Charges (SMRHN.PlusU1 n).toACCSystemCharges) : ((SMνACCs.cubeTriLin (SMRHN.PlusU1.Y n).val) S) S = 6 * SMνACCs.accQuad S"} +{"name":"SMRHN.PlusU1.Y.add_AF_cube","declaration":"theorem SMRHN.PlusU1.Y.add_AF_cube {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (a : ℚ) (b : ℚ) : SMνACCs.accCube (a • S.val + b • (SMRHN.PlusU1.Y n).val) = 0"} +{"name":"SMRHN.PlusU1.Y.add_AFL_quad","declaration":"theorem SMRHN.PlusU1.Y.add_AFL_quad {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (a : ℚ) (b : ℚ) : SMνACCs.accQuad (a • S.val + b • (SMRHN.PlusU1.Y n).val) = a ^ 2 * SMνACCs.accQuad S.val"} +{"name":"SMRHN.PlusU1.Y.add_AFL_cube","declaration":"theorem SMRHN.PlusU1.Y.add_AFL_cube {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (a : ℚ) (b : ℚ) : SMνACCs.accCube (a • S.val + b • (SMRHN.PlusU1.Y n).val) =\n a ^ 2 * (a * SMνACCs.accCube S.val + 3 * b * ((SMνACCs.cubeTriLin S.val) S.val) (SMRHN.PlusU1.Y n).val)"} +{"name":"SMRHN.PlusU1.Y₁_val","declaration":"theorem SMRHN.PlusU1.Y₁_val (i : Fin (SMRHN.PlusU1 1).numberCharges) : SMRHN.PlusU1.Y₁.val i =\n match i with\n | 0 => 1\n | 1 => -4\n | 2 => 2\n | 3 => -3\n | 4 => 6\n | 5 => 0"} +{"name":"SMRHN.PlusU1.Y.on_cubeTriLin'_ALQ","declaration":"theorem SMRHN.PlusU1.Y.on_cubeTriLin'_ALQ {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : ((SMνACCs.cubeTriLin (SMRHN.PlusU1.Y n).val) S.val) S.val = 0"} +{"name":"SMRHN.PlusU1.Y.add_AFQ_cube","declaration":"theorem SMRHN.PlusU1.Y.add_AFQ_cube {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) (b : ℚ) : SMνACCs.accCube (a • S.val + b • (SMRHN.PlusU1.Y n).val) = a ^ 3 * SMνACCs.accCube S.val"} +{"name":"SMRHN.PlusU1.Y.on_cubeTriLin_AFL","declaration":"theorem SMRHN.PlusU1.Y.on_cubeTriLin_AFL {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : ((SMνACCs.cubeTriLin (SMRHN.PlusU1.Y n).val) (SMRHN.PlusU1.Y n).val) S.val = 0"} +{"name":"SMRHN.PlusU1.Y.add_quad","declaration":"theorem SMRHN.PlusU1.Y.add_quad {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) (b : ℚ) : SMνACCs.accQuad (a • S.val + b • (SMRHN.PlusU1.Y n).val) = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ebe6f51b89bc7cecb787545ed7ece0a3d5d36ea3 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols.jsonl @@ -0,0 +1,36 @@ +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f1","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f1 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 1 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₁₀","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₁₀ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f10","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f10 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 10 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₂","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₂ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.Bi_Bj_quad","declaration":"theorem SMRHN.PlusU1.ElevenPlane.Bi_Bj_quad {i : Fin 11} {j : Fin 11} (hi : i ≠ j) : (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i)) (SMRHN.PlusU1.ElevenPlane.B j) = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₉","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₉ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f3","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f3 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 3 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f6","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f6 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 6 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.on_accQuad","declaration":"theorem SMRHN.PlusU1.ElevenPlane.on_accQuad (f : Fin 11 → ℚ) : SMνACCs.accQuad (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) =\n Finset.sum Finset.univ fun i => SMRHN.PlusU1.ElevenPlane.quadCoeff i * f i ^ 2"} +{"name":"SMRHN.PlusU1.eleven_dim_plane_of_no_sols_exists","declaration":"theorem SMRHN.PlusU1.eleven_dim_plane_of_no_sols_exists : ∃ B,\n LinearIndependent ℚ B ∧\n ∀ (f : Fin 11 → ℚ),\n ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • B i) →\n (Finset.sum Finset.univ fun i => f i • B i) = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f8","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f8 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 8 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₆","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₆ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f2","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f2 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 2 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f9","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f9 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 9 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f0","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f0 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 0 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f4","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f4 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 4 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B","declaration":"/-- The charge assignment forming a basis of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B : Fin 11 → ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f_zero","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f_zero (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) (k : Fin 11) : f k = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f7","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f7 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 7 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_sum_part'","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_sum_part' (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) =\n f 9 • SMRHN.PlusU1.ElevenPlane.B₉ + (-3 * f 9) • SMRHN.PlusU1.ElevenPlane.B₁₀"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₅","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₅ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_quadCoeff_f_sq_zero","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) (k : Fin 11) : SMRHN.PlusU1.ElevenPlane.quadCoeff k * f k ^ 2 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₈","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₈ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_sum_part","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_sum_part (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) =\n f 9 • SMRHN.PlusU1.ElevenPlane.B₉ + f 10 • SMRHN.PlusU1.ElevenPlane.B₁₀"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_grav","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_grav (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 10 = -3 * f 9"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_only_if_zero","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i) = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₄","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₄ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.basis_linear_independent","declaration":"theorem SMRHN.PlusU1.ElevenPlane.basis_linear_independent : LinearIndependent ℚ SMRHN.PlusU1.ElevenPlane.B"} +{"name":"SMRHN.PlusU1.ElevenPlane.Bi_sum_quad","declaration":"theorem SMRHN.PlusU1.ElevenPlane.Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) : (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i))\n (Finset.sum Finset.univ fun k => f k • SMRHN.PlusU1.ElevenPlane.B k) =\n f i * (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i)) (SMRHN.PlusU1.ElevenPlane.B i)"} +{"name":"SMRHN.PlusU1.ElevenPlane.quadCoeff","declaration":"/-- The coefficents of the quadratic equation in our basis. -/\ndef SMRHN.PlusU1.ElevenPlane.quadCoeff : Fin 11 → ℚ"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₃","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₃ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.quadCoeff_eq_bilinear","declaration":"theorem SMRHN.PlusU1.ElevenPlane.quadCoeff_eq_bilinear (i : Fin 11) : SMRHN.PlusU1.ElevenPlane.quadCoeff i = (SMνACCs.quadBiLin (SMRHN.PlusU1.ElevenPlane.B i)) (SMRHN.PlusU1.ElevenPlane.B i)"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₀","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₀ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.isSolution_f5","declaration":"theorem SMRHN.PlusU1.ElevenPlane.isSolution_f5 (f : Fin 11 → ℚ) (hS : ACCSystem.IsSolution (SMRHN.PlusU1 3) (Finset.sum Finset.univ fun i => f i • SMRHN.PlusU1.ElevenPlane.B i)) : f 5 = 0"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₇","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₇ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} +{"name":"SMRHN.PlusU1.ElevenPlane.B₁","declaration":"/-- A charge assignment forming one of the basis elements of the plane. -/\ndef SMRHN.PlusU1.ElevenPlane.B₁ : ACCSystemCharges.Charges (SMRHN.PlusU1 3).toACCSystemCharges"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSol.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSol.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7f0aa72dfee7199e93b2831763612240dd5944b1 --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSol.jsonl @@ -0,0 +1,19 @@ +{"name":"SMRHN.PlusU1.QuadSol.toQuadInv_fst","declaration":"theorem SMRHN.PlusU1.QuadSol.toQuadInv_fst {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : (SMRHN.PlusU1.QuadSol.toQuadInv C S).1 = S.toLinSols"} +{"name":"SMRHN.PlusU1.QuadSol.special_on_quad","declaration":"theorem SMRHN.PlusU1.QuadSol.special_on_quad {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (h1 : SMRHN.PlusU1.QuadSol.α₁ C S.toLinSols = 0) : SMRHN.PlusU1.QuadSol.specialToQuad C S.toLinSols 1 0 h1 ⋯ = S"} +{"name":"SMRHN.PlusU1.QuadSol.toQuadInv_generic","declaration":"theorem SMRHN.PlusU1.QuadSol.toQuadInv_generic {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (h : SMRHN.PlusU1.QuadSol.α₁ C S.toLinSols ≠ 0) : (SMRHN.PlusU1.QuadSol.toQuadInv C S).2.1 • SMRHN.PlusU1.QuadSol.genericToQuad C (SMRHN.PlusU1.QuadSol.toQuadInv C S).1 =\n S"} +{"name":"SMRHN.PlusU1.QuadSol.toQuadInv_special","declaration":"theorem SMRHN.PlusU1.QuadSol.toQuadInv_special {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (h : SMRHN.PlusU1.QuadSol.α₁ C S.toLinSols = 0) : SMRHN.PlusU1.QuadSol.specialToQuad C (SMRHN.PlusU1.QuadSol.toQuadInv C S).1 (SMRHN.PlusU1.QuadSol.toQuadInv C S).2.1\n (SMRHN.PlusU1.QuadSol.toQuadInv C S).2.2 ⋯ ⋯ =\n S"} +{"name":"SMRHN.PlusU1.QuadSol.genericToQuad_neq_zero","declaration":"theorem SMRHN.PlusU1.QuadSol.genericToQuad_neq_zero {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (h : SMRHN.PlusU1.QuadSol.α₁ C S.toLinSols ≠ 0) : (SMRHN.PlusU1.QuadSol.α₁ C S.toLinSols)⁻¹ • SMRHN.PlusU1.QuadSol.genericToQuad C S.toLinSols = S"} +{"name":"SMRHN.PlusU1.QuadSol.α₂_AFQ","declaration":"theorem SMRHN.PlusU1.QuadSol.α₂_AFQ {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : SMRHN.PlusU1.QuadSol.α₂ S.toLinSols = 0"} +{"name":"SMRHN.PlusU1.QuadSol.genericToQuad","declaration":"/-- The construction of a `QuadSol` from a `LinSols` in the generic case. -/\ndef SMRHN.PlusU1.QuadSol.genericToQuad {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} +{"name":"SMRHN.PlusU1.QuadSol.accQuad_α₁_α₂_zero","declaration":"theorem SMRHN.PlusU1.QuadSol.accQuad_α₁_α₂_zero {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (h1 : SMRHN.PlusU1.QuadSol.α₁ C S = 0) (h2 : SMRHN.PlusU1.QuadSol.α₂ S = 0) (a : ℚ) (b : ℚ) : SMνACCs.accQuad (a • S + b • C.toLinSols).val = 0"} +{"name":"SMRHN.PlusU1.QuadSol.genericToQuad_on_quad","declaration":"theorem SMRHN.PlusU1.QuadSol.genericToQuad_on_quad {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : SMRHN.PlusU1.QuadSol.genericToQuad C S.toLinSols = SMRHN.PlusU1.QuadSol.α₁ C S.toLinSols • S"} +{"name":"SMRHN.PlusU1.QuadSol.α₂","declaration":"/-- A helper function for what comes later. -/\ndef SMRHN.PlusU1.QuadSol.α₂ {n : ℕ} (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : ℚ"} +{"name":"SMRHN.PlusU1.QuadSol.accQuad_α₁_α₂","declaration":"theorem SMRHN.PlusU1.QuadSol.accQuad_α₁_α₂ {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : SMνACCs.accQuad (SMRHN.PlusU1.QuadSol.α₁ C S • S + SMRHN.PlusU1.QuadSol.α₂ S • C.toLinSols).val = 0"} +{"name":"SMRHN.PlusU1.QuadSol.toQuad_surjective","declaration":"theorem SMRHN.PlusU1.QuadSol.toQuad_surjective {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : Function.Surjective (SMRHN.PlusU1.QuadSol.toQuad C)"} +{"name":"SMRHN.PlusU1.QuadSol.toQuad_rightInverse","declaration":"theorem SMRHN.PlusU1.QuadSol.toQuad_rightInverse {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : Function.RightInverse (SMRHN.PlusU1.QuadSol.toQuadInv C) (SMRHN.PlusU1.QuadSol.toQuad C)"} +{"name":"SMRHN.PlusU1.QuadSol.toQuadInv","declaration":"/-- A function from `QuadSols` to `LinSols × ℚ × ℚ` which is a right inverse to `toQuad`. -/\ndef SMRHN.PlusU1.QuadSol.toQuadInv {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad →\n ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear × ℚ × ℚ"} +{"name":"SMRHN.PlusU1.QuadSol.specialToQuad","declaration":"/-- The construction of a `QuadSol` from a `LinSols` in the special case when `α₁ C S = 0` and\n`α₂ S = 0`. -/\ndef SMRHN.PlusU1.QuadSol.specialToQuad {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (a : ℚ) (b : ℚ) (h1 : SMRHN.PlusU1.QuadSol.α₁ C S = 0) (h2 : SMRHN.PlusU1.QuadSol.α₂ S = 0) : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} +{"name":"SMRHN.PlusU1.QuadSol.add_AFL_quad","declaration":"theorem SMRHN.PlusU1.QuadSol.add_AFL_quad {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) (a : ℚ) (b : ℚ) : SMνACCs.accQuad (a • S.val + b • C.val) = a * (a * SMνACCs.accQuad S.val + 2 * b * (SMνACCs.quadBiLin S.val) C.val)"} +{"name":"SMRHN.PlusU1.QuadSol.toQuad","declaration":"/-- The construction of a `QuadSols` from a `LinSols` and two rationals taking account of the\ngeneric and special cases. This function is a surjection. -/\ndef SMRHN.PlusU1.QuadSol.toQuad {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear × ℚ × ℚ →\n ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad"} +{"name":"SMRHN.PlusU1.QuadSol.α₁","declaration":"/-- A helper function for what comes later. -/\ndef SMRHN.PlusU1.QuadSol.α₁ {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemLinear.LinSols (SMRHN.PlusU1 n).toACCSystemLinear) : ℚ"} +{"name":"SMRHN.PlusU1.QuadSol.toQuadInv_α₁_α₂","declaration":"theorem SMRHN.PlusU1.QuadSol.toQuadInv_α₁_α₂ {n : ℕ} (C : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : SMRHN.PlusU1.QuadSol.α₁ C S.toLinSols = 0 ↔\n SMRHN.PlusU1.QuadSol.α₁ C (SMRHN.PlusU1.QuadSol.toQuadInv C S).1 = 0 ∧\n SMRHN.PlusU1.QuadSol.α₂ (SMRHN.PlusU1.QuadSol.toQuadInv C S).1 = 0"} diff --git a/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSolToSol.jsonl b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSolToSol.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c5e1f12afb285f8335329071cc6c680ed3e3700d --- /dev/null +++ b/hep-declarations/HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSolToSol.jsonl @@ -0,0 +1,20 @@ +{"name":"SMRHN.PlusU1.QuadSolToSol.generic","declaration":"/-- The construction of a `Sol` from a `QuadSol` in the generic case. -/\ndef SMRHN.PlusU1.QuadSolToSol.generic {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.QuadSolToSol.BL_add_α₁_α₂_cube","declaration":"theorem SMRHN.PlusU1.QuadSolToSol.BL_add_α₁_α₂_cube {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : SMνACCs.accCube (SMRHN.PlusU1.BL.addQuad S (SMRHN.PlusU1.QuadSolToSol.α₁ S) (SMRHN.PlusU1.QuadSolToSol.α₂ S)).val = 0"} +{"name":"SMRHN.PlusU1.QuadSolToSol.α₂","declaration":"/-- A helper function for what follows. -/\ndef SMRHN.PlusU1.QuadSolToSol.α₂ {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : ℚ"} +{"name":"SMRHN.PlusU1.quadSolToSol","declaration":"/-- A map from `QuadSols × ℚ × ℚ` to `Sols` taking account of the special and generic cases.\nWe will show that this map is a surjection. -/\ndef SMRHN.PlusU1.quadSolToSol {n : ℕ} : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad × ℚ × ℚ → ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.quadSolToSolInv_α₁_α₂_neq_zero","declaration":"theorem SMRHN.PlusU1.quadSolToSolInv_α₁_α₂_neq_zero {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (h : SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols ≠ 0) : ¬(SMRHN.PlusU1.QuadSolToSol.α₁ (SMRHN.PlusU1.quadSolToSolInv S).1 = 0 ∧\n SMRHN.PlusU1.QuadSolToSol.α₂ (SMRHN.PlusU1.quadSolToSolInv S).1 = 0)"} +{"name":"SMRHN.PlusU1.QuadSolToSol.generic_on_AF","declaration":"theorem SMRHN.PlusU1.QuadSolToSol.generic_on_AF {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) : SMRHN.PlusU1.QuadSolToSol.generic S.toQuadSols = SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols • S"} +{"name":"SMRHN.PlusU1.QuadSolToSol.generic_on_AF_α₁_ne_zero","declaration":"theorem SMRHN.PlusU1.QuadSolToSol.generic_on_AF_α₁_ne_zero {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (h : SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols ≠ 0) : (SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols)⁻¹ • SMRHN.PlusU1.QuadSolToSol.generic S.toQuadSols = S"} +{"name":"SMRHN.PlusU1.QuadSolToSol.α₁","declaration":"/-- A helper function for what follows. -/\ndef SMRHN.PlusU1.QuadSolToSol.α₁ {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) : ℚ"} +{"name":"SMRHN.PlusU1.QuadSolToSol.BL_add_α₁_α₂_AF","declaration":"theorem SMRHN.PlusU1.QuadSolToSol.BL_add_α₁_α₂_AF {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) : SMRHN.PlusU1.BL.addQuad S.toQuadSols (SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols)\n (SMRHN.PlusU1.QuadSolToSol.α₂ S.toQuadSols) =\n SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols • S.toQuadSols"} +{"name":"SMRHN.PlusU1.quadSolToSolInv_rightInverse","declaration":"theorem SMRHN.PlusU1.quadSolToSolInv_rightInverse {n : ℕ} : Function.RightInverse SMRHN.PlusU1.quadSolToSolInv SMRHN.PlusU1.quadSolToSol"} +{"name":"SMRHN.PlusU1.QuadSolToSol.special","declaration":"/-- The construction of a `Sol` from a `QuadSol` in the case when `α₁ S = 0` and `α₂ S = 0`. -/\ndef SMRHN.PlusU1.QuadSolToSol.special {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) (b : ℚ) (h1 : SMRHN.PlusU1.QuadSolToSol.α₁ S = 0) (h2 : SMRHN.PlusU1.QuadSolToSol.α₂ S = 0) : ACCSystem.Sols (SMRHN.PlusU1 n)"} +{"name":"SMRHN.PlusU1.QuadSolToSol.special_on_AF","declaration":"theorem SMRHN.PlusU1.QuadSolToSol.special_on_AF {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (h1 : SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols = 0) : SMRHN.PlusU1.QuadSolToSol.special S.toQuadSols 1 0 h1 ⋯ = S"} +{"name":"SMRHN.PlusU1.quadSolToSolInv_generic","declaration":"theorem SMRHN.PlusU1.quadSolToSolInv_generic {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (h : SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols ≠ 0) : (SMRHN.PlusU1.quadSolToSolInv S).2.1 • SMRHN.PlusU1.QuadSolToSol.generic (SMRHN.PlusU1.quadSolToSolInv S).1 = S"} +{"name":"SMRHN.PlusU1.quadSolToSolInv","declaration":"/-- A map from `Sols` to `QuadSols × ℚ × ℚ` which forms a right-inverse to `quadSolToSol`, as\nshown in `quadSolToSolInv_rightInverse`. -/\ndef SMRHN.PlusU1.quadSolToSolInv {n : ℕ} : ACCSystem.Sols (SMRHN.PlusU1 n) → ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad × ℚ × ℚ"} +{"name":"SMRHN.PlusU1.quadSolToSolInv_1","declaration":"theorem SMRHN.PlusU1.quadSolToSolInv_1 {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) : (SMRHN.PlusU1.quadSolToSolInv S).1 = S.toQuadSols"} +{"name":"SMRHN.PlusU1.quadSolToSolInv_special","declaration":"theorem SMRHN.PlusU1.quadSolToSolInv_special {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (h : SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols = 0) : SMRHN.PlusU1.QuadSolToSol.special (SMRHN.PlusU1.quadSolToSolInv S).1 (SMRHN.PlusU1.quadSolToSolInv S).2.1\n (SMRHN.PlusU1.quadSolToSolInv S).2.2 ⋯ ⋯ =\n S"} +{"name":"SMRHN.PlusU1.QuadSolToSol.cube_α₁_α₂_zero","declaration":"theorem SMRHN.PlusU1.QuadSolToSol.cube_α₁_α₂_zero {n : ℕ} (S : ACCSystemQuad.QuadSols (SMRHN.PlusU1 n).toACCSystemQuad) (a : ℚ) (b : ℚ) (h1 : SMRHN.PlusU1.QuadSolToSol.α₁ S = 0) (h2 : SMRHN.PlusU1.QuadSolToSol.α₂ S = 0) : SMνACCs.accCube (SMRHN.PlusU1.BL.addQuad S a b).val = 0"} +{"name":"SMRHN.PlusU1.QuadSolToSol.α₂_AF","declaration":"theorem SMRHN.PlusU1.QuadSolToSol.α₂_AF {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) : SMRHN.PlusU1.QuadSolToSol.α₂ S.toQuadSols = 0"} +{"name":"SMRHN.PlusU1.quadSolToSol_surjective","declaration":"theorem SMRHN.PlusU1.quadSolToSol_surjective {n : ℕ} : Function.Surjective SMRHN.PlusU1.quadSolToSol"} +{"name":"SMRHN.PlusU1.quadSolToSolInv_α₁_α₂_zero","declaration":"theorem SMRHN.PlusU1.quadSolToSolInv_α₁_α₂_zero {n : ℕ} (S : ACCSystem.Sols (SMRHN.PlusU1 n)) (h : SMRHN.PlusU1.QuadSolToSol.α₁ S.toQuadSols = 0) : SMRHN.PlusU1.QuadSolToSol.α₁ (SMRHN.PlusU1.quadSolToSolInv S).1 = 0 ∧\n SMRHN.PlusU1.QuadSolToSol.α₂ (SMRHN.PlusU1.quadSolToSolInv S).1 = 0"} diff --git a/hep-declarations/HepLean.BeyondTheStandardModel.PatiSalam.Basic.jsonl b/hep-declarations/HepLean.BeyondTheStandardModel.PatiSalam.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ef8da8c2abe94a3f9cf283ed0a3ceb2f2e5b7439 --- /dev/null +++ b/hep-declarations/HepLean.BeyondTheStandardModel.PatiSalam.Basic.jsonl @@ -0,0 +1,9 @@ +{"name":"PatiSalam.sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup","declaration":"/-- An informal lemma. -/\ndef PatiSalam.sm_ℤ₆_factor_through_gaugeGroupℤ₂SubGroup : InformalLemma"} +{"name":"PatiSalam.inclSM_ker","declaration":"/-- An informal lemma. -/\ndef PatiSalam.inclSM_ker : InformalLemma"} +{"name":"PatiSalam.gaugeGroupℤ₂SubGroup","declaration":"/-- An informal definition. -/\ndef PatiSalam.gaugeGroupℤ₂SubGroup : InformalDefinition"} +{"name":"PatiSalam.gaugeGroupISpinEquiv","declaration":"/-- An informal definition. -/\ndef PatiSalam.gaugeGroupISpinEquiv : InformalDefinition"} +{"name":"PatiSalam.GaugeGroupℤ₂","declaration":"/-- An informal definition. -/\ndef PatiSalam.GaugeGroupℤ₂ : InformalDefinition"} +{"name":"PatiSalam.inclSM","declaration":"/-- An informal definition. -/\ndef PatiSalam.inclSM : InformalDefinition"} +{"name":"PatiSalam.embedSMℤ₆Toℤ₂","declaration":"/-- An informal definition. -/\ndef PatiSalam.embedSMℤ₆Toℤ₂ : InformalDefinition"} +{"name":"PatiSalam.GaugeGroupI","declaration":"/-- An informal definition. -/\ndef PatiSalam.GaugeGroupI : InformalDefinition"} +{"name":"PatiSalam.embedSMℤ₃","declaration":"/-- An informal definition. -/\ndef PatiSalam.embedSMℤ₃ : InformalDefinition"} diff --git a/hep-declarations/HepLean.BeyondTheStandardModel.Spin10.Basic.jsonl b/hep-declarations/HepLean.BeyondTheStandardModel.Spin10.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b61af204b1519d4d44a995dbc10b85ff00e8e290 --- /dev/null +++ b/hep-declarations/HepLean.BeyondTheStandardModel.Spin10.Basic.jsonl @@ -0,0 +1,3 @@ +{"name":"Spin10Model.GaugeGroupI","declaration":"/-- An informal definition. -/\ndef Spin10Model.GaugeGroupI : InformalDefinition"} +{"name":"Spin10Model.inclPatiSalam","declaration":"/-- An informal definition. -/\ndef Spin10Model.inclPatiSalam : InformalDefinition"} +{"name":"Spin10Model.inclSM","declaration":"/-- An informal definition. -/\ndef Spin10Model.inclSM : InformalDefinition"} diff --git a/hep-declarations/HepLean.BeyondTheStandardModel.TwoHDM.Basic.jsonl b/hep-declarations/HepLean.BeyondTheStandardModel.TwoHDM.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..58b146055fb2edddb8ad57bc04615a294c6e52e8 --- /dev/null +++ b/hep-declarations/HepLean.BeyondTheStandardModel.TwoHDM.Basic.jsonl @@ -0,0 +1,29 @@ +{"name":"TwoHDM.Potential.neg_left","declaration":"/-- Negating `Φ₁` is equivalent to negating `m₁₂2`, `𝓵₆` and `𝓵₇`. -/\ntheorem TwoHDM.Potential.neg_left (P : TwoHDM.Potential) (Φ1 : StandardModel.HiggsField) (Φ2 : StandardModel.HiggsField) : TwoHDM.Potential.toFun P (-Φ1) Φ2 =\n TwoHDM.Potential.toFun\n { m₁₁2 := P.m₁₁2, m₂₂2 := P.m₂₂2, m₁₂2 := -P.m₁₂2, 𝓵₁ := P.𝓵₁, 𝓵₂ := P.𝓵₂, 𝓵₃ := P.𝓵₃, 𝓵₄ := P.𝓵₄, 𝓵₅ := P.𝓵₅,\n 𝓵₆ := -P.𝓵₆, 𝓵₇ := -P.𝓵₇ }\n Φ1 Φ2"} +{"name":"TwoHDM.Potential.𝓵₆","declaration":"/-- The parameter corresponding to `λ₆` in the 2HDM potential. -/\ndef TwoHDM.Potential.𝓵₆ (self : TwoHDM.Potential) : ℂ"} +{"name":"TwoHDM.Potential.toFun","declaration":"/-- The potential of the two Higgs doublet model. -/\ndef TwoHDM.Potential.toFun (P : TwoHDM.Potential) (Φ1 : StandardModel.HiggsField) (Φ2 : StandardModel.HiggsField) (x : SpaceTime) : ℝ"} +{"name":"TwoHDM.Potential.isBounded_of_left_eq_right","declaration":"theorem TwoHDM.Potential.isBounded_of_left_eq_right {P : TwoHDM.Potential} (h : TwoHDM.Potential.IsBounded P) : 0 ≤ P.𝓵₁ / 2 + P.𝓵₂ / 2 + P.𝓵₃ + P.𝓵₄ + P.𝓵₅.re + 2 * P.𝓵₆.re + 2 * P.𝓵₇.re"} +{"name":"TwoHDM.Potential.m₁₁2","declaration":"/-- The parameter corresponding to `m₁₁²` in the 2HDM potential. -/\ndef TwoHDM.Potential.m₁₁2 (self : TwoHDM.Potential) : ℝ"} +{"name":"TwoHDM.Potential.𝓵₂","declaration":"/-- The parameter corresponding to `λ₂` in the 2HDM potential. -/\ndef TwoHDM.Potential.𝓵₂ (self : TwoHDM.Potential) : ℝ"} +{"name":"TwoHDM.Potential.zero","declaration":"/-- The potential where all parameters are zero. -/\ndef TwoHDM.Potential.zero : TwoHDM.Potential"} +{"name":"TwoHDM.Potential.mk","declaration":"ctor TwoHDM.Potential.mk (m₁₁2 : ℝ) (m₂₂2 : ℝ) (m₁₂2 : ℂ) (𝓵₁ : ℝ) (𝓵₂ : ℝ) (𝓵₃ : ℝ) (𝓵₄ : ℝ) (𝓵₅ : ℂ) (𝓵₆ : ℂ) (𝓵₇ : ℂ) : TwoHDM.Potential"} +{"name":"TwoHDM.Potential.left_eq_right","declaration":"theorem TwoHDM.Potential.left_eq_right (P : TwoHDM.Potential) (Φ1 : StandardModel.HiggsField) : TwoHDM.Potential.toFun P Φ1 Φ1 =\n StandardModel.HiggsField.Potential.toFun\n { μ2 := -P.m₁₁2 - P.m₂₂2 + 2 * P.m₁₂2.re,\n 𝓵 := P.𝓵₁ / 2 + P.𝓵₂ / 2 + P.𝓵₃ + P.𝓵₄ + P.𝓵₅.re + 2 * P.𝓵₆.re + 2 * P.𝓵₇.re }\n Φ1"} +{"name":"TwoHDM.Potential.m₁₂2","declaration":"/-- The parameter corresponding to `m₁₂²` in the 2HDM potential. -/\ndef TwoHDM.Potential.m₁₂2 (self : TwoHDM.Potential) : ℂ"} +{"name":"TwoHDM.Potential.IsBounded","declaration":"/-- The proposition on the coefficents for a potential to be bounded. -/\ndef TwoHDM.Potential.IsBounded (P : TwoHDM.Potential) : Prop"} +{"name":"TwoHDM.Potential.𝓵₄","declaration":"/-- The parameter corresponding to `λ₄` in the 2HDM potential. -/\ndef TwoHDM.Potential.𝓵₄ (self : TwoHDM.Potential) : ℝ"} +{"name":"TwoHDM.Potential.left_eq_neg_right","declaration":"theorem TwoHDM.Potential.left_eq_neg_right (P : TwoHDM.Potential) (Φ1 : StandardModel.HiggsField) : TwoHDM.Potential.toFun P Φ1 (-Φ1) =\n StandardModel.HiggsField.Potential.toFun\n { μ2 := -P.m₁₁2 - P.m₂₂2 - 2 * P.m₁₂2.re,\n 𝓵 := P.𝓵₁ / 2 + P.𝓵₂ / 2 + P.𝓵₃ + P.𝓵₄ + P.𝓵₅.re - 2 * P.𝓵₆.re - 2 * P.𝓵₇.re }\n Φ1"} +{"name":"TwoHDM.Potential.isBounded_left_zero","declaration":"theorem TwoHDM.Potential.isBounded_left_zero {P : TwoHDM.Potential} (h : TwoHDM.Potential.IsBounded P) : StandardModel.HiggsField.Potential.IsBounded { μ2 := -P.m₂₂2, 𝓵 := P.𝓵₂ / 2 }"} +{"name":"TwoHDM.Potential.isBounded_𝓵₁_nonneg","declaration":"theorem TwoHDM.Potential.isBounded_𝓵₁_nonneg {P : TwoHDM.Potential} (h : TwoHDM.Potential.IsBounded P) : 0 ≤ P.𝓵₁"} +{"name":"TwoHDM.Potential.swap_fields","declaration":"theorem TwoHDM.Potential.swap_fields (P : TwoHDM.Potential) (Φ1 : StandardModel.HiggsField) (Φ2 : StandardModel.HiggsField) : TwoHDM.Potential.toFun P Φ1 Φ2 =\n TwoHDM.Potential.toFun\n { m₁₁2 := P.m₂₂2, m₂₂2 := P.m₁₁2, m₁₂2 := (starRingEnd ℂ) P.m₁₂2, 𝓵₁ := P.𝓵₂, 𝓵₂ := P.𝓵₁, 𝓵₃ := P.𝓵₃, 𝓵₄ := P.𝓵₄,\n 𝓵₅ := (starRingEnd ℂ) P.𝓵₅, 𝓵₆ := (starRingEnd ℂ) P.𝓵₇, 𝓵₇ := (starRingEnd ℂ) P.𝓵₆ }\n Φ2 Φ1"} +{"name":"TwoHDM.Potential.left_zero","declaration":"/-- If `Φ₁` is zero the potential reduces to the Higgs potential on `Φ₂`. -/\ntheorem TwoHDM.Potential.left_zero (P : TwoHDM.Potential) (Φ2 : StandardModel.HiggsField) : TwoHDM.Potential.toFun P 0 Φ2 = StandardModel.HiggsField.Potential.toFun { μ2 := -P.m₂₂2, 𝓵 := P.𝓵₂ / 2 } Φ2"} +{"name":"TwoHDM.Potential.𝓵₁","declaration":"/-- The parameter corresponding to `λ₁` in the 2HDM potential. -/\ndef TwoHDM.Potential.𝓵₁ (self : TwoHDM.Potential) : ℝ"} +{"name":"TwoHDM.Potential.isBounded_of_left_eq_neg_right","declaration":"theorem TwoHDM.Potential.isBounded_of_left_eq_neg_right {P : TwoHDM.Potential} (h : TwoHDM.Potential.IsBounded P) : 0 ≤ P.𝓵₁ / 2 + P.𝓵₂ / 2 + P.𝓵₃ + P.𝓵₄ + P.𝓵₅.re - 2 * P.𝓵₆.re - 2 * P.𝓵₇.re"} +{"name":"TwoHDM.Potential.𝓵₅","declaration":"/-- The parameter corresponding to `λ₅` in the 2HDM potential. -/\ndef TwoHDM.Potential.𝓵₅ (self : TwoHDM.Potential) : ℂ"} +{"name":"TwoHDM.Potential","declaration":"/-- The parameters of the Two Higgs doublet model potential. -/\nstructure TwoHDM.Potential : Type"} +{"name":"TwoHDM.Potential.𝓵₃","declaration":"/-- The parameter corresponding to `λ₃` in the 2HDM potential. -/\ndef TwoHDM.Potential.𝓵₃ (self : TwoHDM.Potential) : ℝ"} +{"name":"TwoHDM.Potential.right_zero","declaration":"/-- If `Φ₂` is zero the potential reduces to the Higgs potential on `Φ₁`. -/\ntheorem TwoHDM.Potential.right_zero (P : TwoHDM.Potential) (Φ1 : StandardModel.HiggsField) : TwoHDM.Potential.toFun P Φ1 0 = StandardModel.HiggsField.Potential.toFun { μ2 := -P.m₁₁2, 𝓵 := P.𝓵₁ / 2 } Φ1"} +{"name":"TwoHDM.Potential.isBounded_𝓵₂_nonneg","declaration":"theorem TwoHDM.Potential.isBounded_𝓵₂_nonneg {P : TwoHDM.Potential} (h : TwoHDM.Potential.IsBounded P) : 0 ≤ P.𝓵₂"} +{"name":"TwoHDM.Potential.𝓵₇","declaration":"/-- The parameter corresponding to `λ₇` in the 2HDM potential. -/\ndef TwoHDM.Potential.𝓵₇ (self : TwoHDM.Potential) : ℂ"} +{"name":"TwoHDM.Potential.isBounded_right_zero","declaration":"theorem TwoHDM.Potential.isBounded_right_zero {P : TwoHDM.Potential} (h : TwoHDM.Potential.IsBounded P) : StandardModel.HiggsField.Potential.IsBounded { μ2 := -P.m₁₁2, 𝓵 := P.𝓵₁ / 2 }"} +{"name":"TwoHDM.Potential.m₂₂2","declaration":"/-- The parameter corresponding to `m₂₂²` in the 2HDM potential. -/\ndef TwoHDM.Potential.m₂₂2 (self : TwoHDM.Potential) : ℝ"} +{"name":"TwoHDM.Potential.neg_right","declaration":"/-- Negating `Φ₁` is equivalent to negating `m₁₂2`, `𝓵₆` and `𝓵₇`. -/\ntheorem TwoHDM.Potential.neg_right (P : TwoHDM.Potential) (Φ1 : StandardModel.HiggsField) (Φ2 : StandardModel.HiggsField) : TwoHDM.Potential.toFun P Φ1 (-Φ2) =\n TwoHDM.Potential.toFun\n { m₁₁2 := P.m₁₁2, m₂₂2 := P.m₂₂2, m₁₂2 := -P.m₁₂2, 𝓵₁ := P.𝓵₁, 𝓵₂ := P.𝓵₂, 𝓵₃ := P.𝓵₃, 𝓵₄ := P.𝓵₄, 𝓵₅ := P.𝓵₅,\n 𝓵₆ := -P.𝓵₆, 𝓵₇ := -P.𝓵₇ }\n Φ1 Φ2"} +{"name":"TwoHDM.Potential.nonneg_sum_𝓵₁_to_𝓵₅_of_isBounded","declaration":"theorem TwoHDM.Potential.nonneg_sum_𝓵₁_to_𝓵₅_of_isBounded {P : TwoHDM.Potential} (h : TwoHDM.Potential.IsBounded P) : 0 ≤ P.𝓵₁ / 2 + P.𝓵₂ / 2 + P.𝓵₃ + P.𝓵₄ + P.𝓵₅.re"} diff --git a/hep-declarations/HepLean.FeynmanDiagrams.Basic.jsonl b/hep-declarations/HepLean.FeynmanDiagrams.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..199c2eaea46cd489bad5aed22c7b9bb46c4b65d8 --- /dev/null +++ b/hep-declarations/HepLean.FeynmanDiagrams.Basic.jsonl @@ -0,0 +1,166 @@ +{"name":"FeynmanDiagram.Hom.comp","declaration":"/-- The composition of two morphisms of Feynman diagrams. -/\ndef FeynmanDiagram.Hom.comp {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} {H : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) (g : FeynmanDiagram.Hom G H) : FeynmanDiagram.Hom F H"} +{"name":"FeynmanDiagram.AdjRelation","declaration":"/-- A relation on the vertices of Feynman diagrams. The proposition is true if the two\nvertices are not equal and are connected by a single edge. -/\ndef FeynmanDiagram.AdjRelation {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 F → Prop"} +{"name":"FeynmanDiagram.instFintype𝓔","declaration":"instance FeynmanDiagram.instFintype𝓔 {P : PreFeynmanRule} {F : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] : Fintype (FeynmanDiagram.𝓔 F)"} +{"name":"FeynmanDiagram.Hom.𝓔","declaration":"/-- The map `F.𝓔 → G.𝓔` induced by a homomorphism of Feynman diagrams. -/\ndef FeynmanDiagram.Hom.𝓔 {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G"} +{"name":"FeynmanDiagram.Hom","declaration":"/-- A morphism of Feynman diagrams. -/\nstructure FeynmanDiagram.Hom {P : PreFeynmanRule} (F : FeynmanDiagram P) (G : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram.Hom.comp_𝓔𝓞_left","declaration":"theorem FeynmanDiagram.Hom.comp_𝓔𝓞_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} {H : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) (g : FeynmanDiagram.Hom G H) : ∀ (a : F.𝓔𝓞.left), (FeynmanDiagram.Hom.comp f g).𝓔𝓞.left a = g.𝓔𝓞.left (f.𝓔𝓞.left a)"} +{"name":"FeynmanDiagram.func𝓥","declaration":"/-- The functor from Feynman diagrams to `Type` landing on vertices. -/\ndef FeynmanDiagram.func𝓥 {P : PreFeynmanRule} : CategoryTheory.Functor (FeynmanDiagram P) Type"} +{"name":"FeynmanDiagram.IsFiniteDiagram.𝓔Fintype","declaration":"/-- The type of edges is finite. -/\ndef FeynmanDiagram.IsFiniteDiagram.𝓔Fintype {P : PreFeynmanRule} {F : FeynmanDiagram P} [self : FeynmanDiagram.IsFiniteDiagram F] : Fintype (FeynmanDiagram.𝓔 F)"} +{"name":"FeynmanDiagram.instDecidableRel𝓥AdjToSimpleGraph","declaration":"instance FeynmanDiagram.instDecidableRel𝓥AdjToSimpleGraph {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : DecidableRel (FeynmanDiagram.toSimpleGraph F).Adj"} +{"name":"PreFeynmanRule.preimageType'_map_coe","declaration":"theorem PreFeynmanRule.preimageType'_map_coe {𝓥 : Type} (v : 𝓥) {f : CategoryTheory.Over 𝓥} {g : CategoryTheory.Over 𝓥} (F : f ⟶ g) (x : ↑(f.hom ⁻¹' {v})) : ↑((PreFeynmanRule.preimageType' v).map F x) = F.left ↑x"} +{"name":"FeynmanDiagram.edgeVertexFunc_obj_right","declaration":"theorem FeynmanDiagram.edgeVertexFunc_obj_right {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type)\n (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F))) : ((FeynmanDiagram.edgeVertexFunc 𝓔 𝓥).obj X).right = X.right"} +{"name":"FeynmanDiagram.Hom.mk'_𝓱𝓔To𝓔𝓥_left","declaration":"theorem FeynmanDiagram.Hom.mk'_𝓱𝓔To𝓔𝓥_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G) (C : FeynmanDiagram.Hom.Cond 𝓔 𝓥 𝓱𝓔) : ∀ (a : FeynmanDiagram.𝓱𝓔 F), (FeynmanDiagram.Hom.mk' 𝓔 𝓥 𝓱𝓔 C).𝓱𝓔To𝓔𝓥.left a = 𝓱𝓔 a"} +{"name":"FeynmanDiagram.edgeVertexFunc_map_left","declaration":"theorem FeynmanDiagram.edgeVertexFunc_map_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type)\n (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F))}\n (f : X ⟶ Y) (a : X.left), ((FeynmanDiagram.edgeVertexFunc 𝓔 𝓥).map f).left a = f.left a"} +{"name":"FeynmanDiagram.Hom.cond_symm","declaration":"theorem FeynmanDiagram.Hom.cond_symm {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ≃ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ≃ FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F ≃ FeynmanDiagram.𝓱𝓔 G) (h : FeynmanDiagram.Hom.Cond ⇑𝓔 ⇑𝓥 ⇑𝓱𝓔) : FeynmanDiagram.Hom.Cond ⇑𝓔.symm ⇑𝓥.symm ⇑𝓱𝓔.symm"} +{"name":"PreFeynmanRule.instFintypeLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelVertexLabelMap","declaration":"instance PreFeynmanRule.instFintypeLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelVertexLabelMap (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.VertexLabel) : Fintype (P.vertexLabelMap v).left"} +{"name":"FeynmanDiagram.𝓔Cond","declaration":"/-- Each edge has the correct type of half edges. -/\ndef FeynmanDiagram.𝓔Cond {P : PreFeynmanRule} (self : FeynmanDiagram P) : PreFeynmanRule.DiagramEdgeProp P self.𝓱𝓔To𝓔𝓥 self.𝓔𝓞.hom"} +{"name":"PreFeynmanRule.preimageVertexMapFintype","declaration":"instance PreFeynmanRule.preimageVertexMapFintype (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] {𝓔 : Type} {𝓥 : Type} [DecidableEq 𝓥] (v : 𝓥) (f : 𝓥 ⟶ P.VertexLabel) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [Fintype F.left] : Fintype ((P.vertexLabelMap (f v)).left → ((PreFeynmanRule.preimageVertex P v).obj F).left)"} +{"name":"FeynmanDiagram.fintypeProdHalfEdgeLabel𝓔𝓥","declaration":"instance FeynmanDiagram.fintypeProdHalfEdgeLabel𝓔𝓥 {P : PreFeynmanRule} {F : FeynmanDiagram P} [PreFeynmanRule.IsFinitePreFeynmanRule P] [FeynmanDiagram.IsFiniteDiagram F] : DecidableEq (P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F)"} +{"name":"FeynmanDiagram.edgeVertexFunc_map_right","declaration":"theorem FeynmanDiagram.edgeVertexFunc_map_right {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type)\n (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F))}\n (f : X ⟶ Y), ((FeynmanDiagram.edgeVertexFunc 𝓔 𝓥).map f).right = CategoryTheory.CategoryStruct.id X.right"} +{"name":"FeynmanDiagram.edgeVertexEquiv","declaration":"/-- Given equivalences `F.𝓔 ≃ G.𝓔` and `F.𝓥 ≃ G.𝓥`, the induced equivalence between\n`P.HalfEdgeLabel × F.𝓔 × F.𝓥` and `P.HalfEdgeLabel × G.𝓔 × G.𝓥 ` -/\ndef FeynmanDiagram.edgeVertexEquiv {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ≃ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ≃ FeynmanDiagram.𝓥 G) : P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F ≃ P.HalfEdgeLabel × FeynmanDiagram.𝓔 G × FeynmanDiagram.𝓥 G"} +{"name":"FeynmanDiagram.mk'_self","declaration":"theorem FeynmanDiagram.mk'_self {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.mk' F.𝓔𝓞.hom F.𝓥𝓞.hom F.𝓱𝓔To𝓔𝓥.hom ⋯ = F"} +{"name":"FeynmanDiagram.instDecidableRel𝓥AdjRelationOfIsFiniteDiagram","declaration":"instance FeynmanDiagram.instDecidableRel𝓥AdjRelationOfIsFiniteDiagram {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : DecidableRel (FeynmanDiagram.AdjRelation F)"} +{"name":"PreFeynmanRule.preimageEdgeDecidable","declaration":"instance PreFeynmanRule.preimageEdgeDecidable (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (v : 𝓔) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] : DecidableEq ((PreFeynmanRule.preimageEdge P v).obj F).left"} +{"name":"FeynmanDiagram.𝓥𝓞","declaration":"/-- The type of vertices in the Feynman diagram, labelled by their type. -/\ndef FeynmanDiagram.𝓥𝓞 {P : PreFeynmanRule} (self : FeynmanDiagram P) : CategoryTheory.Over P.VertexLabel"} +{"name":"PreFeynmanRule.instDecidableEqLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelEdgeLabelMap","declaration":"instance PreFeynmanRule.instDecidableEqLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelEdgeLabelMap (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.EdgeLabel) : DecidableEq (P.edgeLabelMap v).left"} +{"name":"PreFeynmanRule.preFeynmanRuleDecEq𝓔","declaration":"instance PreFeynmanRule.preFeynmanRuleDecEq𝓔 (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] : DecidableEq P.EdgeLabel"} +{"name":"PreFeynmanRule.instDecidableEqLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelVertexLabelMap","declaration":"instance PreFeynmanRule.instDecidableEqLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelVertexLabelMap (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.VertexLabel) : DecidableEq (P.vertexLabelMap v).left"} +{"name":"FeynmanDiagram.toSimpleGraph","declaration":"/-- From a Feynman diagram the simple graph showing those vertices which are connected. -/\ndef FeynmanDiagram.toSimpleGraph {P : PreFeynmanRule} (F : FeynmanDiagram P) : SimpleGraph (FeynmanDiagram.𝓥 F)"} +{"name":"FeynmanDiagram.𝓔","declaration":"/-- The type of edges. -/\ndef FeynmanDiagram.𝓔 {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"PreFeynmanRule.preFeynmanRuleDecEq𝓥","declaration":"instance PreFeynmanRule.preFeynmanRuleDecEq𝓥 (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] : DecidableEq P.VertexLabel"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.vertexLabelDecidable","declaration":"/-- The type of vertex labels is decidable. -/\ndef PreFeynmanRule.IsFinitePreFeynmanRule.vertexLabelDecidable {P : PreFeynmanRule} [self : PreFeynmanRule.IsFinitePreFeynmanRule P] : DecidableEq P.VertexLabel"} +{"name":"PreFeynmanRule.instFintypeLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelEdgeLabelMap","declaration":"instance PreFeynmanRule.instFintypeLeftTypeTypesDiscretePUnitDiscreteCategoryIdFromPUnitHalfEdgeLabelEdgeLabelMap (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.EdgeLabel) : Fintype (P.edgeLabelMap v).left"} +{"name":"FeynmanDiagram.func𝓱𝓔","declaration":"/-- The functor from Feynman diagrams to `Type` landing on half-edges. -/\ndef FeynmanDiagram.func𝓱𝓔 {P : PreFeynmanRule} : CategoryTheory.Functor (FeynmanDiagram P) Type"} +{"name":"FeynmanDiagram.Hom.𝓥","declaration":"/-- The map `F.𝓥 → G.𝓥` induced by a homomorphism of Feynman diagrams. -/\ndef FeynmanDiagram.Hom.𝓥 {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G"} +{"name":"PreFeynmanRule.preFeynmanRuleDecEq𝓱𝓔","declaration":"instance PreFeynmanRule.preFeynmanRuleDecEq𝓱𝓔 (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] : DecidableEq P.HalfEdgeLabel"} +{"name":"FeynmanDiagram.Hom.instDecidableCond","declaration":"instance FeynmanDiagram.Hom.instDecidableCond {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] [FeynmanDiagram.IsFiniteDiagram G] [PreFeynmanRule.IsFinitePreFeynmanRule P] (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G) : Decidable (FeynmanDiagram.Hom.Cond 𝓔 𝓥 𝓱𝓔)"} +{"name":"FeynmanDiagram.instDecidableEq𝓥OfIsFiniteDiagram","declaration":"instance FeynmanDiagram.instDecidableEq𝓥OfIsFiniteDiagram {P : PreFeynmanRule} {F : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] : DecidableEq (FeynmanDiagram.𝓥 F)"} +{"name":"FeynmanDiagram.Connected","declaration":"/-- A Feynman diagram is connected if its simple graph is connected. -/\ndef FeynmanDiagram.Connected {P : PreFeynmanRule} (F : FeynmanDiagram P) : Prop"} +{"name":"FeynmanDiagram.SymmetryType","declaration":"/-- The type of isomorphisms of a Feynman diagram. -/\ndef FeynmanDiagram.SymmetryType {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram","declaration":"/-- The type of Feynman diagrams given a `PreFeynmanRule`. -/\nstructure FeynmanDiagram (P : PreFeynmanRule) : Type 1"} +{"name":"PreFeynmanRule.toVertex","declaration":"/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)`\nto `Over (P.VertexLabel)` induced by projections on products. -/\ndef PreFeynmanRule.toVertex (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : CategoryTheory.Functor (CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (CategoryTheory.Over 𝓥)"} +{"name":"FeynmanDiagram.instCategoryFeynmanDiagram_comp_𝓥𝓞_left","declaration":"theorem FeynmanDiagram.instCategoryFeynmanDiagram_comp_𝓥𝓞_left {P : PreFeynmanRule} : ∀ {X Y Z : FeynmanDiagram P} (f : FeynmanDiagram.Hom X Y) (g : FeynmanDiagram.Hom Y Z) (a : X.𝓥𝓞.left),\n (CategoryTheory.CategoryStruct.comp f g).𝓥𝓞.left a = g.𝓥𝓞.left (f.𝓥𝓞.left a)"} +{"name":"PreFeynmanRule.toHalfEdge_map_left","declaration":"theorem PreFeynmanRule.toHalfEdge_map_left (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))}\n (f : X ⟶ Y) (a : X.left), ((PreFeynmanRule.toHalfEdge P).map f).left a = f.left a"} +{"name":"FeynmanDiagram.func𝓱𝓔𝓞","declaration":"/-- The functor from Feynman diagrams to category over half-edge labels. -/\ndef FeynmanDiagram.func𝓱𝓔𝓞 {P : PreFeynmanRule} : CategoryTheory.Functor (FeynmanDiagram P) (CategoryTheory.Over P.HalfEdgeLabel)"} +{"name":"PreFeynmanRule.VertexLabel","declaration":"/-- A type labelling the different types of vertices. -/\ndef PreFeynmanRule.VertexLabel (self : PreFeynmanRule) : Type"} +{"name":"PreFeynmanRule.preimageVertexDecidablePred","declaration":"instance PreFeynmanRule.preimageVertexDecidablePred (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} [DecidableEq 𝓥] (v : 𝓥) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) : DecidablePred fun x => x ∈ ((PreFeynmanRule.toVertex P).obj F).hom ⁻¹' {v}"} +{"name":"PreFeynmanRule.toEdge_obj_left","declaration":"theorem PreFeynmanRule.toEdge_obj_left (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ((PreFeynmanRule.toEdge P).obj X).left = X.left"} +{"name":"PreFeynmanRule.preimageType'","declaration":"/-- The functor from `Over P.VertexLabel` to `Type` induced by pull-back along insertion of\n`v : P.VertexLabel`. -/\ndef PreFeynmanRule.preimageType' {𝓥 : Type} (v : 𝓥) : CategoryTheory.Functor (CategoryTheory.Over 𝓥) Type"} +{"name":"FeynmanDiagram.Hom.comp_𝓥𝓞_left","declaration":"theorem FeynmanDiagram.Hom.comp_𝓥𝓞_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} {H : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) (g : FeynmanDiagram.Hom G H) : ∀ (a : F.𝓥𝓞.left), (FeynmanDiagram.Hom.comp f g).𝓥𝓞.left a = g.𝓥𝓞.left (f.𝓥𝓞.left a)"} +{"name":"FeynmanDiagram.Hom.𝓱𝓔","declaration":"/-- The map `F.𝓱𝓔 → G.𝓱𝓔` induced by a homomorphism of Feynman diagrams. -/\ndef FeynmanDiagram.Hom.𝓱𝓔 {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G"} +{"name":"FeynmanDiagram.symmetryFactor_eq_cardSymmetryFactor","declaration":"theorem FeynmanDiagram.symmetryFactor_eq_cardSymmetryFactor {P : PreFeynmanRule} (F : FeynmanDiagram P) [PreFeynmanRule.IsFinitePreFeynmanRule P] [FeynmanDiagram.IsFiniteDiagram F] : ↑(FeynmanDiagram.symmetryFactor F) = FeynmanDiagram.cardSymmetryFactor F"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule","declaration":"/-- A set of conditions on `PreFeynmanRule` for it to be considered finite. -/\nclass PreFeynmanRule.IsFinitePreFeynmanRule (P : PreFeynmanRule) : Type"} +{"name":"FeynmanDiagram.symmetryTypeEquiv","declaration":"/-- An equivalence between `SymmetryType` and permutation of edges, vertices and half-edges\nsatisfying `Hom.Cond`. -/\ndef FeynmanDiagram.symmetryTypeEquiv {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.SymmetryType F ≃ { S // FeynmanDiagram.Hom.Cond ⇑S.1 ⇑S.2.1 ⇑S.2.2 }"} +{"name":"FeynmanDiagram.mkIso","declaration":"/-- An isomorphism of Feynman diagrams from isomorphisms of edges, vertices and half-edges. -/\ndef FeynmanDiagram.mkIso {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ≃ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ≃ FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F ≃ FeynmanDiagram.𝓱𝓔 G) (C : FeynmanDiagram.Hom.Cond ⇑𝓔 ⇑𝓥 ⇑𝓱𝓔) : F ≅ G"} +{"name":"FeynmanDiagram.mk'","declaration":"/-- Making a Feynman diagram from maps of edges, vertices and half-edges. -/\ndef FeynmanDiagram.mk' {P : PreFeynmanRule} {𝓔 : Type} {𝓥 : Type} {𝓱𝓔 : Type} (𝓔𝓞 : 𝓔 → P.EdgeLabel) (𝓥𝓞 : 𝓥 → P.VertexLabel) (𝓱𝓔To𝓔𝓥 : 𝓱𝓔 → P.HalfEdgeLabel × 𝓔 × 𝓥) (C : FeynmanDiagram.Cond 𝓔𝓞 𝓥𝓞 𝓱𝓔To𝓔𝓥) : FeynmanDiagram P"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.vertexMapDecidable","declaration":"/-- The type of half-edges associated with a vertex is decidable. -/\ndef PreFeynmanRule.IsFinitePreFeynmanRule.vertexMapDecidable {P : PreFeynmanRule} [self : PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.VertexLabel) : DecidableEq (P.vertexLabelMap v).left"} +{"name":"PreFeynmanRule.toVertex_map_right","declaration":"theorem PreFeynmanRule.toVertex_map_right (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))}\n (f : X ⟶ Y), ((PreFeynmanRule.toVertex P).map f).right = CategoryTheory.CategoryStruct.id X.right"} +{"name":"FeynmanDiagram.instIsFiniteDiagramMk'","declaration":"instance FeynmanDiagram.instIsFiniteDiagramMk' {P : PreFeynmanRule} {𝓔 : Type} {𝓥 : Type} {𝓱𝓔 : Type} [h1 : Fintype 𝓔] [h2 : DecidableEq 𝓔] [h3 : Fintype 𝓥] [h4 : DecidableEq 𝓥] [h5 : Fintype 𝓱𝓔] [h6 : DecidableEq 𝓱𝓔] (𝓔𝓞 : 𝓔 → P.EdgeLabel) (𝓥𝓞 : 𝓥 → P.VertexLabel) (𝓱𝓔To𝓔𝓥 : 𝓱𝓔 → P.HalfEdgeLabel × 𝓔 × 𝓥) (C : FeynmanDiagram.Cond 𝓔𝓞 𝓥𝓞 𝓱𝓔To𝓔𝓥) : FeynmanDiagram.IsFiniteDiagram (FeynmanDiagram.mk' 𝓔𝓞 𝓥𝓞 𝓱𝓔To𝓔𝓥 C)"} +{"name":"PreFeynmanRule.externalDecidable","declaration":"instance PreFeynmanRule.externalDecidable (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.VertexLabel) : Decidable (PreFeynmanRule.External v)"} +{"name":"FeynmanDiagram.instCategoryFeynmanDiagram_comp_𝓔𝓞_left","declaration":"theorem FeynmanDiagram.instCategoryFeynmanDiagram_comp_𝓔𝓞_left {P : PreFeynmanRule} : ∀ {X Y Z : FeynmanDiagram P} (f : FeynmanDiagram.Hom X Y) (g : FeynmanDiagram.Hom Y Z) (a : X.𝓔𝓞.left),\n (CategoryTheory.CategoryStruct.comp f g).𝓔𝓞.left a = g.𝓔𝓞.left (f.𝓔𝓞.left a)"} +{"name":"FeynmanDiagram.Hom.instDecidableForAll𝓥EqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnitVertexLabelRightId𝓥𝓞Hom","declaration":"instance FeynmanDiagram.Hom.instDecidableForAll𝓥EqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnitVertexLabelRightId𝓥𝓞Hom {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] [PreFeynmanRule.IsFinitePreFeynmanRule P] (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) : Decidable (∀ (x : FeynmanDiagram.𝓥 F), G.𝓥𝓞.hom (𝓥 x) = F.𝓥𝓞.hom x)"} +{"name":"FeynmanDiagram.Hom.instDecidableForAll𝓔EqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnitEdgeLabelRightId𝓔𝓞Hom","declaration":"instance FeynmanDiagram.Hom.instDecidableForAll𝓔EqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnitEdgeLabelRightId𝓔𝓞Hom {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] [PreFeynmanRule.IsFinitePreFeynmanRule P] (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) : Decidable (∀ (x : FeynmanDiagram.𝓔 F), G.𝓔𝓞.hom (𝓔 x) = F.𝓔𝓞.hom x)"} +{"name":"FeynmanDiagram.𝓱𝓔𝓞","declaration":"/-- The object in Over P.HalfEdgeLabel generated by a Feynman diagram. -/\ndef FeynmanDiagram.𝓱𝓔𝓞 {P : PreFeynmanRule} (F : FeynmanDiagram P) : CategoryTheory.Over P.HalfEdgeLabel"} +{"name":"PreFeynmanRule.External","declaration":"/-- A vertex is external if it has a single half-edge associated to it. -/\ndef PreFeynmanRule.External {P : PreFeynmanRule} (v : P.VertexLabel) : Prop"} +{"name":"FeynmanDiagram.IsFiniteDiagram.𝓱𝓔DecidableEq","declaration":"/-- The type of half-edges is decidable. -/\ndef FeynmanDiagram.IsFiniteDiagram.𝓱𝓔DecidableEq {P : PreFeynmanRule} {F : FeynmanDiagram P} [self : FeynmanDiagram.IsFiniteDiagram F] : DecidableEq (FeynmanDiagram.𝓱𝓔 F)"} +{"name":"PreFeynmanRule.toHalfEdge_obj_right","declaration":"theorem PreFeynmanRule.toHalfEdge_obj_right (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ((PreFeynmanRule.toHalfEdge P).obj X).right = X.right"} +{"name":"PreFeynmanRule.DiagramEdgeProp","declaration":"/-- The proposition on edges which must be satisfied by an object\n`F : Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)` for it to be a Feynman diagram.\nThis condition corresponds to the edges of `F` having the correct half-edges associated\nwith them. -/\ndef PreFeynmanRule.DiagramEdgeProp (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (f : 𝓔 ⟶ P.EdgeLabel) : Prop"} +{"name":"FeynmanDiagram.func𝓥𝓞","declaration":"/-- The functor from Feynman diagrams to category over vertex labels. -/\ndef FeynmanDiagram.func𝓥𝓞 {P : PreFeynmanRule} : CategoryTheory.Functor (FeynmanDiagram P) (CategoryTheory.Over P.VertexLabel)"} +{"name":"FeynmanDiagram.Hom.𝓔𝓞","declaration":"/-- The morphism of edge objects. -/\ndef FeynmanDiagram.Hom.𝓔𝓞 {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (self : FeynmanDiagram.Hom F G) : F.𝓔𝓞 ⟶ G.𝓔𝓞"} +{"name":"FeynmanDiagram.𝓱𝓔","declaration":"/-- The type of half-edges. -/\ndef FeynmanDiagram.𝓱𝓔 {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.mk","declaration":"ctor PreFeynmanRule.IsFinitePreFeynmanRule.mk {P : PreFeynmanRule} (edgeLabelDecidable : DecidableEq P.EdgeLabel) (vertexLabelDecidable : DecidableEq P.VertexLabel) (halfEdgeLabelDecidable : DecidableEq P.HalfEdgeLabel) (vertexMapFintype : (v : P.VertexLabel) → Fintype (P.vertexLabelMap v).left) (vertexMapDecidable : (v : P.VertexLabel) → DecidableEq (P.vertexLabelMap v).left) (edgeMapFintype : (v : P.EdgeLabel) → Fintype (P.edgeLabelMap v).left) (edgeMapDecidable : (v : P.EdgeLabel) → DecidableEq (P.edgeLabelMap v).left) : PreFeynmanRule.IsFinitePreFeynmanRule P"} +{"name":"PreFeynmanRule.toEdge","declaration":"/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)`\nto `Over (P.EdgeLabel)` induced by projections on products. -/\ndef PreFeynmanRule.toEdge (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : CategoryTheory.Functor (CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (CategoryTheory.Over 𝓔)"} +{"name":"FeynmanDiagram.instDecidableEqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnit𝓔RightId𝓱𝓔To𝓔Hom","declaration":"instance FeynmanDiagram.instDecidableEqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnit𝓔RightId𝓱𝓔To𝓔Hom {P : PreFeynmanRule} {F : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] (i : FeynmanDiagram.𝓱𝓔 F) (j : FeynmanDiagram.𝓔 F) : Decidable ((FeynmanDiagram.𝓱𝓔To𝓔 F).hom i = j)"} +{"name":"FeynmanDiagram.Hom.Cond","declaration":"/-- The condition on maps of edges, vertices and half-edges for it to be lifted to a\nmorphism of Feynman diagrams. -/\ndef FeynmanDiagram.Hom.Cond {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G) : Prop"} +{"name":"PreFeynmanRule.mk","declaration":"ctor PreFeynmanRule.mk (HalfEdgeLabel : Type) (EdgeLabel : Type) (VertexLabel : Type) (edgeLabelMap : EdgeLabel → CategoryTheory.Over HalfEdgeLabel) (vertexLabelMap : VertexLabel → CategoryTheory.Over HalfEdgeLabel) : PreFeynmanRule"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.edgeLabelDecidable","declaration":"/-- The type of edge labels is decidable. -/\ndef PreFeynmanRule.IsFinitePreFeynmanRule.edgeLabelDecidable {P : PreFeynmanRule} [self : PreFeynmanRule.IsFinitePreFeynmanRule P] : DecidableEq P.EdgeLabel"} +{"name":"FeynmanDiagram.instDecidableConnected","declaration":"instance FeynmanDiagram.instDecidableConnected {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : Decidable (FeynmanDiagram.Connected F)"} +{"name":"FeynmanDiagram.Hom.mk'_𝓔𝓞_left","declaration":"theorem FeynmanDiagram.Hom.mk'_𝓔𝓞_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G) (C : FeynmanDiagram.Hom.Cond 𝓔 𝓥 𝓱𝓔) : ∀ (a : FeynmanDiagram.𝓔 F), (FeynmanDiagram.Hom.mk' 𝓔 𝓥 𝓱𝓔 C).𝓔𝓞.left a = 𝓔 a"} +{"name":"FeynmanDiagram.symmetryFactor","declaration":"/-- The symmetry factor of a Finite Feynman diagram, as a natural number. -/\ndef FeynmanDiagram.symmetryFactor {P : PreFeynmanRule} (F : FeynmanDiagram P) [PreFeynmanRule.IsFinitePreFeynmanRule P] [FeynmanDiagram.IsFiniteDiagram F] : ℕ"} +{"name":"FeynmanDiagram.Hom.𝓥𝓞","declaration":"/-- The morphism of vertex objects. -/\ndef FeynmanDiagram.Hom.𝓥𝓞 {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (self : FeynmanDiagram.Hom F G) : F.𝓥𝓞 ⟶ G.𝓥𝓞"} +{"name":"FeynmanDiagram.𝓱𝓔To𝓔","declaration":"/-- The map `F.𝓱𝓔 → F.𝓔` as an object in `Over F.𝓔 `. -/\ndef FeynmanDiagram.𝓱𝓔To𝓔 {P : PreFeynmanRule} (F : FeynmanDiagram P) : CategoryTheory.Over (FeynmanDiagram.𝓔 F)"} +{"name":"FeynmanDiagram.Hom.cond_satisfied","declaration":"theorem FeynmanDiagram.Hom.cond_satisfied {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) : FeynmanDiagram.Hom.Cond (FeynmanDiagram.Hom.𝓔 f) (FeynmanDiagram.Hom.𝓥 f) (FeynmanDiagram.Hom.𝓱𝓔 f)"} +{"name":"FeynmanDiagram.edgeVertexFunc_obj_left","declaration":"theorem FeynmanDiagram.edgeVertexFunc_obj_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type)\n (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F))) : ((FeynmanDiagram.edgeVertexFunc 𝓔 𝓥).obj X).left = X.left"} +{"name":"PreFeynmanRule.preimageVertexDecidable","declaration":"instance PreFeynmanRule.preimageVertexDecidable (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (v : 𝓥) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] : DecidableEq ((PreFeynmanRule.preimageVertex P v).obj F).left"} +{"name":"FeynmanDiagram.Hom.mk'","declaration":"/-- Making a Feynman diagram from maps of edges, vertices and half-edges. -/\ndef FeynmanDiagram.Hom.mk' {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G) (C : FeynmanDiagram.Hom.Cond 𝓔 𝓥 𝓱𝓔) : FeynmanDiagram.Hom F G"} +{"name":"FeynmanDiagram.Hom.𝓱𝓔To𝓔𝓥","declaration":"/-- The morphism of half-edge objects. -/\ndef FeynmanDiagram.Hom.𝓱𝓔To𝓔𝓥 {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (self : FeynmanDiagram.Hom F G) : (FeynmanDiagram.edgeVertexFunc self.𝓔𝓞.left self.𝓥𝓞.left).obj F.𝓱𝓔To𝓔𝓥 ⟶ G.𝓱𝓔To𝓔𝓥"} +{"name":"FeynmanDiagram.Cond","declaration":"/-- The condition which must be satisfied by maps to form a Feynman diagram. -/\ndef FeynmanDiagram.Cond {P : PreFeynmanRule} {𝓔 : Type} {𝓥 : Type} {𝓱𝓔 : Type} (𝓔𝓞 : 𝓔 → P.EdgeLabel) (𝓥𝓞 : 𝓥 → P.VertexLabel) (𝓱𝓔To𝓔𝓥 : 𝓱𝓔 → P.HalfEdgeLabel × 𝓔 × 𝓥) : Prop"} +{"name":"FeynmanDiagram.instCategoryFeynmanDiagram_comp_𝓱𝓔To𝓔𝓥_left","declaration":"theorem FeynmanDiagram.instCategoryFeynmanDiagram_comp_𝓱𝓔To𝓔𝓥_left {P : PreFeynmanRule} : ∀ {X Y Z : FeynmanDiagram P} (f : FeynmanDiagram.Hom X Y) (g : FeynmanDiagram.Hom Y Z)\n (a :\n ((FeynmanDiagram.edgeVertexFunc (CategoryTheory.CategoryStruct.comp f.𝓔𝓞 g.𝓔𝓞).left\n (CategoryTheory.CategoryStruct.comp f.𝓥𝓞 g.𝓥𝓞).left).obj\n X.𝓱𝓔To𝓔𝓥).left),\n (CategoryTheory.CategoryStruct.comp f g).𝓱𝓔To𝓔𝓥.left a = g.𝓱𝓔To𝓔𝓥.left (f.𝓱𝓔To𝓔𝓥.left a)"} +{"name":"FeynmanDiagram.instDecidableEq𝓱𝓔OfIsFiniteDiagram","declaration":"instance FeynmanDiagram.instDecidableEq𝓱𝓔OfIsFiniteDiagram {P : PreFeynmanRule} {F : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] : DecidableEq (FeynmanDiagram.𝓱𝓔 F)"} +{"name":"CategoryTheory.Iso.isIso_hom","declaration":"theorem CategoryTheory.Iso.isIso_hom {C : Type u} [CategoryTheory.Category.{v, u} C] {X : C} {Y : C} (e : X ≅ Y) : CategoryTheory.IsIso e.hom"} +{"name":"FeynmanDiagram.Hom.id","declaration":"/-- The identity morphism for a Feynman diagram. -/\ndef FeynmanDiagram.Hom.id {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.Hom F F"} +{"name":"PreFeynmanRule.toEdge_map_left","declaration":"theorem PreFeynmanRule.toEdge_map_left (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))}\n (f : X ⟶ Y) (a : X.left), ((PreFeynmanRule.toEdge P).map f).left a = f.left a"} +{"name":"FeynmanDiagram.edgeVertexFunc_obj_hom","declaration":"theorem FeynmanDiagram.edgeVertexFunc_obj_hom {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type)\n (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F))) : ∀ (a : (CategoryTheory.Functor.id Type).obj X.left),\n ((FeynmanDiagram.edgeVertexFunc 𝓔 𝓥).obj X).hom a = FeynmanDiagram.edgeVertexMap 𝓔 𝓥 (X.hom a)"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.halfEdgeLabelDecidable","declaration":"/-- The type of half-edge labels is decidable. -/\ndef PreFeynmanRule.IsFinitePreFeynmanRule.halfEdgeLabelDecidable {P : PreFeynmanRule} [self : PreFeynmanRule.IsFinitePreFeynmanRule P] : DecidableEq P.HalfEdgeLabel"} +{"name":"FeynmanDiagram.Hom.ext","declaration":"theorem FeynmanDiagram.Hom.ext {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} {f : FeynmanDiagram.Hom F G} {g : FeynmanDiagram.Hom F G} (h𝓔 : FeynmanDiagram.Hom.𝓔 f = FeynmanDiagram.Hom.𝓔 g) (h𝓥 : FeynmanDiagram.Hom.𝓥 f = FeynmanDiagram.Hom.𝓥 g) (h𝓱𝓔 : FeynmanDiagram.Hom.𝓱𝓔 f = FeynmanDiagram.Hom.𝓱𝓔 g) : f = g"} +{"name":"PreFeynmanRule.HalfEdgeLabel","declaration":"/-- The type labelling the different half-edges. -/\ndef PreFeynmanRule.HalfEdgeLabel (self : PreFeynmanRule) : Type"} +{"name":"FeynmanDiagram.edgeVertexMap_fst","declaration":"theorem FeynmanDiagram.edgeVertexMap_fst {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) : ∀ (a : P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F), (FeynmanDiagram.edgeVertexMap 𝓔 𝓥 a).1 = a.1"} +{"name":"FeynmanDiagram.Hom.𝓱𝓔𝓞","declaration":"/-- The morphism `F.𝓱𝓔𝓞 ⟶ G.𝓱𝓔𝓞` induced by a homomorphism of Feynman diagrams. -/\ndef FeynmanDiagram.Hom.𝓱𝓔𝓞 {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) : FeynmanDiagram.𝓱𝓔𝓞 F ⟶ FeynmanDiagram.𝓱𝓔𝓞 G"} +{"name":"FeynmanDiagram.𝓱𝓔To𝓔𝓥","declaration":"/-- The type of half-edges in the Feynman diagram, labelled by their type, the edge it\nbelongs to, and the vertex they belong to. -/\ndef FeynmanDiagram.𝓱𝓔To𝓔𝓥 {P : PreFeynmanRule} (self : FeynmanDiagram P) : CategoryTheory.Over (P.HalfEdgeLabel × self.𝓔𝓞.left × self.𝓥𝓞.left)"} +{"name":"FeynmanDiagram.instDecidableEq𝓔OfIsFiniteDiagram","declaration":"instance FeynmanDiagram.instDecidableEq𝓔OfIsFiniteDiagram {P : PreFeynmanRule} {F : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] : DecidableEq (FeynmanDiagram.𝓔 F)"} +{"name":"PreFeynmanRule.diagramEdgePropDecidable","declaration":"instance PreFeynmanRule.diagramEdgePropDecidable (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] {𝓔 : Type} {𝓥 : Type} [Fintype 𝓔] [DecidableEq 𝓔] (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] [Fintype F.left] (f : 𝓔 ⟶ P.EdgeLabel) : Decidable (PreFeynmanRule.DiagramEdgeProp P F f)"} +{"name":"FeynmanDiagram.instFintype𝓱𝓔","declaration":"instance FeynmanDiagram.instFintype𝓱𝓔 {P : PreFeynmanRule} {F : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] : Fintype (FeynmanDiagram.𝓱𝓔 F)"} +{"name":"PreFeynmanRule.diagramVertexPropDecidable","declaration":"instance PreFeynmanRule.diagramVertexPropDecidable (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] {𝓔 : Type} {𝓥 : Type} [Fintype 𝓥] [DecidableEq 𝓥] (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] [Fintype F.left] (f : 𝓥 ⟶ P.VertexLabel) : Decidable (PreFeynmanRule.DiagramVertexProp P F f)"} +{"name":"FeynmanDiagram.instFintype𝓥","declaration":"instance FeynmanDiagram.instFintype𝓥 {P : PreFeynmanRule} {F : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] : Fintype (FeynmanDiagram.𝓥 F)"} +{"name":"FeynmanDiagram.Hom.instDecidableForAll𝓱𝓔EqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnitProdHalfEdgeLabelLeftIdEdgeLabel𝓔𝓞VertexLabel𝓥𝓞Right𝓱𝓔To𝓔𝓥HomEdgeVertexMap","declaration":"instance FeynmanDiagram.Hom.instDecidableForAll𝓱𝓔EqObjDiscretePUnitToQuiverToCategoryStructDiscreteCategoryTypeToQuiverToCategoryStructTypesToPrefunctorFromPUnitProdHalfEdgeLabelLeftIdEdgeLabel𝓔𝓞VertexLabel𝓥𝓞Right𝓱𝓔To𝓔𝓥HomEdgeVertexMap {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} [FeynmanDiagram.IsFiniteDiagram F] [FeynmanDiagram.IsFiniteDiagram G] [PreFeynmanRule.IsFinitePreFeynmanRule P] (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G) : Decidable (∀ (x : FeynmanDiagram.𝓱𝓔 F), G.𝓱𝓔To𝓔𝓥.hom (𝓱𝓔 x) = FeynmanDiagram.edgeVertexMap 𝓔 𝓥 (F.𝓱𝓔To𝓔𝓥.hom x))"} +{"name":"FeynmanDiagram.IsFiniteDiagram.𝓥Fintype","declaration":"/-- The type of vertices is finite. -/\ndef FeynmanDiagram.IsFiniteDiagram.𝓥Fintype {P : PreFeynmanRule} {F : FeynmanDiagram P} [self : FeynmanDiagram.IsFiniteDiagram F] : Fintype (FeynmanDiagram.𝓥 F)"} +{"name":"FeynmanDiagram.func𝓔","declaration":"/-- The functor from Feynman diagrams to `Type` landing on edges. -/\ndef FeynmanDiagram.func𝓔 {P : PreFeynmanRule} : CategoryTheory.Functor (FeynmanDiagram P) Type"} +{"name":"PreFeynmanRule.diagramEdgeProp_iff","declaration":"theorem PreFeynmanRule.diagramEdgeProp_iff (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (f : 𝓔 ⟶ P.EdgeLabel) : PreFeynmanRule.DiagramEdgeProp P F f ↔\n ∀ (v : 𝓔), ∃ κ, Function.Bijective κ ∧ ((PreFeynmanRule.preimageEdge P v).obj F).hom ∘ κ = (P.edgeLabelMap (f v)).hom"} +{"name":"PreFeynmanRule.toVertex_map_left","declaration":"theorem PreFeynmanRule.toVertex_map_left (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))}\n (f : X ⟶ Y) (a : X.left), ((PreFeynmanRule.toVertex P).map f).left a = f.left a"} +{"name":"FeynmanDiagram.IsFiniteDiagram","declaration":"/-- A set of conditions on a Feynman diagram for it to be considered finite. -/\nclass FeynmanDiagram.IsFiniteDiagram {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram.edgeVertexMap_snd","declaration":"theorem FeynmanDiagram.edgeVertexMap_snd {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) : ∀ (a : P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F),\n (FeynmanDiagram.edgeVertexMap 𝓔 𝓥 a).2 = (𝓔 a.2.1, 𝓥 a.2.2)"} +{"name":"FeynmanDiagram.cardSymmetryFactor","declaration":"/-- The symmetry factor can be defined as the cardinal of the symmetry type.\nIn general this is not a finite number. -/\ndef FeynmanDiagram.cardSymmetryFactor {P : PreFeynmanRule} (F : FeynmanDiagram P) : Cardinal.{0}"} +{"name":"FeynmanDiagram.Hom.comp_𝓱𝓔To𝓔𝓥_left","declaration":"theorem FeynmanDiagram.Hom.comp_𝓱𝓔To𝓔𝓥_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} {H : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) (g : FeynmanDiagram.Hom G H) : ∀\n (a :\n ((FeynmanDiagram.edgeVertexFunc (CategoryTheory.CategoryStruct.comp f.𝓔𝓞 g.𝓔𝓞).left\n (CategoryTheory.CategoryStruct.comp f.𝓥𝓞 g.𝓥𝓞).left).obj\n F.𝓱𝓔To𝓔𝓥).left),\n (FeynmanDiagram.Hom.comp f g).𝓱𝓔To𝓔𝓥.left a = g.𝓱𝓔To𝓔𝓥.left (f.𝓱𝓔To𝓔𝓥.left a)"} +{"name":"PreFeynmanRule.preimageVertex","declaration":"/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)` to\n`Over P.HalfEdgeLabel` induced by pull-back along insertion of `v : P.VertexLabel`. -/\ndef PreFeynmanRule.preimageVertex (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (v : 𝓥) : CategoryTheory.Functor (CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (CategoryTheory.Over P.HalfEdgeLabel)"} +{"name":"FeynmanDiagram.edgeVertexMap","declaration":"/-- Given two maps `F.𝓔 ⟶ G.𝓔` and `F.𝓥 ⟶ G.𝓥` the corresponding map\n`P.HalfEdgeLabel × F.𝓔 × F.𝓥 → P.HalfEdgeLabel × G.𝓔 × G.𝓥`. -/\ndef FeynmanDiagram.edgeVertexMap {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) : P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F → P.HalfEdgeLabel × FeynmanDiagram.𝓔 G × FeynmanDiagram.𝓥 G"} +{"name":"FeynmanDiagram.edgeVertexFunc","declaration":"/-- The functor of over-categories generated by `edgeVertexMap`. -/\ndef FeynmanDiagram.edgeVertexFunc {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F ⟶ FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F ⟶ FeynmanDiagram.𝓥 G) : CategoryTheory.Functor (CategoryTheory.Over (P.HalfEdgeLabel × FeynmanDiagram.𝓔 F × FeynmanDiagram.𝓥 F))\n (CategoryTheory.Over (P.HalfEdgeLabel × FeynmanDiagram.𝓔 G × FeynmanDiagram.𝓥 G))"} +{"name":"PreFeynmanRule.toVertex_obj_left","declaration":"theorem PreFeynmanRule.toVertex_obj_left (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ((PreFeynmanRule.toVertex P).obj X).left = X.left"} +{"name":"PreFeynmanRule.toEdge_map_right","declaration":"theorem PreFeynmanRule.toEdge_map_right (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))}\n (f : X ⟶ Y), ((PreFeynmanRule.toEdge P).map f).right = CategoryTheory.CategoryStruct.id X.right"} +{"name":"PreFeynmanRule.toHalfEdge_obj_left","declaration":"theorem PreFeynmanRule.toHalfEdge_obj_left (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ((PreFeynmanRule.toHalfEdge P).obj X).left = X.left"} +{"name":"FeynmanDiagram.mk","declaration":"ctor FeynmanDiagram.mk {P : PreFeynmanRule} (𝓔𝓞 : CategoryTheory.Over P.EdgeLabel) (𝓥𝓞 : CategoryTheory.Over P.VertexLabel) (𝓱𝓔To𝓔𝓥 : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔𝓞.left × 𝓥𝓞.left)) (𝓔Cond : PreFeynmanRule.DiagramEdgeProp P 𝓱𝓔To𝓔𝓥 𝓔𝓞.hom) (𝓥Cond : PreFeynmanRule.DiagramVertexProp P 𝓱𝓔To𝓔𝓥 𝓥𝓞.hom) : FeynmanDiagram P"} +{"name":"FeynmanDiagram.func𝓔𝓞","declaration":"/-- The functor from Feynman diagrams to category over edge labels. -/\ndef FeynmanDiagram.func𝓔𝓞 {P : PreFeynmanRule} : CategoryTheory.Functor (FeynmanDiagram P) (CategoryTheory.Over P.EdgeLabel)"} +{"name":"FeynmanDiagram.instCategoryFeynmanDiagram_id_𝓱𝓔To𝓔𝓥_left","declaration":"theorem FeynmanDiagram.instCategoryFeynmanDiagram_id_𝓱𝓔To𝓔𝓥_left {P : PreFeynmanRule} (F : FeynmanDiagram P) : ∀ (a : F.𝓱𝓔To𝓔𝓥.left), (CategoryTheory.CategoryStruct.id F).𝓱𝓔To𝓔𝓥.left a = a"} +{"name":"FeynmanDiagram.instDecidableAndPreconnected𝓥ToSimpleGraphNonempty","declaration":"instance FeynmanDiagram.instDecidableAndPreconnected𝓥ToSimpleGraphNonempty {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : Decidable (SimpleGraph.Preconnected (FeynmanDiagram.toSimpleGraph F) ∧ Nonempty (FeynmanDiagram.𝓥 F))"} +{"name":"PreFeynmanRule.DiagramVertexProp","declaration":"/-- The proposition on vertices which must be satisfied by an object\n`F : Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)` for it to be a Feynman diagram.\nThis condition corresponds to the vertices of `F` having the correct half-edges associated\nwith them. -/\ndef PreFeynmanRule.DiagramVertexProp (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (f : 𝓥 ⟶ P.VertexLabel) : Prop"} +{"name":"PreFeynmanRule.diagramVertexProp_iff","declaration":"theorem PreFeynmanRule.diagramVertexProp_iff (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (f : 𝓥 ⟶ P.VertexLabel) : PreFeynmanRule.DiagramVertexProp P F f ↔\n ∀ (v : 𝓥),\n ∃ κ, Function.Bijective κ ∧ ((PreFeynmanRule.preimageVertex P v).obj F).hom ∘ κ = (P.vertexLabelMap (f v)).hom"} +{"name":"FeynmanDiagram.IsFiniteDiagram.𝓥DecidableEq","declaration":"/-- The type of vertices is decidable. -/\ndef FeynmanDiagram.IsFiniteDiagram.𝓥DecidableEq {P : PreFeynmanRule} {F : FeynmanDiagram P} [self : FeynmanDiagram.IsFiniteDiagram F] : DecidableEq (FeynmanDiagram.𝓥 F)"} +{"name":"PreFeynmanRule.preimageEdge","declaration":"/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)` to\n`Over P.HalfEdgeLabel` induced by pull-back along insertion of `v : P.EdgeLabel`. -/\ndef PreFeynmanRule.preimageEdge (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (v : 𝓔) : CategoryTheory.Functor (CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (CategoryTheory.Over P.HalfEdgeLabel)"} +{"name":"FeynmanDiagram.instFintypeSymmetryType","declaration":"instance FeynmanDiagram.instFintypeSymmetryType {P : PreFeynmanRule} (F : FeynmanDiagram P) [PreFeynmanRule.IsFinitePreFeynmanRule P] [FeynmanDiagram.IsFiniteDiagram F] : Fintype (FeynmanDiagram.SymmetryType F)"} +{"name":"PreFeynmanRule.toHalfEdge","declaration":"/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)`\nto `Over (P.HalfEdgeLabel)` induced by projections on products. -/\ndef PreFeynmanRule.toHalfEdge (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : CategoryTheory.Functor (CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) (CategoryTheory.Over P.HalfEdgeLabel)"} +{"name":"PreFeynmanRule.toEdge_obj_right","declaration":"theorem PreFeynmanRule.toEdge_obj_right (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ((PreFeynmanRule.toEdge P).obj X).right = X.right"} +{"name":"PreFeynmanRule.preimageType'_obj","declaration":"theorem PreFeynmanRule.preimageType'_obj {𝓥 : Type} (v : 𝓥) (f : CategoryTheory.Over 𝓥) : (PreFeynmanRule.preimageType' v).obj f = ↑(f.hom ⁻¹' {v})"} +{"name":"PreFeynmanRule.preimageEdgeDecidablePred","declaration":"instance PreFeynmanRule.preimageEdgeDecidablePred (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} [DecidableEq 𝓔] (v : 𝓔) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) : DecidablePred fun x => x ∈ ((PreFeynmanRule.toEdge P).obj F).hom ⁻¹' {v}"} +{"name":"PreFeynmanRule.toVertex_obj_hom","declaration":"theorem PreFeynmanRule.toVertex_obj_hom (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ∀ (a : (CategoryTheory.Functor.id Type).obj X.left), ((PreFeynmanRule.toVertex P).obj X).hom a = (X.hom a).2.2"} +{"name":"FeynmanDiagram.IsFiniteDiagram.𝓱𝓔Fintype","declaration":"/-- The type of half-edges is finite. -/\ndef FeynmanDiagram.IsFiniteDiagram.𝓱𝓔Fintype {P : PreFeynmanRule} {F : FeynmanDiagram P} [self : FeynmanDiagram.IsFiniteDiagram F] : Fintype (FeynmanDiagram.𝓱𝓔 F)"} +{"name":"PreFeynmanRule.external_iff_exists_bijection","declaration":"theorem PreFeynmanRule.external_iff_exists_bijection {P : PreFeynmanRule} (v : P.VertexLabel) : PreFeynmanRule.External v ↔ ∃ κ, Function.Bijective κ"} +{"name":"FeynmanDiagram.instCategoryFeynmanDiagram_id_𝓥𝓞_left","declaration":"theorem FeynmanDiagram.instCategoryFeynmanDiagram_id_𝓥𝓞_left {P : PreFeynmanRule} (F : FeynmanDiagram P) : ∀ (a : F.𝓥𝓞.left), (CategoryTheory.CategoryStruct.id F).𝓥𝓞.left a = a"} +{"name":"FeynmanDiagram.𝓥Cond","declaration":"/-- Each vertex has the correct sort of half edges. -/\ndef FeynmanDiagram.𝓥Cond {P : PreFeynmanRule} (self : FeynmanDiagram P) : PreFeynmanRule.DiagramVertexProp P self.𝓱𝓔To𝓔𝓥 self.𝓥𝓞.hom"} +{"name":"PreFeynmanRule.toEdge_obj_hom","declaration":"theorem PreFeynmanRule.toEdge_obj_hom (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ∀ (a : (CategoryTheory.Functor.id Type).obj X.left), ((PreFeynmanRule.toEdge P).obj X).hom a = (X.hom a).2.1"} +{"name":"FeynmanDiagram.cond_self","declaration":"theorem FeynmanDiagram.cond_self {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.Cond F.𝓔𝓞.hom F.𝓥𝓞.hom F.𝓱𝓔To𝓔𝓥.hom"} +{"name":"PreFeynmanRule.edgeLabelMap","declaration":"/-- A function taking `EdgeLabels` to the half edges it contains. -/\ndef PreFeynmanRule.edgeLabelMap (self : PreFeynmanRule) : self.EdgeLabel → CategoryTheory.Over self.HalfEdgeLabel"} +{"name":"PreFeynmanRule.EdgeLabel","declaration":"/-- A type labelling the different types of edges. -/\ndef PreFeynmanRule.EdgeLabel (self : PreFeynmanRule) : Type"} +{"name":"FeynmanDiagram.𝓔𝓞","declaration":"/-- The type of edges in the Feynman diagram, labelled by their type. -/\ndef FeynmanDiagram.𝓔𝓞 {P : PreFeynmanRule} (self : FeynmanDiagram P) : CategoryTheory.Over P.EdgeLabel"} +{"name":"Subtype.instDecidableEq","declaration":"instance Subtype.instDecidableEq {α : Type u} {p : α → Prop} [DecidableEq α] : DecidableEq { x // p x }"} +{"name":"FeynmanDiagram.instCategoryFeynmanDiagram_id_𝓔𝓞_left","declaration":"theorem FeynmanDiagram.instCategoryFeynmanDiagram_id_𝓔𝓞_left {P : PreFeynmanRule} (F : FeynmanDiagram P) : ∀ (a : F.𝓔𝓞.left), (CategoryTheory.CategoryStruct.id F).𝓔𝓞.left a = a"} +{"name":"PreFeynmanRule.preimageEdgeMapFintype","declaration":"instance PreFeynmanRule.preimageEdgeMapFintype (P : PreFeynmanRule) [PreFeynmanRule.IsFinitePreFeynmanRule P] {𝓔 : Type} {𝓥 : Type} [DecidableEq 𝓔] (v : 𝓔) (f : 𝓔 ⟶ P.EdgeLabel) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [Fintype F.left] : Fintype ((P.edgeLabelMap (f v)).left → ((PreFeynmanRule.preimageEdge P v).obj F).left)"} +{"name":"FeynmanDiagram.Hom.ext'","declaration":"theorem FeynmanDiagram.Hom.ext' {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} {f : FeynmanDiagram.Hom F G} {g : FeynmanDiagram.Hom F G} (h𝓔 : f.𝓔𝓞 = g.𝓔𝓞) (h𝓥 : f.𝓥𝓞 = g.𝓥𝓞) (h𝓱𝓔 : FeynmanDiagram.Hom.𝓱𝓔 f = FeynmanDiagram.Hom.𝓱𝓔 g) : f = g"} +{"name":"PreFeynmanRule.preimageEdgeFintype","declaration":"instance PreFeynmanRule.preimageEdgeFintype (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} [DecidableEq 𝓔] (v : 𝓔) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [h : Fintype F.left] : Fintype ((PreFeynmanRule.preimageEdge P v).obj F).left"} +{"name":"FeynmanDiagram.instCategoryFeynmanDiagram","declaration":"/-- Feynman diagrams form a category. -/\ninstance FeynmanDiagram.instCategoryFeynmanDiagram {P : PreFeynmanRule} : CategoryTheory.Category.{0, 1} (FeynmanDiagram P)"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.edgeMapFintype","declaration":"/-- The type of half-edges associated with an edge is finite. -/\ndef PreFeynmanRule.IsFinitePreFeynmanRule.edgeMapFintype {P : PreFeynmanRule} [self : PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.EdgeLabel) : Fintype (P.edgeLabelMap v).left"} +{"name":"FeynmanDiagram.instDecidableConnected𝓥ToSimpleGraphOfIsFiniteDiagram","declaration":"instance FeynmanDiagram.instDecidableConnected𝓥ToSimpleGraphOfIsFiniteDiagram {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : Decidable (SimpleGraph.Connected (FeynmanDiagram.toSimpleGraph F))"} +{"name":"FeynmanDiagram.Hom.mk'_𝓥𝓞_left","declaration":"theorem FeynmanDiagram.Hom.mk'_𝓥𝓞_left {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔 : FeynmanDiagram.𝓔 F → FeynmanDiagram.𝓔 G) (𝓥 : FeynmanDiagram.𝓥 F → FeynmanDiagram.𝓥 G) (𝓱𝓔 : FeynmanDiagram.𝓱𝓔 F → FeynmanDiagram.𝓱𝓔 G) (C : FeynmanDiagram.Hom.Cond 𝓔 𝓥 𝓱𝓔) : ∀ (a : FeynmanDiagram.𝓥 F), (FeynmanDiagram.Hom.mk' 𝓔 𝓥 𝓱𝓔 C).𝓥𝓞.left a = 𝓥 a"} +{"name":"PreFeynmanRule","declaration":"/-- A `PreFeynmanRule` is a set of rules specifying the allowed half-edges,\nedges and vertices in a diagram. (It does not specify how to turn the diagram\ninto an algebraic expression.) -/\nstructure PreFeynmanRule : Type 1"} +{"name":"FeynmanDiagram.IsFiniteDiagram.𝓔DecidableEq","declaration":"/-- The type of edges is decidable. -/\ndef FeynmanDiagram.IsFiniteDiagram.𝓔DecidableEq {P : PreFeynmanRule} {F : FeynmanDiagram P} [self : FeynmanDiagram.IsFiniteDiagram F] : DecidableEq (FeynmanDiagram.𝓔 F)"} +{"name":"PreFeynmanRule.toHalfEdge_map_right","declaration":"theorem PreFeynmanRule.toHalfEdge_map_right (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} : ∀\n {X Y :\n CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))}\n (f : X ⟶ Y), ((PreFeynmanRule.toHalfEdge P).map f).right = CategoryTheory.CategoryStruct.id X.right"} +{"name":"FeynmanDiagram.𝓱𝓔To𝓥","declaration":"/-- The map `F.𝓱𝓔 → F.𝓥` as an object in `Over F.𝓥 `. -/\ndef FeynmanDiagram.𝓱𝓔To𝓥 {P : PreFeynmanRule} (F : FeynmanDiagram P) : CategoryTheory.Over (FeynmanDiagram.𝓥 F)"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.vertexMapFintype","declaration":"/-- The type of half-edges associated with a vertex is finite. -/\ndef PreFeynmanRule.IsFinitePreFeynmanRule.vertexMapFintype {P : PreFeynmanRule} [self : PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.VertexLabel) : Fintype (P.vertexLabelMap v).left"} +{"name":"FeynmanDiagram.Hom.mk","declaration":"ctor FeynmanDiagram.Hom.mk {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (𝓔𝓞 : F.𝓔𝓞 ⟶ G.𝓔𝓞) (𝓥𝓞 : F.𝓥𝓞 ⟶ G.𝓥𝓞) (𝓱𝓔To𝓔𝓥 : (FeynmanDiagram.edgeVertexFunc 𝓔𝓞.left 𝓥𝓞.left).obj F.𝓱𝓔To𝓔𝓥 ⟶ G.𝓱𝓔To𝓔𝓥) : FeynmanDiagram.Hom F G"} +{"name":"PreFeynmanRule.toVertex_obj_right","declaration":"theorem PreFeynmanRule.toVertex_obj_right (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ((PreFeynmanRule.toVertex P).obj X).right = X.right"} +{"name":"FeynmanDiagram.CondDecidable","declaration":"/-- `Cond` is decidable. -/\ninstance FeynmanDiagram.CondDecidable {P : PreFeynmanRule} [PreFeynmanRule.IsFinitePreFeynmanRule P] {𝓔 : Type} {𝓥 : Type} {𝓱𝓔 : Type} (𝓔𝓞 : 𝓔 → P.EdgeLabel) (𝓥𝓞 : 𝓥 → P.VertexLabel) (𝓱𝓔To𝓔𝓥 : 𝓱𝓔 → P.HalfEdgeLabel × 𝓔 × 𝓥) [Fintype 𝓥] [DecidableEq 𝓥] [Fintype 𝓔] [DecidableEq 𝓔] [h : Fintype 𝓱𝓔] [DecidableEq 𝓱𝓔] : Decidable (FeynmanDiagram.Cond 𝓔𝓞 𝓥𝓞 𝓱𝓔To𝓔𝓥)"} +{"name":"PreFeynmanRule.vertexLabelMap","declaration":"/-- A function taking `VertexLabels` to the half edges it contains. -/\ndef PreFeynmanRule.vertexLabelMap (self : PreFeynmanRule) : self.VertexLabel → CategoryTheory.Over self.HalfEdgeLabel"} +{"name":"PreFeynmanRule.preimageVertexFintype","declaration":"instance PreFeynmanRule.preimageVertexFintype (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} [DecidableEq 𝓥] (v : 𝓥) (F : CategoryTheory.Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [h : Fintype F.left] : Fintype ((PreFeynmanRule.preimageVertex P v).obj F).left"} +{"name":"FeynmanDiagram.Hom.mk'_self","declaration":"theorem FeynmanDiagram.Hom.mk'_self {P : PreFeynmanRule} {F : FeynmanDiagram P} {G : FeynmanDiagram P} (f : FeynmanDiagram.Hom F G) : FeynmanDiagram.Hom.mk' (FeynmanDiagram.Hom.𝓔 f) (FeynmanDiagram.Hom.𝓥 f) (FeynmanDiagram.Hom.𝓱𝓔 f) ⋯ = f"} +{"name":"PreFeynmanRule.IsFinitePreFeynmanRule.edgeMapDecidable","declaration":"/-- The type of half-edges associated with an edge is decidable. -/\ndef PreFeynmanRule.IsFinitePreFeynmanRule.edgeMapDecidable {P : PreFeynmanRule} [self : PreFeynmanRule.IsFinitePreFeynmanRule P] (v : P.EdgeLabel) : DecidableEq (P.edgeLabelMap v).left"} +{"name":"FeynmanDiagram.𝓥","declaration":"/-- The type of vertices. -/\ndef FeynmanDiagram.𝓥 {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram.IsFiniteDiagram.mk","declaration":"ctor FeynmanDiagram.IsFiniteDiagram.mk {P : PreFeynmanRule} {F : FeynmanDiagram P} (𝓔Fintype : Fintype (FeynmanDiagram.𝓔 F)) (𝓔DecidableEq : DecidableEq (FeynmanDiagram.𝓔 F)) (𝓥Fintype : Fintype (FeynmanDiagram.𝓥 F)) (𝓥DecidableEq : DecidableEq (FeynmanDiagram.𝓥 F)) (𝓱𝓔Fintype : Fintype (FeynmanDiagram.𝓱𝓔 F)) (𝓱𝓔DecidableEq : DecidableEq (FeynmanDiagram.𝓱𝓔 F)) : FeynmanDiagram.IsFiniteDiagram F"} +{"name":"PreFeynmanRule.toHalfEdge_obj_hom","declaration":"theorem PreFeynmanRule.toHalfEdge_obj_hom (P : PreFeynmanRule) {𝓔 : Type} {𝓥 : Type} (X : CategoryTheory.Comma (CategoryTheory.Functor.id Type) (CategoryTheory.Functor.fromPUnit (P.HalfEdgeLabel × 𝓔 × 𝓥))) : ∀ (a : (CategoryTheory.Functor.id Type).obj X.left), ((PreFeynmanRule.toHalfEdge P).obj X).hom a = (X.hom a).1"} diff --git a/hep-declarations/HepLean.FeynmanDiagrams.Instances.ComplexScalar.jsonl b/hep-declarations/HepLean.FeynmanDiagrams.Instances.ComplexScalar.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..43d80f51378fb1527d4dc1354cd4dd63046a055f --- /dev/null +++ b/hep-declarations/HepLean.FeynmanDiagrams.Instances.ComplexScalar.jsonl @@ -0,0 +1,10 @@ +{"name":"PhiFour.instIsFinitePreFeynmanRuleComplexScalarFeynmanRules","declaration":"instance PhiFour.instIsFinitePreFeynmanRuleComplexScalarFeynmanRules : PreFeynmanRule.IsFinitePreFeynmanRule PhiFour.complexScalarFeynmanRules"} +{"name":"PhiFour.instOfNatVertexLabelComplexScalarFeynmanRules","declaration":"instance PhiFour.instOfNatVertexLabelComplexScalarFeynmanRules (a : ℕ) : OfNat PhiFour.complexScalarFeynmanRules.VertexLabel a"} +{"name":"PhiFour.complexScalarFeynmanRules_VertexLabel","declaration":"theorem PhiFour.complexScalarFeynmanRules_VertexLabel : PhiFour.complexScalarFeynmanRules.VertexLabel = Fin 3"} +{"name":"PhiFour.complexScalarFeynmanRules_HalfEdgeLabel","declaration":"theorem PhiFour.complexScalarFeynmanRules_HalfEdgeLabel : PhiFour.complexScalarFeynmanRules.HalfEdgeLabel = Fin 2"} +{"name":"PhiFour.instOfNatHalfEdgeLabelComplexScalarFeynmanRules","declaration":"instance PhiFour.instOfNatHalfEdgeLabelComplexScalarFeynmanRules (a : ℕ) : OfNat PhiFour.complexScalarFeynmanRules.HalfEdgeLabel a"} +{"name":"PhiFour.complexScalarFeynmanRules_vertexLabelMap","declaration":"theorem PhiFour.complexScalarFeynmanRules_vertexLabelMap (x : Fin 3) : PhiFour.complexScalarFeynmanRules.vertexLabelMap x =\n match x with\n | 0 => CategoryTheory.Over.mk ![0]\n | 1 => CategoryTheory.Over.mk ![1]\n | 2 => CategoryTheory.Over.mk ![0, 0, 1, 1]"} +{"name":"PhiFour.complexScalarFeynmanRules_EdgeLabel","declaration":"theorem PhiFour.complexScalarFeynmanRules_EdgeLabel : PhiFour.complexScalarFeynmanRules.EdgeLabel = Fin 1"} +{"name":"PhiFour.instOfNatEdgeLabelComplexScalarFeynmanRules","declaration":"instance PhiFour.instOfNatEdgeLabelComplexScalarFeynmanRules (a : ℕ) : OfNat PhiFour.complexScalarFeynmanRules.EdgeLabel a"} +{"name":"PhiFour.complexScalarFeynmanRules_edgeLabelMap","declaration":"theorem PhiFour.complexScalarFeynmanRules_edgeLabelMap (x : Fin 1) : PhiFour.complexScalarFeynmanRules.edgeLabelMap x =\n match x with\n | 0 => CategoryTheory.Over.mk ![0, 1]"} +{"name":"PhiFour.complexScalarFeynmanRules","declaration":"/-- The pre-Feynman rules for a complex scalar theory. -/\ndef PhiFour.complexScalarFeynmanRules : PreFeynmanRule"} diff --git a/hep-declarations/HepLean.FeynmanDiagrams.Instances.Phi4.jsonl b/hep-declarations/HepLean.FeynmanDiagrams.Instances.Phi4.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e314cdb75c5eff623e8f424d826626ccec58f0bd --- /dev/null +++ b/hep-declarations/HepLean.FeynmanDiagrams.Instances.Phi4.jsonl @@ -0,0 +1,13 @@ +{"name":"PhiFour.phi4PreFeynmanRules","declaration":"/-- The pre-Feynman rules for `Phi^4` theory. -/\ndef PhiFour.phi4PreFeynmanRules : PreFeynmanRule"} +{"name":"PhiFour.figureEight_symmetryFactor","declaration":"/-- The symmetry factor of `figureEight` is 8. We can get this from\n`#eval symmetryFactor figureEight`. -/\ntheorem PhiFour.figureEight_symmetryFactor : FeynmanDiagram.symmetryFactor PhiFour.figureEight = 8"} +{"name":"PhiFour.instOfNatHalfEdgeLabelPhi4PreFeynmanRules","declaration":"instance PhiFour.instOfNatHalfEdgeLabelPhi4PreFeynmanRules (a : ℕ) : OfNat PhiFour.phi4PreFeynmanRules.HalfEdgeLabel a"} +{"name":"PhiFour.phi4PreFeynmanRules_HalfEdgeLabel","declaration":"theorem PhiFour.phi4PreFeynmanRules_HalfEdgeLabel : PhiFour.phi4PreFeynmanRules.HalfEdgeLabel = Fin 1"} +{"name":"PhiFour.instOfNatVertexLabelPhi4PreFeynmanRules","declaration":"instance PhiFour.instOfNatVertexLabelPhi4PreFeynmanRules (a : ℕ) : OfNat PhiFour.phi4PreFeynmanRules.VertexLabel a"} +{"name":"PhiFour.instOfNatEdgeLabelPhi4PreFeynmanRules","declaration":"instance PhiFour.instOfNatEdgeLabelPhi4PreFeynmanRules (a : ℕ) : OfNat PhiFour.phi4PreFeynmanRules.EdgeLabel a"} +{"name":"PhiFour.instIsFinitePreFeynmanRulePhi4PreFeynmanRules","declaration":"instance PhiFour.instIsFinitePreFeynmanRulePhi4PreFeynmanRules : PreFeynmanRule.IsFinitePreFeynmanRule PhiFour.phi4PreFeynmanRules"} +{"name":"PhiFour.phi4PreFeynmanRules_edgeLabelMap","declaration":"theorem PhiFour.phi4PreFeynmanRules_edgeLabelMap (x : Fin 1) : PhiFour.phi4PreFeynmanRules.edgeLabelMap x =\n match x with\n | 0 => CategoryTheory.Over.mk ![0, 0]"} +{"name":"PhiFour.figureEight","declaration":"/-- The figure eight Feynman diagram. -/\ndef PhiFour.figureEight : FeynmanDiagram PhiFour.phi4PreFeynmanRules"} +{"name":"PhiFour.figureEight_connected","declaration":"/-- `figureEight` is connected. We can get this from\n`#eval Connected figureEight`. -/\ntheorem PhiFour.figureEight_connected : FeynmanDiagram.Connected PhiFour.figureEight"} +{"name":"PhiFour.phi4PreFeynmanRules_vertexLabelMap","declaration":"theorem PhiFour.phi4PreFeynmanRules_vertexLabelMap (x : Fin 2) : PhiFour.phi4PreFeynmanRules.vertexLabelMap x =\n match x with\n | 0 => CategoryTheory.Over.mk ![0]\n | 1 => CategoryTheory.Over.mk ![0, 0, 0, 0]"} +{"name":"PhiFour.phi4PreFeynmanRules_EdgeLabel","declaration":"theorem PhiFour.phi4PreFeynmanRules_EdgeLabel : PhiFour.phi4PreFeynmanRules.EdgeLabel = Fin 1"} +{"name":"PhiFour.phi4PreFeynmanRules_VertexLabel","declaration":"theorem PhiFour.phi4PreFeynmanRules_VertexLabel : PhiFour.phi4PreFeynmanRules.VertexLabel = Fin 2"} diff --git a/hep-declarations/HepLean.FeynmanDiagrams.Momentum.jsonl b/hep-declarations/HepLean.FeynmanDiagrams.Momentum.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7c43faeb23fddc9dacc76c5d2efbafd111a1313c --- /dev/null +++ b/hep-declarations/HepLean.FeynmanDiagrams.Momentum.jsonl @@ -0,0 +1,24 @@ +{"name":"FeynmanDiagram.euclidInner","declaration":"/-- The Euclidean inner product on `F.HalfEdgeMomenta`. -/\ndef FeynmanDiagram.euclidInner {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : FeynmanDiagram.HalfEdgeMomenta F →ₗ[ℝ] FeynmanDiagram.HalfEdgeMomenta F →ₗ[ℝ] ℝ"} +{"name":"FeynmanDiagram.EdgeVertexMomentaMap","declaration":"/-- The map from `Fin 2` to `Type` landing on `EdgeMomenta` and `VertexMomenta`. -/\ndef FeynmanDiagram.EdgeVertexMomentaMap {P : PreFeynmanRule} (F : FeynmanDiagram P) : Fin 2 → Type"} +{"name":"FeynmanDiagram.vertexToHalfEdgeMomenta","declaration":"/-- The linear map from `F.VertexMomenta` to `F.HalfEdgeMomenta` induced by\nthe map `F.𝓱𝓔To𝓥.hom`. -/\ndef FeynmanDiagram.vertexToHalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.VertexMomenta F →ₗ[ℝ] FeynmanDiagram.HalfEdgeMomenta F"} +{"name":"FeynmanDiagram.instAddCommGroupEdgeMomenta","declaration":"instance FeynmanDiagram.instAddCommGroupEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : AddCommGroup (FeynmanDiagram.EdgeMomenta F)"} +{"name":"FeynmanDiagram.euclidInnerAux","declaration":"/-- An auxiliary function used to define the Euclidean inner product on `F.HalfEdgeMomenta`. -/\ndef FeynmanDiagram.euclidInnerAux {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] (x : FeynmanDiagram.HalfEdgeMomenta F) : FeynmanDiagram.HalfEdgeMomenta F →ₗ[ℝ] ℝ"} +{"name":"FeynmanDiagram.instAddCommGroupEdgeVertexMomenta","declaration":"instance FeynmanDiagram.instAddCommGroupEdgeVertexMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : AddCommGroup (FeynmanDiagram.EdgeVertexMomenta F)"} +{"name":"FeynmanDiagram.instModuleRealHalfEdgeMomentaSemiringToAddCommMonoidInstAddCommGroupHalfEdgeMomenta","declaration":"instance FeynmanDiagram.instModuleRealHalfEdgeMomentaSemiringToAddCommMonoidInstAddCommGroupHalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Module ℝ (FeynmanDiagram.HalfEdgeMomenta F)"} +{"name":"FeynmanDiagram.edgeToHalfEdgeMomenta","declaration":"/-- The linear map from `F.EdgeMomenta` to `F.HalfEdgeMomenta` induced by\nthe map `F.𝓱𝓔To𝓔.hom`. -/\ndef FeynmanDiagram.edgeToHalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.EdgeMomenta F →ₗ[ℝ] FeynmanDiagram.HalfEdgeMomenta F"} +{"name":"FeynmanDiagram.orthogHalfEdgeMomenta","declaration":"/-- The submodule of `F.HalfEdgeMomenta` corresponding to the range of\n`F.edgeVertexToHalfEdgeMomenta`. -/\ndef FeynmanDiagram.orthogHalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Submodule ℝ (FeynmanDiagram.HalfEdgeMomenta F)"} +{"name":"FeynmanDiagram.VertexMomenta","declaration":"/-- The type which assocaites to each ege a `1`-dimensional vector space.\nCorresponding to that spanned by its total inflowing momentum. -/\ndef FeynmanDiagram.VertexMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram.euclidInnerAux_symm","declaration":"theorem FeynmanDiagram.euclidInnerAux_symm {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] (x : FeynmanDiagram.HalfEdgeMomenta F) (y : FeynmanDiagram.HalfEdgeMomenta F) : (FeynmanDiagram.euclidInnerAux F x) y = (FeynmanDiagram.euclidInnerAux F y) x"} +{"name":"FeynmanDiagram.instModuleRealEdgeVertexMomentaSemiringToAddCommMonoidInstAddCommGroupEdgeVertexMomenta","declaration":"instance FeynmanDiagram.instModuleRealEdgeVertexMomentaSemiringToAddCommMonoidInstAddCommGroupEdgeVertexMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Module ℝ (FeynmanDiagram.EdgeVertexMomenta F)"} +{"name":"FeynmanDiagram.instModuleRealEdgeVertexMomentaMapSemiringToAddCommMonoidInstAddCommGroupEdgeVertexMomentaMap","declaration":"instance FeynmanDiagram.instModuleRealEdgeVertexMomentaMapSemiringToAddCommMonoidInstAddCommGroupEdgeVertexMomentaMap {P : PreFeynmanRule} (F : FeynmanDiagram P) (i : Fin 2) : Module ℝ (FeynmanDiagram.EdgeVertexMomentaMap F i)"} +{"name":"FeynmanDiagram.instAddCommGroupEdgeVertexMomentaMap","declaration":"instance FeynmanDiagram.instAddCommGroupEdgeVertexMomentaMap {P : PreFeynmanRule} (F : FeynmanDiagram P) (i : Fin 2) : AddCommGroup (FeynmanDiagram.EdgeVertexMomentaMap F i)"} +{"name":"FeynmanDiagram.edgeVertexToHalfEdgeMomenta","declaration":"/-- The linear map from `F.EdgeVertexMomenta` to `F.HalfEdgeMomenta` induced by\n`F.edgeToHalfEdgeMomenta` and `F.vertexToHalfEdgeMomenta`. -/\ndef FeynmanDiagram.edgeVertexToHalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : FeynmanDiagram.EdgeVertexMomenta F →ₗ[ℝ] FeynmanDiagram.HalfEdgeMomenta F"} +{"name":"FeynmanDiagram.instAddCommGroupVertexMomenta","declaration":"instance FeynmanDiagram.instAddCommGroupVertexMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : AddCommGroup (FeynmanDiagram.VertexMomenta F)"} +{"name":"FeynmanDiagram.instAddCommGroupHalfEdgeMomenta","declaration":"instance FeynmanDiagram.instAddCommGroupHalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : AddCommGroup (FeynmanDiagram.HalfEdgeMomenta F)"} +{"name":"FeynmanDiagram.numberOfLoops","declaration":"/-- The number of loops of a Feynman diagram. Defined as the dimension\nof the space of allowed Half-loop momenta. -/\ndef FeynmanDiagram.numberOfLoops {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : ℕ"} +{"name":"FeynmanDiagram.instModuleRealEdgeMomenta","declaration":"instance FeynmanDiagram.instModuleRealEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Module ℝ (FeynmanDiagram.EdgeMomenta F)"} +{"name":"FeynmanDiagram.EdgeMomenta","declaration":"/-- The type which associates to each edge a `1`-dimensional vector space.\nCorresponding to that spanned by its total outflowing momentum. -/\ndef FeynmanDiagram.EdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram.allowedHalfEdgeMomenta","declaration":"/-- The submodule of `F.HalfEdgeMomenta` corresponding to the allowed momenta. -/\ndef FeynmanDiagram.allowedHalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) [FeynmanDiagram.IsFiniteDiagram F] : Submodule ℝ (FeynmanDiagram.HalfEdgeMomenta F)"} +{"name":"FeynmanDiagram.EdgeVertexMomenta","declaration":"/-- The direct sum of `EdgeMomenta` and `VertexMomenta`. -/\ndef FeynmanDiagram.EdgeVertexMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram.HalfEdgeMomenta","declaration":"/-- The type which assocaites to each half-edge a `1`-dimensional vector space.\nCorresponding to that spanned by its momentum. -/\ndef FeynmanDiagram.HalfEdgeMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Type"} +{"name":"FeynmanDiagram.instModuleRealVertexMomenta","declaration":"instance FeynmanDiagram.instModuleRealVertexMomenta {P : PreFeynmanRule} (F : FeynmanDiagram P) : Module ℝ (FeynmanDiagram.VertexMomenta F)"} diff --git a/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Basic.jsonl b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..485176703f91b5ede7b5fcecb7fb44a69839a99a --- /dev/null +++ b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Basic.jsonl @@ -0,0 +1,62 @@ +{"name":"phaseShiftApply.td","declaration":"theorem phaseShiftApply.td (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 2 0 = Complex.exp (↑c * Complex.I + ↑d * Complex.I) * ↑V 2 0"} +{"name":"CKMMatrix.Rcdcb","declaration":"/-- The ratio of the `cd` and `cb` elements of a CKM matrix. -/\ndef CKMMatrix.Rcdcb (V : CKMMatrix) : ℂ"} +{"name":"phaseShiftRelation_refl","declaration":"theorem phaseShiftRelation_refl (U : ↥(Matrix.unitaryGroup (Fin 3) ℂ)) : PhaseShiftRelation U U"} +{"name":"CKMMatrix.ud_us_ratio","declaration":"/-- The ratio of the `ud` and `us` elements of a CKM matrix. -/\ndef CKMMatrix.ud_us_ratio : Lean.ParserDescr"} +{"name":"phaseShiftRelation_trans","declaration":"theorem phaseShiftRelation_trans {U : ↥(Matrix.unitaryGroup (Fin 3) ℂ)} {V : ↥(Matrix.unitaryGroup (Fin 3) ℂ)} {W : ↥(Matrix.unitaryGroup (Fin 3) ℂ)} : PhaseShiftRelation U V → PhaseShiftRelation V W → PhaseShiftRelation U W"} +{"name":"CKMMatrix.cs_element","declaration":"/-- The `cs`th element of the CKM matrix. -/\ndef CKMMatrix.cs_element : Lean.ParserDescr"} +{"name":"CKMMatrix.us_ud_ratio","declaration":"/-- The ratio of the `us` and `ud` elements of a CKM matrix. -/\ndef CKMMatrix.us_ud_ratio : Lean.ParserDescr"} +{"name":"phaseShiftApply","declaration":"/-- The matrix obtained from `V` by shifting the phases of the fermions. -/\ndef phaseShiftApply (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : CKMMatrix"} +{"name":"CKMMatrix.cb_element","declaration":"/-- The `cb`th element of the CKM matrix. -/\ndef CKMMatrix.cb_element : Lean.ParserDescr"} +{"name":"VAbs'_equiv","declaration":"theorem VAbs'_equiv (i : Fin 3) (j : Fin 3) (V : CKMMatrix) (U : CKMMatrix) (h : V ≈ U) : VAbs' V i j = VAbs' U i j"} +{"name":"VtsAbs","declaration":"/-- The absolute value of the `ts`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VtsAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"VudAbs","declaration":"/-- The absolute value of the `ud`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VudAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"CKMMatrix.us_element","declaration":"/-- The `us`th element of the CKM matrix. -/\ndef CKMMatrix.us_element : Lean.ParserDescr"} +{"name":"CKMMatrix.ub_ud_ratio","declaration":"/-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/\ndef CKMMatrix.ub_ud_ratio : Lean.ParserDescr"} +{"name":"CKMMatrixSetoid","declaration":"instance CKMMatrixSetoid : Setoid CKMMatrix"} +{"name":"CKMMatrix.td_element","declaration":"/-- The `td`th element of the CKM matrix. -/\ndef CKMMatrix.td_element : Lean.ParserDescr"} +{"name":"CKMMatrix.ud_element","declaration":"/-- The `ud`th element of the CKM matrix. -/\ndef CKMMatrix.ud_element : Lean.ParserDescr"} +{"name":"phaseShiftApply.us","declaration":"theorem phaseShiftApply.us (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 0 1 = Complex.exp (↑a * Complex.I + ↑e * Complex.I) * ↑V 0 1"} +{"name":"phaseShiftApply.cs","declaration":"theorem phaseShiftApply.cs (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 1 1 = Complex.exp (↑b * Complex.I + ↑e * Complex.I) * ↑V 1 1"} +{"name":"CKMMatrix.Rcdcb_mul_cb","declaration":"theorem CKMMatrix.Rcdcb_mul_cb {V : CKMMatrix} (h : ↑V 1 2 ≠ 0) : ↑V 1 0 = CKMMatrix.Rcdcb V * ↑V 1 2"} +{"name":"phaseShiftApply.ub","declaration":"theorem phaseShiftApply.ub (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 0 2 = Complex.exp (↑a * Complex.I + ↑f * Complex.I) * ↑V 0 2"} +{"name":"CKMMatrix.Rcscb","declaration":"/-- The ratio of the `cs` and `cb` elements of a CKM matrix. -/\ndef CKMMatrix.Rcscb (V : CKMMatrix) : ℂ"} +{"name":"VusAbs","declaration":"/-- The absolute value of the `us`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VusAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"VcsAbs","declaration":"/-- The absolute value of the `cs`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VcsAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"VAbs'","declaration":"/-- The aboslute value of the `(i,j)`th element of `V`. -/\ndef VAbs' (V : ↥(Matrix.unitaryGroup (Fin 3) ℂ)) (i : Fin 3) (j : Fin 3) : ℝ"} +{"name":"CKMMatrix.Rcscb_mul_cb","declaration":"theorem CKMMatrix.Rcscb_mul_cb {V : CKMMatrix} (h : ↑V 1 2 ≠ 0) : ↑V 1 1 = CKMMatrix.Rcscb V * ↑V 1 2"} +{"name":"phaseShiftApply.ts","declaration":"theorem phaseShiftApply.ts (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 2 1 = Complex.exp (↑c * Complex.I + ↑e * Complex.I) * ↑V 2 1"} +{"name":"CKMMatrix.cd_cb_ratio","declaration":"/-- The ratio of the `cd` and `cb` elements of a CKM matrix. -/\ndef CKMMatrix.cd_cb_ratio : Lean.ParserDescr"} +{"name":"VcbAbs","declaration":"/-- The absolute value of the `cb`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VcbAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"phaseShiftRelation_symm","declaration":"theorem phaseShiftRelation_symm {U : ↥(Matrix.unitaryGroup (Fin 3) ℂ)} {V : ↥(Matrix.unitaryGroup (Fin 3) ℂ)} : PhaseShiftRelation U V → PhaseShiftRelation V U"} +{"name":"CKMMatrix.ts_element","declaration":"/-- The `ts`th element of the CKM matrix. -/\ndef CKMMatrix.ts_element : Lean.ParserDescr"} +{"name":"phaseShiftMatrix_one","declaration":"theorem phaseShiftMatrix_one : phaseShiftMatrix 0 0 0 = 1"} +{"name":"phaseShiftMatrix_star","declaration":"theorem phaseShiftMatrix_star (a : ℝ) (b : ℝ) (c : ℝ) : Matrix.conjTranspose (phaseShiftMatrix a b c) = phaseShiftMatrix (-a) (-b) (-c)"} +{"name":"phaseShiftApply_coe","declaration":"theorem phaseShiftApply_coe (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) =\n ![![Complex.exp (Complex.I * ↑a), 0, 0], ![0, Complex.exp (Complex.I * ↑b), 0],\n ![0, 0, Complex.exp (Complex.I * ↑c)]] *\n ↑V *\n ![![Complex.exp (Complex.I * ↑d), 0, 0], ![0, Complex.exp (Complex.I * ↑e), 0],\n ![0, 0, Complex.exp (Complex.I * ↑f)]]"} +{"name":"phaseShiftApply.tb","declaration":"theorem phaseShiftApply.tb (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 2 2 = Complex.exp (↑c * Complex.I + ↑f * Complex.I) * ↑V 2 2"} +{"name":"phaseShiftApply.cb","declaration":"theorem phaseShiftApply.cb (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 1 2 = Complex.exp (↑b * Complex.I + ↑f * Complex.I) * ↑V 1 2"} +{"name":"CKMMatrix_ext","declaration":"theorem CKMMatrix_ext {U : CKMMatrix} {V : CKMMatrix} (h : ↑U = ↑V) : U = V"} +{"name":"VAbs","declaration":"/-- The absolute value of the `(i,j)`th any representative of `⟦V⟧`. -/\ndef VAbs (i : Fin 3) (j : Fin 3) : Quotient CKMMatrixSetoid → ℝ"} +{"name":"phaseShiftRelation_equiv","declaration":"theorem phaseShiftRelation_equiv : Equivalence PhaseShiftRelation"} +{"name":"CKMMatrix.Rubud","declaration":"/-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/\ndef CKMMatrix.Rubud (V : CKMMatrix) : ℂ"} +{"name":"CKMMatrix.ub_element","declaration":"/-- The `ub`th element of the CKM matrix. -/\ndef CKMMatrix.ub_element : Lean.ParserDescr"} +{"name":"VcdAbs","declaration":"/-- The absolute value of the `cd`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VcdAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"VtdAbs","declaration":"/-- The absolute value of the `td`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VtdAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"CKMMatrix.Rubus","declaration":"/-- The ratio of the `ub` and `us` elements of a CKM matrix. -/\ndef CKMMatrix.Rubus (V : CKMMatrix) : ℂ"} +{"name":"phaseShiftMatrix","declaration":"/-- Given three real numbers `a b c` the complex matrix with `exp (I * a)` etc on the\nleading diagonal. -/\ndef phaseShiftMatrix (a : ℝ) (b : ℝ) (c : ℝ) : Matrix (Fin 3) (Fin 3) ℂ"} +{"name":"CKMMatrix.Rudus","declaration":"/-- The ratio of the `ud` and `us` elements of a CKM matrix. -/\ndef CKMMatrix.Rudus (V : CKMMatrix) : ℂ"} +{"name":"CKMMatrix","declaration":"/-- The type of CKM matrices. -/\ndef CKMMatrix : Type"} +{"name":"PhaseShiftRelation","declaration":"/-- The equivalence relation between CKM matrices. -/\ndef PhaseShiftRelation (U : ↥(Matrix.unitaryGroup (Fin 3) ℂ)) (V : ↥(Matrix.unitaryGroup (Fin 3) ℂ)) : Prop"} +{"name":"phaseShift_coe","declaration":"theorem phaseShift_coe (a : ℝ) (b : ℝ) (c : ℝ) : ↑(phaseShift a b c) =\n ![![Complex.exp (Complex.I * ↑a), 0, 0], ![0, Complex.exp (Complex.I * ↑b), 0], ![0, 0, Complex.exp (Complex.I * ↑c)]]"} +{"name":"phaseShiftMatrix_mul","declaration":"theorem phaseShiftMatrix_mul (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : phaseShiftMatrix a b c * phaseShiftMatrix d e f = phaseShiftMatrix (a + d) (b + e) (c + f)"} +{"name":"phaseShiftApply.ud","declaration":"theorem phaseShiftApply.ud (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 0 0 = Complex.exp (↑a * Complex.I + ↑d * Complex.I) * ↑V 0 0"} +{"name":"VubAbs","declaration":"/-- The absolute value of the `ub`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VubAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"phaseShiftApply.equiv","declaration":"theorem phaseShiftApply.equiv (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : V ≈ phaseShiftApply V a b c d e f"} +{"name":"CKMMatrix.Rusud","declaration":"/-- The ratio of the `us` and `ud` elements of a CKM matrix. -/\ndef CKMMatrix.Rusud (V : CKMMatrix) : ℂ"} +{"name":"CKMMatrix.ub_us_ratio","declaration":"/-- The ratio of the `ub` and `us` elements of a CKM matrix. -/\ndef CKMMatrix.ub_us_ratio : Lean.ParserDescr"} +{"name":"phaseShift","declaration":"/-- Given three real numbers `a b c` the unitary matrix with `exp (I * a)` etc on the\nleading diagonal. -/\ndef phaseShift (a : ℝ) (b : ℝ) (c : ℝ) : ↥(Matrix.unitaryGroup (Fin 3) ℂ)"} +{"name":"CKMMatrix.cd_element","declaration":"/-- The `cd`th element of the CKM matrix. -/\ndef CKMMatrix.cd_element : Lean.ParserDescr"} +{"name":"CKMMatrix.cs_cb_ratio","declaration":"/-- The ratio of the `cs` and `cb` elements of a CKM matrix. -/\ndef CKMMatrix.cs_cb_ratio : Lean.ParserDescr"} +{"name":"phaseShift_coe_matrix","declaration":"theorem phaseShift_coe_matrix (a : ℝ) (b : ℝ) (c : ℝ) : ↑(phaseShift a b c) = phaseShiftMatrix a b c"} +{"name":"VtbAbs","declaration":"/-- The absolute value of the `tb`th element of a representative of an equivalence class of\nCKM matrices. -/\ndef VtbAbs : Quotient CKMMatrixSetoid → ℝ"} +{"name":"CKMMatrix.tb_element","declaration":"/-- The `tb`th element of the CKM matrix. -/\ndef CKMMatrix.tb_element : Lean.ParserDescr"} +{"name":"phaseShiftApply.cd","declaration":"theorem phaseShiftApply.cd (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : ↑(phaseShiftApply V a b c d e f) 1 0 = Complex.exp (↑b * Complex.I + ↑d * Complex.I) * ↑V 1 0"} diff --git a/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Invariants.jsonl b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Invariants.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7a77a3f8d132cb586c914cab6e5aeb5391af3c5e --- /dev/null +++ b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Invariants.jsonl @@ -0,0 +1,7 @@ +{"name":"Invariant.jarlskogℂCKM_re","declaration":"theorem Invariant.jarlskogℂCKM_re (V : CKMMatrix) : (Invariant.jarlskogℂCKM V).re =\n (((↑V 0 1).re * (↑V 1 2).re - (↑V 0 1).im * (↑V 1 2).im) * (↑V 0 2).re +\n ((↑V 0 1).re * (↑V 1 2).im + (↑V 0 1).im * (↑V 1 2).re) * (↑V 0 2).im) *\n (↑V 1 1).re +\n (-(((↑V 0 1).re * (↑V 1 2).re - (↑V 0 1).im * (↑V 1 2).im) * (↑V 0 2).im) +\n ((↑V 0 1).re * (↑V 1 2).im + (↑V 0 1).im * (↑V 1 2).re) * (↑V 0 2).re) *\n (↑V 1 1).im"} +{"name":"Invariant.jarlskogℂCKM_equiv","declaration":"theorem Invariant.jarlskogℂCKM_equiv (V : CKMMatrix) (U : CKMMatrix) (h : V ≈ U) : Invariant.jarlskogℂCKM V = Invariant.jarlskogℂCKM U"} +{"name":"Invariant.VusVubVcdSq","declaration":"/-- An invariant for CKM mtrices corresponding to the square of the absolute values\nof the `us`, `ub` and `cb` elements multipled together divied by `(VudAbs V ^ 2 + VusAbs V ^2)` .\n-/\ndef Invariant.VusVubVcdSq (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"Invariant.jarlskogℂCKM","declaration":"/-- The complex jarlskog invariant for a CKM matrix. -/\ndef Invariant.jarlskogℂCKM (V : CKMMatrix) : ℂ"} +{"name":"Invariant.jarlskogℂ","declaration":"/-- The complex jarlskog invariant for an equivalence class of CKM matrices. -/\ndef Invariant.jarlskogℂ : Quotient CKMMatrixSetoid → ℂ"} +{"name":"Invariant.jarlskogℂCKM_im","declaration":"theorem Invariant.jarlskogℂCKM_im (V : CKMMatrix) : (Invariant.jarlskogℂCKM V).im =\n -((((↑V 0 1).re * (↑V 1 2).re - (↑V 0 1).im * (↑V 1 2).im) * (↑V 0 2).re +\n ((↑V 0 1).re * (↑V 1 2).im + (↑V 0 1).im * (↑V 1 2).re) * (↑V 0 2).im) *\n (↑V 1 1).im) +\n (-(((↑V 0 1).re * (↑V 1 2).re - (↑V 0 1).im * (↑V 1 2).im) * (↑V 0 2).im) +\n ((↑V 0 1).re * (↑V 1 2).im + (↑V 0 1).im * (↑V 1 2).re) * (↑V 0 2).re) *\n (↑V 1 1).re"} +{"name":"Invariant.mulExpδ₁₃","declaration":"/-- An invariant for CKM matrices. The argument of this invariant is `δ₁₃` in the\nstandard parameterization. -/\ndef Invariant.mulExpδ₁₃ (V : Quotient CKMMatrixSetoid) : ℂ"} diff --git a/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom.jsonl b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..03b1984d40c43129810efac992e3c430338e4afe --- /dev/null +++ b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom.jsonl @@ -0,0 +1,16 @@ +{"name":"CKMMatrix.ubOnePhaseCond_hold_up_to_equiv_of_ub_one","declaration":"theorem CKMMatrix.ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)) (hV : CKMMatrix.FstRowThdColRealCond V) : ∃ U, V ≈ U ∧ CKMMatrix.ubOnePhaseCond U"} +{"name":"CKMMatrix.cd_of_fstRowThdColRealCond","declaration":"theorem CKMMatrix.cd_of_fstRowThdColRealCond {V : CKMMatrix} (hb : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) (hV : CKMMatrix.FstRowThdColRealCond V) : ↑V 1 0 =\n -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +\n -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *\n Complex.exp (-↑(Complex.arg (↑V 0 2)) * Complex.I)"} +{"name":"CKMMatrix.shift_ub_phase_zero","declaration":"theorem CKMMatrix.shift_ub_phase_zero (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) {V : CKMMatrix} (h1 : u + b = -Complex.arg (↑V 0 2)) : ↑(phaseShiftApply V u c t d s b) 0 2 = ↑(VubAbs ⟦V⟧)"} +{"name":"CKMMatrix.cs_of_fstRowThdColRealCond","declaration":"theorem CKMMatrix.cs_of_fstRowThdColRealCond {V : CKMMatrix} (hb : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) (hV : CKMMatrix.FstRowThdColRealCond V) : ↑V 1 1 =\n ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +\n -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *\n Complex.exp (-↑(Complex.arg (↑V 0 2)) * Complex.I)"} +{"name":"CKMMatrix.shift_ud_phase_zero","declaration":"theorem CKMMatrix.shift_ud_phase_zero (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) (V : CKMMatrix) (h1 : u + d = -Complex.arg (↑V 0 0)) : ↑(phaseShiftApply V u c t d s b) 0 0 = ↑(VudAbs ⟦V⟧)"} +{"name":"CKMMatrix.shift_us_phase_zero","declaration":"theorem CKMMatrix.shift_us_phase_zero (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) {V : CKMMatrix} (h1 : u + s = -Complex.arg (↑V 0 1)) : ↑(phaseShiftApply V u c t d s b) 0 1 = ↑(VusAbs ⟦V⟧)"} +{"name":"CKMMatrix.fstRowThdColRealCond_shift_solution","declaration":"theorem CKMMatrix.fstRowThdColRealCond_shift_solution (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) {τ : ℝ} {V : CKMMatrix} (h1 : a + d = -Complex.arg (↑V 0 0)) (h2 : a + e = -Complex.arg (↑V 0 1)) (h3 : b + f = -Complex.arg (↑V 1 2)) (h4 : c + f = -Complex.arg (↑V 2 2)) (h5 : τ = -a - b - c - d - e - f) : b = -τ + Complex.arg (↑V 0 0) + Complex.arg (↑V 0 1) + Complex.arg (↑V 2 2) + a ∧\n c = -τ + Complex.arg (↑V 1 2) + Complex.arg (↑V 0 0) + Complex.arg (↑V 0 1) + a ∧\n d = -Complex.arg (↑V 0 0) - a ∧\n e = -Complex.arg (↑V 0 1) - a ∧\n f = τ - Complex.arg (↑V 0 0) - Complex.arg (↑V 0 1) - Complex.arg (↑V 1 2) - Complex.arg (↑V 2 2) - a"} +{"name":"CKMMatrix.FstRowThdColRealCond","declaration":"/-- A proposition which is satisfied by a CKM matrix if its `ud`, `us`, `cb` and `tb` elements\nare positive and real, and there is no phase difference between the `t`th-row and\nthe cross product of the conjugates of the `u`th and `c`th rows. -/\ndef CKMMatrix.FstRowThdColRealCond (U : CKMMatrix) : Prop"} +{"name":"CKMMatrix.shift_cd_phase_pi","declaration":"theorem CKMMatrix.shift_cd_phase_pi (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) {V : CKMMatrix} (h1 : c + d = Real.pi - Complex.arg (↑V 1 0)) : ↑(phaseShiftApply V u c t d s b) 1 0 = -↑(VcdAbs ⟦V⟧)"} +{"name":"CKMMatrix.shift_tb_phase_zero","declaration":"theorem CKMMatrix.shift_tb_phase_zero (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) {V : CKMMatrix} (h1 : t + b = -Complex.arg (↑V 2 2)) : ↑(phaseShiftApply V u c t d s b) 2 2 = ↑(VtbAbs ⟦V⟧)"} +{"name":"CKMMatrix.ubOnePhaseCond_shift_solution","declaration":"theorem CKMMatrix.ubOnePhaseCond_shift_solution (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) {V : CKMMatrix} (h1 : a + f = -Complex.arg (↑V 0 2)) (h2 : 0 = -a - b - c - d - e - f) (h3 : b + d = Real.pi - Complex.arg (↑V 1 0)) (h5 : b + e = -Complex.arg (↑V 1 1)) : c = -Real.pi + Complex.arg (↑V 1 0) + Complex.arg (↑V 1 1) + Complex.arg (↑V 0 2) + b ∧\n d = Real.pi - Complex.arg (↑V 1 0) - b ∧ e = -Complex.arg (↑V 1 1) - b ∧ f = -Complex.arg (↑V 0 2) - a"} +{"name":"CKMMatrix.shift_cross_product_phase_zero","declaration":"theorem CKMMatrix.shift_cross_product_phase_zero (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) {V : CKMMatrix} {τ : ℝ} (hτ : Complex.exp (↑τ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V)) =\n CKMMatrix.tRow V) (h1 : τ = -u - c - t - d - s - b) : CKMMatrix.tRow (phaseShiftApply V u c t d s b) =\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow (phaseShiftApply V u c t d s b))))\n ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow (phaseShiftApply V u c t d s b)))"} +{"name":"CKMMatrix.fstRowThdColRealCond_holds_up_to_equiv","declaration":"theorem CKMMatrix.fstRowThdColRealCond_holds_up_to_equiv (V : CKMMatrix) : ∃ U, V ≈ U ∧ CKMMatrix.FstRowThdColRealCond U"} +{"name":"CKMMatrix.shift_cs_phase_zero","declaration":"theorem CKMMatrix.shift_cs_phase_zero (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) {V : CKMMatrix} (h1 : c + s = -Complex.arg (↑V 1 1)) : ↑(phaseShiftApply V u c t d s b) 1 1 = ↑(VcsAbs ⟦V⟧)"} +{"name":"CKMMatrix.shift_cb_phase_zero","declaration":"theorem CKMMatrix.shift_cb_phase_zero (u : ℝ) (c : ℝ) (t : ℝ) (d : ℝ) (s : ℝ) (b : ℝ) {V : CKMMatrix} (h1 : c + b = -Complex.arg (↑V 1 2)) : ↑(phaseShiftApply V u c t d s b) 1 2 = ↑(VcbAbs ⟦V⟧)"} +{"name":"CKMMatrix.ubOnePhaseCond","declaration":"/-- A proposition which is satisfied by a CKM matrix `ub` is one, `ud`, `us` and `cb` are zero,\n there is no phase difference between the `t`th-row and\nthe cross product of the conjugates of the `u`th and `c`th rows, and the `cd`th and `cs`th\nelements are real and related in a set way.\n-/\ndef CKMMatrix.ubOnePhaseCond (U : CKMMatrix) : Prop"} diff --git a/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Relations.jsonl b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Relations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a002c0cd5b0f28ca909c04ccd2eae189d0c0bed8 --- /dev/null +++ b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Relations.jsonl @@ -0,0 +1,37 @@ +{"name":"CKMMatrix.normSq_Vud_plus_normSq_Vus","declaration":"theorem CKMMatrix.normSq_Vud_plus_normSq_Vus (V : CKMMatrix) : Complex.normSq (↑V 0 0) + Complex.normSq (↑V 0 1) = 1 - Complex.normSq (↑V 0 2)"} +{"name":"CKMMatrix.VAbs_sum_sq_row_eq_one","declaration":"theorem CKMMatrix.VAbs_sum_sq_row_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) : VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2 + VAbs i 2 V ^ 2 = 1"} +{"name":"CKMMatrix.VAbs_thd_eq_one_fst_eq_zero","declaration":"theorem CKMMatrix.VAbs_thd_eq_one_fst_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs i 2 V = 1) : VAbs i 0 V = 0"} +{"name":"CKMMatrix.conj_Vtb_cross_product","declaration":"theorem CKMMatrix.conj_Vtb_cross_product {V : CKMMatrix} {τ : ℝ} (hτ : CKMMatrix.tRow V =\n Complex.exp (↑τ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) : (starRingEnd ℂ) (↑V 2 2) = Complex.exp (-↑τ * Complex.I) * (↑V 1 1 * ↑V 0 0 - ↑V 0 1 * ↑V 1 0)"} +{"name":"CKMMatrix.fst_row_normalized_abs","declaration":"theorem CKMMatrix.fst_row_normalized_abs (V : CKMMatrix) : Complex.abs (↑V 0 0) ^ 2 + Complex.abs (↑V 0 1) ^ 2 + Complex.abs (↑V 0 2) ^ 2 = 1"} +{"name":"CKMMatrix.snd_row_normalized_abs","declaration":"theorem CKMMatrix.snd_row_normalized_abs (V : CKMMatrix) : Complex.abs (↑V 1 0) ^ 2 + Complex.abs (↑V 1 1) ^ 2 + Complex.abs (↑V 1 2) ^ 2 = 1"} +{"name":"CKMMatrix.Vabs_sq_add_neq_zero","declaration":"theorem CKMMatrix.Vabs_sq_add_neq_zero {V : CKMMatrix} (hb : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) : ↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0"} +{"name":"CKMMatrix.cs_of_ud_us_ub_cb_tb","declaration":"theorem CKMMatrix.cs_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) {τ : ℝ} (hτ : CKMMatrix.tRow V =\n Complex.exp (↑τ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) : ↑V 1 1 =\n (-(starRingEnd ℂ) (↑V 0 2) * ↑V 0 1 * ↑V 1 2 +\n Complex.exp (↑τ * Complex.I) * (starRingEnd ℂ) (↑V 2 2) * (starRingEnd ℂ) (↑V 0 0)) /\n (↑(Complex.normSq (↑V 0 0)) + ↑(Complex.normSq (↑V 0 1)))"} +{"name":"CKMMatrix.thd_col_normalized_normSq","declaration":"theorem CKMMatrix.thd_col_normalized_normSq (V : CKMMatrix) : Complex.normSq (↑V 0 2) + Complex.normSq (↑V 1 2) + Complex.normSq (↑V 2 2) = 1"} +{"name":"CKMMatrix.cb_tb_neq_zero_iff_ub_neq_one","declaration":"theorem CKMMatrix.cb_tb_neq_zero_iff_ub_neq_one (V : CKMMatrix) : ↑V 1 2 ≠ 0 ∨ ↑V 2 2 ≠ 0 ↔ Complex.abs (↑V 0 2) ≠ 1"} +{"name":"CKMMatrix.ud_us_neq_zero_iff_ub_neq_one","declaration":"theorem CKMMatrix.ud_us_neq_zero_iff_ub_neq_one (V : CKMMatrix) : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0 ↔ Complex.abs (↑V 0 2) ≠ 1"} +{"name":"CKMMatrix.VAbs_leq_one","declaration":"theorem CKMMatrix.VAbs_leq_one (i : Fin 3) (j : Fin 3) (V : Quotient CKMMatrixSetoid) : VAbs i j V ≤ 1"} +{"name":"CKMMatrix.Vcs_mul_conj_Vus","declaration":"theorem CKMMatrix.Vcs_mul_conj_Vus (V : CKMMatrix) : ↑V 1 1 * (starRingEnd ℂ) (↑V 0 1) = -↑V 1 0 * (starRingEnd ℂ) (↑V 0 0) - ↑V 1 2 * (starRingEnd ℂ) (↑V 0 2)"} +{"name":"CKMMatrix.VcbAbs_sq_add_VtbAbs_sq","declaration":"theorem CKMMatrix.VcbAbs_sq_add_VtbAbs_sq (V : Quotient CKMMatrixSetoid) : VcbAbs V ^ 2 + VtbAbs V ^ 2 = 1 - VubAbs V ^ 2"} +{"name":"CKMMatrix.normSq_Vud_plus_normSq_Vus_neq_zero_ℝ","declaration":"theorem CKMMatrix.normSq_Vud_plus_normSq_Vus_neq_zero_ℝ {V : CKMMatrix} (hb : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) : Complex.normSq (↑V 0 0) + Complex.normSq (↑V 0 1) ≠ 0"} +{"name":"CKMMatrix.conj_Vtb_mul_Vus","declaration":"theorem CKMMatrix.conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ} (hτ : CKMMatrix.tRow V =\n Complex.exp (↑τ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) : Complex.exp (↑τ * Complex.I) * (starRingEnd ℂ) (↑V 2 2) * (starRingEnd ℂ) (↑V 0 1) =\n -(↑V 1 0 * (↑(Complex.normSq (↑V 0 0)) + ↑(Complex.normSq (↑V 0 1))) + ↑V 1 2 * (starRingEnd ℂ) (↑V 0 2) * ↑V 0 0)"} +{"name":"CKMMatrix.VAbs_thd_eq_one_snd_eq_zero","declaration":"theorem CKMMatrix.VAbs_thd_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs i 2 V = 1) : VAbs i 1 V = 0"} +{"name":"CKMMatrix.VAbs_ge_zero","declaration":"theorem CKMMatrix.VAbs_ge_zero (i : Fin 3) (j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i j V"} +{"name":"CKMMatrix.VAbs_fst_col_eq_one_snd_eq_zero","declaration":"theorem CKMMatrix.VAbs_fst_col_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs 0 i V = 1) : VAbs 1 i V = 0"} +{"name":"CKMMatrix.Vcd_mul_conj_Vud","declaration":"theorem CKMMatrix.Vcd_mul_conj_Vud (V : CKMMatrix) : ↑V 1 0 * (starRingEnd ℂ) (↑V 0 0) = -↑V 1 1 * (starRingEnd ℂ) (↑V 0 1) - ↑V 1 2 * (starRingEnd ℂ) (↑V 0 2)"} +{"name":"CKMMatrix.cb_eq_zero_of_ud_us_zero","declaration":"theorem CKMMatrix.cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h : ↑V 0 0 = 0 ∧ ↑V 0 1 = 0) : ↑V 1 2 = 0"} +{"name":"CKMMatrix.VAbsub_neq_zero_Vud_Vus_neq_zero","declaration":"theorem CKMMatrix.VAbsub_neq_zero_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid} (hV : VAbs 0 2 V ≠ 1) : VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0"} +{"name":"CKMMatrix.fst_row_normalized_normSq","declaration":"theorem CKMMatrix.fst_row_normalized_normSq (V : CKMMatrix) : Complex.normSq (↑V 0 0) + Complex.normSq (↑V 0 1) + Complex.normSq (↑V 0 2) = 1"} +{"name":"CKMMatrix.VudAbs_sq_add_VusAbs_sq","declaration":"theorem CKMMatrix.VudAbs_sq_add_VusAbs_sq {V : Quotient CKMMatrixSetoid} : VudAbs V ^ 2 + VusAbs V ^ 2 = 1 - VubAbs V ^ 2"} +{"name":"CKMMatrix.cd_of_ud_us_ub_cb_tb","declaration":"theorem CKMMatrix.cd_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) {τ : ℝ} (hτ : CKMMatrix.tRow V =\n Complex.exp (↑τ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) : ↑V 1 0 =\n -((starRingEnd ℂ) (↑V 0 2) * ↑V 0 0 * ↑V 1 2 +\n Complex.exp (↑τ * Complex.I) * (starRingEnd ℂ) (↑V 2 2) * (starRingEnd ℂ) (↑V 0 1)) /\n (↑(Complex.normSq (↑V 0 0)) + ↑(Complex.normSq (↑V 0 1)))"} +{"name":"CKMMatrix.VAbs_fst_col_eq_one_thd_eq_zero","declaration":"theorem CKMMatrix.VAbs_fst_col_eq_one_thd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs 0 i V = 1) : VAbs 2 i V = 0"} +{"name":"CKMMatrix.conj_Vtb_mul_Vud","declaration":"theorem CKMMatrix.conj_Vtb_mul_Vud {V : CKMMatrix} {τ : ℝ} (hτ : CKMMatrix.tRow V =\n Complex.exp (↑τ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) : Complex.exp (↑τ * Complex.I) * (starRingEnd ℂ) (↑V 2 2) * (starRingEnd ℂ) (↑V 0 0) =\n ↑V 1 1 * (↑(Complex.normSq (↑V 0 0)) + ↑(Complex.normSq (↑V 0 1))) + ↑V 1 2 * (starRingEnd ℂ) (↑V 0 2) * ↑V 0 1"} +{"name":"CKMMatrix.thd_row_normalized_normSq","declaration":"theorem CKMMatrix.thd_row_normalized_normSq (V : CKMMatrix) : Complex.normSq (↑V 2 0) + Complex.normSq (↑V 2 1) + Complex.normSq (↑V 2 2) = 1"} +{"name":"CKMMatrix.VAbs_sum_sq_col_eq_one","declaration":"theorem CKMMatrix.VAbs_sum_sq_col_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) : VAbs 0 i V ^ 2 + VAbs 1 i V ^ 2 + VAbs 2 i V ^ 2 = 1"} +{"name":"CKMMatrix.snd_row_normalized_normSq","declaration":"theorem CKMMatrix.snd_row_normalized_normSq (V : CKMMatrix) : Complex.normSq (↑V 1 0) + Complex.normSq (↑V 1 1) + Complex.normSq (↑V 1 2) = 1"} +{"name":"CKMMatrix.cs_of_ud_us_zero","declaration":"theorem CKMMatrix.cs_of_ud_us_zero {V : CKMMatrix} (ha : ¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)) : VcsAbs ⟦V⟧ = Real.sqrt (1 - VcdAbs ⟦V⟧ ^ 2)"} +{"name":"CKMMatrix.fst_row_orthog_snd_row","declaration":"theorem CKMMatrix.fst_row_orthog_snd_row (V : CKMMatrix) : ↑V 1 0 * (starRingEnd ℂ) (↑V 0 0) + ↑V 1 1 * (starRingEnd ℂ) (↑V 0 1) + ↑V 1 2 * (starRingEnd ℂ) (↑V 0 2) = 0"} +{"name":"CKMMatrix.normSq_Vud_plus_normSq_Vus_neq_zero_ℂ","declaration":"theorem CKMMatrix.normSq_Vud_plus_normSq_Vus_neq_zero_ℂ {V : CKMMatrix} (hb : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) : ↑(Complex.normSq (↑V 0 0)) + ↑(Complex.normSq (↑V 0 1)) ≠ 0"} +{"name":"CKMMatrix.fst_row_orthog_thd_row","declaration":"theorem CKMMatrix.fst_row_orthog_thd_row (V : CKMMatrix) : ↑V 2 0 * (starRingEnd ℂ) (↑V 0 0) + ↑V 2 1 * (starRingEnd ℂ) (↑V 0 1) + ↑V 2 2 * (starRingEnd ℂ) (↑V 0 2) = 0"} +{"name":"CKMMatrix.thd_col_normalized_abs","declaration":"theorem CKMMatrix.thd_col_normalized_abs (V : CKMMatrix) : Complex.abs (↑V 0 2) ^ 2 + Complex.abs (↑V 1 2) ^ 2 + Complex.abs (↑V 2 2) ^ 2 = 1"} +{"name":"CKMMatrix.thd_row_normalized_abs","declaration":"theorem CKMMatrix.thd_row_normalized_abs (V : CKMMatrix) : Complex.abs (↑V 2 0) ^ 2 + Complex.abs (↑V 2 1) ^ 2 + Complex.abs (↑V 2 2) ^ 2 = 1"} +{"name":"CKMMatrix.VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero","declaration":"theorem CKMMatrix.VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid} (hV : VAbs 0 2 V ≠ 1) : Real.sqrt (VudAbs V ^ 2 + VusAbs V ^ 2) ≠ 0"} diff --git a/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Rows.jsonl b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Rows.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..19973fea929ad2a3c2bb370ee8dd16a3c43dc47a --- /dev/null +++ b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.Rows.jsonl @@ -0,0 +1,35 @@ +{"name":"CKMMatrix.rowBasis","declaration":"/-- The rows of a CKM matrix as a basis of `ℂ³`. -/\ndef CKMMatrix.rowBasis (V : CKMMatrix) : Basis (Fin 3) ℂ (Fin 3 → ℂ)"} +{"name":"phaseShiftApply.uRow_mul","declaration":"theorem phaseShiftApply.uRow_mul (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) : CKMMatrix.uRow (phaseShiftApply V a b c 0 0 0) = Complex.exp (↑a * Complex.I) • CKMMatrix.uRow V"} +{"name":"CKMMatrix.tRow","declaration":"/-- The `t`th row of the CKM matrix. -/\ndef CKMMatrix.tRow (V : CKMMatrix) : Fin 3 → ℂ"} +{"name":"CKMMatrix.c_row","declaration":"/-- The `c`th row of the CKM matrix. -/\ndef CKMMatrix.c_row : Lean.ParserDescr"} +{"name":"CKMMatrix.uRow_normalized","declaration":"theorem CKMMatrix.uRow_normalized (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V)) (CKMMatrix.uRow V) = 1"} +{"name":"CKMMatrix.cRow_cross_tRow_eq_uRow","declaration":"theorem CKMMatrix.cRow_cross_tRow_eq_uRow (V : CKMMatrix) : ∃ κ,\n CKMMatrix.uRow V =\n Complex.exp (↑κ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.tRow V))"} +{"name":"CKMMatrix.tRow_uRow_orthog","declaration":"theorem CKMMatrix.tRow_uRow_orthog (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.tRow V)) (CKMMatrix.uRow V) = 0"} +{"name":"phaseShiftApply.cRow_mul","declaration":"theorem phaseShiftApply.cRow_mul (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) : CKMMatrix.cRow (phaseShiftApply V a b c 0 0 0) = Complex.exp (↑b * Complex.I) • CKMMatrix.cRow V"} +{"name":"CKMMatrix.cRow_normalized","declaration":"theorem CKMMatrix.cRow_normalized (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V)) (CKMMatrix.cRow V) = 1"} +{"name":"CKMMatrix.rows","declaration":"/-- A map from `Fin 3` to each row of a CKM matrix. -/\ndef CKMMatrix.rows (V : CKMMatrix) : Fin 3 → Fin 3 → ℂ"} +{"name":"CKMMatrix.uRow_cross_cRow_normalized","declaration":"theorem CKMMatrix.uRow_cross_cRow_normalized (V : CKMMatrix) : Matrix.dotProduct\n ((starRingEnd (Fin 3 → ℂ))\n ((crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))))\n ((crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) =\n 1"} +{"name":"CKMMatrix.u_row","declaration":"/-- The `u`th row of the CKM matrix. -/\ndef CKMMatrix.u_row : Lean.ParserDescr"} +{"name":"CKMMatrix.rowBasis_repr_symm_apply","declaration":"theorem CKMMatrix.rowBasis_repr_symm_apply (V : CKMMatrix) (a : Fin 3 →₀ ℂ) : ∀ (a_1 : Fin 3),\n (LinearEquiv.symm (CKMMatrix.rowBasis V).repr) a a_1 = (Finsupp.total (Fin 3) (Fin 3 → ℂ) ℂ (CKMMatrix.rows V)) a a_1"} +{"name":"CKMMatrix.cRow_tRow_orthog","declaration":"theorem CKMMatrix.cRow_tRow_orthog (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V)) (CKMMatrix.tRow V) = 0"} +{"name":"CKMMatrix.tRow_normalized","declaration":"theorem CKMMatrix.tRow_normalized (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.tRow V)) (CKMMatrix.tRow V) = 1"} +{"name":"CKMMatrix.cRow_cross_tRow_normalized","declaration":"theorem CKMMatrix.cRow_cross_tRow_normalized (V : CKMMatrix) : Matrix.dotProduct\n ((starRingEnd (Fin 3 → ℂ))\n ((crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.tRow V))))\n ((crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.tRow V))) =\n 1"} +{"name":"CKMMatrix.uRow","declaration":"/-- The `u`th row of the CKM matrix. -/\ndef CKMMatrix.uRow (V : CKMMatrix) : Fin 3 → ℂ"} +{"name":"CKMMatrix.cRow_uRow_orthog","declaration":"theorem CKMMatrix.cRow_uRow_orthog (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V)) (CKMMatrix.uRow V) = 0"} +{"name":"phaseShiftApply.ucCross_thd","declaration":"theorem phaseShiftApply.ucCross_thd (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) (V : CKMMatrix) : phaseShiftApply.ucCross V a b c d e f 2 =\n Complex.exp (-↑a * Complex.I + -↑b * Complex.I + -↑d * Complex.I + -↑e * Complex.I) *\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V)) 2"} +{"name":"CKMMatrix.cRow_cross_tRow_conj","declaration":"theorem CKMMatrix.cRow_cross_tRow_conj (V : CKMMatrix) : (starRingEnd (Fin 3 → ℂ))\n ((crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.tRow V))) =\n (crossProduct (CKMMatrix.cRow V)) (CKMMatrix.tRow V)"} +{"name":"CKMMatrix.rowBasis_repr_apply","declaration":"theorem CKMMatrix.rowBasis_repr_apply (V : CKMMatrix) : ∀ (a : Fin 3 → ℂ),\n (CKMMatrix.rowBasis V).repr a =\n (LinearIndependent.repr ⋯)\n ((LinearMap.codRestrict (Submodule.span ℂ (Set.range (CKMMatrix.rows V))) LinearMap.id ⋯) a)"} +{"name":"phaseShiftApply.tRow_mul","declaration":"theorem phaseShiftApply.tRow_mul (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) : CKMMatrix.tRow (phaseShiftApply V a b c 0 0 0) = Complex.exp (↑c * Complex.I) • CKMMatrix.tRow V"} +{"name":"CKMMatrix.t_row","declaration":"/-- The `t`th row of the CKM matrix. -/\ndef CKMMatrix.t_row : Lean.ParserDescr"} +{"name":"CKMMatrix.tRow_cRow_orthog","declaration":"theorem CKMMatrix.tRow_cRow_orthog (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.tRow V)) (CKMMatrix.cRow V) = 0"} +{"name":"CKMMatrix.uRow_cRow_orthog","declaration":"theorem CKMMatrix.uRow_cRow_orthog (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V)) (CKMMatrix.cRow V) = 0"} +{"name":"CKMMatrix.rows_card","declaration":"theorem CKMMatrix.rows_card : Fintype.card (Fin 3) = FiniteDimensional.finrank ℂ (Fin 3 → ℂ)"} +{"name":"CKMMatrix.uRow_cross_cRow_conj","declaration":"theorem CKMMatrix.uRow_cross_cRow_conj (V : CKMMatrix) : (starRingEnd (Fin 3 → ℂ))\n ((crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))) =\n (crossProduct (CKMMatrix.uRow V)) (CKMMatrix.cRow V)"} +{"name":"phaseShiftApply.ucCross","declaration":"/-- The cross product of the conjugate of the `u` and `c` rows of a CKM matrix. -/\ndef phaseShiftApply.ucCross (V : CKMMatrix) (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) : Fin 3 → ℂ"} +{"name":"CKMMatrix.cRow","declaration":"/-- The `c`th row of the CKM matrix. -/\ndef CKMMatrix.cRow (V : CKMMatrix) : Fin 3 → ℂ"} +{"name":"CKMMatrix.ext_Rows","declaration":"theorem CKMMatrix.ext_Rows {U : CKMMatrix} {V : CKMMatrix} (hu : CKMMatrix.uRow U = CKMMatrix.uRow V) (hc : CKMMatrix.cRow U = CKMMatrix.cRow V) (ht : CKMMatrix.tRow U = CKMMatrix.tRow V) : U = V"} +{"name":"phaseShiftApply.ucCross_snd","declaration":"theorem phaseShiftApply.ucCross_snd (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) (V : CKMMatrix) : phaseShiftApply.ucCross V a b c d e f 1 =\n Complex.exp (-↑a * Complex.I + -↑b * Complex.I + -↑d * Complex.I + -↑f * Complex.I) *\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V)) 1"} +{"name":"CKMMatrix.uRow_tRow_orthog","declaration":"theorem CKMMatrix.uRow_tRow_orthog (V : CKMMatrix) : Matrix.dotProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V)) (CKMMatrix.tRow V) = 0"} +{"name":"phaseShiftApply.ucCross_fst","declaration":"theorem phaseShiftApply.ucCross_fst (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (e : ℝ) (f : ℝ) (V : CKMMatrix) : phaseShiftApply.ucCross V a b c d e f 0 =\n Complex.exp (-↑a * Complex.I + -↑b * Complex.I + -↑e * Complex.I + -↑f * Complex.I) *\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V)) 0"} +{"name":"CKMMatrix.uRow_cross_cRow_eq_tRow","declaration":"theorem CKMMatrix.uRow_cross_cRow_eq_tRow (V : CKMMatrix) : ∃ τ,\n CKMMatrix.tRow V =\n Complex.exp (↑τ * Complex.I) •\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow V))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow V))"} +{"name":"CKMMatrix.rows_linearly_independent","declaration":"theorem CKMMatrix.rows_linearly_independent (V : CKMMatrix) : LinearIndependent ℂ (CKMMatrix.rows V)"} diff --git a/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic.jsonl b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f96d1fe59a84d6cff004761559171b79e75793b3 --- /dev/null +++ b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic.jsonl @@ -0,0 +1,8 @@ +{"name":"standParam.VusVubVcdSq_eq","declaration":"theorem standParam.VusVubVcdSq_eq (θ₁₂ : ℝ) (θ₁₃ : ℝ) (θ₂₃ : ℝ) (δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂) (h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) : Invariant.VusVubVcdSq ⟦standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =\n Real.sin θ₁₂ ^ 2 * Real.cos θ₁₃ ^ 2 * Real.sin θ₁₃ ^ 2 * Real.sin θ₂₃ ^ 2"} +{"name":"standParam.eq_exp_of_phases","declaration":"theorem standParam.eq_exp_of_phases (θ₁₂ : ℝ) (θ₁₃ : ℝ) (θ₂₃ : ℝ) (δ₁₃ : ℝ) (δ₁₃' : ℝ) (h : Complex.exp (↑δ₁₃ * Complex.I) = Complex.exp (↑δ₁₃' * Complex.I)) : standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃'"} +{"name":"standParam.cross_product_t","declaration":"theorem standParam.cross_product_t (θ₁₂ : ℝ) (θ₁₃ : ℝ) (θ₂₃ : ℝ) (δ₁₃ : ℝ) : CKMMatrix.tRow (standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃) =\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow (standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃))))\n ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow (standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃)))"} +{"name":"standParam.eq_rows","declaration":"theorem standParam.eq_rows (U : CKMMatrix) {θ₁₂ : ℝ} {θ₁₃ : ℝ} {θ₂₃ : ℝ} {δ₁₃ : ℝ} (hu : CKMMatrix.uRow U = CKMMatrix.uRow (standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃)) (hc : CKMMatrix.cRow U = CKMMatrix.cRow (standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃)) (hU : CKMMatrix.tRow U =\n (crossProduct ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.uRow U))) ((starRingEnd (Fin 3 → ℂ)) (CKMMatrix.cRow U))) : U = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃"} +{"name":"standParam.mulExpδ₁₃_eq","declaration":"theorem standParam.mulExpδ₁₃_eq (θ₁₂ : ℝ) (θ₁₃ : ℝ) (θ₂₃ : ℝ) (δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂) (h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) : Invariant.mulExpδ₁₃ ⟦standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =\n Complex.sin ↑θ₁₂ * Complex.cos ↑θ₁₃ ^ 2 * Complex.sin ↑θ₂₃ * Complex.sin ↑θ₁₃ * Complex.cos ↑θ₁₂ * Complex.cos ↑θ₂₃ *\n Complex.exp (Complex.I * ↑δ₁₃)"} +{"name":"standParamAsMatrix_unitary","declaration":"theorem standParamAsMatrix_unitary (θ₁₂ : ℝ) (θ₁₃ : ℝ) (θ₂₃ : ℝ) (δ₁₃ : ℝ) : Matrix.conjTranspose (standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) * standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃ = 1"} +{"name":"standParam","declaration":"/-- A CKM Matrix from four reals `θ₁₂`, `θ₁₃`, `θ₂₃`, and `δ₁₃`. This is the standard\nparameterization of CKM matrices. -/\ndef standParam (θ₁₂ : ℝ) (θ₁₃ : ℝ) (θ₂₃ : ℝ) (δ₁₃ : ℝ) : CKMMatrix"} +{"name":"standParamAsMatrix","declaration":"/-- Given four reals `θ₁₂ θ₁₃ θ₂₃ δ₁₃` the standard parameterization of the CKM matrix\nas a `3×3` complex matrix. -/\ndef standParamAsMatrix (θ₁₂ : ℝ) (θ₁₃ : ℝ) (θ₂₃ : ℝ) (δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ"} diff --git a/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters.jsonl b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..fecaec3a9a69a85acd030f9a759ae592d2d496f1 --- /dev/null +++ b/hep-declarations/HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters.jsonl @@ -0,0 +1,65 @@ +{"name":"C₁₂_of_Vub_one","declaration":"theorem C₁₂_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : C₁₂ V = 1"} +{"name":"θ₁₂","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `θ₁₂` in the\nstandard parameterization. --/\ndef θ₁₂ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"complexAbs_sin_θ₁₂","declaration":"theorem complexAbs_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : ↑(Complex.abs (Complex.sin ↑(θ₁₂ V))) = Complex.sin ↑(θ₁₂ V)"} +{"name":"C₁₂_eq_ℂcos_θ₁₂","declaration":"theorem C₁₂_eq_ℂcos_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.cos ↑(θ₁₂ V) = ↑(C₁₂ V)"} +{"name":"C₁₃_eq_add_sq","declaration":"theorem C₁₃_eq_add_sq (V : Quotient CKMMatrixSetoid) : C₁₃ V = Real.sqrt (VudAbs V ^ 2 + VusAbs V ^ 2)"} +{"name":"standParam.mulExpδ₁₃_on_param_abs","declaration":"theorem standParam.mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) : ↑(Complex.abs (Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧)) =\n Complex.sin ↑(θ₁₂ ⟦V⟧) * Complex.cos ↑(θ₁₃ ⟦V⟧) ^ 2 * Complex.sin ↑(θ₂₃ ⟦V⟧) * Complex.sin ↑(θ₁₃ ⟦V⟧) *\n Complex.cos ↑(θ₁₂ ⟦V⟧) *\n Complex.cos ↑(θ₂₃ ⟦V⟧)"} +{"name":"S₁₃_nonneg","declaration":"theorem S₁₃_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₁₃ V"} +{"name":"standParam.on_param_sin_θ₁₂_eq_zero","declaration":"theorem standParam.on_param_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₂ ⟦V⟧) = 0) : standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0"} +{"name":"S₁₂_eq_sin_θ₁₂","declaration":"theorem S₁₂_eq_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₂ V) = S₁₂ V"} +{"name":"Vs_zero_iff_cos_sin_zero","declaration":"theorem Vs_zero_iff_cos_sin_zero (V : CKMMatrix) : VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 ↔\n Real.cos (θ₁₂ ⟦V⟧) = 0 ∨\n Real.cos (θ₁₃ ⟦V⟧) = 0 ∨\n Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0"} +{"name":"standParam.mulExpδ₁₃_on_param_neq_zero_arg","declaration":"theorem standParam.mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ) (h1 : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0) : Complex.exp (↑(Complex.arg (Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧)) * Complex.I) =\n Complex.exp (↑δ₁₃ * Complex.I)"} +{"name":"VtbAbs_eq_C₂₃_mul_C₁₃","declaration":"theorem VtbAbs_eq_C₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VtbAbs V = C₂₃ V * C₁₃ V"} +{"name":"S₁₂_of_Vub_one","declaration":"theorem S₁₂_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₁₂ V = 0"} +{"name":"S₁₃_of_Vub_one","declaration":"theorem S₁₃_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₁₃ V = 1"} +{"name":"S₂₃_eq_ℂsin_θ₂₃","declaration":"theorem S₂₃_eq_ℂsin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.sin ↑(θ₂₃ V) = ↑(S₂₃ V)"} +{"name":"VubAbs_eq_S₁₃","declaration":"theorem VubAbs_eq_S₁₃ (V : Quotient CKMMatrixSetoid) : VubAbs V = S₁₃ V"} +{"name":"standParam.exists_for_CKMatrix","declaration":"theorem standParam.exists_for_CKMatrix (V : CKMMatrix) : ∃ θ₁₂ θ₁₃ θ₂₃ δ₁₃, V ≈ standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃"} +{"name":"standParam.on_param_cos_θ₁₂_eq_zero","declaration":"theorem standParam.on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₂ ⟦V⟧) = 0) : standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0"} +{"name":"S₂₃_nonneg","declaration":"theorem S₂₃_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₂₃ V"} +{"name":"θ₁₃","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `θ₁₃` in the\nstandard parameterization. --/\ndef θ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"C₂₃_eq_ℂcos_θ₂₃","declaration":"theorem C₂₃_eq_ℂcos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.cos ↑(θ₂₃ V) = ↑(C₂₃ V)"} +{"name":"complexAbs_sin_θ₂₃","declaration":"theorem complexAbs_sin_θ₂₃ (V : Quotient CKMMatrixSetoid) : ↑(Complex.abs (Complex.sin ↑(θ₂₃ V))) = Complex.sin ↑(θ₂₃ V)"} +{"name":"S₁₂_leq_one","declaration":"theorem S₁₂_leq_one (V : Quotient CKMMatrixSetoid) : S₁₂ V ≤ 1"} +{"name":"standParam.on_param_cos_θ₂₃_eq_zero","declaration":"theorem standParam.on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₂₃ ⟦V⟧) = 0) : standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0"} +{"name":"C₂₃_of_Vub_neq_one","declaration":"theorem C₂₃_of_Vub_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) : C₂₃ V = VtbAbs V / Real.sqrt (VudAbs V ^ 2 + VusAbs V ^ 2)"} +{"name":"VcbAbs_eq_S₂₃_mul_C₁₃","declaration":"theorem VcbAbs_eq_S₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VcbAbs V = S₂₃ V * C₁₃ V"} +{"name":"C₁₂","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₂` in the\nstandard parameterization. --/\ndef C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"complexAbs_sin_θ₁₃","declaration":"theorem complexAbs_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : ↑(Complex.abs (Complex.sin ↑(θ₁₃ V))) = Complex.sin ↑(θ₁₃ V)"} +{"name":"complexAbs_cos_θ₂₃","declaration":"theorem complexAbs_cos_θ₂₃ (V : Quotient CKMMatrixSetoid) : ↑(Complex.abs (Complex.cos ↑(θ₂₃ V))) = Complex.cos ↑(θ₂₃ V)"} +{"name":"S₁₂","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₂` in the\nstandard parameterization. --/\ndef S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"VudAbs_eq_C₁₂_mul_C₁₃","declaration":"theorem VudAbs_eq_C₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VudAbs V = C₁₂ V * C₁₃ V"} +{"name":"S₂₃_sq_add_C₂₃_sq","declaration":"theorem S₂₃_sq_add_C₂₃_sq (V : Quotient CKMMatrixSetoid) : S₂₃ V ^ 2 + C₂₃ V ^ 2 = 1"} +{"name":"standParam.on_param_sin_θ₁₃_eq_zero","declaration":"theorem standParam.on_param_sin_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₃ ⟦V⟧) = 0) : standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0"} +{"name":"θ₂₃","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `θ₂₃` in the\nstandard parameterization. --/\ndef θ₂₃ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"S₁₂_nonneg","declaration":"theorem S₁₂_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₁₂ V"} +{"name":"standParam.mulExpδ₁₃_on_param_eq_zero_iff","declaration":"theorem standParam.mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔\n VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0"} +{"name":"C₁₃_eq_ℂcos_θ₁₃","declaration":"theorem C₁₃_eq_ℂcos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.cos ↑(θ₁₃ V) = ↑(C₁₃ V)"} +{"name":"complexAbs_cos_θ₁₂","declaration":"theorem complexAbs_cos_θ₁₂ (V : Quotient CKMMatrixSetoid) : ↑(Complex.abs (Complex.cos ↑(θ₁₂ V))) = Complex.cos ↑(θ₁₂ V)"} +{"name":"standParam.eq_standardParameterization_δ₃","declaration":"theorem standParam.eq_standardParameterization_δ₃ (V : CKMMatrix) : V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧)"} +{"name":"C₁₃","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₃` in the\nstandard parameterization. --/\ndef C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"S₁₃_sq_add_C₁₃_sq","declaration":"theorem S₁₃_sq_add_C₁₃_sq (V : Quotient CKMMatrixSetoid) : S₁₃ V ^ 2 + C₁₃ V ^ 2 = 1"} +{"name":"S₁₂_sq_add_C₁₂_sq","declaration":"theorem S₁₂_sq_add_C₁₂_sq (V : Quotient CKMMatrixSetoid) : S₁₂ V ^ 2 + C₁₂ V ^ 2 = 1"} +{"name":"standParam.on_param_cos_θ₁₃_eq_zero","declaration":"theorem standParam.on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) : standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0"} +{"name":"S₂₃","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the\nstandard parameterization. --/\ndef S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"S₂₃_leq_one","declaration":"theorem S₂₃_leq_one (V : Quotient CKMMatrixSetoid) : S₂₃ V ≤ 1"} +{"name":"C₁₂_eq_Vud_div_sqrt","declaration":"theorem C₁₂_eq_Vud_div_sqrt {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) : C₁₂ V = VudAbs V / Real.sqrt (VudAbs V ^ 2 + VusAbs V ^ 2)"} +{"name":"standParam.eq_standParam_of_fstRowThdColRealCond","declaration":"theorem standParam.eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : ↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0) (hV : CKMMatrix.FstRowThdColRealCond V) : V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-Complex.arg (↑V 0 2))"} +{"name":"S₁₃_leq_one","declaration":"theorem S₁₃_leq_one (V : Quotient CKMMatrixSetoid) : S₁₃ V ≤ 1"} +{"name":"C₁₃_of_Vub_eq_one","declaration":"theorem C₁₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : C₁₃ V = 0"} +{"name":"standParam.eq_standParam_of_ubOnePhaseCond","declaration":"theorem standParam.eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : CKMMatrix.ubOnePhaseCond V) : V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0"} +{"name":"S₁₃_eq_ℂsin_θ₁₃","declaration":"theorem S₁₃_eq_ℂsin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.sin ↑(θ₁₃ V) = ↑(S₁₃ V)"} +{"name":"VusAbs_eq_S₁₂_mul_C₁₃","declaration":"theorem VusAbs_eq_S₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VusAbs V = S₁₂ V * C₁₃ V"} +{"name":"δ₁₃","declaration":"/-- Given a CKM matrix `V` the real number corresponding to the phase `δ₁₃` in the\nstandard parameterization. --/\ndef δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"S₁₃_eq_sin_θ₁₃","declaration":"theorem S₁₃_eq_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₃ V) = S₁₃ V"} +{"name":"S₁₃","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₃` in the\nstandard parameterization. --/\ndef S₁₃ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"S₁₂_eq_ℂsin_θ₁₂","declaration":"theorem S₁₂_eq_ℂsin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.sin ↑(θ₁₂ V) = ↑(S₁₂ V)"} +{"name":"VubAbs_of_cos_θ₁₃_zero","declaration":"theorem VubAbs_of_cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃ V) = 0) : VubAbs V = 1"} +{"name":"standParam.exists_δ₁₃","declaration":"theorem standParam.exists_δ₁₃ (V : CKMMatrix) : ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃"} +{"name":"complexAbs_cos_θ₁₃","declaration":"theorem complexAbs_cos_θ₁₃ (V : Quotient CKMMatrixSetoid) : ↑(Complex.abs (Complex.cos ↑(θ₁₃ V))) = Complex.cos ↑(θ₁₃ V)"} +{"name":"C₂₃","declaration":"/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the\nstandard parameterization. --/\ndef C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ"} +{"name":"S₂₃_eq_sin_θ₂₃","declaration":"theorem S₂₃_eq_sin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₂₃ V) = S₂₃ V"} +{"name":"standParam.mulExpδ₁₃_on_param_δ₁₃","declaration":"theorem standParam.mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =\n Complex.sin ↑(θ₁₂ ⟦V⟧) * Complex.cos ↑(θ₁₃ ⟦V⟧) ^ 2 * Complex.sin ↑(θ₂₃ ⟦V⟧) * Complex.sin ↑(θ₁₃ ⟦V⟧) *\n Complex.cos ↑(θ₁₂ ⟦V⟧) *\n Complex.cos ↑(θ₂₃ ⟦V⟧) *\n Complex.exp (Complex.I * ↑δ₁₃)"} +{"name":"standParam.on_param_sin_θ₂₃_eq_zero","declaration":"theorem standParam.on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₂₃ ⟦V⟧) = 0) : standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0"} +{"name":"S₂₃_of_Vub_eq_one","declaration":"theorem S₂₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₂₃ V = VcdAbs V"} +{"name":"S₂₃_of_Vub_neq_one","declaration":"theorem S₂₃_of_Vub_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) : S₂₃ V = VcbAbs V / Real.sqrt (VudAbs V ^ 2 + VusAbs V ^ 2)"} diff --git a/hep-declarations/HepLean.Mathematics.LinearMaps.jsonl b/hep-declarations/HepLean.Mathematics.LinearMaps.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..680d0dac3ea5c242b9ede6aea7ef66ab7140ce14 --- /dev/null +++ b/hep-declarations/HepLean.Mathematics.LinearMaps.jsonl @@ -0,0 +1,59 @@ +{"name":"BiLinearSymm.toLinear₁_apply","declaration":"theorem BiLinearSymm.toLinear₁_apply {V : Type} [AddCommMonoid V] [Module ℚ V] (f : BiLinearSymm V) (S : V) (T : V) : (f S) T = (BiLinearSymm.toLinear₁ f T) S"} +{"name":"BiLinearSymm.toHomogeneousQuad_apply","declaration":"theorem BiLinearSymm.toHomogeneousQuad_apply {V : Type} [AddCommMonoid V] [Module ℚ V] (τ : BiLinearSymm V) (S : V) : (BiLinearSymm.toHomogeneousQuad τ) S = (τ S) S"} +{"name":"BiLinearSymm.mk₂_toFun_apply","declaration":"theorem BiLinearSymm.mk₂_toFun_apply {V : Type} [AddCommMonoid V] [Module ℚ V] (f : V × V → ℚ) (map_smul : ∀ (a : ℚ) (S T : V), f (a • S, T) = a * f (S, T)) (map_add : ∀ (S1 S2 T : V), f (S1 + S2, T) = f (S1, T) + f (S2, T)) (swap : ∀ (S T : V), f (S, T) = f (T, S)) (S : V) (T : V) : ((BiLinearSymm.mk₂ f map_smul map_add swap) S) T = f (S, T)"} +{"name":"MulActionHom'","declaration":"/-- Equivariant functions. -/\nstructure MulActionHom' {M : Type u_3} {N : Type u_4} (φ : M → N) (X : Type u_1) [SMul M X] (Y : Type u_2) [SMul N Y] : Type (max u_1 u_2)"} +{"name":"HomogeneousQuadratic.map_smul","declaration":"theorem HomogeneousQuadratic.map_smul {V : Type} [AddCommMonoid V] [Module ℚ V] (f : HomogeneousQuadratic V) (a : ℚ) (S : V) : f (a • S) = a ^ 2 * f S"} +{"name":"BiLinearSymm.toLinear₁","declaration":"/-- Fixing the second input vectors, the resulting linear map. -/\ndef BiLinearSymm.toLinear₁ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ"} +{"name":"BiLinearSymm.swap'","declaration":"def BiLinearSymm.swap' {V : Type} [AddCommMonoid V] [Module ℚ V] (self : BiLinearSymm V) (S : V) (T : V) : (self.toFun S) T = (self.toFun T) S"} +{"name":"TriLinearSymm.swap₁'","declaration":"def TriLinearSymm.swap₁' {V : Type} [AddCommMonoid V] [Module ℚ V] (self : TriLinearSymm V) (S : V) (T : V) (L : V) : ((self.toFun S) T) L = ((self.toFun T) S) L"} +{"name":"BiLinearSymm.map_smul₁","declaration":"theorem BiLinearSymm.map_smul₁ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : BiLinearSymm V) (a : ℚ) (S : V) (T : V) : (f (a • S)) T = a * (f S) T"} +{"name":"IsSymmetric","declaration":"/-- A symmetric bilinear function. -/\nclass IsSymmetric {V : Type} [AddCommMonoid V] [Module ℚ V] (f : V →ₗ[ℚ] V →ₗ[ℚ] ℚ) : Prop"} +{"name":"TriLinearSymm.toCubic","declaration":"/-- The homogenous cubic equation obtainable from a symmetric trilinear function. -/\ndef TriLinearSymm.toCubic {charges : Type} [AddCommMonoid charges] [Module ℚ charges] (τ : TriLinearSymm charges) : HomogeneousCubic charges"} +{"name":"TriLinearSymm.toCubic_toFun","declaration":"theorem TriLinearSymm.toCubic_toFun {charges : Type} [AddCommMonoid charges] [Module ℚ charges] (τ : TriLinearSymm charges) (S : charges) : (TriLinearSymm.toCubic τ).toFun S = ((τ S) S) S"} +{"name":"BiLinearSymm.map_add₁","declaration":"theorem BiLinearSymm.map_add₁ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : BiLinearSymm V) (S1 : V) (S2 : V) (T : V) : (f (S1 + S2)) T = (f S1) T + (f S2) T"} +{"name":"IsSymmetric.swap","declaration":"def IsSymmetric.swap {V : Type} [AddCommMonoid V] [Module ℚ V] {f : V →ₗ[ℚ] V →ₗ[ℚ] ℚ} [self : IsSymmetric f] (S : V) (T : V) : (f S) T = (f T) S"} +{"name":"TriLinearSymm.swap₃","declaration":"theorem TriLinearSymm.swap₃ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (T : V) (L : V) : ((f S) T) L = ((f L) T) S"} +{"name":"TriLinearSymm.swap₂","declaration":"theorem TriLinearSymm.swap₂ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (T : V) (L : V) : ((f S) T) L = ((f S) L) T"} +{"name":"TriLinearSymm.mk","declaration":"ctor TriLinearSymm.mk {V : Type} [AddCommMonoid V] [Module ℚ V] (toLinearMap : V →ₗ[ℚ] V →ₗ[ℚ] V →ₗ[ℚ] ℚ) (swap₁' : ∀ (S T L : V), ((toLinearMap.toFun S) T) L = ((toLinearMap.toFun T) S) L) (swap₂' : ∀ (S T L : V), ((toLinearMap.toFun S) T) L = ((toLinearMap.toFun S) L) T) : TriLinearSymm V"} +{"name":"BiLinearSymm.map_smul₂","declaration":"theorem BiLinearSymm.map_smul₂ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : BiLinearSymm V) (a : ℚ) (S : V) (T : V) : (f S) (a • T) = a * (f S) T"} +{"name":"TriLinearSymm.map_smul₁","declaration":"theorem TriLinearSymm.map_smul₁ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (a : ℚ) (S : V) (T : V) (L : V) : ((f (a • S)) T) L = a * ((f S) T) L"} +{"name":"BiLinearSymm.swap","declaration":"theorem BiLinearSymm.swap {V : Type} [AddCommMonoid V] [Module ℚ V] (f : BiLinearSymm V) (S : V) (T : V) : (f S) T = (f T) S"} +{"name":"TriLinearSymm.map_sum₃","declaration":"theorem TriLinearSymm.map_sum₃ {V : Type} [AddCommMonoid V] [Module ℚ V] {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) : ((f T) L) (Finset.sum Finset.univ fun i => S i) = Finset.sum Finset.univ fun i => ((f T) L) (S i)"} +{"name":"BiLinearSymm.mk","declaration":"ctor BiLinearSymm.mk {V : Type} [AddCommMonoid V] [Module ℚ V] (toLinearMap : V →ₗ[ℚ] V →ₗ[ℚ] ℚ) (swap' : ∀ (S T : V), (toLinearMap.toFun S) T = (toLinearMap.toFun T) S) : BiLinearSymm V"} +{"name":"HomogeneousCubic.map_smul","declaration":"theorem HomogeneousCubic.map_smul {V : Type} [AddCommMonoid V] [Module ℚ V] (f : HomogeneousCubic V) (a : ℚ) (S : V) : f (a • S) = a ^ 3 * f S"} +{"name":"TriLinearSymm.map_sum₁","declaration":"theorem TriLinearSymm.map_sum₁ {V : Type} [AddCommMonoid V] [Module ℚ V] {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) : ((f (Finset.sum Finset.univ fun i => S i)) T) L = Finset.sum Finset.univ fun i => ((f (S i)) T) L"} +{"name":"TriLinearSymm","declaration":"/-- The structure of a symmetric trilinear function. -/\nstructure TriLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] : Type"} +{"name":"HomogeneousCubic","declaration":"/-- The structure of a homogeneous cubic equation. -/\ndef HomogeneousCubic (V : Type) [AddCommMonoid V] [Module ℚ V] : Type"} +{"name":"MulActionHom'.toFun","declaration":"/-- The underlying function. -/\ndef MulActionHom'.toFun {M : Type u_3} {N : Type u_4} {φ : M → N} {X : Type u_1} [SMul M X] {Y : Type u_2} [SMul N Y] (self : X →ₑ[φ] Y) : X → Y"} +{"name":"TriLinearSymm.toCubic_apply","declaration":"theorem TriLinearSymm.toCubic_apply {charges : Type} [AddCommMonoid charges] [Module ℚ charges] (τ : TriLinearSymm charges) (S : charges) : (TriLinearSymm.toCubic τ) S = ((τ S) S) S"} +{"name":"MulActionHom'.mk","declaration":"ctor MulActionHom'.mk {M : Type u_3} {N : Type u_4} {φ : M → N} {X : Type u_1} [SMul M X] {Y : Type u_2} [SMul N Y] (toFun : X → Y) (map_smul' : ∀ (m : M) (x : X), toFun (m • x) = φ m • toFun x) : X →ₑ[φ] Y"} +{"name":"BiLinearSymm.mk₂","declaration":"/-- The construction of a symmetric bilinear map from `smul` and `map_add` in the first factor,\nand swap. -/\ndef BiLinearSymm.mk₂ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : V × V → ℚ) (map_smul : ∀ (a : ℚ) (S T : V), f (a • S, T) = a * f (S, T)) (map_add : ∀ (S1 S2 T : V), f (S1 + S2, T) = f (S1, T) + f (S2, T)) (swap : ∀ (S T : V), f (S, T) = f (T, S)) : BiLinearSymm V"} +{"name":"TriLinearSymm.map_add₃","declaration":"theorem TriLinearSymm.map_add₃ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (T : V) (L1 : V) (L2 : V) : ((f S) T) (L1 + L2) = ((f S) T) L1 + ((f S) T) L2"} +{"name":"TriLinearSymm.map_sum₂","declaration":"theorem TriLinearSymm.map_sum₂ {V : Type} [AddCommMonoid V] [Module ℚ V] {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) : ((f T) (Finset.sum Finset.univ fun i => S i)) L = Finset.sum Finset.univ fun i => ((f T) (S i)) L"} +{"name":"HomogeneousCubic.instFun","declaration":"instance HomogeneousCubic.instFun {V : Type} [AddCommMonoid V] [Module ℚ V] : FunLike (HomogeneousCubic V) V ℚ"} +{"name":"TriLinearSymm.swap₁","declaration":"theorem TriLinearSymm.swap₁ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (T : V) (L : V) : ((f S) T) L = ((f T) S) L"} +{"name":"BiLinearSymm.toHomogeneousQuad_toFun","declaration":"theorem BiLinearSymm.toHomogeneousQuad_toFun {V : Type} [AddCommMonoid V] [Module ℚ V] (τ : BiLinearSymm V) (S : V) : (BiLinearSymm.toHomogeneousQuad τ).toFun S = (τ S) S"} +{"name":"BiLinearSymm","declaration":"/-- The structure of a symmetric bilinear function. -/\nstructure BiLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] : Type"} +{"name":"HomogeneousQuadratic.instFun","declaration":"instance HomogeneousQuadratic.instFun {V : Type} [AddCommMonoid V] [Module ℚ V] : FunLike (HomogeneousQuadratic V) V ℚ"} +{"name":"«MulActionHom'Local≺»","declaration":"/-- `φ`-equivariant functions `X → Y`,\nwhere `φ : M → N`, where `M` and `N` act on `X` and `Y` respectively -/\ndef «MulActionHom'Local≺» : Lean.TrailingParserDescr"} +{"name":"BiLinearSymm.instFun","declaration":"instance BiLinearSymm.instFun (V : Type) [AddCommMonoid V] [Module ℚ V] : FunLike (BiLinearSymm V) V (V →ₗ[ℚ] ℚ)"} +{"name":"TriLinearSymm.toCubic_add","declaration":"theorem TriLinearSymm.toCubic_add {charges : Type} [AddCommMonoid charges] [Module ℚ charges] (τ : TriLinearSymm charges) (S : charges) (T : charges) : (TriLinearSymm.toCubic τ) (S + T) =\n (TriLinearSymm.toCubic τ) S + (TriLinearSymm.toCubic τ) T + 3 * ((τ S) S) T + 3 * ((τ T) T) S"} +{"name":"HomogeneousQuadratic","declaration":"/-- The structure defining a homogeneous quadratic equation. -/\ndef HomogeneousQuadratic (V : Type) [AddCommMonoid V] [Module ℚ V] : Type"} +{"name":"BiLinearSymm.map_add₂","declaration":"theorem BiLinearSymm.map_add₂ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : BiLinearSymm V) (S : V) (T1 : V) (T2 : V) : (f S) (T1 + T2) = (f S) T1 + (f S) T2"} +{"name":"TriLinearSymm.map_add₁","declaration":"theorem TriLinearSymm.map_add₁ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S1 : V) (S2 : V) (T : V) (L : V) : ((f (S1 + S2)) T) L = ((f S1) T) L + ((f S2) T) L"} +{"name":"BiLinearSymm.map_sum₂","declaration":"theorem BiLinearSymm.map_sum₂ {V : Type} [AddCommMonoid V] [Module ℚ V] {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) : (f T) (Finset.sum Finset.univ fun i => S i) = Finset.sum Finset.univ fun i => (f T) (S i)"} +{"name":"TriLinearSymm.mk₃","declaration":"/-- The construction of a symmetric trilinear map from `smul` and `map_add` in the first factor,\nand two swap. -/\ndef TriLinearSymm.mk₃ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : V × V × V → ℚ) (map_smul : ∀ (a : ℚ) (S T L : V), f (a • S, T, L) = a * f (S, T, L)) (map_add : ∀ (S1 S2 T L : V), f (S1 + S2, T, L) = f (S1, T, L) + f (S2, T, L)) (swap₁ : ∀ (S T L : V), f (S, T, L) = f (T, S, L)) (swap₂ : ∀ (S T L : V), f (S, T, L) = f (S, L, T)) : TriLinearSymm V"} +{"name":"TriLinearSymm.instFun","declaration":"instance TriLinearSymm.instFun {V : Type} [AddCommMonoid V] [Module ℚ V] : FunLike (TriLinearSymm V) V (V →ₗ[ℚ] V →ₗ[ℚ] ℚ)"} +{"name":"TriLinearSymm.map_smul₃","declaration":"theorem TriLinearSymm.map_smul₃ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (T : V) (a : ℚ) (L : V) : ((f S) T) (a • L) = a * ((f S) T) L"} +{"name":"TriLinearSymm.map_smul₂","declaration":"theorem TriLinearSymm.map_smul₂ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (a : ℚ) (T : V) (L : V) : ((f S) (a • T)) L = a * ((f S) T) L"} +{"name":"MulActionHom'.map_smul'","declaration":"/-- The proposition that the function preserves the action. -/\ndef MulActionHom'.map_smul' {M : Type u_3} {N : Type u_4} {φ : M → N} {X : Type u_1} [SMul M X] {Y : Type u_2} [SMul N Y] (self : X →ₑ[φ] Y) (m : M) (x : X) : self.toFun (m • x) = φ m • self.toFun x"} +{"name":"BiLinearSymm.map_sum₁","declaration":"theorem BiLinearSymm.map_sum₁ {V : Type} [AddCommMonoid V] [Module ℚ V] {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) : (f (Finset.sum Finset.univ fun i => S i)) T = Finset.sum Finset.univ fun i => (f (S i)) T"} +{"name":"BiLinearSymm.toHomogeneousQuad_add","declaration":"theorem BiLinearSymm.toHomogeneousQuad_add {V : Type} [AddCommMonoid V] [Module ℚ V] (τ : BiLinearSymm V) (S : V) (T : V) : (BiLinearSymm.toHomogeneousQuad τ) (S + T) =\n (BiLinearSymm.toHomogeneousQuad τ) S + (BiLinearSymm.toHomogeneousQuad τ) T + 2 * (τ S) T"} +{"name":"TriLinearSymm.map_sum₁₂₃","declaration":"theorem TriLinearSymm.map_sum₁₂₃ {V : Type} [AddCommMonoid V] [Module ℚ V] {n1 : ℕ} {n2 : ℕ} {n3 : ℕ} (f : TriLinearSymm V) (S : Fin n1 → V) (T : Fin n2 → V) (L : Fin n3 → V) : ((f (Finset.sum Finset.univ fun i => S i)) (Finset.sum Finset.univ fun i => T i))\n (Finset.sum Finset.univ fun i => L i) =\n Finset.sum Finset.univ fun i =>\n Finset.sum Finset.univ fun k => Finset.sum Finset.univ fun l => ((f (S i)) (T k)) (L l)"} +{"name":"IsSymmetric.mk","declaration":"ctor IsSymmetric.mk {V : Type} [AddCommMonoid V] [Module ℚ V] {f : V →ₗ[ℚ] V →ₗ[ℚ] ℚ} (swap : ∀ (S T : V), (f S) T = (f T) S) : IsSymmetric f"} +{"name":"BiLinearSymm.toHomogeneousQuad","declaration":"/-- The homogenous quadratic equation obtainable from a bilinear function. -/\ndef BiLinearSymm.toHomogeneousQuad {V : Type} [AddCommMonoid V] [Module ℚ V] (τ : BiLinearSymm V) : HomogeneousQuadratic V"} +{"name":"TriLinearSymm.swap₂'","declaration":"def TriLinearSymm.swap₂' {V : Type} [AddCommMonoid V] [Module ℚ V] (self : TriLinearSymm V) (S : V) (T : V) (L : V) : ((self.toFun S) T) L = ((self.toFun S) L) T"} +{"name":"TriLinearSymm.toLinear₁","declaration":"/-- Fixing the second and third input vectors, the resulting linear map. -/\ndef TriLinearSymm.toLinear₁ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (T : V) (L : V) : V →ₗ[ℚ] ℚ"} +{"name":"TriLinearSymm.toLinear₁_apply","declaration":"theorem TriLinearSymm.toLinear₁_apply {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (T : V) (L : V) : ((f S) T) L = (TriLinearSymm.toLinear₁ f T L) S"} +{"name":"TriLinearSymm.mk₃_toFun_apply_apply","declaration":"theorem TriLinearSymm.mk₃_toFun_apply_apply {V : Type} [AddCommMonoid V] [Module ℚ V] (f : V × V × V → ℚ) (map_smul : ∀ (a : ℚ) (S T L : V), f (a • S, T, L) = a * f (S, T, L)) (map_add : ∀ (S1 S2 T L : V), f (S1 + S2, T, L) = f (S1, T, L) + f (S2, T, L)) (swap₁ : ∀ (S T L : V), f (S, T, L) = f (T, S, L)) (swap₂ : ∀ (S T L : V), f (S, T, L) = f (S, L, T)) (S : V) (S : V) (T : V) : (((TriLinearSymm.mk₃ f map_smul map_add swap₁ swap₂) S✝) S) T = f (S✝, S, T)"} +{"name":"TriLinearSymm.map_add₂","declaration":"theorem TriLinearSymm.map_add₂ {V : Type} [AddCommMonoid V] [Module ℚ V] (f : TriLinearSymm V) (S : V) (T1 : V) (T2 : V) (L : V) : ((f S) (T1 + T2)) L = ((f S) T1) L + ((f S) T2) L"} diff --git a/hep-declarations/HepLean.Mathematics.SO3.Basic.jsonl b/hep-declarations/HepLean.Mathematics.SO3.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d1824e805b0d5ede857527a2d439f7d330b15eb7 --- /dev/null +++ b/hep-declarations/HepLean.Mathematics.SO3.Basic.jsonl @@ -0,0 +1,28 @@ +{"name":"GroupTheory.SO3.instTopologicalGroupSO3InstTopologicalSpaceSO3SO3Group","declaration":"instance GroupTheory.SO3.instTopologicalGroupSO3InstTopologicalSpaceSO3SO3Group : TopologicalGroup GroupTheory.SO3"} +{"name":"GroupTheory.SO3","declaration":"/-- The group of `3×3` real matrices with determinant 1 and `A * Aᵀ = 1`. -/\ndef GroupTheory.SO3 : Type"} +{"name":"GroupTheory.SO3.toGL","declaration":"/-- The inclusion of `SO(3)` into `GL (Fin 3) ℝ`. -/\ndef GroupTheory.SO3.toGL : GroupTheory.SO3 →* GL (Fin 3) ℝ"} +{"name":"GroupTheory.SO3Group_div","declaration":"theorem GroupTheory.SO3Group_div (a : GroupTheory.SO3) (b : GroupTheory.SO3) : a / b = DivInvMonoid.div' a b"} +{"name":"GroupTheory.SO3.exists_stationary_vec","declaration":"theorem GroupTheory.SO3.exists_stationary_vec (A : GroupTheory.SO3) : ∃ v, Orthonormal ℝ (Set.restrict {0} fun x => v) ∧ (GroupTheory.SO3.toEnd A) v = v"} +{"name":"GroupTheory.SO3.toEnd","declaration":"/-- The endomorphism of `EuclideanSpace ℝ (Fin 3)` associated to a element of `SO(3)`. -/\ndef GroupTheory.SO3.toEnd (A : GroupTheory.SO3) : Module.End ℝ (EuclideanSpace ℝ (Fin 3))"} +{"name":"GroupTheory.SO3.toProd_continuous","declaration":"theorem GroupTheory.SO3.toProd_continuous : Continuous ⇑GroupTheory.SO3.toProd"} +{"name":"GroupTheory.SO3.toProd_eq_transpose","declaration":"theorem GroupTheory.SO3.toProd_eq_transpose {A : GroupTheory.SO3} : GroupTheory.SO3.toProd A = (↑A, { unop' := Matrix.transpose ↑A })"} +{"name":"GroupTheory.SO3Group_inv","declaration":"theorem GroupTheory.SO3Group_inv (A : GroupTheory.SO3) : A⁻¹ = { val := Matrix.transpose ↑A, property := ⋯ }"} +{"name":"GroupTheory.SO3Group","declaration":"instance GroupTheory.SO3Group : Group GroupTheory.SO3"} +{"name":"GroupTheory.SO3.one_in_spectrum","declaration":"theorem GroupTheory.SO3.one_in_spectrum (A : GroupTheory.SO3) : 1 ∈ spectrum ℝ ↑A"} +{"name":"GroupTheory.SO3.det_id_minus","declaration":"theorem GroupTheory.SO3.det_id_minus (A : GroupTheory.SO3) : Matrix.det (1 - ↑A) = 0"} +{"name":"GroupTheory.SO3.toProd_injective","declaration":"theorem GroupTheory.SO3.toProd_injective : Function.Injective ⇑GroupTheory.SO3.toProd"} +{"name":"GroupTheory.SO3.exists_basis_preserved","declaration":"theorem GroupTheory.SO3.exists_basis_preserved (A : GroupTheory.SO3) : ∃ b, (GroupTheory.SO3.toEnd A) (b 0) = b 0"} +{"name":"GroupTheory.instTopologicalSpaceSO3","declaration":"/-- SO3 has the subtype topology. -/\ninstance GroupTheory.instTopologicalSpaceSO3 : TopologicalSpace GroupTheory.SO3"} +{"name":"GroupTheory.SO3.toProd_embedding","declaration":"/-- The embedding of `SO(3)` into the monoid of matrices times the opposite of\nthe monoid of matrices. -/\ntheorem GroupTheory.SO3.toProd_embedding : Embedding ⇑GroupTheory.SO3.toProd"} +{"name":"GroupTheory.SO3.one_is_eigenvalue","declaration":"theorem GroupTheory.SO3.one_is_eigenvalue (A : GroupTheory.SO3) : Module.End.HasEigenvalue (GroupTheory.SO3.toEnd A) 1"} +{"name":"GroupTheory.SO3.toGL_embedding","declaration":"/-- The embedding of `SO(3)` into `GL (Fin 3) ℝ`. -/\ntheorem GroupTheory.SO3.toGL_embedding : Embedding GroupTheory.SO3.toGL.toFun"} +{"name":"GroupTheory.SO3Group_mul_coe","declaration":"theorem GroupTheory.SO3Group_mul_coe (A : GroupTheory.SO3) (B : GroupTheory.SO3) : ↑(A * B) = ↑A * ↑B"} +{"name":"GroupTheory.SO3.toProd_apply","declaration":"theorem GroupTheory.SO3.toProd_apply : ∀ (a : GroupTheory.SO3),\n GroupTheory.SO3.toProd a = (↑(GroupTheory.SO3.toGL a), (MulOpposite.op ↑(GroupTheory.SO3.toGL a))⁻¹)"} +{"name":"GroupTheory.SO3.subtype_val_eq_toGL","declaration":"theorem GroupTheory.SO3.subtype_val_eq_toGL : Subtype.val = Units.val ∘ GroupTheory.SO3.toGL.toFun"} +{"name":"GroupTheory.SO3.det_minus_id","declaration":"theorem GroupTheory.SO3.det_minus_id (A : GroupTheory.SO3) : Matrix.det (↑A - 1) = 0"} +{"name":"GroupTheory.SO3.toProd","declaration":"/-- The inclusion of `SO(3)` into the monoid of matrices times the opposite of\nthe monoid of matrices. -/\ndef GroupTheory.SO3.toProd : GroupTheory.SO3 →* Matrix (Fin 3) (Fin 3) ℝ × (Matrix (Fin 3) (Fin 3) ℝ)ᵐᵒᵖ"} +{"name":"GroupTheory.SO3.coe_inv","declaration":"theorem GroupTheory.SO3.coe_inv (A : GroupTheory.SO3) : ↑A⁻¹ = (↑A)⁻¹"} +{"name":"GroupTheory.SO3.toEnd_apply","declaration":"theorem GroupTheory.SO3.toEnd_apply (A : GroupTheory.SO3) : ∀ (a : EuclideanSpace ℝ (Fin 3)),\n (GroupTheory.SO3.toEnd A) a =\n (LinearEquiv.symm (EuclideanSpace.basisFun (Fin 3) ℝ).repr.toLinearEquiv)\n (Matrix.mulVec ↑A ⇑((OrthonormalBasis.toBasis (EuclideanSpace.basisFun (Fin 3) ℝ)).repr a))"} +{"name":"GroupTheory.SO3.toGL_injective","declaration":"theorem GroupTheory.SO3.toGL_injective : Function.Injective ⇑GroupTheory.SO3.toGL"} +{"name":"GroupTheory.SO3Group_one_coe","declaration":"theorem GroupTheory.SO3Group_one_coe : ↑1 = 1"} +{"name":"GroupTheory.SO3_notation","declaration":"/-- Notation for the group `SO3`. -/\ndef GroupTheory.SO3_notation : Lean.ParserDescr"} diff --git a/hep-declarations/HepLean.Meta.AllFilePaths.jsonl b/hep-declarations/HepLean.Meta.AllFilePaths.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..080b8291209b502030f679a5f5b1fce41470a019 --- /dev/null +++ b/hep-declarations/HepLean.Meta.AllFilePaths.jsonl @@ -0,0 +1,2 @@ +{"name":"allFilePaths","declaration":"/-- Gets an array of all file paths in `HepLean`. -/\ndef allFilePaths : IO (Array System.FilePath)"} +{"name":"allFilePaths.go","declaration":"/-- The recursive function underlying `allFilePaths`. -/\nopaque allFilePaths.go (prev : Array System.FilePath) (root : Lean.Name) (path : System.FilePath) : IO (Array System.FilePath)"} diff --git a/hep-declarations/HepLean.Meta.Informal.jsonl b/hep-declarations/HepLean.Meta.Informal.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f0d0f5df673e2d9661724fbc031ce6b4e1cd7bae --- /dev/null +++ b/hep-declarations/HepLean.Meta.Informal.jsonl @@ -0,0 +1,26 @@ +{"name":"InformalLemma.name","declaration":"/-- The name of the informal lemma. This is autogenerated. -/\ndef InformalLemma.name (self : InformalLemma) : Lean.Name"} +{"name":"Informal.isInformal","declaration":"/-- Is true if and only if a `ConstantInfo` corresponds to an `InformalLemma` or a\n`InformalDefinition`. -/\ndef Informal.isInformal (c : Lean.ConstantInfo) : Bool"} +{"name":"Informal.commandInformal_lemma_Where_","declaration":"/-- The macro turning `informal_lemma ... where ...` into a definition. -/\ndef Informal.commandInformal_lemma_Where_ : Lean.ParserDescr"} +{"name":"InformalDefinition.math","declaration":"/-- The mathematical description of the definition. -/\ndef InformalDefinition.math (self : InformalDefinition) : String"} +{"name":"InformalLemma.math","declaration":"/-- The mathematical description of the lemma. -/\ndef InformalLemma.math (self : InformalLemma) : String"} +{"name":"Lean.Parser.Category.informalAssignment","declaration":"/-- The Parser.Category we will use for assignments. -/\ndef Lean.Parser.Category.informalAssignment : Lean.Parser.Category"} +{"name":"Informal.isInformalLemma","declaration":"/-- Is true if and only if a `ConstantInfo` corresponds to an `InformalLemma`. -/\ndef Informal.isInformalLemma (c : Lean.ConstantInfo) : Bool"} +{"name":"InformalLemma.proof","declaration":"/-- A description of the proof. -/\ndef InformalLemma.proof (self : InformalLemma) : String"} +{"name":"InformalDefinition","declaration":"/-- The structure representating an informal definition. -/\nstructure InformalDefinition : Type"} +{"name":"Informal.commandInformal_definition_Where_","declaration":"/-- The macro turning `informal_definition ... where ...` into a definition. -/\ndef Informal.commandInformal_definition_Where_ : Lean.ParserDescr"} +{"name":"Informal.isInformalDef","declaration":"/-- Is true if and only if a `ConstantInfo` corresponds to an `InformalDefinition`. -/\ndef Informal.isInformalDef (c : Lean.ConstantInfo) : Bool"} +{"name":"Informal.informal_definition","declaration":"/-- The syntax for the command informal_definition. -/\ndef Informal.informal_definition : Lean.ParserDescr"} +{"name":"InformalDefinition.ref","declaration":"/-- References. -/\ndef InformalDefinition.ref (self : InformalDefinition) : String"} +{"name":"InformalLemma.mk","declaration":"ctor InformalLemma.mk (name : Lean.Name) (math : String) (physics : String) (proof : String) (ref : String) (dependencies : List Lean.Name) : InformalLemma"} +{"name":"InformalDefinition.mk","declaration":"ctor InformalDefinition.mk (name : Lean.Name) (math : String) (physics : String) (ref : String) (dependencies : List Lean.Name) : InformalDefinition"} +{"name":"Informal.informal_lemma","declaration":"/-- The syntax for the command `informal_lemma`. -/\ndef Informal.informal_lemma : Lean.ParserDescr"} +{"name":"InformalLemma.physics","declaration":"/-- The physics description of the lemma. -/\ndef InformalLemma.physics (self : InformalLemma) : String"} +{"name":"InformalDefinition.name","declaration":"/-- The name of the informal definition. This is autogenerated. -/\ndef InformalDefinition.name (self : InformalDefinition) : Lean.Name"} +{"name":"InformalLemma","declaration":"/-- The structure representating an informal proof. -/\nstructure InformalLemma : Type"} +{"name":"Informal.informalAssignment.quot","declaration":"def Informal.informalAssignment.quot : Lean.ParserDescr"} +{"name":"InformalDefinition.physics","declaration":"/-- The physics description of the definition. -/\ndef InformalDefinition.physics (self : InformalDefinition) : String"} +{"name":"InformalLemma.ref","declaration":"/-- References. -/\ndef InformalLemma.ref (self : InformalLemma) : String"} +{"name":"InformalLemma.dependencies","declaration":"/-- The names of top-level commands we expect this lemma to depend on. -/\ndef InformalLemma.dependencies (self : InformalLemma) : List Lean.Name"} +{"name":"InformalDefinition.dependencies","declaration":"/-- The names of top-level commands we expect this definition to depend on. -/\ndef InformalDefinition.dependencies (self : InformalDefinition) : List Lean.Name"} +{"name":"Informal.informalAssignmentDeps","declaration":"/-- The syntax describing an informal assignment of `ident` to a list. -/\ndef Informal.informalAssignmentDeps : Lean.ParserDescr"} +{"name":"Informal.informalAssignment","declaration":"/-- The syntax describing an informal assignment of `ident` to a string. -/\ndef Informal.informalAssignment : Lean.ParserDescr"} diff --git a/hep-declarations/HepLean.Meta.TransverseTactics.jsonl b/hep-declarations/HepLean.Meta.TransverseTactics.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..159cbe33c76032d64eebf79970dd37d47880b049 --- /dev/null +++ b/hep-declarations/HepLean.Meta.TransverseTactics.jsonl @@ -0,0 +1,4 @@ +{"name":"transverseTactics.traverseForest","declaration":"/-- Applies `visitInfo` to each node of the info trees. -/\ndef transverseTactics.traverseForest (file : System.FilePath) (visitTacticInfo : System.FilePath → Lean.Elab.ContextInfo → Lean.Elab.TacticInfo → Lean.MetaM Unit) (steps : List (Lean.Environment × Lean.Elab.InfoState)) : List (IO Unit)"} +{"name":"transverseTactics","declaration":"/-- Applies `visitTacticInfo` to each tactic in a file. -/\ndef transverseTactics (file : System.FilePath) (visitTacticInfo : System.FilePath → Lean.Elab.ContextInfo → Lean.Elab.TacticInfo → Lean.MetaM Unit) : IO Unit"} +{"name":"transverseTactics.visitInfo","declaration":"/-- Tests if a given `info` is `ofTacticInfo` and if so runs `visitTacticInfo`. -/\ndef transverseTactics.visitInfo (file : System.FilePath) (env : Lean.Environment) (visitTacticInfo : System.FilePath → Lean.Elab.ContextInfo → Lean.Elab.TacticInfo → Lean.MetaM Unit) (ci : Lean.Elab.ContextInfo) (info : Lean.Elab.Info) (acc : List (IO Unit)) : List (IO Unit)"} +{"name":"transverseTactics.processCommands","declaration":"/-- Takes in a file and returns the infostates of commands and the corresponding\nenviroment before the command is processed. -/\nopaque transverseTactics.processCommands : Lean.Elab.Frontend.FrontendM (List (Lean.Environment × Lean.Elab.InfoState))"} diff --git a/hep-declarations/HepLean.SpaceTime.Basic.jsonl b/hep-declarations/HepLean.SpaceTime.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0c576b31613330f827318adf15e9bdc05b9f6ebc --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.Basic.jsonl @@ -0,0 +1,19 @@ +{"name":"SpaceTime.stdBasis_not_eq","declaration":"theorem SpaceTime.stdBasis_not_eq {μ : Fin 4} {ν : Fin 4} (h : μ ≠ ν) : SpaceTime.stdBasis μ ν = 0"} +{"name":"SpaceTime.smul_apply","declaration":"theorem SpaceTime.smul_apply (x : SpaceTime) (a : ℝ) (i : Fin 4) : (a • x) i = a * x i"} +{"name":"SpaceTime.asSmoothManifold","declaration":"/-- The structure of a smooth manifold on spacetime. -/\ndef SpaceTime.asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime"} +{"name":"SpaceTime.stdBasis_1","declaration":"theorem SpaceTime.stdBasis_1 : SpaceTime.stdBasis 1 = ![0, 1, 0, 0]"} +{"name":"SpaceTime.instChartedSpaceSpaceTimeToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedAddCommGroupEuclideanNormedAddCommGroup","declaration":"instance SpaceTime.instChartedSpaceSpaceTimeToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedAddCommGroupEuclideanNormedAddCommGroup : ChartedSpace SpaceTime SpaceTime"} +{"name":"euclideanNormedSpace","declaration":"instance euclideanNormedSpace : NormedSpace ℝ SpaceTime"} +{"name":"SpaceTime.explicit","declaration":"theorem SpaceTime.explicit (x : SpaceTime) : x = ![x 0, x 1, x 2, x 3]"} +{"name":"SpaceTime.space","declaration":"/-- The space part of spacetime. -/\ndef SpaceTime.space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3)"} +{"name":"SpaceTime.stdBasis_3","declaration":"theorem SpaceTime.stdBasis_3 : SpaceTime.stdBasis 3 = ![0, 0, 0, 1]"} +{"name":"SpaceTime.stdBasis_0","declaration":"theorem SpaceTime.stdBasis_0 : SpaceTime.stdBasis 0 = ![1, 0, 0, 0]"} +{"name":"SpaceTime.stdBasis_2","declaration":"theorem SpaceTime.stdBasis_2 : SpaceTime.stdBasis 2 = ![0, 0, 1, 0]"} +{"name":"instModuleRealSpaceTimeSemiringInstAddCommMonoidSpaceTime","declaration":"/-- Give spacetime the structure of a module over the reals. -/\ninstance instModuleRealSpaceTimeSemiringInstAddCommMonoidSpaceTime : Module ℝ SpaceTime"} +{"name":"SpaceTime.stdBasis","declaration":"/-- The standard basis for spacetime. -/\ndef SpaceTime.stdBasis : Basis (Fin 4) ℝ SpaceTime"} +{"name":"euclideanNormedAddCommGroup","declaration":"instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime"} +{"name":"SpaceTime.stdBasis_apply","declaration":"theorem SpaceTime.stdBasis_apply (μ : Fin 4) (ν : Fin 4) : SpaceTime.stdBasis μ ν = if μ = ν then 1 else 0"} +{"name":"SpaceTime.add_apply","declaration":"theorem SpaceTime.add_apply (x : SpaceTime) (y : SpaceTime) (i : Fin 4) : (x + y) i = x i + y i"} +{"name":"SpaceTime","declaration":"/-- The space-time -/\ndef SpaceTime : Type"} +{"name":"SpaceTime.stdBasis_mulVec","declaration":"theorem SpaceTime.stdBasis_mulVec (μ : Fin 4) (ν : Fin 4) (Λ : Matrix (Fin 4) (Fin 4) ℝ) : Matrix.mulVec Λ (SpaceTime.stdBasis μ) ν = Λ ν μ"} +{"name":"instAddCommMonoidSpaceTime","declaration":"/-- Give spacetime the structure of an additive commutative monoid. -/\ninstance instAddCommMonoidSpaceTime : AddCommMonoid SpaceTime"} diff --git a/hep-declarations/HepLean.SpaceTime.CliffordAlgebra.jsonl b/hep-declarations/HepLean.SpaceTime.CliffordAlgebra.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ff38c29c97c5b75c144b6765357acc4d58e43c18 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.CliffordAlgebra.jsonl @@ -0,0 +1,11 @@ +{"name":"spaceTime.γ5","declaration":"/-- The γ⁵ gamma matrix in the Dirac representation. -/\ndef spaceTime.γ5 : Matrix (Fin 4) (Fin 4) ℂ"} +{"name":"spaceTime.γ1","declaration":"/-- The γ¹ gamma matrix in the Dirac representation. -/\ndef spaceTime.γ1 : Matrix (Fin 4) (Fin 4) ℂ"} +{"name":"spaceTime.γ2","declaration":"/-- The γ² gamma matrix in the Dirac representation. -/\ndef spaceTime.γ2 : Matrix (Fin 4) (Fin 4) ℂ"} +{"name":"spaceTime.γ.γ_in_γSet","declaration":"theorem spaceTime.γ.γ_in_γSet (μ : Fin 4) : spaceTime.γ μ ∈ spaceTime.γ.γSet"} +{"name":"spaceTime.γ","declaration":"/-- The γ gamma matrices in the Dirac representation. -/\ndef spaceTime.γ : Fin 4 → Matrix (Fin 4) (Fin 4) ℂ"} +{"name":"spaceTime.γ.γSet","declaration":"/-- The subset of `Matrix (Fin 4) (Fin 4) ℂ` formed by the gamma matrices in the Dirac\nrepresentation. -/\ndef spaceTime.γ.γSet : Set (Matrix (Fin 4) (Fin 4) ℂ)"} +{"name":"spaceTime.γ3","declaration":"/-- The γ³ gamma matrix in the Dirac representation. -/\ndef spaceTime.γ3 : Matrix (Fin 4) (Fin 4) ℂ"} +{"name":"spaceTime.γ.γ_in_diracAlgebra","declaration":"theorem spaceTime.γ.γ_in_diracAlgebra (μ : Fin 4) : spaceTime.γ μ ∈ spaceTime.γ.diracAlgebra"} +{"name":"spaceTime.γ.diracAlgebra","declaration":"/-- The algebra generated by the gamma matrices in the Dirac representation. -/\ndef spaceTime.γ.diracAlgebra : Subalgebra ℝ (Matrix (Fin 4) (Fin 4) ℂ)"} +{"name":"spaceTime.γ.γSet_subset_diracAlgebra","declaration":"theorem spaceTime.γ.γSet_subset_diracAlgebra : spaceTime.γ.γSet ⊆ ↑spaceTime.γ.diracAlgebra"} +{"name":"spaceTime.γ0","declaration":"/-- The γ⁰ gamma matrix in the Dirac representation. -/\ndef spaceTime.γ0 : Matrix (Fin 4) (Fin 4) ℂ"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..fa4fd7c4c941af374686c50bd6cf5878a7aef19f --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl @@ -0,0 +1,11 @@ +{"name":"SpaceTime.lorentzAlgebra","declaration":"/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/\ndef SpaceTime.lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ)"} +{"name":"SpaceTime.lorentzAlgebra.mem_iff","declaration":"theorem SpaceTime.lorentzAlgebra.mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} : A ∈ SpaceTime.lorentzAlgebra ↔ Matrix.transpose A * minkowskiMatrix = -minkowskiMatrix * A"} +{"name":"SpaceTime.lorentzAlgebra.diag_comp","declaration":"theorem SpaceTime.lorentzAlgebra.diag_comp (Λ : ↥SpaceTime.lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : ↑Λ μ μ = 0"} +{"name":"SpaceTime.lorentzVectorAsLieRingModule_bracket","declaration":"theorem SpaceTime.lorentzVectorAsLieRingModule_bracket (Λ : ↥SpaceTime.lorentzAlgebra) (x : LorentzVector 3) : ∀ (a : Fin 1 ⊕ Fin 3), ⁅Λ, x⁆ a = Matrix.mulVec (↑Λ) x a"} +{"name":"SpaceTime.lorentzAlgebra.space_comps","declaration":"theorem SpaceTime.lorentzAlgebra.space_comps (Λ : ↥SpaceTime.lorentzAlgebra) (i : Fin 3) (j : Fin 3) : ↑Λ (Sum.inr i) (Sum.inr j) = -↑Λ (Sum.inr j) (Sum.inr i)"} +{"name":"SpaceTime.lorentzAlgebra.mem_of_transpose_eta_eq_eta_mul_self","declaration":"theorem SpaceTime.lorentzAlgebra.mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} (h : Matrix.transpose A * minkowskiMatrix = -minkowskiMatrix * A) : A ∈ SpaceTime.lorentzAlgebra"} +{"name":"SpaceTime.lorentzAlgebra.time_comps","declaration":"theorem SpaceTime.lorentzAlgebra.time_comps (Λ : ↥SpaceTime.lorentzAlgebra) (i : Fin 3) : ↑Λ (Sum.inr i) (Sum.inl 0) = ↑Λ (Sum.inl 0) (Sum.inr i)"} +{"name":"SpaceTime.lorentzVectorAsLieRingModule","declaration":"instance SpaceTime.lorentzVectorAsLieRingModule : LieRingModule (↥SpaceTime.lorentzAlgebra) (LorentzVector 3)"} +{"name":"SpaceTime.lorentzAlgebra.mem_iff'","declaration":"theorem SpaceTime.lorentzAlgebra.mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) : A ∈ SpaceTime.lorentzAlgebra ↔ A = -minkowskiMatrix * Matrix.transpose A * minkowskiMatrix"} +{"name":"SpaceTime.spaceTimeAsLieModule","declaration":"instance SpaceTime.spaceTimeAsLieModule : LieModule ℝ (↥SpaceTime.lorentzAlgebra) (LorentzVector 3)"} +{"name":"SpaceTime.lorentzAlgebra.transpose_eta","declaration":"theorem SpaceTime.lorentzAlgebra.transpose_eta (A : ↥SpaceTime.lorentzAlgebra) : Matrix.transpose ↑A * minkowskiMatrix = -minkowskiMatrix * ↑A"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzAlgebra.Basis.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzAlgebra.Basis.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/hep-declarations/HepLean.SpaceTime.LorentzGroup.Basic.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cfeaf7f9f50fe0caf12f94cf3fed460a6e31116b --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Basic.jsonl @@ -0,0 +1,40 @@ +{"name":"LorentzGroup.instTopologicalGroupElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroupInstTopologicalSpaceElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroupLorentzGroupIsGroup","declaration":"instance LorentzGroup.instTopologicalGroupElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroupInstTopologicalSpaceElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroupLorentzGroupIsGroup {d : ℕ} : TopologicalGroup ↑(LorentzGroup d)"} +{"name":"LorentzGroup.coe_inv","declaration":"theorem LorentzGroup.coe_inv {d : ℕ} {Λ : ↑(LorentzGroup d)} : ↑Λ⁻¹ = (↑Λ)⁻¹"} +{"name":"LorentzGroup.toNormOneLorentzVector","declaration":"/-- The first column of a Lorentz matrix as a `NormOneLorentzVector`. -/\ndef LorentzGroup.toNormOneLorentzVector {d : ℕ} (Λ : ↑(LorentzGroup d)) : ↑(NormOneLorentzVector d)"} +{"name":"LorentzGroup.mul_minkowskiMatrix_mul_transpose","declaration":"theorem LorentzGroup.mul_minkowskiMatrix_mul_transpose {d : ℕ} {Λ : ↑(LorentzGroup d)} : ↑Λ * minkowskiMatrix * Matrix.transpose ↑Λ = minkowskiMatrix"} +{"name":"LorentzGroup.toGL_injective","declaration":"theorem LorentzGroup.toGL_injective {d : ℕ} : Function.Injective ⇑LorentzGroup.toGL"} +{"name":"Matrix.mulVec_single_one","declaration":"theorem Matrix.mulVec_single_one {n : Type u_1} {R : Type u_2} {m : Type u_3} [Fintype n] [DecidableEq n] [NonAssocSemiring R] (M : Matrix m n R) (j : n) : Matrix.mulVec M (Pi.single j 1) = Matrix.transpose M j"} +{"name":"lorentzGroupIsGroup_div","declaration":"theorem lorentzGroupIsGroup_div {d : ℕ} (a : ↑(LorentzGroup d)) (b : ↑(LorentzGroup d)) : a / b = DivInvMonoid.div' a b"} +{"name":"LorentzGroup.toProd_embedding","declaration":"/-- The embedding from the Lorentz Group into the monoid of matrices times the opposite of\nthe monoid of matrices. -/\ntheorem LorentzGroup.toProd_embedding {d : ℕ} : Embedding ⇑LorentzGroup.toProd"} +{"name":"LorentzGroup.dual_mem","declaration":"theorem LorentzGroup.dual_mem {d : ℕ} {Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} (h : Λ ∈ LorentzGroup d) : minkowskiMetric.dual Λ ∈ LorentzGroup d"} +{"name":"LorentzGroup.lorentzGroup_notation","declaration":"/-- Notation for the Lorentz group. -/\ndef LorentzGroup.lorentzGroup_notation : Lean.ParserDescr"} +{"name":"LorentzGroup.toProd","declaration":"/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of\nthe monoid of matrices. -/\ndef LorentzGroup.toProd {d : ℕ} : ↑(LorentzGroup d) →* Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ × (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)ᵐᵒᵖ"} +{"name":"LorentzGroup.mem_iff_self_mul_dual","declaration":"theorem LorentzGroup.mem_iff_self_mul_dual {d : ℕ} {Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} : Λ ∈ LorentzGroup d ↔ Λ * minkowskiMetric.dual Λ = 1"} +{"name":"lorentzGroupIsGroup_mul_coe","declaration":"theorem lorentzGroupIsGroup_mul_coe {d : ℕ} (A : ↑(LorentzGroup d)) (B : ↑(LorentzGroup d)) : ↑(A * B) = ↑A * ↑B"} +{"name":"LorentzGroup.toProd_injective","declaration":"theorem LorentzGroup.toProd_injective {d : ℕ} : Function.Injective ⇑LorentzGroup.toProd"} +{"name":"LorentzGroup.toProd_apply","declaration":"theorem LorentzGroup.toProd_apply {d : ℕ} : ∀ (a : ↑(LorentzGroup d)), LorentzGroup.toProd a = (↑(LorentzGroup.toGL a), (MulOpposite.op ↑(LorentzGroup.toGL a))⁻¹)"} +{"name":"LorentzGroup.timeComp","declaration":"/-- The time like element of a Lorentz matrix. -/\ndef LorentzGroup.timeComp {d : ℕ} (Λ : ↑(LorentzGroup d)) : ℝ"} +{"name":"LorentzGroup.subtype_inv_mul","declaration":"theorem LorentzGroup.subtype_inv_mul {d : ℕ} {Λ : ↑(LorentzGroup d)} : (↑Λ)⁻¹ * ↑Λ = 1"} +{"name":"lorentzGroupIsGroup_one_coe","declaration":"theorem lorentzGroupIsGroup_one_coe {d : ℕ} : ↑1 = 1"} +{"name":"LorentzGroup.transpose_mul_minkowskiMatrix_mul_self","declaration":"theorem LorentzGroup.transpose_mul_minkowskiMatrix_mul_self {d : ℕ} {Λ : ↑(LorentzGroup d)} : Matrix.transpose ↑Λ * minkowskiMatrix * ↑Λ = minkowskiMatrix"} +{"name":"LorentzGroup.toGL_embedding","declaration":"/-- The embedding from the Lorentz Group into `GL (Fin 4) ℝ`. -/\ntheorem LorentzGroup.toGL_embedding {d : ℕ} : Embedding LorentzGroup.toGL.toFun"} +{"name":"LorentzGroup.transpose_mul","declaration":"theorem LorentzGroup.transpose_mul {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} : LorentzGroup.transpose (Λ * Λ') = LorentzGroup.transpose Λ' * LorentzGroup.transpose Λ"} +{"name":"LorentzGroup.mem_iff_transpose","declaration":"theorem LorentzGroup.mem_iff_transpose {d : ℕ} {Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} : Λ ∈ LorentzGroup d ↔ Matrix.transpose Λ ∈ LorentzGroup d"} +{"name":"LorentzGroup.timeComp_mul","declaration":"theorem LorentzGroup.timeComp_mul {d : ℕ} (Λ : ↑(LorentzGroup d)) (Λ' : ↑(LorentzGroup d)) : LorentzGroup.timeComp (Λ * Λ') =\n (minkowskiMetric ↑(LorentzGroup.toNormOneLorentzVector (LorentzGroup.transpose Λ)))\n (LorentzVector.spaceReflection ↑(LorentzGroup.toNormOneLorentzVector Λ'))"} +{"name":"LorentzGroup.toGL","declaration":"/-- The homomorphism of the Lorentz group into `GL (Fin 4) ℝ`. -/\ndef LorentzGroup.toGL {d : ℕ} : ↑(LorentzGroup d) →* GL (Fin 1 ⊕ Fin d) ℝ"} +{"name":"LorentzGroup.one_mem","declaration":"theorem LorentzGroup.one_mem {d : ℕ} : 1 ∈ LorentzGroup d"} +{"name":"LorentzGroup.transpose_one","declaration":"theorem LorentzGroup.transpose_one {d : ℕ} : LorentzGroup.transpose 1 = 1"} +{"name":"LorentzGroup.toNormOneLorentzVector_coe","declaration":"theorem LorentzGroup.toNormOneLorentzVector_coe {d : ℕ} (Λ : ↑(LorentzGroup d)) : ∀ (a : Fin 1 ⊕ Fin d), ↑(LorentzGroup.toNormOneLorentzVector Λ) a = Matrix.mulVec (↑Λ) LorentzVector.timeVec a"} +{"name":"instTopologicalSpaceElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroup","declaration":"/-- `LorentzGroup` has the subtype topology. -/\ninstance instTopologicalSpaceElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroup {d : ℕ} : TopologicalSpace ↑(LorentzGroup d)"} +{"name":"lorentzGroupIsGroup","declaration":"instance lorentzGroupIsGroup {d : ℕ} : Group ↑(LorentzGroup d)"} +{"name":"LorentzGroup.toProd_eq_transpose_η","declaration":"theorem LorentzGroup.toProd_eq_transpose_η {d : ℕ} {Λ : ↑(LorentzGroup d)} : LorentzGroup.toProd Λ = (↑Λ, MulOpposite.op (minkowskiMetric.dual ↑Λ))"} +{"name":"LorentzGroup.subtype_mul_inv","declaration":"theorem LorentzGroup.subtype_mul_inv {d : ℕ} {Λ : ↑(LorentzGroup d)} : ↑Λ * (↑Λ)⁻¹ = 1"} +{"name":"LorentzGroup.toProd_continuous","declaration":"theorem LorentzGroup.toProd_continuous {d : ℕ} : Continuous ⇑LorentzGroup.toProd"} +{"name":"LorentzGroup.timeComp_eq_toNormOneLorentzVector","declaration":"theorem LorentzGroup.timeComp_eq_toNormOneLorentzVector {d : ℕ} {Λ : ↑(LorentzGroup d)} : LorentzGroup.timeComp Λ = LorentzVector.time ↑(LorentzGroup.toNormOneLorentzVector Λ)"} +{"name":"LorentzGroup.mem_mul","declaration":"theorem LorentzGroup.mem_mul {d : ℕ} {Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} {Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} (hΛ : Λ ∈ LorentzGroup d) (hΛ' : Λ' ∈ LorentzGroup d) : Λ * Λ' ∈ LorentzGroup d"} +{"name":"LorentzGroup.transpose","declaration":"/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef LorentzGroup.transpose {d : ℕ} (Λ : ↑(LorentzGroup d)) : ↑(LorentzGroup d)"} +{"name":"LorentzGroup.mem_iff_norm","declaration":"theorem LorentzGroup.mem_iff_norm {d : ℕ} {Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} : Λ ∈ LorentzGroup d ↔\n ∀ (x : LorentzVector d), (minkowskiMetric (Matrix.mulVec Λ x)) (Matrix.mulVec Λ x) = (minkowskiMetric x) x"} +{"name":"LorentzGroup.mem_iff_on_right","declaration":"theorem LorentzGroup.mem_iff_on_right {d : ℕ} {Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} : Λ ∈ LorentzGroup d ↔\n ∀ (x y : LorentzVector d), (minkowskiMetric x) (Matrix.mulVec (minkowskiMetric.dual Λ * Λ) y) = (minkowskiMetric x) y"} +{"name":"LorentzGroup","declaration":"/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : ℕ) : Set (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)"} +{"name":"lorentzGroupIsGroup_inv","declaration":"theorem lorentzGroupIsGroup_inv {d : ℕ} (A : ↑(LorentzGroup d)) : A⁻¹ = { val := minkowskiMetric.dual ↑A, property := ⋯ }"} +{"name":"LorentzGroup.mem_iff_dual_mul_self","declaration":"theorem LorentzGroup.mem_iff_dual_mul_self {d : ℕ} {Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ} : Λ ∈ LorentzGroup d ↔ minkowskiMetric.dual Λ * Λ = 1"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzGroup.Boosts.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Boosts.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..99a84606063b234b4e9fe9d5b6731e9e0f2c9653 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Boosts.jsonl @@ -0,0 +1,14 @@ +{"name":"LorentzGroup.genBoost.toLorentz_in_connected_component_1","declaration":"theorem LorentzGroup.genBoost.toLorentz_in_connected_component_1 {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzGroup.genBoost.toLorentz u v ∈ connectedComponent 1"} +{"name":"LorentzGroup.genBoost.self","declaration":"/-- This lemma states that for a given four-velocity `u`, the general boost\ntransformation `genBoost u u` is equal to the identity linear map `LinearMap.id`.\n-/\ntheorem LorentzGroup.genBoost.self {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzGroup.genBoost u u = LinearMap.id"} +{"name":"LorentzGroup.genBoost.toMatrix_mulVec","declaration":"theorem LorentzGroup.genBoost.toMatrix_mulVec {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) (x : LorentzVector d) : Matrix.mulVec (LorentzGroup.genBoost.toMatrix u v) x = (LorentzGroup.genBoost u v) x"} +{"name":"LorentzGroup.genBoost.isProper","declaration":"theorem LorentzGroup.genBoost.isProper {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzGroup.IsProper (LorentzGroup.genBoost.toLorentz u v)"} +{"name":"LorentzGroup.genBoost.toMatrix_continuous","declaration":"theorem LorentzGroup.genBoost.toMatrix_continuous {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) : Continuous (LorentzGroup.genBoost.toMatrix u)"} +{"name":"LorentzGroup.genBoost.toLorentz","declaration":"/-- A generalised boost as an element of the Lorentz Group. -/\ndef LorentzGroup.genBoost.toLorentz {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : ↑(LorentzGroup d)"} +{"name":"LorentzGroup.genBoost.toMatrix","declaration":"/-- A generalised boost as a matrix. -/\ndef LorentzGroup.genBoost.toMatrix {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ"} +{"name":"LorentzGroup.genBoost","declaration":"/-- An generalised boost. This is a Lorentz transformation which takes the four velocity `u`\nto `v`. -/\ndef LorentzGroup.genBoost {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzVector d →ₗ[ℝ] LorentzVector d"} +{"name":"LorentzGroup.genBoostAux₂","declaration":"/-- An auxillary linear map used in the definition of a genearlised boost. -/\ndef LorentzGroup.genBoostAux₂ {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzVector d →ₗ[ℝ] LorentzVector d"} +{"name":"LorentzGroup.genBoost.toLorentz_joined_to_1","declaration":"theorem LorentzGroup.genBoost.toLorentz_joined_to_1 {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : Joined 1 (LorentzGroup.genBoost.toLorentz u v)"} +{"name":"LorentzGroup.genBoostAux₁","declaration":"/-- An auxillary linear map used in the definition of a generalised boost. -/\ndef LorentzGroup.genBoostAux₁ {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzVector d →ₗ[ℝ] LorentzVector d"} +{"name":"LorentzGroup.genBoost.toMatrix_in_lorentzGroup","declaration":"theorem LorentzGroup.genBoost.toMatrix_in_lorentzGroup {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzGroup.genBoost.toMatrix u v ∈ LorentzGroup d"} +{"name":"LorentzGroup.genBoost.toLorentz_continuous","declaration":"theorem LorentzGroup.genBoost.toLorentz_continuous {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) : Continuous (LorentzGroup.genBoost.toLorentz u)"} +{"name":"LorentzGroup.genBoost.toMatrix_apply","declaration":"theorem LorentzGroup.genBoost.toMatrix_apply {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) (v : ↑(NormOneLorentzVector.FuturePointing d)) (μ : Fin 1 ⊕ Fin d) (ν : Fin 1 ⊕ Fin d) : LorentzGroup.genBoost.toMatrix u v μ ν =\n minkowskiMatrix μ μ *\n ((minkowskiMetric (LorentzVector.stdBasis μ)) (LorentzVector.stdBasis ν) +\n 2 * (minkowskiMetric (LorentzVector.stdBasis ν)) ↑↑u * (minkowskiMetric (LorentzVector.stdBasis μ)) ↑↑v -\n (minkowskiMetric (LorentzVector.stdBasis μ)) (↑↑u + ↑↑v) *\n (minkowskiMetric (LorentzVector.stdBasis ν)) (↑↑u + ↑↑v) /\n (1 + (minkowskiMetric ↑↑u) ↑↑v))"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..69ded3acaa6987592e62093eeb40ba1caa7de158 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl @@ -0,0 +1,21 @@ +{"name":"LorentzGroup.orthchroMapReal_minus_one_or_one","declaration":"theorem LorentzGroup.orthchroMapReal_minus_one_or_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.orthchroMapReal Λ = -1 ∨ LorentzGroup.orthchroMapReal Λ = 1"} +{"name":"LorentzGroup.IsOrthochronous_iff_ge_one","declaration":"theorem LorentzGroup.IsOrthochronous_iff_ge_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.IsOrthochronous Λ ↔ 1 ≤ LorentzGroup.timeComp Λ"} +{"name":"LorentzGroup.orthchroMap_not_IsOrthochronous","declaration":"theorem LorentzGroup.orthchroMap_not_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : ¬LorentzGroup.IsOrthochronous Λ) : LorentzGroup.orthchroMap Λ = Additive.toMul 1"} +{"name":"LorentzGroup.timeCompCont","declaration":"/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/\ndef LorentzGroup.timeCompCont {d : ℕ} : C(↑(LorentzGroup d), ℝ)"} +{"name":"LorentzGroup.mul_othchron_of_othchron_othchron","declaration":"theorem LorentzGroup.mul_othchron_of_othchron_othchron {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : LorentzGroup.IsOrthochronous Λ) (h' : LorentzGroup.IsOrthochronous Λ') : LorentzGroup.IsOrthochronous (Λ * Λ')"} +{"name":"LorentzGroup.mul_not_othchron_of_othchron_not_othchron","declaration":"theorem LorentzGroup.mul_not_othchron_of_othchron_not_othchron {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : LorentzGroup.IsOrthochronous Λ) (h' : ¬LorentzGroup.IsOrthochronous Λ') : ¬LorentzGroup.IsOrthochronous (Λ * Λ')"} +{"name":"LorentzGroup.mul_not_othchron_of_not_othchron_othchron","declaration":"theorem LorentzGroup.mul_not_othchron_of_not_othchron_othchron {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : ¬LorentzGroup.IsOrthochronous Λ) (h' : LorentzGroup.IsOrthochronous Λ') : ¬LorentzGroup.IsOrthochronous (Λ * Λ')"} +{"name":"LorentzGroup.orthchroMap","declaration":"/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/\ndef LorentzGroup.orthchroMap {d : ℕ} : C(↑(LorentzGroup d), Multiplicative (ZMod 2))"} +{"name":"LorentzGroup.stepFunction","declaration":"/-- An auxillary function used in the definition of `orthchroMapReal`. -/\ndef LorentzGroup.stepFunction : ℝ → ℝ"} +{"name":"LorentzGroup.orthchroRep","declaration":"/-- The homomorphism from `LorentzGroup` to `ℤ₂` whose kernel are the Orthochronous elements. -/\ndef LorentzGroup.orthchroRep {d : ℕ} : ↑(LorentzGroup d) →* Multiplicative (ZMod 2)"} +{"name":"LorentzGroup.stepFunction_continuous","declaration":"theorem LorentzGroup.stepFunction_continuous : Continuous LorentzGroup.stepFunction"} +{"name":"LorentzGroup.orthchroMap_IsOrthochronous","declaration":"theorem LorentzGroup.orthchroMap_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : LorentzGroup.IsOrthochronous Λ) : LorentzGroup.orthchroMap Λ = 1"} +{"name":"LorentzGroup.IsOrthochronous_iff_transpose","declaration":"theorem LorentzGroup.IsOrthochronous_iff_transpose {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.IsOrthochronous Λ ↔ LorentzGroup.IsOrthochronous (LorentzGroup.transpose Λ)"} +{"name":"LorentzGroup.orthchroMapReal_on_not_IsOrthochronous","declaration":"theorem LorentzGroup.orthchroMapReal_on_not_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : ¬LorentzGroup.IsOrthochronous Λ) : LorentzGroup.orthchroMapReal Λ = -1"} +{"name":"LorentzGroup.IsOrthochronous_iff_futurePointing","declaration":"theorem LorentzGroup.IsOrthochronous_iff_futurePointing {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.IsOrthochronous Λ ↔ LorentzGroup.toNormOneLorentzVector Λ ∈ NormOneLorentzVector.FuturePointing d"} +{"name":"LorentzGroup.not_orthochronous_iff_le_neg_one","declaration":"theorem LorentzGroup.not_orthochronous_iff_le_neg_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : ¬LorentzGroup.IsOrthochronous Λ ↔ LorentzGroup.timeComp Λ ≤ -1"} +{"name":"LorentzGroup.not_orthochronous_iff_le_zero","declaration":"theorem LorentzGroup.not_orthochronous_iff_le_zero {d : ℕ} (Λ : ↑(LorentzGroup d)) : ¬LorentzGroup.IsOrthochronous Λ ↔ LorentzGroup.timeComp Λ ≤ 0"} +{"name":"LorentzGroup.orthchroMapReal","declaration":"/-- The continuous map from `lorentzGroup` to `ℝ` wh\ntaking Orthochronous elements to `1` and non-orthochronous to `-1`. -/\ndef LorentzGroup.orthchroMapReal {d : ℕ} : C(↑(LorentzGroup d), ℝ)"} +{"name":"LorentzGroup.orthchroMapReal_on_IsOrthochronous","declaration":"theorem LorentzGroup.orthchroMapReal_on_IsOrthochronous {d : ℕ} {Λ : ↑(LorentzGroup d)} (h : LorentzGroup.IsOrthochronous Λ) : LorentzGroup.orthchroMapReal Λ = 1"} +{"name":"LorentzGroup.mul_othchron_of_not_othchron_not_othchron","declaration":"theorem LorentzGroup.mul_othchron_of_not_othchron_not_othchron {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : ¬LorentzGroup.IsOrthochronous Λ) (h' : ¬LorentzGroup.IsOrthochronous Λ') : LorentzGroup.IsOrthochronous (Λ * Λ')"} +{"name":"LorentzGroup.IsOrthochronous","declaration":"/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/\ndef LorentzGroup.IsOrthochronous {d : ℕ} (Λ : ↑(LorentzGroup d)) : Prop"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzGroup.Proper.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Proper.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e292fe7beb44c67c6687e1fb8ba05305b405703e --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Proper.jsonl @@ -0,0 +1,23 @@ +{"name":"LorentzGroup.id_IsProper","declaration":"theorem LorentzGroup.id_IsProper {d : ℕ} : LorentzGroup.IsProper 1"} +{"name":"LorentzGroup.instTopologicalSpaceMultiplicativeZModOfNatNatInstOfNatNat","declaration":"instance LorentzGroup.instTopologicalSpaceMultiplicativeZModOfNatNatInstOfNatNat : TopologicalSpace (Multiplicative (ZMod 2))"} +{"name":"LorentzGroup.IsProper","declaration":"/-- A Lorentz Matrix is proper if its determinant is 1. -/\ndef LorentzGroup.IsProper {d : ℕ} (Λ : ↑(LorentzGroup d)) : Prop"} +{"name":"LorentzGroup.det_of_joined","declaration":"theorem LorentzGroup.det_of_joined {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : Joined Λ Λ') : Matrix.det ↑Λ = Matrix.det ↑Λ'"} +{"name":"LorentzGroup.coeForℤ₂_apply","declaration":"theorem LorentzGroup.coeForℤ₂_apply (x : ↑{-1, 1}) : LorentzGroup.coeForℤ₂ x = if x = { val := 1, property := LorentzGroup.coeForℤ₂.proof_1 } then 1 else Additive.toMul 1"} +{"name":"LorentzGroup.instDiscreteTopologyMultiplicativeZModOfNatNatInstOfNatNatInstTopologicalSpaceMultiplicativeZModOfNatNatInstOfNatNat","declaration":"instance LorentzGroup.instDiscreteTopologyMultiplicativeZModOfNatNatInstOfNatNatInstTopologicalSpaceMultiplicativeZModOfNatNatInstOfNatNat : DiscreteTopology (Multiplicative (ZMod 2))"} +{"name":"LorentzGroup.detRep_apply","declaration":"theorem LorentzGroup.detRep_apply {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.detRep Λ = LorentzGroup.detContinuous Λ"} +{"name":"Finite.instDiscreteTopology","declaration":"instance Finite.instDiscreteTopology {X : Type u_1} [TopologicalSpace X] [T1Space X] [Finite X] : DiscreteTopology X"} +{"name":"LorentzGroup.instTopologicalGroupMultiplicativeZModOfNatNatInstOfNatNatInstTopologicalSpaceMultiplicativeZModOfNatNatInstOfNatNatGroupToAddGroupToAddGroupWithOneToRingToDivisionRingInstFieldZModFact_prime_two","declaration":"instance LorentzGroup.instTopologicalGroupMultiplicativeZModOfNatNatInstOfNatNatInstTopologicalSpaceMultiplicativeZModOfNatNatInstOfNatNatGroupToAddGroupToAddGroupWithOneToRingToDivisionRingInstFieldZModFact_prime_two : TopologicalGroup (Multiplicative (ZMod 2))"} +{"name":"LorentzGroup.IsProper_iff","declaration":"theorem LorentzGroup.IsProper_iff {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.IsProper Λ ↔ LorentzGroup.detRep Λ = 1"} +{"name":"LorentzGroup.detRep_on_connected_component","declaration":"theorem LorentzGroup.detRep_on_connected_component {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : Λ' ∈ connectedComponent Λ) : LorentzGroup.detRep Λ = LorentzGroup.detRep Λ'"} +{"name":"LorentzGroup.detContinuous","declaration":"/-- The continuous map taking a Lorentz matrix to its determinant. -/\ndef LorentzGroup.detContinuous {d : ℕ} : C(↑(LorentzGroup d), Multiplicative (ZMod 2))"} +{"name":"LorentzGroup.detContinuous_eq_one","declaration":"theorem LorentzGroup.detContinuous_eq_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.detContinuous Λ = Additive.toMul 0 ↔ Matrix.det ↑Λ = 1"} +{"name":"toMul_eq_one","declaration":"theorem toMul_eq_one {α : Type u_1} [One α] {x : Additive α} : Additive.toMul x = 1 ↔ x = 0"} +{"name":"LorentzGroup.det_on_connected_component","declaration":"theorem LorentzGroup.det_on_connected_component {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : Λ' ∈ connectedComponent Λ) : Matrix.det ↑Λ = Matrix.det ↑Λ'"} +{"name":"LorentzGroup.isProper_on_connected_component","declaration":"theorem LorentzGroup.isProper_on_connected_component {d : ℕ} {Λ : ↑(LorentzGroup d)} {Λ' : ↑(LorentzGroup d)} (h : Λ' ∈ connectedComponent Λ) : LorentzGroup.IsProper Λ ↔ LorentzGroup.IsProper Λ'"} +{"name":"LorentzGroup.det_eq_one_or_neg_one","declaration":"/-- The determinant of a member of the Lorentz group is `1` or `-1`. -/\ntheorem LorentzGroup.det_eq_one_or_neg_one {d : ℕ} (Λ : ↑(LorentzGroup d)) : Matrix.det ↑Λ = 1 ∨ Matrix.det ↑Λ = -1"} +{"name":"LorentzGroup.detRep_continuous","declaration":"theorem LorentzGroup.detRep_continuous {d : ℕ} : Continuous ⇑LorentzGroup.detRep"} +{"name":"LorentzGroup.coeForℤ₂","declaration":"/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/\ndef LorentzGroup.coeForℤ₂ : C(↑{-1, 1}, Multiplicative (ZMod 2))"} +{"name":"LorentzGroup.detContinuous_eq_iff_det_eq","declaration":"theorem LorentzGroup.detContinuous_eq_iff_det_eq {d : ℕ} (Λ : ↑(LorentzGroup d)) (Λ' : ↑(LorentzGroup d)) : LorentzGroup.detContinuous Λ = LorentzGroup.detContinuous Λ' ↔ Matrix.det ↑Λ = Matrix.det ↑Λ'"} +{"name":"LorentzGroup.instDecidablePredElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroupIsProper","declaration":"instance LorentzGroup.instDecidablePredElemMatrixSumFinOfNatNatInstOfNatNatRealLorentzGroupIsProper {d : ℕ} : DecidablePred LorentzGroup.IsProper"} +{"name":"LorentzGroup.detRep","declaration":"/-- The representation taking a Lorentz matrix to its determinant. -/\ndef LorentzGroup.detRep {d : ℕ} : ↑(LorentzGroup d) →* Multiplicative (ZMod 2)"} +{"name":"LorentzGroup.detContinuous_eq_zero","declaration":"theorem LorentzGroup.detContinuous_eq_zero {d : ℕ} (Λ : ↑(LorentzGroup d)) : LorentzGroup.detContinuous Λ = Additive.toMul 1 ↔ Matrix.det ↑Λ = -1"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzGroup.Restricted.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Restricted.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..70feda378e68953a80f74ba57cda29f20065bbc0 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Restricted.jsonl @@ -0,0 +1 @@ +{"name":"LorentzGroup.Restricted","declaration":"/-- An informal definition. -/\ndef LorentzGroup.Restricted : InformalDefinition"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzGroup.Rotations.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Rotations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..51c67747f550f84b9b87b0c7a9ca99028501ea5e --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzGroup.Rotations.jsonl @@ -0,0 +1,4 @@ +{"name":"LorentzGroup.SO3ToMatrix","declaration":"/-- Given a element of `SO(3)` the matrix corresponding to a space-time rotation. -/\ndef LorentzGroup.SO3ToMatrix (A : GroupTheory.SO3) : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ"} +{"name":"LorentzGroup.SO3ToMatrix_in_LorentzGroup","declaration":"theorem LorentzGroup.SO3ToMatrix_in_LorentzGroup (A : GroupTheory.SO3) : LorentzGroup.SO3ToMatrix A ∈ LorentzGroup 3"} +{"name":"LorentzGroup.SO3ToLorentz","declaration":"/-- Given a element of `SO(3)` the element of the Lorentz group corresponding to a\nspace-time rotation. -/\ndef LorentzGroup.SO3ToLorentz : GroupTheory.SO3 →* ↑(LorentzGroup 3)"} +{"name":"LorentzGroup.SO3ToMatrix_injective","declaration":"theorem LorentzGroup.SO3ToMatrix_injective : Function.Injective LorentzGroup.SO3ToMatrix"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzTensor.Real.Basic.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzTensor.Real.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0f9dd6c2a01df56be80e7dda387dd29040e93421 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzTensor.Real.Basic.jsonl @@ -0,0 +1,11 @@ +{"name":"realTensorColor.instFintypeColorType","declaration":"instance realTensorColor.instFintypeColorType : Fintype realTensorColor.ColorType"} +{"name":"realTensorColor.colorTypEquivFin1Fin1","declaration":"/-- An equivalence between `ColorType ≃ Fin 1 ⊕ Fin 1`. -/\ndef realTensorColor.colorTypEquivFin1Fin1 : realTensorColor.ColorType ≃ Fin 1 ⊕ Fin 1"} +{"name":"realTensorColor.instDecidableEqColorType","declaration":"instance realTensorColor.instDecidableEqColorType : DecidableEq realTensorColor.ColorType"} +{"name":"realTensorColor.ColorType.down","declaration":"ctor realTensorColor.ColorType.down : realTensorColor.ColorType"} +{"name":"realTensorColor.ColorType","declaration":"/-- The type colors for real Lorentz tensors. -/\ninductive realTensorColor.ColorType : Type"} +{"name":"realLorentzTensor","declaration":"/-- The `TensorStructure` associated with real Lorentz tensors. -/\ndef realLorentzTensor (d : ℕ) : TensorStructure ℝ"} +{"name":"realTensorColor","declaration":"/-- The color structure for real lorentz tensors. -/\ndef realTensorColor : TensorColor"} +{"name":"instMulActionTensorRealInstCommSemiringRealElemMatrixSumFinOfNatNatInstOfNatNatLorentzGroupToMonoidToDivInvMonoidLorentzGroupIsGroupRealLorentzTensor","declaration":"/-- The action of the Lorentz group on real Lorentz tensors. -/\ninstance instMulActionTensorRealInstCommSemiringRealElemMatrixSumFinOfNatNatInstOfNatNatLorentzGroupToMonoidToDivInvMonoidLorentzGroupIsGroupRealLorentzTensor {d : ℕ} : MulActionTensor (↑(LorentzGroup d)) (realLorentzTensor d)"} +{"name":"instDecidableEqColorRealTensorColor","declaration":"instance instDecidableEqColorRealTensorColor : DecidableEq realTensorColor.Color"} +{"name":"instFintypeColorRealTensorColor","declaration":"instance instFintypeColorRealTensorColor : Fintype realTensorColor.Color"} +{"name":"realTensorColor.ColorType.up","declaration":"ctor realTensorColor.ColorType.up : realTensorColor.ColorType"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzTensor.Real.IndexNotation.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzTensor.Real.IndexNotation.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..8e44585938956dd06d4d917bd457629965c83a99 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzTensor.Real.IndexNotation.jsonl @@ -0,0 +1,12 @@ +{"name":"instIndexNotationColorRealTensorColor","declaration":"instance instIndexNotationColorRealTensorColor : IndexNotation realTensorColor.Color"} +{"name":"realLorentzTensor.tacticDualMapTactic","declaration":"/-- A tactic used to prove `boolFin` for real Lornetz tensors. -/\ndef realLorentzTensor.tacticDualMapTactic : Lean.ParserDescr"} +{"name":"realLorentzTensor.decidableEq_eq_color","declaration":"theorem realLorentzTensor.decidableEq_eq_color {d : ℕ} : realLorentzTensor.instDecidableEqColor = instDecidableEqColorRealTensorColor"} +{"name":"realLorentzTensor.tacticProdTactic","declaration":"/-- A tactics used to prove `colorPropBool` for real Lorentz tensors. -/\ndef realLorentzTensor.tacticProdTactic : Lean.ParserDescr"} +{"name":"realLorentzTensor.instIndexNotationColor","declaration":"instance realLorentzTensor.instIndexNotationColor {d : ℕ} : IndexNotation (realLorentzTensor d).Color"} +{"name":"realLorentzTensor.«term_|_»","declaration":"/-- Notation for the construction of a tensor index from a tensor and a string.\nConditions are checked automatically. -/\ndef realLorentzTensor.«term_|_» : Lean.TrailingParserDescr"} +{"name":"realLorentzTensor.instDecidableEqColor","declaration":"instance realLorentzTensor.instDecidableEqColor {d : ℕ} : DecidableEq (realLorentzTensor d).Color"} +{"name":"realLorentzTensor.toTensorColor_eq","declaration":"theorem realLorentzTensor.toTensorColor_eq {d : ℕ} : (realLorentzTensor d).toTensorColor = realTensorColor"} +{"name":"realLorentzTensor.indexNotation_eq_color","declaration":"theorem realLorentzTensor.indexNotation_eq_color {d : ℕ} : realLorentzTensor.instIndexNotationColor = instIndexNotationColorRealTensorColor"} +{"name":"realLorentzTensor.fromIndexStringColor","declaration":"/-- The construction of a tensor index from a tensor and a string satisfying conditions\nwhich can be automatically checked. This is a modified version of\n`TensorStructure.TensorIndex.mkDualMap` specific to real Lorentz tensors. -/\ndef realLorentzTensor.fromIndexStringColor {d : ℕ} {n : ℕ} {cn : Fin n → realTensorColor.Color} (T : TensorStructure.Tensor (realLorentzTensor d) cn) (s : String) (hs : IndexNotation.listCharIsIndexString realTensorColor.Color (String.toList s) = true) (hn : n = IndexNotation.IndexList.length (IndexNotation.IndexString.toIndexList' s hs)) (hD : IndexNotation.IndexList.OnlyUniqueDuals (IndexNotation.IndexString.toIndexList' s hs)) (hC : IndexNotation.IndexList.ColorCond.bool (IndexNotation.IndexString.toIndexList' s hs) = true) (hd : TensorColor.ColorMap.DualMap.boolFin (IndexNotation.IndexList.colorMap (IndexNotation.IndexString.toIndexList' s hs))\n (cn ∘ Fin.cast ⋯) =\n true) : TensorStructure.TensorIndex (realLorentzTensor d)"} +{"name":"realLorentzTensor.realLorentzTensor_color","declaration":"theorem realLorentzTensor.realLorentzTensor_color {d : ℕ} : (realLorentzTensor d).Color = realTensorColor.Color"} +{"name":"realLorentzTensor.«term_⊗ᵀ_»","declaration":"/-- The product of Real Lorentz tensors. Conditions on indices are checked automatically. -/\ndef realLorentzTensor.«term_⊗ᵀ_» : Lean.TrailingParserDescr"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..04bc9d372fc8bf4ce5f520d73625b03ea947ec43 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix.jsonl @@ -0,0 +1,7 @@ +{"name":"SpaceTime.toMatrix","declaration":"/-- A 2×2-complex matrix formed from a space-time point. -/\ndef SpaceTime.toMatrix (x : LorentzVector 3) : Matrix (Fin 2) (Fin 2) ℂ"} +{"name":"SpaceTime.toSelfAdjointMatrix'_coe","declaration":"theorem SpaceTime.toSelfAdjointMatrix'_coe (x : LorentzVector 3) : ↑(SpaceTime.toSelfAdjointMatrix' x) =\n Matrix.of\n ![![↑(x (Sum.inl 0)) + ↑(x (Sum.inr 2)), ↑(x (Sum.inr 0)) - ↑(x (Sum.inr 1)) * Complex.I],\n ![↑(x (Sum.inr 0)) + ↑(x (Sum.inr 1)) * Complex.I, ↑(x (Sum.inl 0)) - ↑(x (Sum.inr 2))]]"} +{"name":"SpaceTime.fromSelfAdjointMatrix'","declaration":"/-- A self-adjoint matrix formed from a space-time point. -/\ndef SpaceTime.fromSelfAdjointMatrix' (x : ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))) : LorentzVector 3"} +{"name":"SpaceTime.toSelfAdjointMatrix'","declaration":"/-- A self-adjoint matrix formed from a space-time point. -/\ndef SpaceTime.toSelfAdjointMatrix' (x : LorentzVector 3) : ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))"} +{"name":"SpaceTime.toSelfAdjointMatrix","declaration":"/-- The linear equivalence between the vector-space `spaceTime` and self-adjoint\n2×2-complex matrices. -/\ndef SpaceTime.toSelfAdjointMatrix : LorentzVector 3 ≃ₗ[ℝ] ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))"} +{"name":"SpaceTime.det_eq_ηLin","declaration":"theorem SpaceTime.det_eq_ηLin (x : LorentzVector 3) : Matrix.det ↑(SpaceTime.toSelfAdjointMatrix x) = ↑((minkowskiMetric x) x)"} +{"name":"SpaceTime.toMatrix_isSelfAdjoint","declaration":"/-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/\ntheorem SpaceTime.toMatrix_isSelfAdjoint (x : LorentzVector 3) : IsSelfAdjoint (SpaceTime.toMatrix x)"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzVector.Basic.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzVector.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..39f030b71868347751c4dce6d2de8aa5cca8a39a --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzVector.Basic.jsonl @@ -0,0 +1,23 @@ +{"name":"LorentzVector.spaceReflection_time","declaration":"theorem LorentzVector.spaceReflection_time {d : ℕ} (v : LorentzVector d) : LorentzVector.time (LorentzVector.spaceReflection v) = LorentzVector.time v"} +{"name":"LorentzVector.stdBasis_repr_apply_support_val","declaration":"theorem LorentzVector.stdBasis_repr_apply_support_val {d : ℕ} : ∀ (a : Fin 1 ⊕ Fin d → ℝ), (LorentzVector.stdBasis.repr a).support.val = Multiset.map Subtype.val Finset.univ.val"} +{"name":"LorentzVector.stdBasis_repr_apply_toFun","declaration":"theorem LorentzVector.stdBasis_repr_apply_toFun {d : ℕ} : ∀ (a : Fin 1 ⊕ Fin d → ℝ) (a_1 : Fin 1 ⊕ Fin d), (LorentzVector.stdBasis.repr a) a_1 = a a_1"} +{"name":"LorentzVector.spaceReflection_space","declaration":"theorem LorentzVector.spaceReflection_space {d : ℕ} (v : LorentzVector d) : LorentzVector.space (LorentzVector.spaceReflection v) = -LorentzVector.space v"} +{"name":"instAddCommMonoidLorentzVector","declaration":"/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance instAddCommMonoidLorentzVector (d : ℕ) : AddCommMonoid (LorentzVector d)"} +{"name":"LorentzVector.timeVec","declaration":"/-- The standard unit time vector. -/\ndef LorentzVector.timeVec {d : ℕ} : LorentzVector d"} +{"name":"LorentzVector.spaceReflection","declaration":"/-- The reflection of space. -/\ndef LorentzVector.spaceReflection {d : ℕ} (v : LorentzVector d) : LorentzVector d"} +{"name":"LorentzVector.decomp_stdBasis'","declaration":"theorem LorentzVector.decomp_stdBasis' {d : ℕ} (v : LorentzVector d) : (v (Sum.inl 0) • LorentzVector.stdBasis (Sum.inl 0) +\n Finset.sum Finset.univ fun a₂ => v (Sum.inr a₂) • LorentzVector.stdBasis (Sum.inr a₂)) =\n v"} +{"name":"LorentzVector.timeVec_space","declaration":"theorem LorentzVector.timeVec_space {d : ℕ} : LorentzVector.space LorentzVector.timeVec = 0"} +{"name":"instAddCommGroupLorentzVector","declaration":"instance instAddCommGroupLorentzVector (d : ℕ) : AddCommGroup (LorentzVector d)"} +{"name":"LorentzVector.stdBasis","declaration":"/-- The standard basis of `LorentzVector` indexed by `Fin 1 ⊕ Fin (d)`. -/\ndef LorentzVector.stdBasis {d : ℕ} : Basis (Fin 1 ⊕ Fin d) ℝ (LorentzVector d)"} +{"name":"instModuleRealLorentzVector","declaration":"/-- An instance of a module on `LorentzVector`. -/\ninstance instModuleRealLorentzVector (d : ℕ) : Module ℝ (LorentzVector d)"} +{"name":"LorentzVector.timeVec_time","declaration":"theorem LorentzVector.timeVec_time {d : ℕ} : LorentzVector.time LorentzVector.timeVec = 1"} +{"name":"LorentzVector.time","declaration":"/-- The time component. -/\ndef LorentzVector.time {d : ℕ} (v : LorentzVector d) : ℝ"} +{"name":"instTopologicalSpaceLorentzVector","declaration":"/-- The structure of a topological space `LorentzVector d`. -/\ninstance instTopologicalSpaceLorentzVector (d : ℕ) : TopologicalSpace (LorentzVector d)"} +{"name":"LorentzVector.spaceReflectionLin_apply","declaration":"theorem LorentzVector.spaceReflectionLin_apply {d : ℕ} (x : LorentzVector d) : ∀ (a : Fin 1 ⊕ Fin d), LorentzVector.spaceReflectionLin x a = Sum.elim (x ∘ Sum.inl) (-x ∘ Sum.inr) a"} +{"name":"LorentzVector.termE","declaration":"/-- Notation for `stdBasis`. -/\ndef LorentzVector.termE : Lean.ParserDescr"} +{"name":"LorentzVector.decomp_stdBasis","declaration":"theorem LorentzVector.decomp_stdBasis {d : ℕ} (v : LorentzVector d) : (Finset.sum Finset.univ fun i => v i • LorentzVector.stdBasis i) = v"} +{"name":"LorentzVector.space","declaration":"/-- The space components. -/\ndef LorentzVector.space {d : ℕ} (v : LorentzVector d) : EuclideanSpace ℝ (Fin d)"} +{"name":"LorentzVector.spaceReflectionLin","declaration":"/-- The reflection of space as a linear map. -/\ndef LorentzVector.spaceReflectionLin {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d"} +{"name":"LorentzVector.stdBasis_repr_symm_apply","declaration":"theorem LorentzVector.stdBasis_repr_symm_apply {d : ℕ} : ∀ (a : Fin 1 ⊕ Fin d →₀ ℝ) (a_1 : Fin 1 ⊕ Fin d), (LinearEquiv.symm LorentzVector.stdBasis.repr) a a_1 = a a_1"} +{"name":"LorentzVector.stdBasis_apply","declaration":"theorem LorentzVector.stdBasis_apply {d : ℕ} (μ : Fin 1 ⊕ Fin d) (ν : Fin 1 ⊕ Fin d) : LorentzVector.stdBasis μ ν = if μ = ν then 1 else 0"} +{"name":"LorentzVector","declaration":"/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector (d : ℕ) : Type"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzVector.Contraction.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzVector.Contraction.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cf63eb1e90a0b8f8ca7c3b9ad291fbe6561b7207 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzVector.Contraction.jsonl @@ -0,0 +1,23 @@ +{"name":"minkowskiMatrix.asCoTenProd_diag","declaration":"theorem minkowskiMatrix.asCoTenProd_diag {d : ℕ} : minkowskiMatrix.asCoTenProd =\n Finset.sum Finset.univ fun μ =>\n minkowskiMatrix μ μ • CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ"} +{"name":"minkowskiMatrix.asTenProd_invariant","declaration":"theorem minkowskiMatrix.asTenProd_invariant {d : ℕ} (g : ↑(LorentzGroup d)) : (TensorProduct.map (LorentzVector.rep g) (LorentzVector.rep g)) minkowskiMatrix.asTenProd = minkowskiMatrix.asTenProd"} +{"name":"LorentzVector.unitDown_rid","declaration":"theorem LorentzVector.unitDown_rid {d : ℕ} (x : CovariantLorentzVector d) : (TensorStructure.contrLeftAux LorentzVector.contrDownUp) (x ⊗ₜ[ℝ] LorentzVector.unitDown) = x"} +{"name":"LorentzVector.unitDown","declaration":"/-- The unit of the contraction of covariant Lorentz vector and a\ncontravariant Lorentz vector. -/\ndef LorentzVector.unitDown {d : ℕ} : TensorProduct ℝ (LorentzVector d) (CovariantLorentzVector d)"} +{"name":"LorentzVector.contrUpDown","declaration":"/-- The linear map defining the contraction of a contravariant Lorentz vector\nand a covariant Lorentz vector. -/\ndef LorentzVector.contrUpDown {d : ℕ} : TensorProduct ℝ (LorentzVector d) (CovariantLorentzVector d) →ₗ[ℝ] ℝ"} +{"name":"minkowskiMatrix.contrLeft_asTenProd","declaration":"theorem minkowskiMatrix.contrLeft_asTenProd {d : ℕ} (x : CovariantLorentzVector d) : (TensorStructure.contrLeftAux LorentzVector.contrDownUp) (x ⊗ₜ[ℝ] minkowskiMatrix.asTenProd) =\n Finset.sum Finset.univ fun μ => (minkowskiMatrix μ μ * x μ) • LorentzVector.stdBasis μ"} +{"name":"minkowskiMatrix.asTenProd_diag","declaration":"theorem minkowskiMatrix.asTenProd_diag {d : ℕ} : minkowskiMatrix.asTenProd =\n Finset.sum Finset.univ fun μ => minkowskiMatrix μ μ • LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ"} +{"name":"LorentzVector.contrUpDown_tmul_eq_dotProduct","declaration":"theorem LorentzVector.contrUpDown_tmul_eq_dotProduct {d : ℕ} {x : LorentzVector d} {y : CovariantLorentzVector d} : LorentzVector.contrUpDown (x ⊗ₜ[ℝ] y) = Matrix.dotProduct x y"} +{"name":"minkowskiMatrix.asTenProd_contr_asCoTenProd","declaration":"theorem minkowskiMatrix.asTenProd_contr_asCoTenProd {d : ℕ} : (TensorStructure.contrMidAux LorentzVector.contrUpDown) (minkowskiMatrix.asTenProd ⊗ₜ[ℝ] minkowskiMatrix.asCoTenProd) =\n (TensorProduct.comm ℝ (CovariantLorentzVector d) (LorentzVector d)) LorentzVector.unitUp"} +{"name":"LorentzVector.unitUp_rid","declaration":"theorem LorentzVector.unitUp_rid {d : ℕ} (x : LorentzVector d) : (TensorStructure.contrLeftAux LorentzVector.contrUpDown) (x ⊗ₜ[ℝ] LorentzVector.unitUp) = x"} +{"name":"LorentzVector.contrUpDown_invariant_lorentzAction","declaration":"theorem LorentzVector.contrUpDown_invariant_lorentzAction {d : ℕ} {g : ↑(LorentzGroup d)} {x : LorentzVector d} {y : CovariantLorentzVector d} : LorentzVector.contrUpDown ((LorentzVector.rep g) x ⊗ₜ[ℝ] (CovariantLorentzVector.rep g) y) =\n LorentzVector.contrUpDown (x ⊗ₜ[ℝ] y)"} +{"name":"minkowskiMatrix.asCoTenProd_contr_asTenProd","declaration":"theorem minkowskiMatrix.asCoTenProd_contr_asTenProd {d : ℕ} : (TensorStructure.contrMidAux LorentzVector.contrDownUp) (minkowskiMatrix.asCoTenProd ⊗ₜ[ℝ] minkowskiMatrix.asTenProd) =\n (TensorProduct.comm ℝ (LorentzVector d) (CovariantLorentzVector d)) LorentzVector.unitDown"} +{"name":"minkowskiMatrix.asCoTenProd_invariant","declaration":"theorem minkowskiMatrix.asCoTenProd_invariant {d : ℕ} (g : ↑(LorentzGroup d)) : (TensorProduct.map (CovariantLorentzVector.rep g) (CovariantLorentzVector.rep g)) minkowskiMatrix.asCoTenProd =\n minkowskiMatrix.asCoTenProd"} +{"name":"LorentzVector.contrUpDown_stdBasis_right","declaration":"theorem LorentzVector.contrUpDown_stdBasis_right {d : ℕ} (x : CovariantLorentzVector d) (i : Fin 1 ⊕ Fin d) : LorentzVector.contrUpDown (LorentzVector.stdBasis i ⊗ₜ[ℝ] x) = x i"} +{"name":"LorentzVector.contrDownUp_tmul_eq_dotProduct","declaration":"theorem LorentzVector.contrDownUp_tmul_eq_dotProduct {d : ℕ} {x : CovariantLorentzVector d} {y : LorentzVector d} : LorentzVector.contrDownUp (x ⊗ₜ[ℝ] y) = Matrix.dotProduct x y"} +{"name":"minkowskiMatrix.asTenProd","declaration":"/-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/\ndef minkowskiMatrix.asTenProd {d : ℕ} : TensorProduct ℝ (LorentzVector d) (LorentzVector d)"} +{"name":"LorentzVector.contrDownUp","declaration":"/-- The linear map defining the contraction of a covariant Lorentz vector\nand a contravariant Lorentz vector. -/\ndef LorentzVector.contrDownUp {d : ℕ} : TensorProduct ℝ (CovariantLorentzVector d) (LorentzVector d) →ₗ[ℝ] ℝ"} +{"name":"minkowskiMatrix.asCoTenProd","declaration":"/-- The metric tensor as an element of `CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d`. -/\ndef minkowskiMatrix.asCoTenProd {d : ℕ} : TensorProduct ℝ (CovariantLorentzVector d) (CovariantLorentzVector d)"} +{"name":"LorentzVector.contrUpDown_stdBasis_left","declaration":"theorem LorentzVector.contrUpDown_stdBasis_left {d : ℕ} (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) : LorentzVector.contrUpDown (x ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i) = x i"} +{"name":"minkowskiMatrix.contrLeft_asCoTenProd","declaration":"theorem minkowskiMatrix.contrLeft_asCoTenProd {d : ℕ} (x : LorentzVector d) : (TensorStructure.contrLeftAux LorentzVector.contrUpDown) (x ⊗ₜ[ℝ] minkowskiMatrix.asCoTenProd) =\n Finset.sum Finset.univ fun μ => (minkowskiMatrix μ μ * x μ) • CovariantLorentzVector.stdBasis μ"} +{"name":"LorentzVector.contrDownUp_invariant_lorentzAction","declaration":"theorem LorentzVector.contrDownUp_invariant_lorentzAction {d : ℕ} {g : ↑(LorentzGroup d)} {x : CovariantLorentzVector d} {y : LorentzVector d} : LorentzVector.contrDownUp ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ] (LorentzVector.rep g) y) =\n LorentzVector.contrDownUp (x ⊗ₜ[ℝ] y)"} +{"name":"LorentzVector.contrUpDownBi","declaration":"/-- The bi-linear map defining the contraction of a contravariant Lorentz vector\nand a covariant Lorentz vector. -/\ndef LorentzVector.contrUpDownBi {d : ℕ} : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ"} +{"name":"LorentzVector.unitUp","declaration":"/-- The unit of the contraction of contravariant Lorentz vector and a\ncovariant Lorentz vector. -/\ndef LorentzVector.unitUp {d : ℕ} : TensorProduct ℝ (CovariantLorentzVector d) (LorentzVector d)"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzVector.Covariant.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzVector.Covariant.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..392691e2baa8256303b0f096370014910df34811 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzVector.Covariant.jsonl @@ -0,0 +1,14 @@ +{"name":"CovariantLorentzVector.decomp_stdBasis","declaration":"theorem CovariantLorentzVector.decomp_stdBasis {d : ℕ} (v : CovariantLorentzVector d) : (Finset.sum Finset.univ fun i => v i • CovariantLorentzVector.stdBasis i) = v"} +{"name":"CovariantLorentzVector.stdBasis_repr_symm_apply","declaration":"theorem CovariantLorentzVector.stdBasis_repr_symm_apply {d : ℕ} : ∀ (a : Fin 1 ⊕ Fin d →₀ ℝ) (a_1 : Fin 1 ⊕ Fin d), (LinearEquiv.symm CovariantLorentzVector.stdBasis.repr) a a_1 = a a_1"} +{"name":"CovariantLorentzVector.stdBasis","declaration":"/-- The standard basis of `LorentzVector` indexed by `Fin 1 ⊕ Fin (d)`. -/\ndef CovariantLorentzVector.stdBasis {d : ℕ} : Basis (Fin 1 ⊕ Fin d) ℝ (CovariantLorentzVector d)"} +{"name":"instAddCommMonoidCovariantLorentzVector","declaration":"/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance instAddCommMonoidCovariantLorentzVector (d : ℕ) : AddCommMonoid (CovariantLorentzVector d)"} +{"name":"CovariantLorentzVector.rep_apply","declaration":"theorem CovariantLorentzVector.rep_apply {d : ℕ} (v : CovariantLorentzVector d) (g : ↑(LorentzGroup d)) : (CovariantLorentzVector.rep g) v = Matrix.mulVec (Matrix.transpose (↑g)⁻¹) v"} +{"name":"CovariantLorentzVector.stdBasis_repr_apply_toFun","declaration":"theorem CovariantLorentzVector.stdBasis_repr_apply_toFun {d : ℕ} : ∀ (a : Fin 1 ⊕ Fin d → ℝ) (a_1 : Fin 1 ⊕ Fin d), (CovariantLorentzVector.stdBasis.repr a) a_1 = a a_1"} +{"name":"instTopologicalSpaceCovariantLorentzVector","declaration":"/-- The structure of a topological space `LorentzVector d`. -/\ninstance instTopologicalSpaceCovariantLorentzVector (d : ℕ) : TopologicalSpace (CovariantLorentzVector d)"} +{"name":"CovariantLorentzVector.rep","declaration":"/-- The representation of the Lorentz group acting on covariant Lorentz vectors. -/\ndef CovariantLorentzVector.rep {d : ℕ} : Representation ℝ (↑(LorentzGroup d)) (CovariantLorentzVector d)"} +{"name":"instModuleRealCovariantLorentzVector","declaration":"/-- An instance of a module on `LorentzVector`. -/\ninstance instModuleRealCovariantLorentzVector (d : ℕ) : Module ℝ (CovariantLorentzVector d)"} +{"name":"CovariantLorentzVector.rep_apply_stdBasis","declaration":"theorem CovariantLorentzVector.rep_apply_stdBasis {d : ℕ} (g : ↑(LorentzGroup d)) (μ : Fin 1 ⊕ Fin d) : (CovariantLorentzVector.rep g) (CovariantLorentzVector.stdBasis μ) =\n Finset.sum Finset.univ fun ν => ↑g⁻¹ μ ν • CovariantLorentzVector.stdBasis ν"} +{"name":"CovariantLorentzVector.stdBasis_repr_apply_support_val","declaration":"theorem CovariantLorentzVector.stdBasis_repr_apply_support_val {d : ℕ} : ∀ (a : Fin 1 ⊕ Fin d → ℝ),\n (CovariantLorentzVector.stdBasis.repr a).support.val = Multiset.map Subtype.val Finset.univ.val"} +{"name":"instAddCommGroupCovariantLorentzVector","declaration":"instance instAddCommGroupCovariantLorentzVector (d : ℕ) : AddCommGroup (CovariantLorentzVector d)"} +{"name":"CovariantLorentzVector","declaration":"/-- The type of covariant Lorentz Vectors in `d`-space dimensions. -/\ndef CovariantLorentzVector (d : ℕ) : Type"} +{"name":"CovariantLorentzVector.decomp_stdBasis'","declaration":"theorem CovariantLorentzVector.decomp_stdBasis' {d : ℕ} (v : CovariantLorentzVector d) : (v (Sum.inl 0) • CovariantLorentzVector.stdBasis (Sum.inl 0) +\n Finset.sum Finset.univ fun a₂ => v (Sum.inr a₂) • CovariantLorentzVector.stdBasis (Sum.inr a₂)) =\n v"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzVector.LorentzAction.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzVector.LorentzAction.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6c74ea8bbb727168af5ae6b8873b1ba8734b54e0 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzVector.LorentzAction.jsonl @@ -0,0 +1,3 @@ +{"name":"LorentzVector.rep_apply_stdBasis","declaration":"theorem LorentzVector.rep_apply_stdBasis {d : ℕ} (g : ↑(LorentzGroup d)) (μ : Fin 1 ⊕ Fin d) : (LorentzVector.rep g) (LorentzVector.stdBasis μ) =\n Finset.sum Finset.univ fun ν => Matrix.transpose (↑g) μ ν • LorentzVector.stdBasis ν"} +{"name":"LorentzVector.rep_apply","declaration":"theorem LorentzVector.rep_apply {d : ℕ} (v : LorentzVector d) (g : ↑(LorentzGroup d)) : (LorentzVector.rep g) v = Matrix.mulVec (↑g) v"} +{"name":"LorentzVector.rep","declaration":"/-- The contravariant action of the Lorentz group on a Lorentz vector. -/\ndef LorentzVector.rep {d : ℕ} : Representation ℝ (↑(LorentzGroup d)) (LorentzVector d)"} diff --git a/hep-declarations/HepLean.SpaceTime.LorentzVector.NormOne.jsonl b/hep-declarations/HepLean.SpaceTime.LorentzVector.NormOne.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f2b10c8850c07e4ec30734d2aa6097d71e82a3ff --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.LorentzVector.NormOne.jsonl @@ -0,0 +1,33 @@ +{"name":"NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_mem","declaration":"theorem NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_mem {d : ℕ} {v : ↑(NormOneLorentzVector d)} {w : ↑(NormOneLorentzVector d)} (h : v ∉ NormOneLorentzVector.FuturePointing d) (hw : w ∈ NormOneLorentzVector.FuturePointing d) : (minkowskiMetric ↑v) (LorentzVector.spaceReflection ↑w) ≤ 0"} +{"name":"NormOneLorentzVector.FuturePointing.pathFromTime","declaration":"/-- A continuous path from `timeVecNormOneFuture` to any other. -/\ndef NormOneLorentzVector.FuturePointing.pathFromTime {d : ℕ} (u : ↑(NormOneLorentzVector.FuturePointing d)) : Path NormOneLorentzVector.FuturePointing.timeVecNormOneFuture u"} +{"name":"NormOneLorentzVector.time_pos_iff","declaration":"theorem NormOneLorentzVector.time_pos_iff {d : ℕ} (v : ↑(NormOneLorentzVector d)) : 0 < LorentzVector.time ↑v ↔ 1 ≤ LorentzVector.time ↑v"} +{"name":"NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg","declaration":"theorem NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg {d : ℕ} (v : ↑(NormOneLorentzVector d)) : v ∈ NormOneLorentzVector.FuturePointing d ↔ 0 ≤ LorentzVector.time ↑v"} +{"name":"instTopologicalSpaceElemLorentzVectorNormOneLorentzVector","declaration":"instance instTopologicalSpaceElemLorentzVectorNormOneLorentzVector {d : ℕ} : TopologicalSpace ↑(NormOneLorentzVector d)"} +{"name":"NormOneLorentzVector.time_nonpos_iff","declaration":"theorem NormOneLorentzVector.time_nonpos_iff {d : ℕ} (v : ↑(NormOneLorentzVector d)) : LorentzVector.time ↑v ≤ 0 ↔ LorentzVector.time ↑v ≤ -1"} +{"name":"NormOneLorentzVector.abs_time_ge_one","declaration":"theorem NormOneLorentzVector.abs_time_ge_one {d : ℕ} (v : ↑(NormOneLorentzVector d)) : 1 ≤ |LorentzVector.time ↑v|"} +{"name":"NormOneLorentzVector","declaration":"/-- The set of Lorentz vectors with norm 1. -/\ndef NormOneLorentzVector (d : ℕ) : Set (LorentzVector d)"} +{"name":"NormOneLorentzVector.time_sq","declaration":"theorem NormOneLorentzVector.time_sq {d : ℕ} (v : ↑(NormOneLorentzVector d)) : LorentzVector.time ↑v ^ 2 = 1 + ‖LorentzVector.space ↑v‖ ^ 2"} +{"name":"NormOneLorentzVector.FuturePointing.timeVecNormOneFuture_coe_coe","declaration":"theorem NormOneLorentzVector.FuturePointing.timeVecNormOneFuture_coe_coe {d : ℕ} : ↑↑NormOneLorentzVector.FuturePointing.timeVecNormOneFuture = LorentzVector.timeVec"} +{"name":"NormOneLorentzVector.time_nonneg_iff","declaration":"theorem NormOneLorentzVector.time_nonneg_iff {d : ℕ} (v : ↑(NormOneLorentzVector d)) : 0 ≤ LorentzVector.time ↑v ↔ 1 ≤ LorentzVector.time ↑v"} +{"name":"NormOneLorentzVector.norm_space_leq_abs_time","declaration":"theorem NormOneLorentzVector.norm_space_leq_abs_time {d : ℕ} (v : ↑(NormOneLorentzVector d)) : ‖LorentzVector.space ↑v‖ ≤ |LorentzVector.time ↑v|"} +{"name":"NormOneLorentzVector.FuturePointing","declaration":"/-- The future pointing Lorentz vectors with Norm one. -/\ndef NormOneLorentzVector.FuturePointing (d : ℕ) : Set ↑(NormOneLorentzVector d)"} +{"name":"NormOneLorentzVector.FuturePointing.metric_reflect_mem_mem","declaration":"theorem NormOneLorentzVector.FuturePointing.metric_reflect_mem_mem {d : ℕ} {v : ↑(NormOneLorentzVector d)} {w : ↑(NormOneLorentzVector d)} (h : v ∈ NormOneLorentzVector.FuturePointing d) (hw : w ∈ NormOneLorentzVector.FuturePointing d) : 0 ≤ (minkowskiMetric ↑v) (LorentzVector.spaceReflection ↑w)"} +{"name":"NormOneLorentzVector.FuturePointing.not_mem_iff_neg","declaration":"theorem NormOneLorentzVector.FuturePointing.not_mem_iff_neg {d : ℕ} (v : ↑(NormOneLorentzVector d)) : v ∉ NormOneLorentzVector.FuturePointing d ↔ NormOneLorentzVector.neg v ∈ NormOneLorentzVector.FuturePointing d"} +{"name":"NormOneLorentzVector.FuturePointing.one_add_metric_non_zero","declaration":"theorem NormOneLorentzVector.FuturePointing.one_add_metric_non_zero {d : ℕ} (f : ↑(NormOneLorentzVector.FuturePointing d)) (f' : ↑(NormOneLorentzVector.FuturePointing d)) : 1 + (minkowskiMetric ↑↑f) ↑↑f' ≠ 0"} +{"name":"NormOneLorentzVector.neg","declaration":"/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef NormOneLorentzVector.neg {d : ℕ} (v : ↑(NormOneLorentzVector d)) : ↑(NormOneLorentzVector d)"} +{"name":"NormOneLorentzVector.FuturePointing.time_eq_sqrt","declaration":"theorem NormOneLorentzVector.FuturePointing.time_eq_sqrt {d : ℕ} (f : ↑(NormOneLorentzVector.FuturePointing d)) : LorentzVector.time ↑↑f = Real.sqrt (1 + ‖LorentzVector.space ↑↑f‖ ^ 2)"} +{"name":"NormOneLorentzVector.time_abs_sub_space_norm","declaration":"theorem NormOneLorentzVector.time_abs_sub_space_norm {d : ℕ} (v : ↑(NormOneLorentzVector d)) (w : ↑(NormOneLorentzVector d)) : 0 ≤ |LorentzVector.time ↑v| * |LorentzVector.time ↑w| - ‖LorentzVector.space ↑v‖ * ‖LorentzVector.space ↑w‖"} +{"name":"NormOneLorentzVector.FuturePointing.metric_nonneg","declaration":"theorem NormOneLorentzVector.FuturePointing.metric_nonneg {d : ℕ} (f : ↑(NormOneLorentzVector.FuturePointing d)) (f' : ↑(NormOneLorentzVector.FuturePointing d)) : 0 ≤ (minkowskiMetric ↑↑f) ↑↑f'"} +{"name":"NormOneLorentzVector.FuturePointing.time_nonneg","declaration":"theorem NormOneLorentzVector.FuturePointing.time_nonneg {d : ℕ} (f : ↑(NormOneLorentzVector.FuturePointing d)) : 0 ≤ LorentzVector.time ↑↑f"} +{"name":"NormOneLorentzVector.FuturePointing.timeVecNormOneFuture","declaration":"/-- The `FuturePointing d` which has all space components zero. -/\ndef NormOneLorentzVector.FuturePointing.timeVecNormOneFuture {d : ℕ} : ↑(NormOneLorentzVector.FuturePointing d)"} +{"name":"NormOneLorentzVector.norm_space_le_abs_time","declaration":"theorem NormOneLorentzVector.norm_space_le_abs_time {d : ℕ} (v : ↑(NormOneLorentzVector d)) : ‖LorentzVector.space ↑v‖ < |LorentzVector.time ↑v|"} +{"name":"NormOneLorentzVector.FuturePointing.metric_reflect_mem_not_mem","declaration":"theorem NormOneLorentzVector.FuturePointing.metric_reflect_mem_not_mem {d : ℕ} {v : ↑(NormOneLorentzVector d)} {w : ↑(NormOneLorentzVector d)} (h : v ∈ NormOneLorentzVector.FuturePointing d) (hw : w ∉ NormOneLorentzVector.FuturePointing d) : (minkowskiMetric ↑v) (LorentzVector.spaceReflection ↑w) ≤ 0"} +{"name":"NormOneLorentzVector.FuturePointing.metric_continuous","declaration":"theorem NormOneLorentzVector.FuturePointing.metric_continuous {d : ℕ} (u : LorentzVector d) : Continuous fun a => (minkowskiMetric u) ↑↑a"} +{"name":"NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem","declaration":"theorem NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem {d : ℕ} {v : ↑(NormOneLorentzVector d)} {w : ↑(NormOneLorentzVector d)} (h : v ∉ NormOneLorentzVector.FuturePointing d) (hw : w ∉ NormOneLorentzVector.FuturePointing d) : 0 ≤ (minkowskiMetric ↑v) (LorentzVector.spaceReflection ↑w)"} +{"name":"NormOneLorentzVector.FuturePointing.mem_iff","declaration":"theorem NormOneLorentzVector.FuturePointing.mem_iff {d : ℕ} (v : ↑(NormOneLorentzVector d)) : v ∈ NormOneLorentzVector.FuturePointing d ↔ 0 < LorentzVector.time ↑v"} +{"name":"NormOneLorentzVector.time_le_minus_one_or_ge_one","declaration":"theorem NormOneLorentzVector.time_le_minus_one_or_ge_one {d : ℕ} (v : ↑(NormOneLorentzVector d)) : LorentzVector.time ↑v ≤ -1 ∨ 1 ≤ LorentzVector.time ↑v"} +{"name":"NormOneLorentzVector.instTopologicalSpaceElemLorentzVectorNormOneLorentzVectorFuturePointing","declaration":"instance NormOneLorentzVector.instTopologicalSpaceElemLorentzVectorNormOneLorentzVectorFuturePointing {d : ℕ} : TopologicalSpace ↑(NormOneLorentzVector.FuturePointing d)"} +{"name":"NormOneLorentzVector.FuturePointing.isPathConnected","declaration":"theorem NormOneLorentzVector.FuturePointing.isPathConnected {d : ℕ} : IsPathConnected Set.univ"} +{"name":"NormOneLorentzVector.mem_iff","declaration":"theorem NormOneLorentzVector.mem_iff {d : ℕ} {x : LorentzVector d} : x ∈ NormOneLorentzVector d ↔ (minkowskiMetric x) x = 1"} +{"name":"NormOneLorentzVector.FuturePointing.abs_time","declaration":"theorem NormOneLorentzVector.FuturePointing.abs_time {d : ℕ} (f : ↑(NormOneLorentzVector.FuturePointing d)) : |LorentzVector.time ↑↑f| = LorentzVector.time ↑↑f"} +{"name":"NormOneLorentzVector.FuturePointing.not_mem_iff","declaration":"theorem NormOneLorentzVector.FuturePointing.not_mem_iff {d : ℕ} (v : ↑(NormOneLorentzVector d)) : v ∉ NormOneLorentzVector.FuturePointing d ↔ LorentzVector.time ↑v ≤ 0"} diff --git a/hep-declarations/HepLean.SpaceTime.MinkowskiMetric.jsonl b/hep-declarations/HepLean.SpaceTime.MinkowskiMetric.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6a9515204592d970c091eabf770a28dfe9bc3df8 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.MinkowskiMetric.jsonl @@ -0,0 +1,42 @@ +{"name":"minkowskiMetric.matrix_apply_eq_iff_sub","declaration":"theorem minkowskiMetric.matrix_apply_eq_iff_sub {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : (minkowskiMetric v) (Matrix.mulVec Λ w) = (minkowskiMetric v) (Matrix.mulVec Λ' w) ↔\n (minkowskiMetric v) (Matrix.mulVec (Λ - Λ') w) = 0"} +{"name":"minkowskiMetric.dual_mul","declaration":"theorem minkowskiMetric.dual_mul {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : minkowskiMetric.dual (Λ * Λ') = minkowskiMetric.dual Λ' * minkowskiMetric.dual Λ"} +{"name":"minkowskiMetric.nondegenerate","declaration":"/-- The metric tensor is non-degenerate. -/\ntheorem minkowskiMetric.nondegenerate {d : ℕ} (v : LorentzVector d) : (∀ (w : LorentzVector d), (minkowskiMetric w) v = 0) ↔ v = 0"} +{"name":"minkowskiMetric.dual","declaration":"/-- The dual of a matrix with respect to the Minkowski metric. -/\ndef minkowskiMetric.dual {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ"} +{"name":"minkowskiMetric.leq_time_sq","declaration":"theorem minkowskiMetric.leq_time_sq {d : ℕ} (v : LorentzVector d) : (minkowskiMetric v) v ≤ LorentzVector.time v ^ 2"} +{"name":"minkowskiMetric.ge_sub_norm","declaration":"theorem minkowskiMetric.ge_sub_norm {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) : LorentzVector.time v * LorentzVector.time w - ‖LorentzVector.space v‖ * ‖LorentzVector.space w‖ ≤ (minkowskiMetric v) w"} +{"name":"minkowskiMetric.on_basis","declaration":"theorem minkowskiMetric.on_basis {d : ℕ} (μ : Fin 1 ⊕ Fin d) (ν : Fin 1 ⊕ Fin d) : (minkowskiMetric (LorentzVector.stdBasis μ)) (LorentzVector.stdBasis ν) = minkowskiMatrix μ ν"} +{"name":"minkowskiMetric.matrix_eq_iff_eq_forall'","declaration":"theorem minkowskiMetric.matrix_eq_iff_eq_forall' {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : (∀ (v : Fin 1 ⊕ Fin d → ℝ), Matrix.mulVec Λ v = Matrix.mulVec Λ' v) ↔\n ∀ (w : Fin 1 ⊕ Fin d → ℝ) (v : LorentzVector d),\n (minkowskiMetric v) (Matrix.mulVec Λ w) = (minkowskiMetric v) (Matrix.mulVec Λ' w)"} +{"name":"minkowskiMatrix.eq_transpose","declaration":"theorem minkowskiMatrix.eq_transpose {d : ℕ} : Matrix.transpose minkowskiMatrix = minkowskiMatrix"} +{"name":"minkowskiMetric.as_sum","declaration":"/-- The Minkowski metric expressed as a sum. -/\ntheorem minkowskiMetric.as_sum {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) : (minkowskiMetric v) w =\n LorentzVector.time v * LorentzVector.time w -\n Finset.sum Finset.univ fun i => LorentzVector.space v i * LorentzVector.space w i"} +{"name":"minkowskiMatrix.as_block","declaration":"theorem minkowskiMatrix.as_block {d : ℕ} : minkowskiMatrix = Matrix.fromBlocks 1 0 0 (-1)"} +{"name":"minkowskiMetric.basis_left","declaration":"theorem minkowskiMetric.basis_left {d : ℕ} (v : LorentzVector d) (μ : Fin 1 ⊕ Fin d) : (minkowskiMetric (LorentzVector.stdBasis μ)) v = minkowskiMatrix μ μ * v μ"} +{"name":"minkowskiMetric.eq_time_minus_inner_prod","declaration":"theorem minkowskiMetric.eq_time_minus_inner_prod {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) : (minkowskiMetric v) w = LorentzVector.time v * LorentzVector.time w - ⟪LorentzVector.space v, LorentzVector.space w⟫_ℝ"} +{"name":"minkowskiMatrix.termη","declaration":"/-- Notation for `minkowskiMatrix`. -/\ndef minkowskiMatrix.termη : Lean.ParserDescr"} +{"name":"minkowskiMetric.as_sum_self","declaration":"/-- The Minkowski metric expressed as a sum for a single vector. -/\ntheorem minkowskiMetric.as_sum_self {d : ℕ} (v : LorentzVector d) : (minkowskiMetric v) v = LorentzVector.time v ^ 2 - ‖LorentzVector.space v‖ ^ 2"} +{"name":"minkowskiMatrix","declaration":"/-- The `d.succ`-dimensional real matrix of the form `diag(1, -1, -1, -1, ...)`. -/\ndef minkowskiMatrix {d : ℕ} : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ"} +{"name":"minkowskiMetric.matrix_eq_id_iff","declaration":"theorem minkowskiMetric.matrix_eq_id_iff {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : Λ = 1 ↔ ∀ (w : Fin 1 ⊕ Fin d → ℝ) (v : LorentzVector d), (minkowskiMetric v) (Matrix.mulVec Λ w) = (minkowskiMetric v) w"} +{"name":"minkowskiMetric.dual_mulVec_right","declaration":"theorem minkowskiMetric.dual_mulVec_right {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) {x : LorentzVector d} {y : Fin 1 ⊕ Fin d → ℝ} : (minkowskiMetric x) (Matrix.mulVec (minkowskiMetric.dual Λ) y) = (minkowskiMetric (Matrix.mulVec Λ x)) y"} +{"name":"minkowskiMetric.matrix_apply_stdBasis","declaration":"theorem minkowskiMetric.matrix_apply_stdBasis {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (ν : Fin 1 ⊕ Fin d) (μ : Fin 1 ⊕ Fin d) : Λ ν μ = minkowskiMatrix ν ν * (minkowskiMetric (LorentzVector.stdBasis ν)) (Matrix.mulVec Λ (LorentzVector.stdBasis μ))"} +{"name":"minkowskiMetric.dual_id","declaration":"theorem minkowskiMetric.dual_id {d : ℕ} : minkowskiMetric.dual 1 = 1"} +{"name":"minkowskiMetric.dual_eta","declaration":"theorem minkowskiMetric.dual_eta {d : ℕ} : minkowskiMetric.dual minkowskiMatrix = minkowskiMatrix"} +{"name":"minkowskiMatrix.η_apply_mul_η_apply_diag","declaration":"theorem minkowskiMatrix.η_apply_mul_η_apply_diag {d : ℕ} (μ : Fin 1 ⊕ Fin d) : minkowskiMatrix μ μ * minkowskiMatrix μ μ = 1"} +{"name":"minkowskiMetric.time_sq_eq_metric_add_space","declaration":"theorem minkowskiMetric.time_sq_eq_metric_add_space {d : ℕ} (v : LorentzVector d) : LorentzVector.time v ^ 2 = (minkowskiMetric v) v + ‖LorentzVector.space v‖ ^ 2"} +{"name":"minkowskiMetric.self_eq_time_minus_norm","declaration":"theorem minkowskiMetric.self_eq_time_minus_norm {d : ℕ} (v : LorentzVector d) : (minkowskiMetric v) v = LorentzVector.time v ^ 2 - ‖LorentzVector.space v‖ ^ 2"} +{"name":"minkowskiMatrix.det_eq_neg_one_pow_d","declaration":"theorem minkowskiMatrix.det_eq_neg_one_pow_d {d : ℕ} : Matrix.det minkowskiMatrix = (-1) ^ d"} +{"name":"minkowskiMetric.self_spaceReflection_eq_zero_iff","declaration":"theorem minkowskiMetric.self_spaceReflection_eq_zero_iff {d : ℕ} (v : LorentzVector d) : (minkowskiMetric v) (LorentzVector.spaceReflection v) = 0 ↔ v = 0"} +{"name":"minkowskiMetric.on_timeVec","declaration":"theorem minkowskiMetric.on_timeVec {d : ℕ} : (minkowskiMetric LorentzVector.timeVec) LorentzVector.timeVec = 1"} +{"name":"minkowskiLinearForm","declaration":"/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/\ndef minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ"} +{"name":"minkowskiMetric.dual_dual","declaration":"theorem minkowskiMetric.dual_dual {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : minkowskiMetric.dual (minkowskiMetric.dual Λ) = Λ"} +{"name":"minkowskiMetric.dual_mulVec_left","declaration":"theorem minkowskiMetric.dual_mulVec_left {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) {x : Fin 1 ⊕ Fin d → ℝ} {y : LorentzVector d} : (minkowskiMetric (Matrix.mulVec (minkowskiMetric.dual Λ) x)) y = (minkowskiMetric x) (Matrix.mulVec Λ y)"} +{"name":"minkowskiMetric.det_dual","declaration":"theorem minkowskiMetric.det_dual {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : Matrix.det (minkowskiMetric.dual Λ) = Matrix.det Λ"} +{"name":"minkowskiMetric.matrix_eq_iff_eq_forall","declaration":"theorem minkowskiMetric.matrix_eq_iff_eq_forall {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : Λ = Λ' ↔\n ∀ (w : Fin 1 ⊕ Fin d → ℝ) (v : LorentzVector d),\n (minkowskiMetric v) (Matrix.mulVec Λ w) = (minkowskiMetric v) (Matrix.mulVec Λ' w)"} +{"name":"minkowskiMetric.symm","declaration":"/-- The Minkowski metric is symmetric in its arguments.. -/\ntheorem minkowskiMetric.symm {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) : (minkowskiMetric v) w = (minkowskiMetric w) v"} +{"name":"minkowskiMetric.right_spaceReflection","declaration":"theorem minkowskiMetric.right_spaceReflection {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) : (minkowskiMetric v) (LorentzVector.spaceReflection w) =\n LorentzVector.time v * LorentzVector.time w + ⟪LorentzVector.space v, LorentzVector.space w⟫_ℝ"} +{"name":"minkowskiMetric.ge_abs_inner_product","declaration":"theorem minkowskiMetric.ge_abs_inner_product {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) : LorentzVector.time v * LorentzVector.time w - ‖⟪LorentzVector.space v, LorentzVector.space w⟫_ℝ‖ ≤ (minkowskiMetric v) w"} +{"name":"minkowskiMetric.on_basis_mulVec","declaration":"theorem minkowskiMetric.on_basis_mulVec {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (μ : Fin 1 ⊕ Fin d) (ν : Fin 1 ⊕ Fin d) : (minkowskiMetric (LorentzVector.stdBasis μ)) (Matrix.mulVec Λ (LorentzVector.stdBasis ν)) = minkowskiMatrix μ μ * Λ μ ν"} +{"name":"minkowskiMetric.«term⟪_,_⟫ₘ»","declaration":"/-- Notation for `minkowskiMetric`. -/\ndef minkowskiMetric.«term⟪_,_⟫ₘ» : Lean.ParserDescr"} +{"name":"minkowskiMetric","declaration":"/-- The Minkowski metric as a bilinear map. -/\ndef minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ"} +{"name":"minkowskiMatrix.sq","declaration":"theorem minkowskiMatrix.sq {d : ℕ} : minkowskiMatrix * minkowskiMatrix = 1"} +{"name":"minkowskiMatrix.off_diag_zero","declaration":"theorem minkowskiMatrix.off_diag_zero {d : ℕ} {μ : Fin 1 ⊕ Fin d} {ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : minkowskiMatrix μ ν = 0"} +{"name":"minkowskiLinearForm_apply","declaration":"theorem minkowskiLinearForm_apply {d : ℕ} (v : LorentzVector d) (w : LorentzVector d) : (minkowskiLinearForm v) w = Matrix.dotProduct v (Matrix.mulVec minkowskiMatrix w)"} +{"name":"minkowskiMetric.dual_transpose","declaration":"theorem minkowskiMetric.dual_transpose {d : ℕ} (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) : minkowskiMetric.dual (Matrix.transpose Λ) = Matrix.transpose (minkowskiMetric.dual Λ)"} diff --git a/hep-declarations/HepLean.SpaceTime.SL2C.Basic.jsonl b/hep-declarations/HepLean.SpaceTime.SL2C.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a3925f8ddfd3fadb2fa3ae50158815b23fee5672 --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.SL2C.Basic.jsonl @@ -0,0 +1,13 @@ +{"name":"SpaceTime.SL2C.toLorentzGroupElem","declaration":"/-- Given an element `M ∈ SL(2, ℂ)` the corresponding element of the Lorentz group. -/\ndef SpaceTime.SL2C.toLorentzGroupElem (M : Matrix.SpecialLinearGroup (Fin 2) ℂ) : ↑(LorentzGroup 3)"} +{"name":"SpaceTime.SL2C.repLorentzVector","declaration":"/-- The representation of `SL(2, ℂ)` on `spaceTime` obtained from `toSelfAdjointMatrix` and\n`repSelfAdjointMatrix`. -/\ndef SpaceTime.SL2C.repLorentzVector : Representation ℝ (Matrix.SpecialLinearGroup (Fin 2) ℂ) (LorentzVector 3)"} +{"name":"SpaceTime.SL2C.toLorentzGroup_det_one","declaration":"/-- An informal lemma. -/\ndef SpaceTime.SL2C.toLorentzGroup_det_one : InformalLemma"} +{"name":"SpaceTime.SL2C.toLorentzGroup_timeComp_nonneg","declaration":"/-- An informal lemma. -/\ndef SpaceTime.SL2C.toLorentzGroup_timeComp_nonneg : InformalLemma"} +{"name":"SpaceTime.SL2C.toLorentzGroup","declaration":"/-- The group homomorphism from ` SL(2, ℂ)` to the Lorentz group `𝓛`. -/\ndef SpaceTime.SL2C.toLorentzGroup : Matrix.SpecialLinearGroup (Fin 2) ℂ →* ↑(LorentzGroup 3)"} +{"name":"SpaceTime.SL2C.toLinearMapSelfAdjointMatrix","declaration":"/-- Given an element `M ∈ SL(2, ℂ)` the linear map from `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` to\nitself defined by `A ↦ M * A * Mᴴ`. -/\ndef SpaceTime.SL2C.toLinearMapSelfAdjointMatrix (M : Matrix.SpecialLinearGroup (Fin 2) ℂ) : ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) →ₗ[ℝ] ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))"} +{"name":"SpaceTime.SL2C.iff_det_selfAdjoint","declaration":"theorem SpaceTime.SL2C.iff_det_selfAdjoint (Λ : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) : Λ ∈ LorentzGroup 3 ↔\n ∀ (x : ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))),\n Matrix.det\n ↑((⇑SpaceTime.toSelfAdjointMatrix ∘\n ⇑((Matrix.toLin LorentzVector.stdBasis LorentzVector.stdBasis) Λ) ∘\n ⇑(LinearEquiv.symm SpaceTime.toSelfAdjointMatrix))\n x) =\n Matrix.det ↑x"} +{"name":"SpaceTime.SL2C.repSelfAdjointMatrix","declaration":"/-- The representation of `SL(2, ℂ)` on `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` given by\n`M A ↦ M * A * Mᴴ`. -/\ndef SpaceTime.SL2C.repSelfAdjointMatrix : Representation ℝ (Matrix.SpecialLinearGroup (Fin 2) ℂ) ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))"} +{"name":"SpaceTime.SL2C.toLorentzGroupElem_coe","declaration":"theorem SpaceTime.SL2C.toLorentzGroupElem_coe (M : Matrix.SpecialLinearGroup (Fin 2) ℂ) : ↑(SpaceTime.SL2C.toLorentzGroupElem M) =\n (LinearMap.toMatrix LorentzVector.stdBasis LorentzVector.stdBasis) (SpaceTime.SL2C.repLorentzVector M)"} +{"name":"SpaceTime.SL2C.toLorentzGroup_apply_coe","declaration":"theorem SpaceTime.SL2C.toLorentzGroup_apply_coe (M : Matrix.SpecialLinearGroup (Fin 2) ℂ) : ↑(SpaceTime.SL2C.toLorentzGroup M) =\n (LinearMap.toMatrix LorentzVector.stdBasis LorentzVector.stdBasis) (SpaceTime.SL2C.repLorentzVector M)"} +{"name":"SpaceTime.SL2C.repSelfAdjointMatrix_apply_apply_coe","declaration":"theorem SpaceTime.SL2C.repSelfAdjointMatrix_apply_apply_coe (M : Matrix.SpecialLinearGroup (Fin 2) ℂ) (A : ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))) : ↑((SpaceTime.SL2C.repSelfAdjointMatrix M) A) = ↑M * ↑A * Matrix.conjTranspose ↑M"} +{"name":"SpaceTime.SL2C.toLinearMapSelfAdjointMatrix_apply_coe","declaration":"theorem SpaceTime.SL2C.toLinearMapSelfAdjointMatrix_apply_coe (M : Matrix.SpecialLinearGroup (Fin 2) ℂ) (A : ↥(selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))) : ↑((SpaceTime.SL2C.toLinearMapSelfAdjointMatrix M) A) = ↑M * ↑A * Matrix.conjTranspose ↑M"} +{"name":"SpaceTime.SL2C.toRestrictedLorentzGroup","declaration":"/-- An informal lemma. -/\ndef SpaceTime.SL2C.toRestrictedLorentzGroup : InformalLemma"} diff --git a/hep-declarations/HepLean.SpaceTime.WeylFermion.Basic.jsonl b/hep-declarations/HepLean.SpaceTime.WeylFermion.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1d04c0fec0667553948175c08621a9d1d71fd4bd --- /dev/null +++ b/hep-declarations/HepLean.SpaceTime.WeylFermion.Basic.jsonl @@ -0,0 +1,18 @@ +{"name":"rightHandedWeylFermionAltEquiv_equivariant","declaration":"/-- An informal lemma. -/\ndef rightHandedWeylFermionAltEquiv_equivariant : InformalLemma"} +{"name":"leftAltWeylFermionContraction_symm_altLeftWeylFermionContraction","declaration":"/-- An informal lemma. -/\ndef leftAltWeylFermionContraction_symm_altLeftWeylFermionContraction : InformalLemma"} +{"name":"altLeftHandedWeylFermion","declaration":"/-- An informal definition. -/\ndef altLeftHandedWeylFermion : InformalDefinition"} +{"name":"leftAltWeylFermionContraction_invariant","declaration":"/-- An informal lemma. -/\ndef leftAltWeylFermionContraction_invariant : InformalLemma"} +{"name":"altRightWeylFermionContraction_invariant","declaration":"/-- An informal lemma. -/\ndef altRightWeylFermionContraction_invariant : InformalLemma"} +{"name":"leftHandedWeylFermionAltEquiv_equivariant","declaration":"/-- An informal lemma. -/\ndef leftHandedWeylFermionAltEquiv_equivariant : InformalLemma"} +{"name":"altRightWeylFermionContraction","declaration":"/-- An informal definition. -/\ndef altRightWeylFermionContraction : InformalDefinition"} +{"name":"rightAltWeylFermionContraction","declaration":"/-- An informal definition. -/\ndef rightAltWeylFermionContraction : InformalDefinition"} +{"name":"leftHandedWeylFermionAltEquiv","declaration":"/-- An informal definition. -/\ndef leftHandedWeylFermionAltEquiv : InformalDefinition"} +{"name":"rightAltWeylFermionContraction_symm_altRightWeylFermionContraction","declaration":"/-- An informal lemma. -/\ndef rightAltWeylFermionContraction_symm_altRightWeylFermionContraction : InformalLemma"} +{"name":"rightAltWeylFermionContraction_invariant","declaration":"/-- An informal lemma. -/\ndef rightAltWeylFermionContraction_invariant : InformalLemma"} +{"name":"altRightHandedWeylFermion","declaration":"/-- An informal definition. -/\ndef altRightHandedWeylFermion : InformalDefinition"} +{"name":"altLeftWeylFermionContraction_invariant","declaration":"/-- An informal lemma. -/\ndef altLeftWeylFermionContraction_invariant : InformalLemma"} +{"name":"rightHandedWeylFermion","declaration":"/-- An informal definition. -/\ndef rightHandedWeylFermion : InformalDefinition"} +{"name":"altLeftWeylFermionContraction","declaration":"/-- An informal definition. -/\ndef altLeftWeylFermionContraction : InformalDefinition"} +{"name":"leftAltWeylFermionContraction","declaration":"/-- An informal definition. -/\ndef leftAltWeylFermionContraction : InformalDefinition"} +{"name":"rightHandedWeylFermionAltEquiv","declaration":"/-- An informal definition. -/\ndef rightHandedWeylFermionAltEquiv : InformalDefinition"} +{"name":"leftHandedWeylFermion","declaration":"/-- An informal definition. -/\ndef leftHandedWeylFermion : InformalDefinition"} diff --git a/hep-declarations/HepLean.StandardModel.Basic.jsonl b/hep-declarations/HepLean.StandardModel.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0db5aa678972d7a15fdab4c4b006e6aebec54e00 --- /dev/null +++ b/hep-declarations/HepLean.StandardModel.Basic.jsonl @@ -0,0 +1,15 @@ +{"name":"StandardModel.gaugeGroupℤ₂SubGroup","declaration":"/-- An informal definition. -/\ndef StandardModel.gaugeGroupℤ₂SubGroup : InformalDefinition"} +{"name":"StandardModel.GaugeGroupℤ₆","declaration":"/-- An informal definition. -/\ndef StandardModel.GaugeGroupℤ₆ : InformalDefinition"} +{"name":"Matrix.specialUnitaryGroup","declaration":"/-- `Matrix.specialUnitaryGroup` is the group of unitary `n` by `n` matrices where the determinant\nis 1. (This definition is only correct if 2 is invertible.)-/\ndef Matrix.specialUnitaryGroup (n : Type u_1) (α : Type u_2) [DecidableEq n] [Fintype n] [CommRing α] [StarRing α] : Submonoid (Matrix n n α)"} +{"name":"StandardModel.GaugeGroupQuot.ℤ₆","declaration":"ctor StandardModel.GaugeGroupQuot.ℤ₆ : StandardModel.GaugeGroupQuot"} +{"name":"StandardModel.gaugeGroupℤ₃SubGroup","declaration":"/-- An informal definition. -/\ndef StandardModel.gaugeGroupℤ₃SubGroup : InformalDefinition"} +{"name":"StandardModel.GaugeGroupℤ₃","declaration":"/-- An informal definition. -/\ndef StandardModel.GaugeGroupℤ₃ : InformalDefinition"} +{"name":"Matrix.mem_specialUnitaryGroup_iff","declaration":"theorem Matrix.mem_specialUnitaryGroup_iff {n : Type u_1} {α : Type u_2} [DecidableEq n] [Fintype n] [CommRing α] [StarRing α] {A : Matrix n n α} : A ∈ Matrix.specialUnitaryGroup n α ↔ A ∈ Matrix.unitaryGroup n α ∧ Matrix.det A = 1"} +{"name":"StandardModel.gaugeGroupℤ₆SubGroup","declaration":"/-- An informal definition. -/\ndef StandardModel.gaugeGroupℤ₆SubGroup : InformalDefinition"} +{"name":"StandardModel.GaugeGroupI","declaration":"/-- The global gauge group of the Standard Model with no discrete quotients.\nThe `I` in the Name is an indication of the statement that this has no discrete quotients. -/\ndef StandardModel.GaugeGroupI : Type"} +{"name":"StandardModel.GaugeGroupQuot.ℤ₂","declaration":"ctor StandardModel.GaugeGroupQuot.ℤ₂ : StandardModel.GaugeGroupQuot"} +{"name":"StandardModel.GaugeGroupℤ₂","declaration":"/-- An informal definition. -/\ndef StandardModel.GaugeGroupℤ₂ : InformalDefinition"} +{"name":"StandardModel.GaugeGroupQuot","declaration":"/-- Specifies the allowed quotients of `SU(3) x SU(2) x U(1)` which give a valid\ngauge group of the Standard Model. -/\ninductive StandardModel.GaugeGroupQuot : Type"} +{"name":"StandardModel.GaugeGroup","declaration":"/-- An informal definition. -/\ndef StandardModel.GaugeGroup : InformalDefinition"} +{"name":"StandardModel.GaugeGroupQuot.I","declaration":"ctor StandardModel.GaugeGroupQuot.I : StandardModel.GaugeGroupQuot"} +{"name":"StandardModel.GaugeGroupQuot.ℤ₃","declaration":"ctor StandardModel.GaugeGroupQuot.ℤ₃ : StandardModel.GaugeGroupQuot"} diff --git a/hep-declarations/HepLean.StandardModel.HiggsBoson.Basic.jsonl b/hep-declarations/HepLean.StandardModel.HiggsBoson.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..43c55c75fc3c7b89e4b1b8164cecb211adb43449 --- /dev/null +++ b/hep-declarations/HepLean.StandardModel.HiggsBoson.Basic.jsonl @@ -0,0 +1,26 @@ +{"name":"StandardModel.HiggsField.toVec_smooth","declaration":"theorem StandardModel.HiggsField.toVec_smooth (φ : StandardModel.HiggsField) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ (Fin 2 → ℂ)) ⇑φ"} +{"name":"StandardModel.HiggsField.toHiggsVec","declaration":"/-- Given a `HiggsField`, the corresponding map from `SpaceTime` to `HiggsVec`. -/\ndef StandardModel.HiggsField.toHiggsVec (φ : StandardModel.HiggsField) : SpaceTime → StandardModel.HiggsVec"} +{"name":"StandardModel.HiggsField.apply_smooth","declaration":"theorem StandardModel.HiggsField.apply_smooth (φ : StandardModel.HiggsField) (i : Fin 2) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ ℂ) fun x => φ x i"} +{"name":"StandardModel.HiggsField.apply_im_smooth","declaration":"theorem StandardModel.HiggsField.apply_im_smooth (φ : StandardModel.HiggsField) (i : Fin 2) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ ℝ) (⇑Complex.imCLM ∘ fun x => φ x i)"} +{"name":"StandardModel.HiggsField","declaration":"/-- A Higgs field is a smooth section of the Higgs bundle. -/\ndef StandardModel.HiggsField : Type"} +{"name":"StandardModel.HiggsVec.ofReal_normSq","declaration":"theorem StandardModel.HiggsVec.ofReal_normSq {a : ℝ} (ha : 0 ≤ a) : ‖StandardModel.HiggsVec.ofReal a‖ ^ 2 = a"} +{"name":"StandardModel.HiggsField.isConst_of_higgsVec","declaration":"theorem StandardModel.HiggsField.isConst_of_higgsVec (φ : StandardModel.HiggsVec) : StandardModel.HiggsField.IsConst (StandardModel.HiggsVec.toField φ)"} +{"name":"StandardModel.HiggsVec.ofReal","declaration":"/-- Generating a Higgs vector from a real number, such that the norm-squared of that Higgs vector\nis the given real number. -/\ndef StandardModel.HiggsVec.ofReal (a : ℝ) : StandardModel.HiggsVec"} +{"name":"StandardModel.HiggsVec.toField","declaration":"/-- Given a vector in `HiggsVec` the constant Higgs field with value equal to that\nsection. -/\ndef StandardModel.HiggsVec.toField (φ : StandardModel.HiggsVec) : StandardModel.HiggsField"} +{"name":"StandardModel.HiggsField.toField_toHiggsVec_apply","declaration":"theorem StandardModel.HiggsField.toField_toHiggsVec_apply (φ : StandardModel.HiggsField) (x : SpaceTime) : (StandardModel.HiggsVec.toField (StandardModel.HiggsField.toHiggsVec φ x)) x = φ x"} +{"name":"StandardModel.HiggsField.IsConst","declaration":"/-- A Higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/\ndef StandardModel.HiggsField.IsConst (Φ : StandardModel.HiggsField) : Prop"} +{"name":"StandardModel.HiggsField.apply_re_smooth","declaration":"theorem StandardModel.HiggsField.apply_re_smooth (φ : StandardModel.HiggsField) (i : Fin 2) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ ℝ) (⇑Complex.reCLM ∘ fun x => φ x i)"} +{"name":"StandardModel.HiggsVec.orthonormBasis","declaration":"/-- An orthonormal basis of `HiggsVec`. -/\ndef StandardModel.HiggsVec.orthonormBasis : OrthonormalBasis (Fin 2) ℂ StandardModel.HiggsVec"} +{"name":"StandardModel.HiggsVec.smooth_toFin2ℂ","declaration":"/-- The map `toFin2ℂ` is smooth. -/\ntheorem StandardModel.HiggsVec.smooth_toFin2ℂ : Smooth (modelWithCornersSelf ℝ StandardModel.HiggsVec) (modelWithCornersSelf ℝ (Fin 2 → ℂ))\n ⇑StandardModel.HiggsVec.toFin2ℂ"} +{"name":"StandardModel.HiggsBundle","declaration":"/-- The trivial vector bundle 𝓡² × ℂ². -/\ndef StandardModel.HiggsBundle : SpaceTime → Type"} +{"name":"StandardModel.HiggsVec.toFin2ℂ","declaration":"/-- The continuous linear map from the vector space `HiggsVec` to `(Fin 2 → ℂ)` achieved by\ncasting vectors. -/\ndef StandardModel.HiggsVec.toFin2ℂ : StandardModel.HiggsVec →L[ℝ] Fin 2 → ℂ"} +{"name":"StandardModel.HiggsField.isConst_iff_of_higgsVec","declaration":"theorem StandardModel.HiggsField.isConst_iff_of_higgsVec (Φ : StandardModel.HiggsField) : StandardModel.HiggsField.IsConst Φ ↔ ∃ φ, Φ = StandardModel.HiggsVec.toField φ"} +{"name":"StandardModel.HiggsField.toHiggsVec_smooth","declaration":"theorem StandardModel.HiggsField.toHiggsVec_smooth (φ : StandardModel.HiggsField) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ StandardModel.HiggsVec)\n (StandardModel.HiggsField.toHiggsVec φ)"} +{"name":"StandardModel.HiggsField.zero","declaration":"/-- The higgs field which is all zero. -/\ndef StandardModel.HiggsField.zero : StandardModel.HiggsField"} +{"name":"StandardModel.HiggsVec.toField_apply","declaration":"theorem StandardModel.HiggsVec.toField_apply (φ : StandardModel.HiggsVec) (x : SpaceTime) : (StandardModel.HiggsVec.toField φ) x = φ"} +{"name":"StandardModel.HiggsField.toFin2ℂ_comp_toHiggsVec","declaration":"theorem StandardModel.HiggsField.toFin2ℂ_comp_toHiggsVec (φ : StandardModel.HiggsField) : ⇑StandardModel.HiggsVec.toFin2ℂ ∘ StandardModel.HiggsField.toHiggsVec φ = ⇑φ"} +{"name":"StandardModel.instSmoothVectorBundleRealSpaceTimeHiggsVecHiggsBundleToNontriviallyNormedFieldDenselyNormedFieldEuclideanNormedAddCommGroupEuclideanNormedSpaceToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedAddCommGroupAsSmoothManifoldInstChartedSpaceSpaceTimeToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedAddCommGroupEuclideanNormedAddCommGroupToAddCommMonoidInstAddCommGroupOfNatENNRealInstOfNatAtLeastTwoToNatCastInstCanonicallyOrderedCommSemiringENNRealProof_1ForAllFinNatInstOfNatNatComplexAddCommGroupAddCommGroupInstModuleSemiringModuleToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingToNonUnitalSeminormedCommRingToSeminormedCommRingToNormedCommRingInstNormedFieldComplexInstModuleToModuleNormedFieldToSeminormedAddCommGroupToNonUnitalSeminormedRingNormedCommRingToNormedSpaceRCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupFact_one_le_two_ennrealFintypeInstNormedAddCommGroupComplexNormedSpaceToNormedFieldComplexToRealInstRCLikeComplexTopologicalSpaceUniformSpaceToPseudoMetricSpaceToSeminormedRingFiberBundleVectorBundle","declaration":"instance StandardModel.instSmoothVectorBundleRealSpaceTimeHiggsVecHiggsBundleToNontriviallyNormedFieldDenselyNormedFieldEuclideanNormedAddCommGroupEuclideanNormedSpaceToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedAddCommGroupAsSmoothManifoldInstChartedSpaceSpaceTimeToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedAddCommGroupEuclideanNormedAddCommGroupToAddCommMonoidInstAddCommGroupOfNatENNRealInstOfNatAtLeastTwoToNatCastInstCanonicallyOrderedCommSemiringENNRealProof_1ForAllFinNatInstOfNatNatComplexAddCommGroupAddCommGroupInstModuleSemiringModuleToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingToNonUnitalSeminormedCommRingToSeminormedCommRingToNormedCommRingInstNormedFieldComplexInstModuleToModuleNormedFieldToSeminormedAddCommGroupToNonUnitalSeminormedRingNormedCommRingToNormedSpaceRCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupFact_one_le_two_ennrealFintypeInstNormedAddCommGroupComplexNormedSpaceToNormedFieldComplexToRealInstRCLikeComplexTopologicalSpaceUniformSpaceToPseudoMetricSpaceToSeminormedRingFiberBundleVectorBundle : SmoothVectorBundle StandardModel.HiggsVec StandardModel.HiggsBundle SpaceTime.asSmoothManifold"} +{"name":"StandardModel.HiggsField.zero_is_zero_section","declaration":"/-- An informal lemma. -/\ndef StandardModel.HiggsField.zero_is_zero_section : InformalLemma"} +{"name":"StandardModel.instNormedAddCommGroupForAllFinOfNatNatInstOfNatNatComplex","declaration":"instance StandardModel.instNormedAddCommGroupForAllFinOfNatNatInstOfNatNatComplex : NormedAddCommGroup (Fin 2 → ℂ)"} +{"name":"StandardModel.HiggsField.ofReal","declaration":"/-- Generating a constant Higgs field from a real number, such that the norm-squared of that Higgs\nvector is the given real number. -/\ndef StandardModel.HiggsField.ofReal (a : ℝ) : StandardModel.HiggsField"} +{"name":"StandardModel.HiggsVec","declaration":"/-- The complex vector space in which the Higgs field takes values. -/\ndef StandardModel.HiggsVec : Type"} diff --git a/hep-declarations/HepLean.StandardModel.HiggsBoson.GaugeAction.jsonl b/hep-declarations/HepLean.StandardModel.HiggsBoson.GaugeAction.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..70291f61f2af0e55e9d16950337bc1e3bc34a0ae --- /dev/null +++ b/hep-declarations/HepLean.StandardModel.HiggsBoson.GaugeAction.jsonl @@ -0,0 +1,23 @@ +{"name":"StandardModel.HiggsVec.rotateMatrix","declaration":"/-- Given a Higgs vector, a rotation matrix which puts the first component of the\nvector to zero, and the second component to a real -/\ndef StandardModel.HiggsVec.rotateMatrix (φ : StandardModel.HiggsVec) : Matrix (Fin 2) (Fin 2) ℂ"} +{"name":"StandardModel.HiggsVec.matrixToLin","declaration":"/-- Takes in a `2×2`-matrix and returns a linear map of `higgsVec`. -/\ndef StandardModel.HiggsVec.matrixToLin : Matrix (Fin 2) (Fin 2) ℂ →* StandardModel.HiggsVec →L[ℂ] StandardModel.HiggsVec"} +{"name":"StandardModel.HiggsVec.rotate_fst_real_snd_zero","declaration":"theorem StandardModel.HiggsVec.rotate_fst_real_snd_zero (φ : StandardModel.HiggsVec) : ∃ g, (StandardModel.HiggsVec.rep g) φ = ![Complex.ofReal ‖φ‖, 0]"} +{"name":"StandardModel.HiggsVec.unitaryToLin","declaration":"/-- The natural homomorphism from unitary `2×2` complex matrices to unitary transformations\nof `higgsVec`. -/\ndef StandardModel.HiggsVec.unitaryToLin : ↥(Matrix.unitaryGroup (Fin 2) ℂ) →* ↥(unitary (StandardModel.HiggsVec →L[ℂ] StandardModel.HiggsVec))"} +{"name":"StandardModel.HiggsVec.rep_apply_apply","declaration":"theorem StandardModel.HiggsVec.rep_apply_apply : ∀ (a : StandardModel.GaugeGroupI) (a_1 : StandardModel.HiggsVec),\n (StandardModel.HiggsVec.rep a) a_1 =\n StandardModel.HiggsVec.unitaryToLin (StandardModel.HiggsVec.higgsRepUnitary a) • a_1"} +{"name":"StandardModel.HiggsVec.rotateGuageGroup","declaration":"/-- Given a Higgs vector, an element of the gauge group which puts the first component of the\nvector to zero, and the second component to a real number. -/\ndef StandardModel.HiggsVec.rotateGuageGroup {φ : StandardModel.HiggsVec} (hφ : φ ≠ 0) : StandardModel.GaugeGroupI"} +{"name":"StandardModel.HiggsVec.stability_group","declaration":"/-- An informal lemma. -/\ndef StandardModel.HiggsVec.stability_group : InformalLemma"} +{"name":"StandardModel.HiggsVec.matrixToLin_unitary","declaration":"theorem StandardModel.HiggsVec.matrixToLin_unitary (g : ↥(Matrix.unitaryGroup (Fin 2) ℂ)) : StandardModel.HiggsVec.matrixToLin ↑g ∈ unitary (StandardModel.HiggsVec →L[ℂ] StandardModel.HiggsVec)"} +{"name":"StandardModel.HiggsVec.higgsRepUnitary","declaration":"/-- The Higgs representation as a homomorphism from the gauge group to unitary `2×2`-matrices. -/\ndef StandardModel.HiggsVec.higgsRepUnitary : StandardModel.GaugeGroupI →* ↥(Matrix.unitaryGroup (Fin 2) ℂ)"} +{"name":"StandardModel.HiggsVec.rep_apply","declaration":"theorem StandardModel.HiggsVec.rep_apply (g : StandardModel.GaugeGroupI) (φ : StandardModel.HiggsVec) : (StandardModel.HiggsVec.rep g) φ = g.2.2 ^ 3 • Matrix.mulVec (↑g.2.1) φ"} +{"name":"StandardModel.HiggsVec.rotateMatrix_det","declaration":"theorem StandardModel.HiggsVec.rotateMatrix_det {φ : StandardModel.HiggsVec} (hφ : φ ≠ 0) : Matrix.det (StandardModel.HiggsVec.rotateMatrix φ) = 1"} +{"name":"StandardModel.HiggsVec.rep","declaration":"/-- The representation of the gauge group acting on `higgsVec`. -/\ndef StandardModel.HiggsVec.rep : Representation ℂ StandardModel.GaugeGroupI StandardModel.HiggsVec"} +{"name":"StandardModel.HiggsVec.unitToLinear","declaration":"/-- The inclusion of unitary transformations on `higgsVec` into all linear transformations. -/\ndef StandardModel.HiggsVec.unitToLinear : ↥(unitary (StandardModel.HiggsVec →L[ℂ] StandardModel.HiggsVec)) →* StandardModel.HiggsVec →ₗ[ℂ] StandardModel.HiggsVec"} +{"name":"StandardModel.HiggsVec.matrixToLin_star","declaration":"/-- `matrixToLin` commutes with the `star` operation. -/\ntheorem StandardModel.HiggsVec.matrixToLin_star (g : Matrix (Fin 2) (Fin 2) ℂ) : StandardModel.HiggsVec.matrixToLin (star g) = star (StandardModel.HiggsVec.matrixToLin g)"} +{"name":"StandardModel.HiggsVec.rotateMatrix_specialUnitary","declaration":"theorem StandardModel.HiggsVec.rotateMatrix_specialUnitary {φ : StandardModel.HiggsVec} (hφ : φ ≠ 0) : StandardModel.HiggsVec.rotateMatrix φ ∈ Matrix.specialUnitaryGroup (Fin 2) ℂ"} +{"name":"StandardModel.HiggsVec.stability_group_single","declaration":"/-- An informal lemma. -/\ndef StandardModel.HiggsVec.stability_group_single : InformalLemma"} +{"name":"StandardModel.HiggsVec.unitToLinear_apply_apply","declaration":"theorem StandardModel.HiggsVec.unitToLinear_apply_apply (s : ↥(unitary (StandardModel.HiggsVec →L[ℂ] StandardModel.HiggsVec))) : ∀ (a : StandardModel.HiggsVec), (StandardModel.HiggsVec.unitToLinear s) a = s • a"} +{"name":"StandardModel.HiggsVec.higgsRepUnitary_mul","declaration":"theorem StandardModel.HiggsVec.higgsRepUnitary_mul (g : StandardModel.GaugeGroupI) (φ : StandardModel.HiggsVec) : Matrix.mulVec (↑(StandardModel.HiggsVec.higgsRepUnitary g)) φ = g.2.2 ^ 3 • Matrix.mulVec (↑g.2.1) φ"} +{"name":"StandardModel.HiggsVec.rotateGuageGroup_apply","declaration":"theorem StandardModel.HiggsVec.rotateGuageGroup_apply {φ : StandardModel.HiggsVec} (hφ : φ ≠ 0) : (StandardModel.HiggsVec.rep (StandardModel.HiggsVec.rotateGuageGroup hφ)) φ = ![0, Complex.ofReal ‖φ‖]"} +{"name":"StandardModel.HiggsVec.rotateMatrix_star","declaration":"theorem StandardModel.HiggsVec.rotateMatrix_star (φ : StandardModel.HiggsVec) : star (StandardModel.HiggsVec.rotateMatrix φ) =\n ![![(starRingEnd ((fun x => ℂ) 1)) (φ 1) / ↑‖φ‖, φ 0 / ↑‖φ‖],\n ![-(starRingEnd ((fun x => ℂ) 0)) (φ 0) / ↑‖φ‖, φ 1 / ↑‖φ‖]]"} +{"name":"StandardModel.HiggsVec.higgsRepUnitary_apply_coe","declaration":"theorem StandardModel.HiggsVec.higgsRepUnitary_apply_coe (g : StandardModel.GaugeGroupI) : ↑(StandardModel.HiggsVec.higgsRepUnitary g) = g.2.2 ^ 3 • ↑g.2.1"} +{"name":"StandardModel.HiggsVec.rotate_fst_zero_snd_real","declaration":"theorem StandardModel.HiggsVec.rotate_fst_zero_snd_real (φ : StandardModel.HiggsVec) : ∃ g, (StandardModel.HiggsVec.rep g) φ = ![0, Complex.ofReal ‖φ‖]"} +{"name":"StandardModel.HiggsVec.rotateMatrix_unitary","declaration":"theorem StandardModel.HiggsVec.rotateMatrix_unitary {φ : StandardModel.HiggsVec} (hφ : φ ≠ 0) : StandardModel.HiggsVec.rotateMatrix φ ∈ Matrix.unitaryGroup (Fin 2) ℂ"} diff --git a/hep-declarations/HepLean.StandardModel.HiggsBoson.PointwiseInnerProd.jsonl b/hep-declarations/HepLean.StandardModel.HiggsBoson.PointwiseInnerProd.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..5a4dcec59de69b799cad3a80b4adcbae9491ab0e --- /dev/null +++ b/hep-declarations/HepLean.StandardModel.HiggsBoson.PointwiseInnerProd.jsonl @@ -0,0 +1,20 @@ +{"name":"StandardModel.HiggsField.innerProd_left_zero","declaration":"theorem StandardModel.HiggsField.innerProd_left_zero (φ : StandardModel.HiggsField) : StandardModel.HiggsField.innerProd 0 φ = 0"} +{"name":"StandardModel.HiggsField.normSq_nonneg","declaration":"theorem StandardModel.HiggsField.normSq_nonneg (φ : StandardModel.HiggsField) (x : SpaceTime) : 0 ≤ StandardModel.HiggsField.normSq φ x"} +{"name":"StandardModel.HiggsField.normSq_smooth","declaration":"theorem StandardModel.HiggsField.normSq_smooth (φ : StandardModel.HiggsField) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ ℝ) (StandardModel.HiggsField.normSq φ)"} +{"name":"StandardModel.HiggsField.innerProd_self_eq_normSq","declaration":"theorem StandardModel.HiggsField.innerProd_self_eq_normSq (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.innerProd φ φ x = ↑(StandardModel.HiggsField.normSq φ x)"} +{"name":"StandardModel.HiggsField.normSq_eq_innerProd_self","declaration":"theorem StandardModel.HiggsField.normSq_eq_innerProd_self (φ : StandardModel.HiggsField) (x : SpaceTime) : ‖φ x‖ ^ 2 = (StandardModel.HiggsField.innerProd φ φ x).re"} +{"name":"StandardModel.HiggsField.«term⟪_,_⟫_H»","declaration":"/-- Notation for the inner product of two Higgs fields. -/\ndef StandardModel.HiggsField.«term⟪_,_⟫_H» : Lean.ParserDescr"} +{"name":"StandardModel.HiggsField.normSq_expand","declaration":"theorem StandardModel.HiggsField.normSq_expand (φ : StandardModel.HiggsField) : StandardModel.HiggsField.normSq φ = fun x =>\n ((starRingEnd ((fun x => ℂ) 0)) (φ x 0) * φ x 0 + (starRingEnd ((fun x => ℂ) 1)) (φ x 1) * φ x 1).re"} +{"name":"StandardModel.HiggsField.normSq_apply_im_zero","declaration":"theorem StandardModel.HiggsField.normSq_apply_im_zero (φ : StandardModel.HiggsField) (x : SpaceTime) : (↑‖φ x‖ ^ 2).im = 0"} +{"name":"StandardModel.HiggsField.innerProd_right_zero","declaration":"theorem StandardModel.HiggsField.innerProd_right_zero (φ : StandardModel.HiggsField) : StandardModel.HiggsField.innerProd φ 0 = 0"} +{"name":"StandardModel.HiggsField.innerProd_neg_right","declaration":"theorem StandardModel.HiggsField.innerProd_neg_right (φ1 : StandardModel.HiggsField) (φ2 : StandardModel.HiggsField) : StandardModel.HiggsField.innerProd φ1 (-φ2) = -StandardModel.HiggsField.innerProd φ1 φ2"} +{"name":"StandardModel.HiggsField.ofReal_normSq","declaration":"theorem StandardModel.HiggsField.ofReal_normSq {a : ℝ} (ha : 0 ≤ a) (x : SpaceTime) : StandardModel.HiggsField.normSq (StandardModel.HiggsField.ofReal a) x = a"} +{"name":"StandardModel.HiggsField.«term‖_‖_H_^_2»","declaration":"/-- Notation for the norm squared of a Higgs field. -/\ndef StandardModel.HiggsField.«term‖_‖_H_^_2» : Lean.ParserDescr"} +{"name":"StandardModel.HiggsField.normSq","declaration":"/-- Given a `HiggsField`, the map `SpaceTime → ℝ` obtained by taking the square norm of the\npointwise Higgs vector. -/\ndef StandardModel.HiggsField.normSq (φ : StandardModel.HiggsField) : SpaceTime → ℝ"} +{"name":"StandardModel.HiggsField.innerProd_expand","declaration":"theorem StandardModel.HiggsField.innerProd_expand (φ1 : StandardModel.HiggsField) (φ2 : StandardModel.HiggsField) : StandardModel.HiggsField.innerProd φ1 φ2 = fun x =>\n (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)\n ((φ1 x 0).re * (φ2 x 0).re + (φ1 x 1).re * (φ2 x 1).re + (φ1 x 0).im * (φ2 x 0).im + (φ1 x 1).im * (φ2 x 1).im,\n (φ1 x 0).re * (φ2 x 0).im + (φ1 x 1).re * (φ2 x 1).im - (φ1 x 0).im * (φ2 x 0).re - (φ1 x 1).im * (φ2 x 1).re)"} +{"name":"StandardModel.HiggsField.innerProd","declaration":"/-- Given two `HiggsField`, the map `SpaceTime → ℂ` obtained by taking their pointwise\ninner product. -/\ndef StandardModel.HiggsField.innerProd (φ1 : StandardModel.HiggsField) (φ2 : StandardModel.HiggsField) : SpaceTime → ℂ"} +{"name":"StandardModel.HiggsField.normSq_apply_re_self","declaration":"theorem StandardModel.HiggsField.normSq_apply_re_self (φ : StandardModel.HiggsField) (x : SpaceTime) : (↑‖φ x‖ ^ 2).re = StandardModel.HiggsField.normSq φ x"} +{"name":"StandardModel.HiggsField.innerProd_neg_left","declaration":"theorem StandardModel.HiggsField.innerProd_neg_left (φ1 : StandardModel.HiggsField) (φ2 : StandardModel.HiggsField) : StandardModel.HiggsField.innerProd (-φ1) φ2 = -StandardModel.HiggsField.innerProd φ1 φ2"} +{"name":"StandardModel.HiggsField.smooth_innerProd","declaration":"theorem StandardModel.HiggsField.smooth_innerProd (φ1 : StandardModel.HiggsField) (φ2 : StandardModel.HiggsField) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ ℂ) (StandardModel.HiggsField.innerProd φ1 φ2)"} +{"name":"StandardModel.HiggsField.normSq_zero","declaration":"theorem StandardModel.HiggsField.normSq_zero (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.normSq φ x = 0 ↔ φ x = 0"} +{"name":"StandardModel.HiggsField.toHiggsVec_norm","declaration":"theorem StandardModel.HiggsField.toHiggsVec_norm (φ : StandardModel.HiggsField) (x : SpaceTime) : ‖φ x‖ = ‖StandardModel.HiggsField.toHiggsVec φ x‖"} diff --git a/hep-declarations/HepLean.StandardModel.HiggsBoson.Potential.jsonl b/hep-declarations/HepLean.StandardModel.HiggsBoson.Potential.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6f91f4bf80ab2989e3cac7a49c76a13864f316d6 --- /dev/null +++ b/hep-declarations/HepLean.StandardModel.HiggsBoson.Potential.jsonl @@ -0,0 +1,37 @@ +{"name":"StandardModel.HiggsField.Potential.quadDiscrim_eq_zero_iff","declaration":"theorem StandardModel.HiggsField.Potential.quadDiscrim_eq_zero_iff (P : StandardModel.HiggsField.Potential) (h : P.𝓵 ≠ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.quadDiscrim P φ x = 0 ↔\n StandardModel.HiggsField.Potential.toFun P φ x = -P.μ2 ^ 2 / (4 * P.𝓵)"} +{"name":"StandardModel.HiggsField.Potential.isMinOn_iff_of_μSq_nonpos_𝓵_pos","declaration":"theorem StandardModel.HiggsField.Potential.isMinOn_iff_of_μSq_nonpos_𝓵_pos (P : StandardModel.HiggsField.Potential) (h𝓵 : 0 < P.𝓵) (hμ2 : P.μ2 ≤ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : IsMinOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun P φ x)\n Set.univ (φ, x) ↔\n StandardModel.HiggsField.Potential.toFun P φ x = 0"} +{"name":"StandardModel.HiggsField.Potential.isBounded_𝓵_nonneg","declaration":"theorem StandardModel.HiggsField.Potential.isBounded_𝓵_nonneg (P : StandardModel.HiggsField.Potential) (h : StandardModel.HiggsField.Potential.IsBounded P) : 0 ≤ P.𝓵"} +{"name":"StandardModel.HiggsField.Potential.isBounded_of_𝓵_pos","declaration":"theorem StandardModel.HiggsField.Potential.isBounded_of_𝓵_pos (P : StandardModel.HiggsField.Potential) (h : 0 < P.𝓵) : StandardModel.HiggsField.Potential.IsBounded P"} +{"name":"StandardModel.HiggsField.Potential.pos_𝓵_sol_exists_iff","declaration":"theorem StandardModel.HiggsField.Potential.pos_𝓵_sol_exists_iff (P : StandardModel.HiggsField.Potential) (h𝓵 : 0 < P.𝓵) (c : ℝ) : (∃ φ x, StandardModel.HiggsField.Potential.toFun P φ x = c) ↔ P.μ2 < 0 ∧ 0 ≤ c ∨ 0 ≤ P.μ2 ∧ -P.μ2 ^ 2 / (4 * P.𝓵) ≤ c"} +{"name":"StandardModel.HiggsField.Potential.pos_𝓵_toFun_pos","declaration":"theorem StandardModel.HiggsField.Potential.pos_𝓵_toFun_pos (P : StandardModel.HiggsField.Potential) (h : 0 < P.𝓵) (φ : StandardModel.HiggsField) (x : SpaceTime) : P.μ2 < 0 ∧ 0 ≤ StandardModel.HiggsField.Potential.toFun P φ x ∨ 0 ≤ P.μ2"} +{"name":"StandardModel.HiggsField.Potential.as_quad","declaration":"theorem StandardModel.HiggsField.Potential.as_quad (P : StandardModel.HiggsField.Potential) (φ : StandardModel.HiggsField) (x : SpaceTime) : P.𝓵 * StandardModel.HiggsField.normSq φ x * StandardModel.HiggsField.normSq φ x +\n -P.μ2 * StandardModel.HiggsField.normSq φ x +\n -StandardModel.HiggsField.Potential.toFun P φ x =\n 0"} +{"name":"StandardModel.HiggsField.Potential.isMinOn_iff_field_of_μSq_nonneg_𝓵_pos","declaration":"theorem StandardModel.HiggsField.Potential.isMinOn_iff_field_of_μSq_nonneg_𝓵_pos (P : StandardModel.HiggsField.Potential) (h𝓵 : 0 < P.𝓵) (hμ2 : 0 ≤ P.μ2) (φ : StandardModel.HiggsField) (x : SpaceTime) : IsMinOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun P φ x)\n Set.univ (φ, x) ↔\n StandardModel.HiggsField.normSq φ x = P.μ2 / (2 * P.𝓵)"} +{"name":"StandardModel.HiggsField.Potential.toFun_smooth","declaration":"/-- The potential is smooth. -/\ntheorem StandardModel.HiggsField.Potential.toFun_smooth (P : StandardModel.HiggsField.Potential) (φ : StandardModel.HiggsField) : Smooth (modelWithCornersSelf ℝ SpaceTime) (modelWithCornersSelf ℝ ℝ) fun x =>\n StandardModel.HiggsField.Potential.toFun P φ x"} +{"name":"StandardModel.HiggsField.Potential.neg_𝓵_toFun_neg","declaration":"theorem StandardModel.HiggsField.Potential.neg_𝓵_toFun_neg (P : StandardModel.HiggsField.Potential) (h : P.𝓵 < 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : 0 < P.μ2 ∧ StandardModel.HiggsField.Potential.toFun P φ x ≤ 0 ∨ P.μ2 ≤ 0"} +{"name":"StandardModel.HiggsField.Potential.quadDiscrim","declaration":"/-- The discrimiant of the quadratic equation formed by the Higgs potential. -/\ndef StandardModel.HiggsField.Potential.quadDiscrim (P : StandardModel.HiggsField.Potential) (φ : StandardModel.HiggsField) (x : SpaceTime) : ℝ"} +{"name":"StandardModel.HiggsField.Potential.pos_𝓵_quadDiscrim_zero_bound","declaration":"theorem StandardModel.HiggsField.Potential.pos_𝓵_quadDiscrim_zero_bound (P : StandardModel.HiggsField.Potential) (h : 0 < P.𝓵) (φ : StandardModel.HiggsField) (x : SpaceTime) : -P.μ2 ^ 2 / (4 * P.𝓵) ≤ StandardModel.HiggsField.Potential.toFun P φ x"} +{"name":"StandardModel.HiggsField.Potential.neg_𝓵_sol_exists_iff","declaration":"theorem StandardModel.HiggsField.Potential.neg_𝓵_sol_exists_iff (P : StandardModel.HiggsField.Potential) (h𝓵 : P.𝓵 < 0) (c : ℝ) : (∃ φ x, StandardModel.HiggsField.Potential.toFun P φ x = c) ↔ 0 < P.μ2 ∧ c ≤ 0 ∨ P.μ2 ≤ 0 ∧ c ≤ -P.μ2 ^ 2 / (4 * P.𝓵)"} +{"name":"StandardModel.HiggsField.Potential.toFun","declaration":"/-- The function corresponding to the Higgs potential. -/\ndef StandardModel.HiggsField.Potential.toFun (P : StandardModel.HiggsField.Potential) (φ : StandardModel.HiggsField) (x : SpaceTime) : ℝ"} +{"name":"StandardModel.HiggsField.Potential.𝓵","declaration":"/-- The quartic coupling of the Higgs boson. Usually denoted λ. -/\ndef StandardModel.HiggsField.Potential.𝓵 (self : StandardModel.HiggsField.Potential) : ℝ"} +{"name":"StandardModel.HiggsField.Potential.eq_zero_iff_of_μSq_nonpos_𝓵_pos","declaration":"theorem StandardModel.HiggsField.Potential.eq_zero_iff_of_μSq_nonpos_𝓵_pos (P : StandardModel.HiggsField.Potential) (h𝓵 : 0 < P.𝓵) (hμ2 : P.μ2 ≤ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.toFun P φ x = 0 ↔ φ x = 0"} +{"name":"StandardModel.HiggsField.Potential.isMaxOn_iff_field_of_𝓵_neg","declaration":"theorem StandardModel.HiggsField.Potential.isMaxOn_iff_field_of_𝓵_neg (P : StandardModel.HiggsField.Potential) (h𝓵 : P.𝓵 < 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : IsMaxOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun P φ x)\n Set.univ (φ, x) ↔\n P.μ2 ≤ 0 ∧ StandardModel.HiggsField.normSq φ x = P.μ2 / (2 * P.𝓵) ∨ 0 < P.μ2 ∧ φ x = 0"} +{"name":"StandardModel.HiggsField.Potential.μ2_neg","declaration":"theorem StandardModel.HiggsField.Potential.μ2_neg (P : StandardModel.HiggsField.Potential) : (StandardModel.HiggsField.Potential.neg P).μ2 = -P.μ2"} +{"name":"StandardModel.HiggsField.Potential.toFun_zero","declaration":"theorem StandardModel.HiggsField.Potential.toFun_zero (P : StandardModel.HiggsField.Potential) (x : SpaceTime) : StandardModel.HiggsField.Potential.toFun P StandardModel.HiggsField.zero x = 0"} +{"name":"StandardModel.HiggsField.Potential.quadDiscrim_eq_zero_iff_normSq","declaration":"theorem StandardModel.HiggsField.Potential.quadDiscrim_eq_zero_iff_normSq (P : StandardModel.HiggsField.Potential) (h : P.𝓵 ≠ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.quadDiscrim P φ x = 0 ↔ StandardModel.HiggsField.normSq φ x = P.μ2 / (2 * P.𝓵)"} +{"name":"StandardModel.HiggsField.Potential","declaration":"/-- The parameters of the Higgs potential. -/\nstructure StandardModel.HiggsField.Potential : Type"} +{"name":"StandardModel.HiggsField.Potential.isMaxOn_iff_isMinOn_neg","declaration":"theorem StandardModel.HiggsField.Potential.isMaxOn_iff_isMinOn_neg (P : StandardModel.HiggsField.Potential) (φ : StandardModel.HiggsField) (x : SpaceTime) : IsMaxOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun P φ x)\n Set.univ (φ, x) ↔\n IsMinOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun (StandardModel.HiggsField.Potential.neg P) φ x)\n Set.univ (φ, x)"} +{"name":"StandardModel.HiggsField.Potential.neg_𝓵_quadDiscrim_zero_bound","declaration":"theorem StandardModel.HiggsField.Potential.neg_𝓵_quadDiscrim_zero_bound (P : StandardModel.HiggsField.Potential) (h : P.𝓵 < 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.toFun P φ x ≤ -P.μ2 ^ 2 / (4 * P.𝓵)"} +{"name":"StandardModel.HiggsField.Potential.mk","declaration":"ctor StandardModel.HiggsField.Potential.mk (μ2 : ℝ) (𝓵 : ℝ) : StandardModel.HiggsField.Potential"} +{"name":"StandardModel.HiggsField.Potential.𝓵_neg","declaration":"theorem StandardModel.HiggsField.Potential.𝓵_neg (P : StandardModel.HiggsField.Potential) : (StandardModel.HiggsField.Potential.neg P).𝓵 = -P.𝓵"} +{"name":"StandardModel.HiggsField.Potential.isMinOn_iff_of_μSq_nonneg_𝓵_pos","declaration":"theorem StandardModel.HiggsField.Potential.isMinOn_iff_of_μSq_nonneg_𝓵_pos (P : StandardModel.HiggsField.Potential) (h𝓵 : 0 < P.𝓵) (hμ2 : 0 ≤ P.μ2) (φ : StandardModel.HiggsField) (x : SpaceTime) : IsMinOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun P φ x)\n Set.univ (φ, x) ↔\n StandardModel.HiggsField.Potential.toFun P φ x = -P.μ2 ^ 2 / (4 * P.𝓵)"} +{"name":"StandardModel.HiggsField.Potential.isMinOn_iff_field_of_𝓵_pos","declaration":"theorem StandardModel.HiggsField.Potential.isMinOn_iff_field_of_𝓵_pos (P : StandardModel.HiggsField.Potential) (h𝓵 : 0 < P.𝓵) (φ : StandardModel.HiggsField) (x : SpaceTime) : IsMinOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun P φ x)\n Set.univ (φ, x) ↔\n 0 ≤ P.μ2 ∧ StandardModel.HiggsField.normSq φ x = P.μ2 / (2 * P.𝓵) ∨ P.μ2 < 0 ∧ φ x = 0"} +{"name":"StandardModel.HiggsField.Potential.toFun_eq_zero_iff","declaration":"theorem StandardModel.HiggsField.Potential.toFun_eq_zero_iff (P : StandardModel.HiggsField.Potential) (h : P.𝓵 ≠ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.toFun P φ x = 0 ↔ φ x = 0 ∨ StandardModel.HiggsField.normSq φ x = P.μ2 / P.𝓵"} +{"name":"StandardModel.HiggsField.Potential.isMinOn_iff_field_of_μSq_nonpos_𝓵_pos","declaration":"theorem StandardModel.HiggsField.Potential.isMinOn_iff_field_of_μSq_nonpos_𝓵_pos (P : StandardModel.HiggsField.Potential) (h𝓵 : 0 < P.𝓵) (hμ2 : P.μ2 ≤ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : IsMinOn\n (fun x =>\n match x with\n | (φ, x) => StandardModel.HiggsField.Potential.toFun P φ x)\n Set.univ (φ, x) ↔\n φ x = 0"} +{"name":"StandardModel.HiggsField.Potential.IsBounded","declaration":"/-- The proposition on the coefficents for a potential to be bounded. -/\ndef StandardModel.HiggsField.Potential.IsBounded (P : StandardModel.HiggsField.Potential) : Prop"} +{"name":"StandardModel.HiggsField.Potential.quadDiscrim_nonneg","declaration":"/-- The discriminant of the quadratic formed by the potential is non-negative. -/\ntheorem StandardModel.HiggsField.Potential.quadDiscrim_nonneg (P : StandardModel.HiggsField.Potential) (h : P.𝓵 ≠ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : 0 ≤ StandardModel.HiggsField.Potential.quadDiscrim P φ x"} +{"name":"StandardModel.HiggsField.Potential.μ2","declaration":"/-- The mass-squared of the Higgs boson. -/\ndef StandardModel.HiggsField.Potential.μ2 (self : StandardModel.HiggsField.Potential) : ℝ"} +{"name":"StandardModel.HiggsField.Potential.toFun_neg","declaration":"theorem StandardModel.HiggsField.Potential.toFun_neg (P : StandardModel.HiggsField.Potential) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.toFun (StandardModel.HiggsField.Potential.neg P) φ x =\n -StandardModel.HiggsField.Potential.toFun P φ x"} +{"name":"StandardModel.HiggsField.Potential.isBounded_iff_of_𝓵_zero","declaration":"/-- An informal lemma. -/\ndef StandardModel.HiggsField.Potential.isBounded_iff_of_𝓵_zero : InformalLemma"} +{"name":"StandardModel.HiggsField.Potential.quadDiscrim_eq_sqrt_mul_sqrt","declaration":"theorem StandardModel.HiggsField.Potential.quadDiscrim_eq_sqrt_mul_sqrt (P : StandardModel.HiggsField.Potential) (h : P.𝓵 ≠ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.quadDiscrim P φ x =\n Real.sqrt (StandardModel.HiggsField.Potential.quadDiscrim P φ x) *\n Real.sqrt (StandardModel.HiggsField.Potential.quadDiscrim P φ x)"} +{"name":"StandardModel.HiggsField.Potential.complete_square","declaration":"theorem StandardModel.HiggsField.Potential.complete_square (P : StandardModel.HiggsField.Potential) (h : P.𝓵 ≠ 0) (φ : StandardModel.HiggsField) (x : SpaceTime) : StandardModel.HiggsField.Potential.toFun P φ x =\n P.𝓵 * (StandardModel.HiggsField.normSq φ x - P.μ2 / (2 * P.𝓵)) ^ 2 - P.μ2 ^ 2 / (4 * P.𝓵)"} +{"name":"StandardModel.HiggsField.Potential.neg","declaration":"/-- The Higgs potential formed by negating the mass squared and the quartic coupling. -/\ndef StandardModel.HiggsField.Potential.neg (P : StandardModel.HiggsField.Potential) : StandardModel.HiggsField.Potential"} diff --git a/hep-declarations/HepLean.StandardModel.Representations.jsonl b/hep-declarations/HepLean.StandardModel.Representations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..eae50d2d717a5ee67bbcdd445e81e52129912688 --- /dev/null +++ b/hep-declarations/HepLean.StandardModel.Representations.jsonl @@ -0,0 +1,7 @@ +{"name":"StandardModel.repU1_fundamentalSU2_commute","declaration":"theorem StandardModel.repU1_fundamentalSU2_commute (u1 : ↥(unitary ℂ)) (g : ↥(Matrix.specialUnitaryGroup (Fin 2) ℂ)) : StandardModel.repU1 u1 * StandardModel.fundamentalSU2 g = StandardModel.fundamentalSU2 g * StandardModel.repU1 u1"} +{"name":"StandardModel.fundamentalSU2_apply_coe","declaration":"theorem StandardModel.fundamentalSU2_apply_coe (g : ↥(Matrix.specialUnitaryGroup (Fin 2) ℂ)) : ↑(StandardModel.fundamentalSU2 g) = ↑g"} +{"name":"StandardModel.repU1Map","declaration":"/-- The 2d representation of U(1) with charge 3 as a map from U(1) to `unitaryGroup (Fin 2) ℂ`. -/\ndef StandardModel.repU1Map (g : ↥(unitary ℂ)) : ↥(Matrix.unitaryGroup (Fin 2) ℂ)"} +{"name":"StandardModel.fundamentalSU2","declaration":"/-- The fundamental representation of SU(2) as a homomorphism to `unitaryGroup (Fin 2) ℂ`. -/\ndef StandardModel.fundamentalSU2 : ↥(Matrix.specialUnitaryGroup (Fin 2) ℂ) →* ↥(Matrix.unitaryGroup (Fin 2) ℂ)"} +{"name":"StandardModel.repU1_apply_coe","declaration":"theorem StandardModel.repU1_apply_coe (g : ↥(unitary ℂ)) : ↑(StandardModel.repU1 g) = g ^ 3 • 1"} +{"name":"StandardModel.repU1Map_coe","declaration":"theorem StandardModel.repU1Map_coe (g : ↥(unitary ℂ)) : ↑(StandardModel.repU1Map g) = g ^ 3 • 1"} +{"name":"StandardModel.repU1","declaration":"/-- The 2d representation of U(1) with charge 3 as a homomorphism\nfrom U(1) to `unitaryGroup (Fin 2) ℂ`. -/\ndef StandardModel.repU1 : ↥(unitary ℂ) →* ↥(Matrix.unitaryGroup (Fin 2) ℂ)"} diff --git a/hep-declarations/HepLean.Tensors.Basic.jsonl b/hep-declarations/HepLean.Tensors.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7966e090583d44ab43c421ef0d0a0601ca564667 --- /dev/null +++ b/hep-declarations/HepLean.Tensors.Basic.jsonl @@ -0,0 +1,81 @@ +{"name":"TensorStructure.instModuleTensorToSemiringInstAddCommMonoidTensor","declaration":"instance TensorStructure.instModuleTensorToSemiringInstAddCommMonoidTensor {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} : Module R (TensorStructure.Tensor 𝓣 cX)"} +{"name":"TensorColor.colorSetoid","declaration":"/-- The structure of a setoid on colors, two colors are related if they are equal,\nor dual. -/\ninstance TensorColor.colorSetoid (𝓒 : TensorColor) : Setoid 𝓒.Color"} +{"name":"TensorStructure.decompEmbed","declaration":"/-- Decomposes a tensor into a tensor product of two tensors\none which has indices in the image of the given embedding, and the other has indices in\nthe complement of that image. -/\ndef TensorStructure.decompEmbed {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} (f : Y ↪ X) : TensorStructure.Tensor 𝓣 cX ≃ₗ[R]\n TensorProduct R (TensorStructure.Tensor 𝓣 (TensorStructure.decompEmbedColorLeft 𝓣 cX f))\n (TensorStructure.Tensor 𝓣 (cX ∘ ⇑f))"} +{"name":"TensorColor.colorQuot","declaration":"/-- A map taking a color to its equivalence class in `colorSetoid`. -/\ndef TensorColor.colorQuot (𝓒 : TensorColor) (μ : 𝓒.Color) : Quotient (TensorColor.colorSetoid 𝓒)"} +{"name":"TensorStructure.elimPureTensor_update_right","declaration":"theorem TensorStructure.elimPureTensor_update_right {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (p : TensorStructure.PureTensor 𝓣 cX) (q : TensorStructure.PureTensor 𝓣 cY) (y : Y) (r : 𝓣.ColorModule (cY y)) : TensorStructure.elimPureTensor 𝓣 p (Function.update q y r) =\n Function.update (TensorStructure.elimPureTensor 𝓣 p q) (Sum.inr y) r"} +{"name":"TensorStructure.mapIso_trans","declaration":"theorem TensorStructure.mapIso_trans {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} (e : X ≃ Y) (e' : Y ≃ Z) (h : TensorColor.ColorMap.MapIso e cX cY) (h' : TensorColor.ColorMap.MapIso e' cY cZ) : TensorStructure.mapIso 𝓣 e h ≪≫ₗ TensorStructure.mapIso 𝓣 e' h' = TensorStructure.mapIso 𝓣 (e.trans e') ⋯"} +{"name":"TensorStructure.PureTensor","declaration":"/-- The type of pure tensors, i.e. of the form `v1 ⊗ v2 ⊗ v3 ⊗ ...`. -/\ndef TensorStructure.PureTensor {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} (c : X → 𝓣.Color) : Type"} +{"name":"TensorStructure.tensoratorEquiv","declaration":"/-- An equivalence formed by taking the tensor product of tensors. -/\ndef TensorStructure.tensoratorEquiv {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] (c : X → 𝓣.Color) (d : Y → 𝓣.Color) : TensorProduct R (TensorStructure.Tensor 𝓣 c) (TensorStructure.Tensor 𝓣 d) ≃ₗ[R] TensorStructure.Tensor 𝓣 (Sum.elim c d)"} +{"name":"TensorStructure.mapIso_tprod","declaration":"theorem TensorStructure.mapIso_tprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {c : TensorColor.ColorMap 𝓣.toTensorColor X} {d : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (h : TensorColor.ColorMap.MapIso e c d) (f : (i : X) → 𝓣.ColorModule (c i)) : (TensorStructure.mapIso 𝓣 e h) ((PiTensorProduct.tprod R) f) =\n (PiTensorProduct.tprod R) fun i => (TensorStructure.colorModuleCast 𝓣 ⋯) (f (e.symm i))"} +{"name":"TensorStructure.mk","declaration":"ctor TensorStructure.mk {R : Type} [CommSemiring R] (toTensorColor : TensorColor) (ColorModule : toTensorColor.Color → Type) (colorModule_addCommMonoid : (μ : toTensorColor.Color) → AddCommMonoid (ColorModule μ)) (colorModule_module : (μ : toTensorColor.Color) → Module R (ColorModule μ)) (contrDual : (μ : toTensorColor.Color) → TensorProduct R (ColorModule μ) (ColorModule (toTensorColor.τ μ)) →ₗ[R] R) (contrDual_symm : ∀ (μ : toTensorColor.Color) (x : ColorModule μ) (y : ColorModule (toTensorColor.τ μ)),\n (contrDual μ) (x ⊗ₜ[R] y) = (contrDual (toTensorColor.τ μ)) (y ⊗ₜ[R] (Equiv.cast ⋯) x)) (unit : (μ : toTensorColor.Color) → TensorProduct R (ColorModule (toTensorColor.τ μ)) (ColorModule μ)) (unit_rid : ∀ (μ : toTensorColor.Color) (x : ColorModule μ), (TensorStructure.contrLeftAux (contrDual μ)) (x ⊗ₜ[R] unit μ) = x) (metric : (μ : toTensorColor.Color) → TensorProduct R (ColorModule μ) (ColorModule μ)) (metric_dual : ∀ (μ : toTensorColor.Color),\n (TensorStructure.contrMidAux (contrDual μ)) (metric μ ⊗ₜ[R] metric (toTensorColor.τ μ)) =\n (TensorProduct.comm R (ColorModule (toTensorColor.τ μ)) (ColorModule μ)) (unit μ)) : TensorStructure R"} +{"name":"TensorColor.ColorMap.MapIso.dual","declaration":"theorem TensorColor.ColorMap.MapIso.dual {X : Type} {Y : Type} {𝓒 : TensorColor} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} {e : X ≃ Y} (h : TensorColor.ColorMap.MapIso e cX cY) : TensorColor.ColorMap.MapIso e (TensorColor.ColorMap.dual cX) (TensorColor.ColorMap.dual cY)"} +{"name":"TensorStructure.metric_dual","declaration":"/-- The metric contracted with its dual is the unit. -/\ndef TensorStructure.metric_dual {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) : (TensorStructure.contrMidAux (self.contrDual μ)) (self.metric μ ⊗ₜ[R] self.metric (self.τ μ)) =\n (TensorProduct.comm R (self.ColorModule (self.τ μ)) (self.ColorModule μ)) (self.unit μ)"} +{"name":"TensorStructure.decompEmbedColorLeft","declaration":"/-- The restriction of a map from an indexing set to the space to the complement of the image\nof an embedding. -/\ndef TensorStructure.decompEmbedColorLeft {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] (c : X → 𝓣.Color) (f : Y ↪ X) : { x // x ∈ (Finset.image (⇑f) Finset.univ)ᶜ } → 𝓣.Color"} +{"name":"TensorStructure.inlPureTensor_update_right","declaration":"theorem TensorStructure.inlPureTensor_update_right {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} [DecidableEq (X ⊕ Y)] (f : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)) (y : Y) (v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inr y))) : TensorStructure.inlPureTensor 𝓣 (Function.update f (Sum.inr y) v1) = TensorStructure.inlPureTensor 𝓣 f"} +{"name":"TensorStructure.contrDual_cast","declaration":"theorem TensorStructure.contrDual_cast {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {μ : 𝓣.Color} {ν : 𝓣.Color} (h : μ = ν) (x : 𝓣.ColorModule μ) (y : 𝓣.ColorModule (𝓣.τ μ)) : (𝓣.contrDual μ) (x ⊗ₜ[R] y) =\n (𝓣.contrDual ν) ((TensorStructure.colorModuleCast 𝓣 h) x ⊗ₜ[R] (TensorStructure.colorModuleCast 𝓣 ⋯) y)"} +{"name":"TensorStructure.domCoprod","declaration":"/-- The multi-linear map taking a pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` and constructing\na vector in `𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY`. -/\ndef TensorStructure.domCoprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} : MultilinearMap R (fun x => 𝓣.ColorModule (Sum.elim cX cY x))\n (TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 cY))"} +{"name":"TensorStructure.mapIso","declaration":"/-- An linear equivalence of tensor spaces given a color-preserving equivalence of indexing sets. -/\ndef TensorStructure.mapIso {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {c : TensorColor.ColorMap 𝓣.toTensorColor X} {d : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (h : TensorColor.ColorMap.MapIso e c d) : TensorStructure.Tensor 𝓣 c ≃ₗ[R] TensorStructure.Tensor 𝓣 d"} +{"name":"TensorStructure.contrDual_symm_contrRightAux","declaration":"theorem TensorStructure.contrDual_symm_contrRightAux {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {μ : 𝓣.Color} {ν : 𝓣.Color} {η : 𝓣.Color} (h : ν = η) : ↑(TensorStructure.colorModuleCast 𝓣 h) ∘ₗ TensorStructure.contrRightAux (𝓣.contrDual μ) =\n TensorStructure.contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ))) ∘ₗ\n ↑(TensorProduct.congr\n (TensorProduct.congr (TensorStructure.colorModuleCast 𝓣 h) (TensorStructure.colorModuleCast 𝓣 ⋯))\n (TensorStructure.colorModuleCast 𝓣 ⋯))"} +{"name":"TensorStructure.isEmptyEquiv_tprod","declaration":"theorem TensorStructure.isEmptyEquiv_tprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} [IsEmpty X] (f : TensorStructure.PureTensor 𝓣 cX) : (TensorStructure.isEmptyEquiv 𝓣) ((PiTensorProduct.tprod R) f) = 1"} +{"name":"TensorStructure.contrRightAux_comp","declaration":"theorem TensorStructure.contrRightAux_comp {R : Type} [CommSemiring R] {V1 : Type} {V2 : Type} {V3 : Type} {V4 : Type} {V5 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3] [AddCommMonoid V4] [AddCommMonoid V5] [Module R V1] [Module R V3] [Module R V2] [Module R V4] [Module R V5] (f : TensorProduct R V2 V3 →ₗ[R] R) (g : TensorProduct R V4 V5 →ₗ[R] R) : TensorStructure.contrRightAux f ∘ₗ TensorProduct.map LinearMap.id (TensorStructure.contrRightAux g) =\n TensorStructure.contrRightAux g ∘ₗ\n TensorProduct.map (TensorStructure.contrMidAux f) LinearMap.id ∘ₗ\n ↑(LinearEquiv.symm (TensorProduct.assoc R (TensorProduct R V1 V2) (TensorProduct R V3 V4) V5))"} +{"name":"TensorColor.τ","declaration":"/-- A map taking every color to its dual color. -/\ndef TensorColor.τ (self : TensorColor) : self.Color → self.Color"} +{"name":"TensorStructure.decompEmbedSet","declaration":"/-- The decomposition of a set into a direct sum based on the image of an injection. -/\ndef TensorStructure.decompEmbedSet {X : Type} {Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] (f : Y ↪ X) : X ≃ { x // x ∈ (Finset.image (⇑f) Finset.univ)ᶜ } ⊕ Y"} +{"name":"TensorStructure.metric","declaration":"/-- The metric for a given color. -/\ndef TensorStructure.metric {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) : TensorProduct R (self.ColorModule μ) (self.ColorModule μ)"} +{"name":"TensorStructure.ColorModule","declaration":"/-- To each color we associate a module. -/\ndef TensorStructure.ColorModule {R : Type} [CommSemiring R] (self : TensorStructure R) : self.Color → Type"} +{"name":"TensorStructure.contrLeftAux","declaration":"/-- An auxillary function to contract the vector space `V1` and `V2` in `V1 ⊗[R] V2 ⊗[R] V3`. -/\ndef TensorStructure.contrLeftAux {R : Type} [CommSemiring R] {V1 : Type} {V2 : Type} {V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3] [Module R V1] [Module R V2] [Module R V3] (f : TensorProduct R V1 V2 →ₗ[R] R) : TensorProduct R V1 (TensorProduct R V2 V3) →ₗ[R] V3"} +{"name":"TensorColor.instDecidableEqQuotientColorColorSetoid","declaration":"instance TensorColor.instDecidableEqQuotientColorColorSetoid (𝓒 : TensorColor) [DecidableEq 𝓒.Color] : DecidableEq (Quotient (TensorColor.colorSetoid 𝓒))"} +{"name":"TensorColor.ColorMap.MapIso.sum","declaration":"theorem TensorColor.ColorMap.MapIso.sum {X : Type} {X' : Type} {Y : Type} {Y' : Type} {𝓒 : TensorColor} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} {cX' : TensorColor.ColorMap 𝓒 X'} {cY' : TensorColor.ColorMap 𝓒 Y'} {eX : X ≃ X'} {eY : Y ≃ Y'} (hX : TensorColor.ColorMap.MapIso eX cX cX') (hY : TensorColor.ColorMap.MapIso eY cY cY') : TensorColor.ColorMap.MapIso (Equiv.sumCongr eX eY) (TensorColor.ColorMap.sum cX cY) (TensorColor.ColorMap.sum cX' cY')"} +{"name":"TensorStructure.instAddCommMonoidTensor","declaration":"instance TensorStructure.instAddCommMonoidTensor {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} : AddCommMonoid (TensorStructure.Tensor 𝓣 cX)"} +{"name":"TensorColor.ColorMap.MapIso","declaration":"/-- A relation, given an equivalence of types, between ColorMap which is true\nif related by composition of the equivalence. -/\ndef TensorColor.ColorMap.MapIso {X : Type} {Y : Type} {𝓒 : TensorColor} (e : X ≃ Y) (cX : TensorColor.ColorMap 𝓒 X) (cY : TensorColor.ColorMap 𝓒 Y) : Prop"} +{"name":"TensorStructure.colorModule_module","declaration":"/-- Each `ColorModule` has the structure of a module over `R`. -/\ndef TensorStructure.colorModule_module {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) : Module R (self.ColorModule μ)"} +{"name":"TensorStructure.contrDual","declaration":"/-- The contraction of a vector with a vector with dual color. -/\ndef TensorStructure.contrDual {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) : TensorProduct R (self.ColorModule μ) (self.ColorModule (self.τ μ)) →ₗ[R] R"} +{"name":"TensorColor.colorRel","declaration":"/-- A relation on colors which is true if the two colors are equal or are duals. -/\ndef TensorColor.colorRel (𝓒 : TensorColor) (μ : 𝓒.Color) (ν : 𝓒.Color) : Prop"} +{"name":"TensorColor.instDecidableEquivColorInstHasEquivColorSetoid","declaration":"instance TensorColor.instDecidableEquivColorInstHasEquivColorSetoid (𝓒 : TensorColor) [DecidableEq 𝓒.Color] (μ : 𝓒.Color) (ν : 𝓒.Color) : Decidable (μ ≈ ν)"} +{"name":"TensorStructure.mapIso_mapIso","declaration":"theorem TensorStructure.mapIso_mapIso {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} (e : X ≃ Y) (e' : Y ≃ Z) (h : TensorColor.ColorMap.MapIso e cX cY) (h' : TensorColor.ColorMap.MapIso e' cY cZ) (T : TensorStructure.Tensor 𝓣 cX) : (TensorStructure.mapIso 𝓣 e' h') ((TensorStructure.mapIso 𝓣 e h) T) = (TensorStructure.mapIso 𝓣 (e.trans e') ⋯) T"} +{"name":"TensorStructure.unit_rid","declaration":"/-- The unit is a right identity. -/\ndef TensorStructure.unit_rid {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) (x : self.ColorModule μ) : (TensorStructure.contrLeftAux (self.contrDual μ)) (x ⊗ₜ[R] self.unit μ) = x"} +{"name":"TensorStructure.isEmptyEquiv","declaration":"/-- The equivalence between `𝓣.Tensor cX` and `R` when the indexing set `X` is empty. -/\ndef TensorStructure.isEmptyEquiv {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} [IsEmpty X] : TensorStructure.Tensor 𝓣 cX ≃ₗ[R] R"} +{"name":"TensorStructure.tensoratorEquiv_symm_tprod","declaration":"theorem TensorStructure.tensoratorEquiv_symm_tprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (f : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)) : (LinearEquiv.symm (TensorStructure.tensoratorEquiv 𝓣 cX cY)) ((PiTensorProduct.tprod R) f) =\n (PiTensorProduct.tprod R) (TensorStructure.inlPureTensor 𝓣 f) ⊗ₜ[R]\n (PiTensorProduct.tprod R) (TensorStructure.inrPureTensor 𝓣 f)"} +{"name":"TensorStructure.elimPureTensor_update_left","declaration":"theorem TensorStructure.elimPureTensor_update_left {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (p : TensorStructure.PureTensor 𝓣 cX) (q : TensorStructure.PureTensor 𝓣 cY) (x : X) (r : 𝓣.ColorModule (cX x)) : TensorStructure.elimPureTensor 𝓣 (Function.update p x r) q =\n Function.update (TensorStructure.elimPureTensor 𝓣 p q) (Sum.inl x) r"} +{"name":"TensorStructure.tensoratorEquiv_mapIso_apply","declaration":"theorem TensorStructure.tensoratorEquiv_mapIso_apply {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} {W : Type} [DecidableEq X] [DecidableEq Y] [DecidableEq Z] [DecidableEq W] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {cW : TensorColor.ColorMap 𝓣.toTensorColor W} (e' : Z ≃ Y) (e'' : W ≃ X) (h' : TensorColor.ColorMap.MapIso e' cZ cY) (h'' : TensorColor.ColorMap.MapIso e'' cW cX) (x : TensorProduct R (TensorStructure.Tensor 𝓣 cW) (TensorStructure.Tensor 𝓣 cZ)) : (TensorStructure.tensoratorEquiv 𝓣 cX cY)\n ((TensorProduct.congr (TensorStructure.mapIso 𝓣 e'' h'') (TensorStructure.mapIso 𝓣 e' h')) x) =\n (TensorStructure.mapIso 𝓣 (Equiv.sumCongr e'' e') ⋯) ((TensorStructure.tensoratorEquiv 𝓣 cW cZ) x)"} +{"name":"TensorColor.Color","declaration":"/-- The allowed colors of indices.\nFor example for a real Lorentz tensor these are `{up, down}`. -/\ndef TensorColor.Color (self : TensorColor) : Type"} +{"name":"TensorStructure.tensoratorEquiv_tmul_tprod","declaration":"theorem TensorStructure.tensoratorEquiv_tmul_tprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (p : TensorStructure.PureTensor 𝓣 cX) (q : TensorStructure.PureTensor 𝓣 cY) : (TensorStructure.tensoratorEquiv 𝓣 cX cY) ((PiTensorProduct.tprod R) p ⊗ₜ[R] (PiTensorProduct.tprod R) q) =\n (PiTensorProduct.tprod R) (TensorStructure.elimPureTensor 𝓣 p q)"} +{"name":"TensorStructure.inrPureTensor_update_left","declaration":"theorem TensorStructure.inrPureTensor_update_left {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} [DecidableEq (X ⊕ Y)] (f : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)) (x : X) (v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inl x))) : TensorStructure.inrPureTensor 𝓣 (Function.update f (Sum.inl x) v1) = TensorStructure.inrPureTensor 𝓣 f"} +{"name":"TensorStructure.mapIso_symm","declaration":"theorem TensorStructure.mapIso_symm {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (h : TensorColor.ColorMap.MapIso e cX cY) : LinearEquiv.symm (TensorStructure.mapIso 𝓣 e h) = TensorStructure.mapIso 𝓣 e.symm ⋯"} +{"name":"TensorColor.ColorMap","declaration":"/-- The types of maps from an `X` to `𝓒.Color`. -/\ndef TensorColor.ColorMap (𝓒 : TensorColor) (X : Type) : Type"} +{"name":"TensorStructure.tensorProd_piTensorProd_ext","declaration":"theorem TensorStructure.tensorProd_piTensorProd_ext {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {M : Type} [AddCommMonoid M] [Module R M] {f : TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 cY) →ₗ[R] M} {g : TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 cY) →ₗ[R] M} (h : ∀ (p : (i : X) → 𝓣.ColorModule (cX i)) (q : (i : Y) → 𝓣.ColorModule (cY i)),\n f ((PiTensorProduct.tprod R) p ⊗ₜ[R] (PiTensorProduct.tprod R) q) =\n g ((PiTensorProduct.tprod R) p ⊗ₜ[R] (PiTensorProduct.tprod R) q)) : f = g"} +{"name":"TensorStructure.decompEmbedColorRight","declaration":"/-- The restriction of a map from an indexing set to the space to the image\nof an embedding. -/\ndef TensorStructure.decompEmbedColorRight {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] (c : X → 𝓣.Color) (f : Y ↪ X) : Y → 𝓣.Color"} +{"name":"TensorStructure.instAddCommMonoidColorModule","declaration":"instance TensorStructure.instAddCommMonoidColorModule {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {μ : 𝓣.Color} : AddCommMonoid (𝓣.ColorModule μ)"} +{"name":"TensorStructure.unit","declaration":"/-- The unit of the contraction. -/\ndef TensorStructure.unit {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) : TensorProduct R (self.ColorModule (self.τ μ)) (self.ColorModule μ)"} +{"name":"TensorColor.ColorMap.sum","declaration":"/-- The sum of two color maps, formed by `Sum.elim`. -/\ndef TensorColor.ColorMap.sum {X : Type} {Y : Type} {𝓒 : TensorColor} (cX : TensorColor.ColorMap 𝓒 X) (cY : TensorColor.ColorMap 𝓒 Y) : TensorColor.ColorMap 𝓒 (X ⊕ Y)"} +{"name":"TensorColor.ColorMap.dual","declaration":"/-- The dual of a color map, formed by composition with `𝓒.τ`. -/\ndef TensorColor.ColorMap.dual {X : Type} {𝓒 : TensorColor} (cX : TensorColor.ColorMap 𝓒 X) : TensorColor.ColorMap 𝓒 X"} +{"name":"TensorStructure.contrRightAux","declaration":"/-- An auxillary function to contract the vector space `V1` and `V2` in `(V3 ⊗[R] V1) ⊗[R] V2`. -/\ndef TensorStructure.contrRightAux {R : Type} [CommSemiring R] {V1 : Type} {V2 : Type} {V3 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3] [Module R V1] [Module R V2] [Module R V3] (f : TensorProduct R V1 V2 →ₗ[R] R) : TensorProduct R (TensorProduct R V3 V1) V2 →ₗ[R] V3"} +{"name":"TensorStructure.mapIso_ext","declaration":"theorem TensorStructure.mapIso_ext {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {c : TensorColor.ColorMap 𝓣.toTensorColor X} {d : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (e' : X ≃ Y) (h : TensorColor.ColorMap.MapIso e c d) (h' : TensorColor.ColorMap.MapIso e' c d) (he : e = e') : TensorStructure.mapIso 𝓣 e h = TensorStructure.mapIso 𝓣 e' h'"} +{"name":"TensorStructure.inrPureTensor_update_right","declaration":"theorem TensorStructure.inrPureTensor_update_right {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} [DecidableEq (X ⊕ Y)] (f : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)) (y : Y) (v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inr y))) : TensorStructure.inrPureTensor 𝓣 (Function.update f (Sum.inr y) v1) =\n Function.update (TensorStructure.inrPureTensor 𝓣 f) y v1"} +{"name":"TensorStructure.tensoratorEquiv_mapIso","declaration":"theorem TensorStructure.tensoratorEquiv_mapIso {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} {W : Type} [DecidableEq X] [DecidableEq Y] [DecidableEq Z] [DecidableEq W] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {cW : TensorColor.ColorMap 𝓣.toTensorColor W} (e' : Z ≃ Y) (e'' : W ≃ X) (h' : TensorColor.ColorMap.MapIso e' cZ cY) (h'' : TensorColor.ColorMap.MapIso e'' cW cX) : TensorProduct.congr (TensorStructure.mapIso 𝓣 e'' h'') (TensorStructure.mapIso 𝓣 e' h') ≪≫ₗ\n TensorStructure.tensoratorEquiv 𝓣 cX cY =\n TensorStructure.tensoratorEquiv 𝓣 cW cZ ≪≫ₗ TensorStructure.mapIso 𝓣 (Equiv.sumCongr e'' e') ⋯"} +{"name":"TensorStructure.contrDual_symm","declaration":"/-- The contraction is symmetric. -/\ndef TensorStructure.contrDual_symm {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) (x : self.ColorModule μ) (y : self.ColorModule (self.τ μ)) : (self.contrDual μ) (x ⊗ₜ[R] y) = (self.contrDual (self.τ μ)) (y ⊗ₜ[R] (Equiv.cast ⋯) x)"} +{"name":"TensorColor.mk","declaration":"ctor TensorColor.mk (Color : Type) (τ : Color → Color) (τ_involutive : Function.Involutive τ) : TensorColor"} +{"name":"TensorStructure.tensoratorSymm","declaration":"/-- The linear map combining two tensors into a single tensor\nvia the tensor product i.e. `v1 v2 ↦ v1 ⊗ v2`. -/\ndef TensorStructure.tensoratorSymm {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} : TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 cY) →ₗ[R]\n TensorStructure.Tensor 𝓣 (Sum.elim cX cY)"} +{"name":"TensorStructure.Tensor","declaration":"/-- The type of tensors given a map from an indexing set `X` to the type of colors,\nspecifying the color of that index. -/\ndef TensorStructure.Tensor {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} (c : TensorColor.ColorMap 𝓣.toTensorColor X) : Type"} +{"name":"TensorStructure.decompEmbed_cond","declaration":"theorem TensorStructure.decompEmbed_cond {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] [DecidableEq X] [Fintype Y] (c : X → 𝓣.Color) (f : Y ↪ X) : c =\n Sum.elim (TensorStructure.decompEmbedColorLeft 𝓣 c f) (TensorStructure.decompEmbedColorRight 𝓣 c f) ∘\n ⇑(TensorStructure.decompEmbedSet f)"} +{"name":"TensorColor","declaration":"/-- The index color data associated with a tensor structure.\nThis corresponds to a type with an involution. -/\nstructure TensorColor : Type 1"} +{"name":"TensorStructure","declaration":"/-- An initial structure specifying a tensor system (e.g. a system in which you can\ndefine real Lorentz tensors or Einstein notation convention). -/\nstructure TensorStructure (R : Type) [CommSemiring R] : Type 1"} +{"name":"TensorColor.instDecidableColorRel","declaration":"instance TensorColor.instDecidableColorRel (𝓒 : TensorColor) [DecidableEq 𝓒.Color] {μ : 𝓒.Color} {ν : 𝓒.Color} : Decidable (TensorColor.colorRel 𝓒 μ ν)"} +{"name":"TensorColor.colorRel_equivalence","declaration":"/-- An equivalence relation on colors which is true if the two colors are equal or are duals. -/\ntheorem TensorColor.colorRel_equivalence (𝓒 : TensorColor) : Equivalence (TensorColor.colorRel 𝓒)"} +{"name":"TensorStructure.elimPureTensorMulLin","declaration":"/-- The multilinear map taking pure tensors a `𝓣.PureTensor cX` and a pure tensor in\n`𝓣.PureTensor cY`, and constructing a tensor in `𝓣.Tensor (Sum.elim cX cY))`. -/\ndef TensorStructure.elimPureTensorMulLin {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} : MultilinearMap R (fun i => 𝓣.ColorModule (cX i))\n (MultilinearMap R (fun x => 𝓣.ColorModule (cY x)) (TensorStructure.Tensor 𝓣 (Sum.elim cX cY)))"} +{"name":"TensorStructure.tensoratorEquiv_mapIso_tmul","declaration":"theorem TensorStructure.tensoratorEquiv_mapIso_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} {W : Type} [DecidableEq X] [DecidableEq Y] [DecidableEq Z] [DecidableEq W] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {cW : TensorColor.ColorMap 𝓣.toTensorColor W} (e' : Z ≃ Y) (e'' : W ≃ X) (h' : TensorColor.ColorMap.MapIso e' cZ cY) (h'' : TensorColor.ColorMap.MapIso e'' cW cX) (x : TensorStructure.Tensor 𝓣 cW) (y : TensorStructure.Tensor 𝓣 cZ) : (TensorStructure.tensoratorEquiv 𝓣 cX cY)\n ((TensorStructure.mapIso 𝓣 e'' h'') x ⊗ₜ[R] (TensorStructure.mapIso 𝓣 e' h') y) =\n (TensorStructure.mapIso 𝓣 (Equiv.sumCongr e'' e') ⋯) ((TensorStructure.tensoratorEquiv 𝓣 cW cZ) (x ⊗ₜ[R] y))"} +{"name":"TensorColor.ColorMap.MapIso.symm'","declaration":"theorem TensorColor.ColorMap.MapIso.symm' {X : Type} {Y : Type} {𝓒 : TensorColor} {e : X ≃ Y} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} : TensorColor.ColorMap.MapIso e cX cY ↔ TensorColor.ColorMap.MapIso e.symm cY cX"} +{"name":"TensorStructure.inlPureTensor_update_left","declaration":"theorem TensorStructure.inlPureTensor_update_left {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} [DecidableEq (X ⊕ Y)] (f : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)) (x : X) (v1 : 𝓣.ColorModule (Sum.elim cX cY (Sum.inl x))) : TensorStructure.inlPureTensor 𝓣 (Function.update f (Sum.inl x) v1) =\n Function.update (TensorStructure.inlPureTensor 𝓣 f) x v1"} +{"name":"TensorStructure.mapIso_refl","declaration":"theorem TensorStructure.mapIso_refl {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} : TensorStructure.mapIso 𝓣 (Equiv.refl X) ⋯ = LinearEquiv.refl R (TensorStructure.Tensor 𝓣 cX)"} +{"name":"TensorStructure.colorModule_addCommMonoid","declaration":"/-- Each `ColorModule` has the structure of an additive commutative monoid. -/\ndef TensorStructure.colorModule_addCommMonoid {R : Type} [CommSemiring R] (self : TensorStructure R) (μ : self.Color) : AddCommMonoid (self.ColorModule μ)"} +{"name":"TensorStructure.tensorator","declaration":"/-- Splitting a tensor in `𝓣.Tensor (Sum.elim cX cY)` into the tensor product of two tensors. -/\ndef TensorStructure.tensorator {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [DecidableEq X] [DecidableEq Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} : TensorStructure.Tensor 𝓣 (Sum.elim cX cY) →ₗ[R]\n TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 cY)"} +{"name":"TensorStructure.instModuleColorModuleToSemiringInstAddCommMonoidColorModule","declaration":"instance TensorStructure.instModuleColorModuleToSemiringInstAddCommMonoidColorModule {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {μ : 𝓣.Color} : Module R (𝓣.ColorModule μ)"} +{"name":"TensorColor.τ_involutive","declaration":"/-- The map `τ` is an involution. -/\ndef TensorColor.τ_involutive (self : TensorColor) : Function.Involutive self.τ"} +{"name":"TensorStructure.elimPureTensor","declaration":"/-- A pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` constructed from a pure tensor\nin `𝓣.PureTensor cX` and a pure tensor in `𝓣.PureTensor cY`. -/\ndef TensorStructure.elimPureTensor {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (p : TensorStructure.PureTensor 𝓣 cX) (q : TensorStructure.PureTensor 𝓣 cY) : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)"} +{"name":"TensorStructure.inlPureTensor","declaration":"/-- The projection of a pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` onto a pure tensor in\n`𝓣.PureTensor cX`. -/\ndef TensorStructure.inlPureTensor {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (p : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)) : TensorStructure.PureTensor 𝓣 cX"} +{"name":"TensorStructure.colorModuleCast","declaration":"/-- Equivalence of `ColorModule` given an equality of colors. -/\ndef TensorStructure.colorModuleCast {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {μ : 𝓣.Color} {ν : 𝓣.Color} (h : μ = ν) : 𝓣.ColorModule μ ≃ₗ[R] 𝓣.ColorModule ν"} +{"name":"TensorStructure.inrPureTensor","declaration":"/-- The projection of a pure tensor in `𝓣.PureTensor (Sum.elim cX cY)` onto a pure tensor in\n`𝓣.PureTensor cY`. -/\ndef TensorStructure.inrPureTensor {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (p : TensorStructure.PureTensor 𝓣 (Sum.elim cX cY)) : TensorStructure.PureTensor 𝓣 cY"} +{"name":"TensorStructure.contrMidAux","declaration":"/-- An auxillary function to contract the vector space `V1` and `V2` in\n`V4 ⊗[R] V1 ⊗[R] V2 ⊗[R] V3`. -/\ndef TensorStructure.contrMidAux {R : Type} [CommSemiring R] {V1 : Type} {V2 : Type} {V3 : Type} {V4 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [AddCommMonoid V3] [AddCommMonoid V4] [Module R V1] [Module R V2] [Module R V3] [Module R V4] (f : TensorProduct R V1 V2 →ₗ[R] R) : TensorProduct R (TensorProduct R V4 V1) (TensorProduct R V2 V3) →ₗ[R] TensorProduct R V4 V3"} +{"name":"TensorStructure.contrDual_symm_contrRightAux_apply_tmul","declaration":"theorem TensorStructure.contrDual_symm_contrRightAux_apply_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {μ : 𝓣.Color} {ν : 𝓣.Color} {η : 𝓣.Color} (h : ν = η) (m : TensorProduct R (𝓣.ColorModule ν) (𝓣.ColorModule μ)) (x : 𝓣.ColorModule (𝓣.τ μ)) : (TensorStructure.colorModuleCast 𝓣 h) ((TensorStructure.contrRightAux (𝓣.contrDual μ)) (m ⊗ₜ[R] x)) =\n (TensorStructure.contrRightAux (𝓣.contrDual (𝓣.τ (𝓣.τ μ))))\n ((TensorProduct.congr (TensorStructure.colorModuleCast 𝓣 h) (TensorStructure.colorModuleCast 𝓣 ⋯)) m ⊗ₜ[R]\n (TensorStructure.colorModuleCast 𝓣 ⋯) x)"} +{"name":"TensorStructure.contrDual_symm'","declaration":"/-- `𝓣.contrDual (𝓣.τ μ)` in terms of `𝓣.contrDual μ`. -/\ntheorem TensorStructure.contrDual_symm' {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) (μ : 𝓣.Color) (x : 𝓣.ColorModule (𝓣.τ μ)) (y : 𝓣.ColorModule (𝓣.τ (𝓣.τ μ))) : (𝓣.contrDual (𝓣.τ μ)) (x ⊗ₜ[R] y) = (𝓣.contrDual μ) ((TensorStructure.colorModuleCast 𝓣 ⋯) y ⊗ₜ[R] x)"} +{"name":"TensorColor.ColorMap.MapIso.trans","declaration":"theorem TensorColor.ColorMap.MapIso.trans {X : Type} {Y : Type} {Z : Type} {𝓒 : TensorColor} {e : X ≃ Y} {e' : Y ≃ Z} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} {cZ : TensorColor.ColorMap 𝓒 Z} (h : TensorColor.ColorMap.MapIso e cX cY) (h' : TensorColor.ColorMap.MapIso e' cY cZ) : TensorColor.ColorMap.MapIso (e.trans e') cX cZ"} +{"name":"TensorColor.ColorMap.MapIso.symm","declaration":"theorem TensorColor.ColorMap.MapIso.symm {X : Type} {Y : Type} {𝓒 : TensorColor} {e : X ≃ Y} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} (h : TensorColor.ColorMap.MapIso e cX cY) : TensorColor.ColorMap.MapIso e.symm cY cX"} diff --git a/hep-declarations/HepLean.Tensors.Contraction.jsonl b/hep-declarations/HepLean.Tensors.Contraction.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d328c39424aef54e285989215a6fa40147cbce95 --- /dev/null +++ b/hep-declarations/HepLean.Tensors.Contraction.jsonl @@ -0,0 +1,38 @@ +{"name":"TensorStructure.contr_cond","declaration":"theorem TensorStructure.contr_cond {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {C : Type} {P : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} (e : (C ⊕ C) ⊕ P ≃ X) : TensorColor.ColorMap.MapIso e.symm cX\n (Sum.elim (Sum.elim (TensorColor.ColorMap.contrLeft e cX) (TensorColor.ColorMap.contrRight e cX))\n (TensorColor.ColorMap.contr e cX))"} +{"name":"TensorStructure.contrAll'_mapIso","declaration":"theorem TensorStructure.contrAll'_mapIso {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] [Fintype Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (h : TensorColor.ColorMap.MapIso e cX cY) : TensorStructure.contrAll' 𝓣 ∘ₗ\n ↑(TensorProduct.congr (TensorStructure.mapIso 𝓣 e h) (LinearEquiv.refl R (TensorStructure.Tensor 𝓣 (𝓣.τ ∘ cY)))) =\n TensorStructure.contrAll' 𝓣 ∘ₗ\n ↑(TensorProduct.congr (LinearEquiv.refl R (TensorStructure.Tensor 𝓣 cX)) (TensorStructure.mapIso 𝓣 e.symm ⋯))"} +{"name":"TensorColor.ColorMap.ContrAll","declaration":"/-- Given an equivalence `e` of types the condition that the color map `cX` is the dual to `cY`\nup to this equivalence. -/\ndef TensorColor.ColorMap.ContrAll {X : Type} {Y : Type} {𝓒 : TensorColor} (e : X ≃ Y) (cX : TensorColor.ColorMap 𝓒 X) (cY : TensorColor.ColorMap 𝓒 Y) : Prop"} +{"name":"TensorStructure.contrAll_mapIso_left_tmul","declaration":"theorem TensorStructure.contrAll_mapIso_left_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} [Fintype X] [Fintype Z] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {e : X ≃ Y} {e' : Z ≃ X} (h : TensorColor.ColorMap.ContrAll e cX cY) (h' : TensorColor.ColorMap.MapIso e' cZ cX) (x : TensorStructure.Tensor 𝓣 cZ) (y : TensorStructure.Tensor 𝓣 cY) : (TensorStructure.contrAll 𝓣 e h) ((TensorStructure.mapIso 𝓣 e' h') x ⊗ₜ[R] y) =\n (TensorStructure.contrAll 𝓣 (e'.trans e) ⋯) (x ⊗ₜ[R] y)"} +{"name":"TensorStructure.contrAll_tmul","declaration":"theorem TensorStructure.contrAll_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (h : TensorColor.ColorMap.ContrAll e cX cY) (x : TensorStructure.Tensor 𝓣 cX) (y : TensorStructure.Tensor 𝓣 cY) : (TensorStructure.contrAll 𝓣 e h) (x ⊗ₜ[R] y) =\n (TensorStructure.contrAll' 𝓣) (x ⊗ₜ[R] (TensorStructure.mapIso 𝓣 e.symm ⋯) y)"} +{"name":"TensorStructure.contrElim","declaration":"/-- The linear map from `𝓣.Tensor (Sum.elim cW cX) ⊗[R] 𝓣.Tensor (Sum.elim cY cZ)`\nto `𝓣.Tensor (Sum.elim cW cZ)` formed by contracting the indices specified by\n`cX` and `cY`, which are assumed to be dual. -/\ndef TensorStructure.contrElim {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} {W : Type} [Fintype X] [DecidableEq X] [DecidableEq Y] [DecidableEq Z] [DecidableEq W] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {cW : TensorColor.ColorMap 𝓣.toTensorColor W} (e : X ≃ Y) (h : TensorColor.ColorMap.ContrAll e cX cY) : TensorProduct R (TensorStructure.Tensor 𝓣 (Sum.elim cW cX)) (TensorStructure.Tensor 𝓣 (Sum.elim cY cZ)) →ₗ[R]\n TensorStructure.Tensor 𝓣 (Sum.elim cW cZ)"} +{"name":"TensorColor.ColorMap.ContrAll.toMapIso","declaration":"theorem TensorColor.ColorMap.ContrAll.toMapIso {X : Type} {Y : Type} {𝓒 : TensorColor} {e : X ≃ Y} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} (h : TensorColor.ColorMap.ContrAll e cX cY) : TensorColor.ColorMap.MapIso e cX (TensorColor.ColorMap.dual cY)"} +{"name":"TensorColor.ColorMap.ContrAll.mapIso_trans","declaration":"theorem TensorColor.ColorMap.ContrAll.mapIso_trans {X : Type} {Y : Type} {Z : Type} {𝓒 : TensorColor} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} {cZ : TensorColor.ColorMap 𝓒 Z} {e : X ≃ Y} {e' : Z ≃ X} (h : TensorColor.ColorMap.ContrAll e cX cY) (h' : TensorColor.ColorMap.MapIso e' cZ cX) : TensorColor.ColorMap.ContrAll (e'.trans e) cZ cY"} +{"name":"TensorStructure.contrAll_comp_mapIso_right","declaration":"theorem TensorStructure.contrAll_comp_mapIso_right {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} [Fintype X] {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {c : TensorColor.ColorMap 𝓣.toTensorColor X} (e : X ≃ Y) (e' : Z ≃ Y) (h : TensorColor.ColorMap.ContrAll e c cY) (h' : TensorColor.ColorMap.MapIso e' cZ cY) : TensorStructure.contrAll 𝓣 e h ∘ₗ\n ↑(TensorProduct.congr (LinearEquiv.refl R (TensorStructure.Tensor 𝓣 c)) (TensorStructure.mapIso 𝓣 e' h')) =\n TensorStructure.contrAll 𝓣 (e.trans e'.symm) ⋯"} +{"name":"TensorStructure.contrAllLeft","declaration":"/-- The linear map from `𝓣.Tensor cX ⊗[R] 𝓣.Tensor cY ⊗[R] 𝓣.Tensor cZ` to\n`𝓣.Tensor cZ` obtained by contracting all indices in `𝓣.Tensor cX` and `𝓣.Tensor cY`,\ngiven a proof that this is possible. -/\ndef TensorStructure.contrAllLeft {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} (e : X ≃ Y) (h : TensorColor.ColorMap.ContrAll e cX cY) : TensorProduct R (TensorStructure.Tensor 𝓣 cX)\n (TensorProduct R (TensorStructure.Tensor 𝓣 cY) (TensorStructure.Tensor 𝓣 cZ)) →ₗ[R]\n TensorStructure.Tensor 𝓣 cZ"} +{"name":"TensorStructure.contr_tprod_isEmpty","declaration":"theorem TensorStructure.contr_tprod_isEmpty {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {C : Type} {P : Type} [Fintype C] [DecidableEq C] [DecidableEq P] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} [IsEmpty C] (e : (C ⊕ C) ⊕ P ≃ X) (h : TensorColor.ColorMap.ContrCond e cX) (f : (i : X) → 𝓣.ColorModule (cX i)) : (TensorStructure.contr 𝓣 e h) ((PiTensorProduct.tprod R) f) = (PiTensorProduct.tprod R) fun p => f (e (Sum.inr p))"} +{"name":"TensorColor.ColorMap.ContrAll.symm","declaration":"theorem TensorColor.ColorMap.ContrAll.symm {X : Type} {Y : Type} {𝓒 : TensorColor} {e : X ≃ Y} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} (h : TensorColor.ColorMap.ContrAll e cX cY) : TensorColor.ColorMap.ContrAll e.symm cY cX"} +{"name":"TensorStructure.pairProd","declaration":"/-- A linear map taking tensors mapped with the same index set to the product of paired tensors. -/\ndef TensorStructure.pairProd {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cX2 : TensorColor.ColorMap 𝓣.toTensorColor X} : TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 cX2) →ₗ[R]\n PiTensorProduct R fun x => TensorProduct R (𝓣.ColorModule (cX x)) (𝓣.ColorModule (cX2 x))"} +{"name":"TensorStructure.contrAll'","declaration":"/-- Given a tensor in `𝓣.Tensor cX` and a tensor in `𝓣.Tensor (𝓣.τ ∘ cX)`, the element of\n`R` formed by contracting all of their indices. -/\ndef TensorStructure.contrAll' {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} : TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 (𝓣.τ ∘ cX)) →ₗ[R] R"} +{"name":"TensorColor.ColorMap.contr","declaration":"/-- Given an equivalence `(C ⊕ C) ⊕ P ≃ X` the restriction of a color map `cX` on to `P`. -/\ndef TensorColor.ColorMap.contr {X : Type} {C : Type} {P : Type} {𝓒 : TensorColor} (e : (C ⊕ C) ⊕ P ≃ X) (cX : TensorColor.ColorMap 𝓒 X) : TensorColor.ColorMap 𝓒 P"} +{"name":"TensorStructure.contrAll_rep_apply","declaration":"theorem TensorStructure.contrAll_rep_apply {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] {G : Type} [Group G] [MulActionTensor G 𝓣] {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ ⇑e) (g : G) (x : TensorProduct R (TensorStructure.Tensor 𝓣 c) (TensorStructure.Tensor 𝓣 d)) : (TensorStructure.contrAll 𝓣 e h) ((TensorProduct.map ((TensorStructure.rep 𝓣) g) ((TensorStructure.rep 𝓣) g)) x) =\n (TensorStructure.contrAll 𝓣 e h) x"} +{"name":"TensorColor.ColorMap.contrLeft","declaration":"/-- Given an equivalence `(C ⊕ C) ⊕ P ≃ X` the restriction of a color map `cX` on `X`\nto the first `C`. -/\ndef TensorColor.ColorMap.contrLeft {X : Type} {C : Type} {P : Type} {𝓒 : TensorColor} (e : (C ⊕ C) ⊕ P ≃ X) (cX : TensorColor.ColorMap 𝓒 X) : TensorColor.ColorMap 𝓒 C"} +{"name":"TensorStructure.pairProd_tmul_tprod_tprod","declaration":"theorem TensorStructure.pairProd_tmul_tprod_tprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cX2 : TensorColor.ColorMap 𝓣.toTensorColor X} (fx : (i : X) → 𝓣.ColorModule (cX i)) (fx2 : (i : X) → 𝓣.ColorModule (cX2 i)) : (TensorStructure.pairProd 𝓣) ((PiTensorProduct.tprod R) fx ⊗ₜ[R] (PiTensorProduct.tprod R) fx2) =\n (PiTensorProduct.tprod R) fun x => fx x ⊗ₜ[R] fx2 x"} +{"name":"LinearEquiv.ofLinear_toLinearMap'","declaration":"theorem LinearEquiv.ofLinear_toLinearMap' {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_1} {R : Type} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] {module_M : Module R M} {module_M₂ : Module R₂ M₂} {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂} (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₁] M) {h₁ : LinearMap.comp f g = LinearMap.id} {h₂ : LinearMap.comp g f = LinearMap.id} : ↑(LinearEquiv.ofLinear f g h₁ h₂) = f"} +{"name":"TensorStructure.contrAll_mapIso_right_tmul","declaration":"theorem TensorStructure.contrAll_mapIso_right_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} [Fintype X] {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {c : TensorColor.ColorMap 𝓣.toTensorColor X} (e : X ≃ Y) (e' : Z ≃ Y) (h : TensorColor.ColorMap.ContrAll e c cY) (h' : TensorColor.ColorMap.MapIso e' cZ cY) (x : TensorStructure.Tensor 𝓣 c) (z : TensorStructure.Tensor 𝓣 cZ) : (TensorStructure.contrAll 𝓣 e h) (x ⊗ₜ[R] (TensorStructure.mapIso 𝓣 e' h') z) =\n (TensorStructure.contrAll 𝓣 (e.trans e'.symm) ⋯) (x ⊗ₜ[R] z)"} +{"name":"TensorColor.ColorMap.ContrCond.to_contrAll","declaration":"theorem TensorColor.ColorMap.ContrCond.to_contrAll {X : Type} {C : Type} {P : Type} {𝓒 : TensorColor} {e : (C ⊕ C) ⊕ P ≃ X} {cX : TensorColor.ColorMap 𝓒 X} (h : TensorColor.ColorMap.ContrCond e cX) : TensorColor.ColorMap.ContrAll (Equiv.refl C) (TensorColor.ColorMap.contrLeft e cX)\n (TensorColor.ColorMap.contrRight e cX)"} +{"name":"TensorStructure.contr_equivariant","declaration":"/-- The contraction of indices via `contr` is equivariant. -/\ntheorem TensorStructure.contr_equivariant {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {C : Type} {P : Type} [Fintype C] [DecidableEq C] [DecidableEq P] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {G : Type} [Group G] [MulActionTensor G 𝓣] (e : (C ⊕ C) ⊕ P ≃ X) (h : TensorColor.ColorMap.ContrCond e cX) (g : G) (x : TensorStructure.Tensor 𝓣 cX) : (TensorStructure.contr 𝓣 e h) (((TensorStructure.rep 𝓣) g) x) =\n ((TensorStructure.rep 𝓣) g) ((TensorStructure.contr 𝓣 e h) x)"} +{"name":"TensorColor.ColorMap.ContrCond","declaration":"/-- Given an equivalence `(C ⊕ C) ⊕ P ≃ X` the condition on `cX` so that we contract\nthe indices of the `C`'s under this equivalence. -/\ndef TensorColor.ColorMap.ContrCond {X : Type} {C : Type} {P : Type} {𝓒 : TensorColor} (e : (C ⊕ C) ⊕ P ≃ X) (cX : TensorColor.ColorMap 𝓒 X) : Prop"} +{"name":"TensorStructure.contrAll","declaration":"/-- The contraction of all the indices of two tensors with dual colors. -/\ndef TensorStructure.contrAll {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (h : TensorColor.ColorMap.ContrAll e cX cY) : TensorProduct R (TensorStructure.Tensor 𝓣 cX) (TensorStructure.Tensor 𝓣 cY) →ₗ[R] R"} +{"name":"TensorStructure.contrAll'_tmul_tprod_tprod","declaration":"theorem TensorStructure.contrAll'_tmul_tprod_tprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} (fx : (i : X) → 𝓣.ColorModule (cX i)) (fy : (i : X) → 𝓣.ColorModule (𝓣.τ (cX i))) : (TensorStructure.contrAll' 𝓣) ((PiTensorProduct.tprod R) fx ⊗ₜ[R] (PiTensorProduct.tprod R) fy) =\n (PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R))\n ((PiTensorProduct.tprod R) fun x => (𝓣.contrDual (cX x)) (fx x ⊗ₜ[R] fy x))"} +{"name":"TensorStructure.contrDualLeft","declaration":"/-- The contraction of a vector in `𝓣.ColorModule ν` with a vector in\n`𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η` to form a vector in `𝓣.ColorModule η`. -/\ndef TensorStructure.contrDualLeft {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {ν : 𝓣.Color} {η : 𝓣.Color} : TensorProduct R (𝓣.ColorModule ν) (TensorProduct R (𝓣.ColorModule (𝓣.τ ν)) (𝓣.ColorModule η)) →ₗ[R] 𝓣.ColorModule η"} +{"name":"TensorStructure.contrAll'_mapIso_tmul","declaration":"theorem TensorStructure.contrAll'_mapIso_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] [Fintype Y] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} (e : X ≃ Y) (h : TensorColor.ColorMap.MapIso e cX cY) (x : TensorStructure.Tensor 𝓣 cX) (y : TensorStructure.Tensor 𝓣 (𝓣.τ ∘ cY)) : (TensorStructure.contrAll' 𝓣) ((TensorStructure.mapIso 𝓣 e h) x ⊗ₜ[R] y) =\n (TensorStructure.contrAll' 𝓣) (x ⊗ₜ[R] (TensorStructure.mapIso 𝓣 e.symm ⋯) y)"} +{"name":"TensorStructure.contrAll_rep_tmul","declaration":"theorem TensorStructure.contrAll_rep_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] {G : Type} [Group G] [MulActionTensor G 𝓣] {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ ⇑e) (g : G) (x : TensorStructure.Tensor 𝓣 c) (y : TensorStructure.Tensor 𝓣 d) : (TensorStructure.contrAll 𝓣 e h) (((TensorStructure.rep 𝓣) g) x ⊗ₜ[R] ((TensorStructure.rep 𝓣) g) y) =\n (TensorStructure.contrAll 𝓣 e h) (x ⊗ₜ[R] y)"} +{"name":"TensorColor.ColorMap.ContrAll.trans_mapIso","declaration":"theorem TensorColor.ColorMap.ContrAll.trans_mapIso {X : Type} {Y : Type} {Z : Type} {𝓒 : TensorColor} {cX : TensorColor.ColorMap 𝓒 X} {cY : TensorColor.ColorMap 𝓒 Y} {cZ : TensorColor.ColorMap 𝓒 Z} {e : X ≃ Y} {e' : Z ≃ Y} (h : TensorColor.ColorMap.ContrAll e cX cY) (h' : TensorColor.ColorMap.MapIso e' cZ cY) : TensorColor.ColorMap.ContrAll (e.trans e'.symm) cX cZ"} +{"name":"TensorStructure.contrAll_rep","declaration":"theorem TensorStructure.contrAll_rep {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {G : Type} [Group G] [MulActionTensor G 𝓣] (e : X ≃ Y) (h : TensorColor.ColorMap.ContrAll e cX cY) (g : G) : TensorStructure.contrAll 𝓣 e h ∘ₗ TensorProduct.map ((TensorStructure.rep 𝓣) g) ((TensorStructure.rep 𝓣) g) =\n TensorStructure.contrAll 𝓣 e h"} +{"name":"TensorStructure.contr_tprod","declaration":"theorem TensorStructure.contr_tprod {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {C : Type} {P : Type} [Fintype C] [DecidableEq C] [DecidableEq P] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} (e : (C ⊕ C) ⊕ P ≃ X) (h : TensorColor.ColorMap.ContrCond e cX) (f : (i : X) → 𝓣.ColorModule (cX i)) : (TensorStructure.contr 𝓣 e h) ((PiTensorProduct.tprod R) f) =\n (TensorStructure.contrAll 𝓣 (Equiv.refl C) ⋯)\n (((PiTensorProduct.tprod R) fun i => f (e (Sum.inl (Sum.inl i)))) ⊗ₜ[R]\n (PiTensorProduct.tprod R) fun i => f (e (Sum.inl (Sum.inr i)))) •\n (PiTensorProduct.tprod R) fun p => f (e (Sum.inr p))"} +{"name":"TensorStructure.contrAll'_isEmpty_tmul","declaration":"theorem TensorStructure.contrAll'_isEmpty_tmul {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} [IsEmpty X] (x : TensorStructure.Tensor 𝓣 cX) (y : TensorStructure.Tensor 𝓣 (𝓣.τ ∘ cX)) : (TensorStructure.contrAll' 𝓣) (x ⊗ₜ[R] y) = (TensorStructure.isEmptyEquiv 𝓣) x * (TensorStructure.isEmptyEquiv 𝓣) y"} +{"name":"TensorStructure.mkPiAlgebra_equiv","declaration":"theorem TensorStructure.mkPiAlgebra_equiv {R : Type} [CommSemiring R] {X : Type} {Y : Type} [Fintype X] [Fintype Y] (e : X ≃ Y) : PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R X R) =\n PiTensorProduct.lift (MultilinearMap.mkPiAlgebra R Y R) ∘ₗ ↑(PiTensorProduct.reindex R (fun i => R) e)"} +{"name":"TensorColor.ColorMap.contrRight","declaration":"/-- Given an equivalence `(C ⊕ C) ⊕ P ≃ X` the restriction of a color map `cX` on `X`\nto the second `C`. -/\ndef TensorColor.ColorMap.contrRight {X : Type} {C : Type} {P : Type} {𝓒 : TensorColor} (e : (C ⊕ C) ⊕ P ≃ X) (cX : TensorColor.ColorMap 𝓒 X) : TensorColor.ColorMap 𝓒 C"} +{"name":"TensorStructure.contr","declaration":"/-- Contraction of indices based on an equivalence `(C ⊕ C) ⊕ P ≃ X`. The indices\nin `C` are contracted pair-wise, whilst the indices in `P` are preserved. -/\ndef TensorStructure.contr {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {C : Type} {P : Type} [Fintype C] [DecidableEq C] [DecidableEq P] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} (e : (C ⊕ C) ⊕ P ≃ X) (h : TensorColor.ColorMap.ContrCond e cX) : TensorStructure.Tensor 𝓣 cX →ₗ[R] TensorStructure.Tensor 𝓣 (TensorColor.ColorMap.contr e cX)"} +{"name":"TensorStructure.contrAll_mapIso_left","declaration":"theorem TensorStructure.contrAll_mapIso_left {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} [Fintype X] [Fintype Z] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {e : X ≃ Y} {e' : Z ≃ X} (h : TensorColor.ColorMap.ContrAll e cX cY) (h' : TensorColor.ColorMap.MapIso e' cZ cX) : TensorStructure.contrAll 𝓣 e h ∘ₗ\n ↑(TensorProduct.congr (TensorStructure.mapIso 𝓣 e' h') (LinearEquiv.refl R (TensorStructure.Tensor 𝓣 cY))) =\n TensorStructure.contrAll 𝓣 (e'.trans e) ⋯"} +{"name":"TensorStructure.contrDualMid","declaration":"/-- The contraction of a vector in `𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule ν` with a vector in\n`𝓣.ColorModule (𝓣.τ ν) ⊗[R] 𝓣.ColorModule η` to form a vector in\n`𝓣.ColorModule μ ⊗[R] 𝓣.ColorModule η`. -/\ndef TensorStructure.contrDualMid {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {μ : 𝓣.Color} {ν : 𝓣.Color} {η : 𝓣.Color} : TensorProduct R (TensorProduct R (𝓣.ColorModule μ) (𝓣.ColorModule ν))\n (TensorProduct R (𝓣.ColorModule (𝓣.τ ν)) (𝓣.ColorModule η)) →ₗ[R]\n TensorProduct R (𝓣.ColorModule μ) (𝓣.ColorModule η)"} +{"name":"TensorStructure.contrAllMid","declaration":"/-- The linear map from `(𝓣.Tensor cW ⊗[R] 𝓣.Tensor cX) ⊗[R] (𝓣.Tensor cY ⊗[R] 𝓣.Tensor cZ)`\nto `𝓣.Tensor cW ⊗[R] 𝓣.Tensor cZ` obtained by contracting all indices of the tensors\nin the middle. -/\ndef TensorStructure.contrAllMid {R : Type} [CommSemiring R] (𝓣 : TensorStructure R) {X : Type} {Y : Type} {Z : Type} {W : Type} [Fintype X] {cX : TensorColor.ColorMap 𝓣.toTensorColor X} {cY : TensorColor.ColorMap 𝓣.toTensorColor Y} {cZ : TensorColor.ColorMap 𝓣.toTensorColor Z} {cW : TensorColor.ColorMap 𝓣.toTensorColor W} (e : X ≃ Y) (h : TensorColor.ColorMap.ContrAll e cX cY) : TensorProduct R (TensorProduct R (TensorStructure.Tensor 𝓣 cW) (TensorStructure.Tensor 𝓣 cX))\n (TensorProduct R (TensorStructure.Tensor 𝓣 cY) (TensorStructure.Tensor 𝓣 cZ)) →ₗ[R]\n TensorProduct R (TensorStructure.Tensor 𝓣 cW) (TensorStructure.Tensor 𝓣 cZ)"} diff --git a/hep-declarations/HepLean.Tensors.EinsteinNotation.Basic.jsonl b/hep-declarations/HepLean.Tensors.EinsteinNotation.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f5595040dc501f7b18a355a8a6cc50e046dcd88c --- /dev/null +++ b/hep-declarations/HepLean.Tensors.EinsteinNotation.Basic.jsonl @@ -0,0 +1,11 @@ +{"name":"einsteinTensor","declaration":"/-- The `TensorStructure` associated with `n`-dimensional tensors. -/\ndef einsteinTensor (R : Type) [CommSemiring R] (n : ℕ) : TensorStructure R"} +{"name":"einsteinTensor.toVec","declaration":"/-- A vector from an Einstein tensor with one index. -/\ndef einsteinTensor.toVec {R : Type} [CommSemiring R] {n : ℕ} : TensorStructure.Tensor (einsteinTensor R n) ![()] ≃ₗ[R] Fin n → R"} +{"name":"einsteinTensorColor","declaration":"/-- Einstein tensors have only one color, corresponding to a `down` index. . -/\ndef einsteinTensorColor : TensorColor"} +{"name":"instFintypeColorEinsteinTensorColor","declaration":"instance instFintypeColorEinsteinTensorColor : Fintype einsteinTensorColor.Color"} +{"name":"einsteinTensor.toMatrix","declaration":"/-- A matrix from an Einstein tensor with two indices. -/\ndef einsteinTensor.toMatrix {R : Type} [CommSemiring R] {n : ℕ} : TensorStructure.Tensor (einsteinTensor R n) ![(), ()] ≃ₗ[R] Matrix (Fin n) (Fin n) R"} +{"name":"Fintype.linearCombination","declaration":"def Fintype.linearCombination {α : Type u_1} {M : Type u_2} (R : Type u_3) [Fintype α] [Semiring R] [AddCommMonoid M] [Module R M] (S : Type u_4) [Semiring S] [Module S M] [SMulCommClass R S M] : (α → M) →ₗ[S] (α → R) →ₗ[R] M"} +{"name":"einsteinTensor.instOfNatColorEinsteinTensorColorOfNatNat","declaration":"instance einsteinTensor.instOfNatColorEinsteinTensorColorOfNatNat : OfNat einsteinTensorColor.Color 0"} +{"name":"Fintype.linearCombination_apply","declaration":"theorem Fintype.linearCombination_apply {α : Type u_1} {M : Type u_2} (R : Type u_3) [Fintype α] [Semiring R] [AddCommMonoid M] [Module R M] {S : Type u_4} [Semiring S] [Module S M] [SMulCommClass R S M] (v : α → M) (f : α → R) : ((Fintype.linearCombination R S) v) f = Finset.sum Finset.univ fun i => f i • v i"} +{"name":"instDecidableEqColorEinsteinTensorColor","declaration":"instance instDecidableEqColorEinsteinTensorColor : DecidableEq einsteinTensorColor.Color"} +{"name":"einsteinTensor.ofNat_inst_eq","declaration":"theorem einsteinTensor.ofNat_inst_eq {R : Type} [CommSemiring R] {n : ℕ} : einsteinTensor.instOfNatColorOfNatNat = einsteinTensor.instOfNatColorEinsteinTensorColorOfNatNat"} +{"name":"einsteinTensor.instOfNatColorOfNatNat","declaration":"instance einsteinTensor.instOfNatColorOfNatNat {R : Type} [CommSemiring R] {n : ℕ} : OfNat (einsteinTensor R n).Color 0"} diff --git a/hep-declarations/HepLean.Tensors.EinsteinNotation.IndexNotation.jsonl b/hep-declarations/HepLean.Tensors.EinsteinNotation.IndexNotation.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..bc82d55f011100b95231c87a77688022e04548e9 --- /dev/null +++ b/hep-declarations/HepLean.Tensors.EinsteinNotation.IndexNotation.jsonl @@ -0,0 +1,15 @@ +{"name":"instIndexNotationColorEinsteinTensorColor","declaration":"instance instIndexNotationColorEinsteinTensorColor : IndexNotation einsteinTensorColor.Color"} +{"name":"einsteinTensor.«term_|_»","declaration":"/-- Notation for the construction of a tensor index from a tensor and a string.\nConditions are checked automatically. -/\ndef einsteinTensor.«term_|_» : Lean.TrailingParserDescr"} +{"name":"einsteinTensor.mem_fin_list","declaration":"theorem einsteinTensor.mem_fin_list (n : ℕ) (i : Fin n) : i ∈ Fin.list n"} +{"name":"einsteinTensor.tacticProdTactic","declaration":"/-- A tactics used to prove `colorPropBool` for real Lorentz tensors. -/\ndef einsteinTensor.tacticProdTactic : Lean.ParserDescr"} +{"name":"einsteinTensor.instDecidableMemFinListInstMembershipListList","declaration":"instance einsteinTensor.instDecidableMemFinListInstMembershipListList (n : ℕ) (i : Fin n) : Decidable (i ∈ Fin.list n)"} +{"name":"einsteinTensor.«term_⊗ᵀ_»","declaration":"/-- The product of Real Lorentz tensors. Conditions on indices are checked automatically. -/\ndef einsteinTensor.«term_⊗ᵀ_» : Lean.TrailingParserDescr"} +{"name":"einsteinTensor.indexNotation_eq_color","declaration":"theorem einsteinTensor.indexNotation_eq_color {R : Type} [CommSemiring R] {n : ℕ} : einsteinTensor.instIndexNotationColor = instIndexNotationColorEinsteinTensorColor"} +{"name":"einsteinTensor.toTensorColor_eq","declaration":"theorem einsteinTensor.toTensorColor_eq {R : Type} [CommSemiring R] {n : ℕ} : (einsteinTensor R n).toTensorColor = einsteinTensorColor"} +{"name":"einsteinTensor.fromIndexStringColor","declaration":"/-- The construction of a tensor index from a tensor and a string satisfying conditions\nwhich can be automatically checked. This is a modified version of\n`TensorStructure.TensorIndex.mkDualMap` specific to real Lorentz tensors. -/\ndef einsteinTensor.fromIndexStringColor {n : ℕ} {m : ℕ} {R : Type} [CommSemiring R] {cn : Fin n → einsteinTensorColor.Color} (T : TensorStructure.Tensor (einsteinTensor R m) cn) (s : String) (hs : IndexNotation.listCharIsIndexString einsteinTensorColor.Color (String.toList s) = true) (hn : n = IndexNotation.IndexList.length (IndexNotation.IndexString.toIndexList' s hs)) (hD : IndexNotation.IndexList.OnlyUniqueDuals (IndexNotation.IndexString.toIndexList' s hs)) (hC : IndexNotation.IndexList.ColorCond.bool (IndexNotation.IndexString.toIndexList' s hs) = true) (hd : TensorColor.ColorMap.DualMap.boolFin' (IndexNotation.IndexList.colorMap (IndexNotation.IndexString.toIndexList' s hs))\n (cn ∘ Fin.cast ⋯) =\n true) : TensorStructure.TensorIndex (einsteinTensor R m)"} +{"name":"einsteinTensor.instIndexNotationColor","declaration":"instance einsteinTensor.instIndexNotationColor {R : Type} [CommSemiring R] {n : ℕ} : IndexNotation (einsteinTensor R n).Color"} +{"name":"einsteinTensor.decidableEq_eq_color","declaration":"theorem einsteinTensor.decidableEq_eq_color {R : Type} [CommSemiring R] {n : ℕ} : einsteinTensor.instDecidableEqColor = instDecidableEqColorEinsteinTensorColor"} +{"name":"einsteinTensor.instDecidableEqColor","declaration":"instance einsteinTensor.instDecidableEqColor {R : Type} [CommSemiring R] {n : ℕ} : DecidableEq (einsteinTensor R n).Color"} +{"name":"einsteinTensor.fromIndexStringColor_indexList","declaration":"theorem einsteinTensor.fromIndexStringColor_indexList {n : ℕ} {m : ℕ} {R : Type} [CommSemiring R] {cn : Fin n → einsteinTensorColor.Color} (T : TensorStructure.Tensor (einsteinTensor R m) cn) (s : String) (hs : IndexNotation.listCharIsIndexString einsteinTensorColor.Color (String.toList s) = true) (hn : n = IndexNotation.IndexList.length (IndexNotation.IndexString.toIndexList' s hs)) (hD : IndexNotation.IndexList.OnlyUniqueDuals (IndexNotation.IndexString.toIndexList' s hs)) (hC : IndexNotation.IndexList.ColorCond.bool (IndexNotation.IndexString.toIndexList' s hs) = true) (hd : TensorColor.ColorMap.DualMap.boolFin' (IndexNotation.IndexList.colorMap (IndexNotation.IndexString.toIndexList' s hs))\n (cn ∘ Fin.cast ⋯) =\n true) : (einsteinTensor.fromIndexStringColor T s hs hn hD hC hd).toIndexList = IndexNotation.IndexString.toIndexList' s hs"} +{"name":"einsteinTensor.einsteinTensor_color","declaration":"theorem einsteinTensor.einsteinTensor_color {R : Type} [CommSemiring R] {n : ℕ} : (einsteinTensor R n).Color = einsteinTensorColor.Color"} +{"name":"einsteinTensor.tacticDualMapTactic","declaration":"/-- A tactic used to prove `boolFin` for real Lornetz tensors. -/\ndef einsteinTensor.tacticDualMapTactic : Lean.ParserDescr"} diff --git a/hep-declarations/HepLean.Tensors.EinsteinNotation.Lemmas.jsonl b/hep-declarations/HepLean.Tensors.EinsteinNotation.Lemmas.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/hep-declarations/HepLean.jsonl b/hep-declarations/HepLean.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/htpi-declarations/HTPILib.Chap3.jsonl b/htpi-declarations/HTPILib.Chap3.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1760d377cbd3dc48f520f6143c073f7387a2f75c --- /dev/null +++ b/htpi-declarations/HTPILib.Chap3.jsonl @@ -0,0 +1,51 @@ +{"name":"HTPI.Exercises.Exercise_3_2_1b","declaration":"theorem HTPI.Exercises.Exercise_3_2_1b (P : Prop) (Q : Prop) (R : Prop) (h1 : ¬R → P → ¬Q) : P → Q → R"} +{"name":"HTPI.Like_Example_3_4_1","declaration":"theorem HTPI.Like_Example_3_4_1 (U : Type) (A : Set U) (B : Set U) (C : Set U) (D : Set U) (h1 : A ⊆ B) (h2 : ¬∃ c, c ∈ C ∩ D) : A ∩ C ⊆ B \\ D"} +{"name":"HTPI.Theorem_3_4_7","declaration":"theorem HTPI.Theorem_3_4_7 (n : ℤ) : 6 ∣ n ↔ 2 ∣ n ∧ 3 ∣ n"} +{"name":"HTPI.Exercises.Exercise_3_6_6b","declaration":"theorem HTPI.Exercises.Exercise_3_6_6b (U : Type) : ∃! A, ∀ (B : Set U), A ∪ B = A"} +{"name":"HTPI.odd","declaration":"def HTPI.odd (n : ℤ) : Prop"} +{"name":"HTPI.Example_3_6_4","declaration":"theorem HTPI.Example_3_6_4 (U : Type) (A : Set U) (B : Set U) (C : Set U) (h1 : ∃ x, x ∈ A ∩ B) (h2 : ∃ x, x ∈ A ∩ C) (h3 : ∃! x, x ∈ A) : ∃ x, x ∈ B ∩ C"} +{"name":"HTPI.Exercises.Exercise_3_6_8a","declaration":"theorem HTPI.Exercises.Exercise_3_6_8a (U : Type) (A : Set U) : ∃! B, ∀ (C : Set U), C \\ A = C ∩ B"} +{"name":"HTPI.Exercises.Exercise_3_3_8","declaration":"theorem HTPI.Exercises.Exercise_3_3_8 (U : Type) (F : Set (Set U)) (A : Set U) (h1 : A ∈ F) : A ⊆ ⋃₀ F"} +{"name":"HTPI.Exercises.Exercise_3_5_9","declaration":"theorem HTPI.Exercises.Exercise_3_5_9 (U : Type) (A : Set U) (B : Set U) (h1 : 𝒫(A ∪ B) = 𝒫 A ∪ 𝒫 B) : A ⊆ B ∨ B ⊆ A"} +{"name":"HTPI.Example_3_6_2","declaration":"theorem HTPI.Example_3_6_2 (U : Type) : ∃! A, ∀ (B : Set U), A ∪ B = B"} +{"name":"HTPI.Exercises.Exercise_3_6_7b","declaration":"theorem HTPI.Exercises.Exercise_3_6_7b (U : Type) : ∃! A, ∀ (B : Set U), A ∩ B = A"} +{"name":"HTPI.Exercises.Exercise_3_5_18","declaration":"theorem HTPI.Exercises.Exercise_3_5_18 (U : Type) (F : Set (Set U)) (G : Set (Set U)) (H : Set (Set U)) (h1 : ∀ A ∈ F, ∀ B ∈ G, A ∪ B ∈ H) : ⋂₀ H ⊆ ⋂₀ F ∪ ⋂₀ G"} +{"name":"HTPI.Example_3_5_2","declaration":"theorem HTPI.Example_3_5_2 (U : Type) (A : Set U) (B : Set U) (C : Set U) : A \\ (B \\ C) ⊆ A \\ B ∪ C"} +{"name":"HTPI.Like_Example_3_2_5","declaration":"theorem HTPI.Like_Example_3_2_5 (U : Type) (A : Set U) (B : Set U) (C : Set U) (a : U) (h1 : a ∈ A) (h2 : a ∉ A \\ B) (h3 : a ∈ B → a ∈ C) : a ∈ C"} +{"name":"HTPI.Exercises.Exercise_3_4_19","declaration":"theorem HTPI.Exercises.Exercise_3_4_19 (U : Type) (F : Set (Set U)) (G : Set (Set U)) : ⋃₀ F ∩ ⋃₀ G ⊆ ⋃₀ (F ∩ G) ↔ ∀ (A B : Set U), A ∈ F → B ∈ G → A ∩ B ⊆ ⋃₀ (F ∩ G)"} +{"name":"HTPI.Exercises.Exercise_3_2_2a","declaration":"theorem HTPI.Exercises.Exercise_3_2_2a (P : Prop) (Q : Prop) (R : Prop) (h1 : P → Q) (h2 : R → ¬Q) : P → ¬R"} +{"name":"HTPI.Exercises.Exercise_3_4_18a","declaration":"theorem HTPI.Exercises.Exercise_3_4_18a (U : Type) (F : Set (Set U)) (G : Set (Set U)) : ⋃₀ (F ∩ G) ⊆ ⋃₀ F ∩ ⋃₀ G"} +{"name":"HTPI.Exercises.Exercise_3_2_2b","declaration":"theorem HTPI.Exercises.Exercise_3_2_2b (P : Prop) (Q : Prop) (h1 : P) : Q → ¬(Q → ¬P)"} +{"name":"HTPI.Exercises.Exercise_3_3_16","declaration":"theorem HTPI.Exercises.Exercise_3_3_16 (U : Type) (B : Set U) (F : Set (Set U)) : F ⊆ 𝒫 B → ⋃₀ F ⊆ B"} +{"name":"HTPI.union_comm","declaration":"theorem HTPI.union_comm {U : Type} (X : Set U) (Y : Set U) : X ∪ Y = Y ∪ X"} +{"name":"HTPI.Exercises.Exercise_3_4_7","declaration":"theorem HTPI.Exercises.Exercise_3_4_7 (U : Type) (A : Set U) (B : Set U) : 𝒫(A ∩ B) = 𝒫 A ∩ 𝒫 B"} +{"name":"HTPI.Exercises.Exercise_3_5_8","declaration":"theorem HTPI.Exercises.Exercise_3_5_8 (U : Type) (A : Set U) (B : Set U) : 𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)"} +{"name":"HTPI.Exercises.Exercise_3_5_7","declaration":"theorem HTPI.Exercises.Exercise_3_5_7 (U : Type) (A : Set U) (B : Set U) (C : Set U) : A ∪ C ⊆ B ∪ C ↔ A \\ C ⊆ B \\ C"} +{"name":"HTPI.Exercises.Exercise_3_4_17","declaration":"theorem HTPI.Exercises.Exercise_3_4_17 (U : Type) (A : Set U) : A = ⋃₀ (𝒫 A)"} +{"name":"HTPI.Exercises.Exercise_3_3_10","declaration":"theorem HTPI.Exercises.Exercise_3_3_10 (U : Type) (B : Set U) (F : Set (Set U)) (h1 : ∀ A ∈ F, B ⊆ A) : B ⊆ ⋂₀ F"} +{"name":"HTPI.Exercises.Like_Exercise_3_7_5","declaration":"theorem HTPI.Exercises.Like_Exercise_3_7_5 (U : Type) (F : Set (Set U)) (h1 : 𝒫⋃₀ F ⊆ ⋃₀ {x | ∃ A ∈ F, 𝒫 A = x}) : ∃ A ∈ F, ∀ B ∈ F, B ⊆ A"} +{"name":"HTPI.Example_3_3_5","declaration":"theorem HTPI.Example_3_3_5 (U : Type) (B : Set U) (F : Set (Set U)) : ⋃₀ F ⊆ B → F ⊆ 𝒫 B"} +{"name":"HTPI.Exercises.Exercise_3_3_17","declaration":"theorem HTPI.Exercises.Exercise_3_3_17 (U : Type) (F : Set (Set U)) (G : Set (Set U)) (h1 : ∀ A ∈ F, ∀ B ∈ G, A ⊆ B) : ⋃₀ F ⊆ ⋂₀ G"} +{"name":"HTPI.Exercises.Exercise_3_3_13","declaration":"theorem HTPI.Exercises.Exercise_3_3_13 (U : Type) (F : Set (Set U)) (G : Set (Set U)) : F ⊆ G → ⋂₀ G ⊆ ⋂₀ F"} +{"name":"HTPI.even","declaration":"def HTPI.even (n : ℤ) : Prop"} +{"name":"HTPI.Example_3_4_5","declaration":"theorem HTPI.Example_3_4_5 (U : Type) (A : Set U) (B : Set U) (C : Set U) : A ∩ (B \\ C) = (A ∩ B) \\ C"} +{"name":"HTPI.Exercises.Exercise_3_3_1","declaration":"theorem HTPI.Exercises.Exercise_3_3_1 (U : Type) (P : HTPI.Pred U) (Q : HTPI.Pred U) (h1 : ∃ x, P x → Q x) : (∀ (x : U), P x) → ∃ x, Q x"} +{"name":"HTPI.Exercises.Exercise_3_5_24a","declaration":"theorem HTPI.Exercises.Exercise_3_5_24a (U : Type) (A : Set U) (B : Set U) (C : Set U) : (A ∪ B) ∆ C ⊆ A ∆ C ∪ B ∆ C"} +{"name":"HTPI.Exercises.Exercise_3_3_18a","declaration":"theorem HTPI.Exercises.Exercise_3_3_18a (a : ℤ) (b : ℤ) (c : ℤ) (h1 : a ∣ b) (h2 : a ∣ c) : a ∣ b + c"} +{"name":"HTPI.Theorem_3_3_7","declaration":"theorem HTPI.Theorem_3_3_7 (a : ℤ) (b : ℤ) (c : ℤ) : a ∣ b → b ∣ c → a ∣ c"} +{"name":"HTPI.Exercises.Exercise_3_5_5","declaration":"theorem HTPI.Exercises.Exercise_3_5_5 (U : Type) (A : Set U) (B : Set U) (C : Set U) (h1 : A ∩ C ⊆ B ∩ C) (h2 : A ∪ C ⊆ B ∪ C) : A ⊆ B"} +{"name":"HTPI.Exercises.Exercise_3_6_10","declaration":"theorem HTPI.Exercises.Exercise_3_6_10 (U : Type) (A : Set U) (h1 : ∀ (F : Set (Set U)), ⋃₀ F = A → A ∈ F) : ∃! x, x ∈ A"} +{"name":"HTPI.Exercises.Exercise_3_4_4","declaration":"theorem HTPI.Exercises.Exercise_3_4_4 (U : Type) (A : Set U) (B : Set U) (C : Set U) (h1 : A ⊆ B) (h2 : ¬A ⊆ C) : ¬B ⊆ C"} +{"name":"HTPI.Exercises.Exercise_3_5_2","declaration":"theorem HTPI.Exercises.Exercise_3_5_2 (U : Type) (A : Set U) (B : Set U) (C : Set U) : (A ∪ B) \\ C ⊆ A ∪ B \\ C"} +{"name":"HTPI.Example_3_2_4_v3","declaration":"theorem HTPI.Example_3_2_4_v3 (P : Prop) (Q : Prop) (R : Prop) (h : P → Q → R) : ¬R → P → ¬Q"} +{"name":"HTPI.empty_union","declaration":"theorem HTPI.empty_union {U : Type} (B : Set U) : ∅ ∪ B = B"} +{"name":"HTPI.Example_3_5_4","declaration":"theorem HTPI.Example_3_5_4 (x : ℝ) (h1 : x ≤ x ^ 2) : x ≤ 0 ∨ 1 ≤ x"} +{"name":"HTPI.Exercises.Exercise_3_4_15","declaration":"theorem HTPI.Exercises.Exercise_3_4_15 (U : Type) (B : Set U) (F : Set (Set U)) : ⋃₀ {X | ∃ A ∈ F, X = A \\ B} ⊆ ⋃₀ (F \\ 𝒫 B)"} +{"name":"HTPI.Exercises.Exercise_3_4_10","declaration":"theorem HTPI.Exercises.Exercise_3_4_10 (x : ℤ) (y : ℤ) (h1 : HTPI.odd x) (h2 : HTPI.odd y) : HTPI.even (x - y)"} +{"name":"HTPI.Exercises.Exercise_3_2_1a","declaration":"theorem HTPI.Exercises.Exercise_3_2_1a (P : Prop) (Q : Prop) (R : Prop) (h1 : P → Q) (h2 : Q → R) : P → R"} +{"name":"HTPI.Exercises.Exercise_3_3_9","declaration":"theorem HTPI.Exercises.Exercise_3_3_9 (U : Type) (F : Set (Set U)) (A : Set U) (h1 : A ∈ F) : ⋂₀ F ⊆ A"} +{"name":"HTPI.Exercises.Exercise_3_4_27a","declaration":"theorem HTPI.Exercises.Exercise_3_4_27a (n : ℤ) : 15 ∣ n ↔ 3 ∣ n ∧ 5 ∣ n"} +{"name":"HTPI.Example_3_2_4_v2","declaration":"theorem HTPI.Example_3_2_4_v2 (P : Prop) (Q : Prop) (R : Prop) (h : P → Q → R) : ¬R → P → ¬Q"} +{"name":"HTPI.Exercises.Exercise_3_4_6","declaration":"theorem HTPI.Exercises.Exercise_3_4_6 (U : Type) (A : Set U) (B : Set U) (C : Set U) : A \\ (B ∩ C) = A \\ B ∪ A \\ C"} +{"name":"HTPI.Exercises.Exercise_3_4_2","declaration":"theorem HTPI.Exercises.Exercise_3_4_2 (U : Type) (A : Set U) (B : Set U) (C : Set U) (h1 : A ⊆ B) (h2 : A ⊆ C) : A ⊆ B ∩ C"} +{"name":"HTPI.Exercises.Exercise_3_5_17b","declaration":"theorem HTPI.Exercises.Exercise_3_5_17b (U : Type) (F : Set (Set U)) (B : Set U) : B ∪ ⋂₀ F = {x | ∀ A ∈ F, x ∈ B ∪ A}"} diff --git a/htpi-declarations/HTPILib.Chap4.jsonl b/htpi-declarations/HTPILib.Chap4.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..783a57bf75ad130c07c31481932052f0b0f7d40f --- /dev/null +++ b/htpi-declarations/HTPILib.Chap4.jsonl @@ -0,0 +1,88 @@ +{"name":"HTPI.Theorem_4_5_4_part_3","declaration":"theorem HTPI.Theorem_4_5_4_part_3 {A : Type} (R : HTPI.BinRel A) (h : HTPI.equiv_rel R) (X : Set A) : X ∈ HTPI.mod A R → ¬HTPI.empty X"} +{"name":"HTPI.Exercises.Lemma_4_5_7_symm","declaration":"theorem HTPI.Exercises.Lemma_4_5_7_symm {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : HTPI.symmetric (HTPI.EqRelFromPart F)"} +{"name":"HTPI.inv_def","declaration":"theorem HTPI.inv_def {A : Type} {B : Type} (R : Set (A × B)) (a : A) (b : B) : (b, a) ∈ HTPI.inv R ↔ (a, b) ∈ R"} +{"name":"HTPI.empty","declaration":"def HTPI.empty {A : Type} (X : Set A) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_2_9b","declaration":"theorem HTPI.Exercises.Exercise_4_2_9b {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) : HTPI.Ran R ⊆ HTPI.Dom S → HTPI.Dom (HTPI.comp S R) = HTPI.Dom R"} +{"name":"HTPI.equiv_rel","declaration":"def HTPI.equiv_rel {A : Type} (R : HTPI.BinRel A) : Prop"} +{"name":"HTPI.Exercises.overlap_implies_equal","declaration":"theorem HTPI.Exercises.overlap_implies_equal {A : Type} (F : Set (Set A)) (h : HTPI.partition F) (X : Set A) : X ∈ F → ∀ Y ∈ F, ∀ x ∈ X, x ∈ Y → X = Y"} +{"name":"HTPI.transitive","declaration":"def HTPI.transitive {A : Type} (R : HTPI.BinRel A) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_5_20b","declaration":"theorem HTPI.Exercises.Exercise_4_5_20b {A : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel A) (h1 : HTPI.equiv_rel R) (h2 : HTPI.equiv_rel S) (x : A) : HTPI.equivClass (HTPI.Exercises.conj R S) x = HTPI.equivClass R x ∩ HTPI.equivClass S x"} +{"name":"HTPI.Theorem_4_5_6","declaration":"theorem HTPI.Theorem_4_5_6 {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : ∃ R, HTPI.equiv_rel R ∧ HTPI.mod A R = F"} +{"name":"HTPI.Exercises.Theorem_4_3_4_1","declaration":"theorem HTPI.Exercises.Theorem_4_3_4_1 {A : Type} (R : HTPI.BinRel A) : HTPI.reflexive R ↔ {(x, y) | x = y} ⊆ HTPI.extension R"} +{"name":"HTPI.Lemma_4_5_8","declaration":"theorem HTPI.Lemma_4_5_8 {A : Type} (F : Set (Set A)) (h : HTPI.partition F) (X : Set A) : X ∈ F → ∀ x ∈ X, HTPI.equivClass (HTPI.EqRelFromPart F) x = X"} +{"name":"HTPI.Exercises.Theorem_4_4_11","declaration":"theorem HTPI.Exercises.Theorem_4_4_11 {A : Type} (F : Set (Set A)) : HTPI.lub (HTPI.sub A) (⋃₀ F) F"} +{"name":"HTPI.upperBd","declaration":"def HTPI.upperBd {A : Type} (R : HTPI.BinRel A) (a : A) (B : Set A) : Prop"} +{"name":"HTPI.overlap_implies_equal","declaration":"theorem HTPI.overlap_implies_equal {A : Type} (F : Set (Set A)) (h : HTPI.partition F) (X : Set A) : X ∈ F → ∀ Y ∈ F, ∀ x ∈ X, x ∈ Y → X = Y"} +{"name":"HTPI.symmetric","declaration":"def HTPI.symmetric {A : Type} (R : HTPI.BinRel A) : Prop"} +{"name":"HTPI.Exercises.conj","declaration":"def HTPI.Exercises.conj {A : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel A) (x : A) (y : A) : Prop"} +{"name":"HTPI.Lemma_4_5_7_trans","declaration":"theorem HTPI.Lemma_4_5_7_trans {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : HTPI.transitive (HTPI.EqRelFromPart F)"} +{"name":"HTPI.Exercises.Exercise_4_4_9_part","declaration":"theorem HTPI.Exercises.Exercise_4_4_9_part {A : Type} {B : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel B) (L : HTPI.BinRel (A × B)) (h1 : HTPI.total_order R) (h2 : HTPI.total_order S) (h3 : ∀ (a a' : A) (b b' : B), L (a, b) (a', b') ↔ R a a' ∧ (a = a' → S b b')) (a : A) (a' : A) (b : B) (b' : B) : L (a, b) (a', b') ∨ L (a', b') (a, b)"} +{"name":"HTPI.Exercises.dot","declaration":"def HTPI.Exercises.dot {A : Type} (F : Set (Set A)) (G : Set (Set A)) : Set (Set A)"} +{"name":"HTPI.Theorem_4_2_5_3","declaration":"theorem HTPI.Theorem_4_2_5_3 {A : Type} {B : Type} (R : Set (A × B)) : HTPI.Ran (HTPI.inv R) = HTPI.Dom R"} +{"name":"HTPI.Exercises.Example_4_4_3_1","declaration":"theorem HTPI.Exercises.Example_4_4_3_1 {A : Type} : HTPI.partial_order (HTPI.sub A)"} +{"name":"HTPI.Exercises.Exercise_4_3_13b","declaration":"theorem HTPI.Exercises.Exercise_4_3_13b {A : Type} (R1 : HTPI.BinRel A) (R2 : HTPI.BinRel A) (h1 : HTPI.symmetric R1) (h2 : HTPI.symmetric R2) : HTPI.symmetric (HTPI.RelFromExt (HTPI.extension R1 ∪ HTPI.extension R2))"} +{"name":"HTPI.Lemma_4_5_5_2","declaration":"theorem HTPI.Lemma_4_5_5_2 {A : Type} (R : HTPI.BinRel A) (h : HTPI.equiv_rel R) (x : A) (y : A) : y ∈ HTPI.equivClass R x ↔ HTPI.equivClass R y = HTPI.equivClass R x"} +{"name":"HTPI.partial_order","declaration":"def HTPI.partial_order {A : Type} (R : HTPI.BinRel A) : Prop"} +{"name":"HTPI.Ran","declaration":"def HTPI.Ran {A : Type} {B : Type} (R : Set (A × B)) : Set B"} +{"name":"HTPI.Lemma_4_5_5_1","declaration":"theorem HTPI.Lemma_4_5_5_1 {A : Type} (R : HTPI.BinRel A) (h : HTPI.equiv_rel R) (x : A) : x ∈ HTPI.equivClass R x"} +{"name":"HTPI.RelFromExt","declaration":"def HTPI.RelFromExt {A : Type} {B : Type} (R : Set (A × B)) (a : A) (b : B) : Prop"} +{"name":"HTPI.Dom","declaration":"def HTPI.Dom {A : Type} {B : Type} (R : Set (A × B)) : Set A"} +{"name":"HTPI.pairwise_disjoint","declaration":"def HTPI.pairwise_disjoint {A : Type} (F : Set (Set A)) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_3_18","declaration":"theorem HTPI.Exercises.Exercise_4_3_18 {A : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel A) (h1 : HTPI.transitive R) (h2 : HTPI.transitive S) (h3 : HTPI.comp (HTPI.extension S) (HTPI.extension R) ⊆ HTPI.comp (HTPI.extension R) (HTPI.extension S)) : HTPI.transitive (HTPI.RelFromExt (HTPI.comp (HTPI.extension R) (HTPI.extension S)))"} +{"name":"HTPI.Exercises.equiv_mod","declaration":"def HTPI.Exercises.equiv_mod (m : ℤ) (x : ℤ) (y : ℤ) : Prop"} +{"name":"HTPI.Theorem_4_5_4","declaration":"theorem HTPI.Theorem_4_5_4 {A : Type} (R : HTPI.BinRel A) (h : HTPI.equiv_rel R) : HTPI.partition (HTPI.mod A R)"} +{"name":"HTPI.Exercises.Exercise_4_2_14c","declaration":"theorem HTPI.Exercises.Exercise_4_2_14c {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) (T : Set (B × C)) : HTPI.comp (S ∩ T) R = HTPI.comp S R ∩ HTPI.comp T R"} +{"name":"HTPI.smallestElt","declaration":"def HTPI.smallestElt {A : Type} (R : HTPI.BinRel A) (b : A) (B : Set A) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_2_9c","declaration":"theorem HTPI.Exercises.Exercise_4_2_9c {___ : Sort u_1} {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) : ___ → HTPI.Ran (HTPI.comp S R) = HTPI.Ran S"} +{"name":"HTPI.Theorem_4_5_4_part_1","declaration":"theorem HTPI.Theorem_4_5_4_part_1 {A : Type} (R : HTPI.BinRel A) (h : HTPI.equiv_rel R) (x : A) : x ∈ ⋃₀ HTPI.mod A R"} +{"name":"HTPI.Exercises.elt_mod_equiv_class_of_elt","declaration":"theorem HTPI.Exercises.elt_mod_equiv_class_of_elt {A : Type} (R : HTPI.BinRel A) (h : HTPI.equiv_rel R) (X : Set A) : X ∈ HTPI.mod A R → ∀ x ∈ X, HTPI.equivClass R x = X"} +{"name":"HTPI.Exercises.Exercise_4_3_13c","declaration":"theorem HTPI.Exercises.Exercise_4_3_13c {A : Type} (R1 : HTPI.BinRel A) (R2 : HTPI.BinRel A) (h1 : HTPI.transitive R1) (h2 : HTPI.transitive R2) : HTPI.transitive (HTPI.RelFromExt (HTPI.extension R1 ∪ HTPI.extension R2))"} +{"name":"HTPI.Exercises.Theorem_4_5_10","declaration":"theorem HTPI.Exercises.Theorem_4_5_10 (m : ℤ) : HTPI.equiv_rel (HTPI.Exercises.equiv_mod m)"} +{"name":"HTPI.Exercises.Exercise_4_2_12a","declaration":"theorem HTPI.Exercises.Exercise_4_2_12a {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) (T : Set (B × C)) : HTPI.comp S R \\ HTPI.comp T R ⊆ HTPI.comp (S \\ T) R"} +{"name":"HTPI.Theorem_4_3_4_2","declaration":"theorem HTPI.Theorem_4_3_4_2 {A : Type} (R : HTPI.BinRel A) : HTPI.symmetric R ↔ HTPI.extension R = HTPI.inv (HTPI.extension R)"} +{"name":"HTPI.Exercises.Exercise_4_3_12c","declaration":"theorem HTPI.Exercises.Exercise_4_3_12c {A : Type} (R : HTPI.BinRel A) (h1 : HTPI.transitive R) : HTPI.transitive (HTPI.RelFromExt (HTPI.inv (HTPI.extension R)))"} +{"name":"HTPI.ext_def","declaration":"theorem HTPI.ext_def {A : Type} {B : Type} (R : Rel A B) (a : A) (b : B) : (a, b) ∈ HTPI.extension R ↔ R a b"} +{"name":"HTPI.Theorem_4_2_5_5","declaration":"theorem HTPI.Theorem_4_2_5_5 {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) : HTPI.inv (HTPI.comp S R) = HTPI.comp (HTPI.inv R) (HTPI.inv S)"} +{"name":"HTPI.Lemma_4_5_7_ref","declaration":"theorem HTPI.Lemma_4_5_7_ref {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : HTPI.reflexive (HTPI.EqRelFromPart F)"} +{"name":"HTPI.Exercises.Exercise_4_3_19","declaration":"theorem HTPI.Exercises.Exercise_4_3_19 {A : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel (Set A)) (h : ∀ (X Y : Set A), S X Y ↔ ∃ x y, x ∈ X ∧ y ∈ Y ∧ R x y) : HTPI.transitive R → HTPI.transitive S"} +{"name":"HTPI.lub","declaration":"def HTPI.lub {A : Type} (R : HTPI.BinRel A) (a : A) (B : Set A) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_3_20","declaration":"theorem HTPI.Exercises.Exercise_4_3_20 {A : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel (Set A)) (h : ∀ (X Y : Set A), S X Y ↔ X ≠ ∅ ∧ Y ≠ ∅ ∧ ∀ (x y : A), x ∈ X → y ∈ Y → R x y) : HTPI.transitive R → HTPI.transitive S"} +{"name":"HTPI.Theorem_4_2_5_2","declaration":"theorem HTPI.Theorem_4_2_5_2 {A : Type} {B : Type} (R : Set (A × B)) : HTPI.Dom (HTPI.inv R) = HTPI.Ran R"} +{"name":"HTPI.Exercises.Exercise_4_4_15a","declaration":"theorem HTPI.Exercises.Exercise_4_4_15a {A : Type} (R1 : HTPI.BinRel A) (R2 : HTPI.BinRel A) (B : Set A) (b : A) (h1 : HTPI.partial_order R1) (h2 : HTPI.partial_order R2) (h3 : HTPI.extension R1 ⊆ HTPI.extension R2) : HTPI.smallestElt R1 b B → HTPI.smallestElt R2 b B"} +{"name":"HTPI.Theorem_4_2_5_1","declaration":"theorem HTPI.Theorem_4_2_5_1 {A : Type} {B : Type} (R : Set (A × B)) : HTPI.inv (HTPI.inv R) = R"} +{"name":"HTPI.EqRelFromPart","declaration":"def HTPI.EqRelFromPart {A : Type} (F : Set (Set A)) (x : A) (y : A) : Prop"} +{"name":"HTPI.comp","declaration":"def HTPI.comp {A : Type} {B : Type} {C : Type} (S : Set (B × C)) (R : Set (A × B)) : Set (A × C)"} +{"name":"HTPI.Exercises.Lemma_4_5_7_ref","declaration":"theorem HTPI.Exercises.Lemma_4_5_7_ref {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : HTPI.reflexive (HTPI.EqRelFromPart F)"} +{"name":"HTPI.Exercises.Exercise_4_5_20c","declaration":"theorem HTPI.Exercises.Exercise_4_5_20c {A : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel A) (h1 : HTPI.equiv_rel R) (h2 : HTPI.equiv_rel S) : HTPI.mod A (HTPI.Exercises.conj R S) = HTPI.Exercises.dot (HTPI.mod A R) (HTPI.mod A S)"} +{"name":"HTPI.Exercises.Exercise_4_2_12b","declaration":"theorem HTPI.Exercises.Exercise_4_2_12b {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) (T : Set (B × C)) : HTPI.comp (S \\ T) R ⊆ HTPI.comp S R \\ HTPI.comp T R"} +{"name":"HTPI.Theorem_4_2_5_4","declaration":"theorem HTPI.Theorem_4_2_5_4 {A : Type} {B : Type} {C : Type} {D : Type} (R : Set (A × B)) (S : Set (B × C)) (T : Set (C × D)) : HTPI.comp T (HTPI.comp S R) = HTPI.comp (HTPI.comp T S) R"} +{"name":"HTPI.Exercises.Exercise_4_2_14d","declaration":"theorem HTPI.Exercises.Exercise_4_2_14d {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) (T : Set (B × C)) : HTPI.comp (S ∪ T) R = HTPI.comp S R ∪ HTPI.comp T R"} +{"name":"HTPI.Exercises.Theorem_4_3_4_3","declaration":"theorem HTPI.Exercises.Theorem_4_3_4_3 {A : Type} (R : HTPI.BinRel A) : HTPI.transitive R ↔ HTPI.comp (HTPI.extension R) (HTPI.extension R) ⊆ HTPI.extension R"} +{"name":"HTPI.Lemma_4_5_7","declaration":"theorem HTPI.Lemma_4_5_7 {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : HTPI.equiv_rel (HTPI.EqRelFromPart F)"} +{"name":"HTPI.Exercises.Exercise_4_4_22","declaration":"theorem HTPI.Exercises.Exercise_4_4_22 {A : Type} (R : HTPI.BinRel A) (B1 : Set A) (B2 : Set A) (x1 : A) (x2 : A) (h1 : HTPI.partial_order R) (h2 : HTPI.lub R x1 B1) (h3 : HTPI.lub R x2 B2) : B1 ⊆ B2 → R x1 x2"} +{"name":"HTPI.elementhood","declaration":"def HTPI.elementhood (A : Type) (a : A) (X : Set A) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_5_20a","declaration":"theorem HTPI.Exercises.Exercise_4_5_20a {A : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel A) (h1 : HTPI.equiv_rel R) (h2 : HTPI.equiv_rel S) : HTPI.equiv_rel (HTPI.Exercises.conj R S)"} +{"name":"HTPI.inv","declaration":"def HTPI.inv {A : Type} {B : Type} (R : Set (A × B)) : Set (B × A)"} +{"name":"HTPI.Exercises.Exercise_4_2_9a","declaration":"theorem HTPI.Exercises.Exercise_4_2_9a {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) : HTPI.Dom (HTPI.comp S R) ⊆ HTPI.Dom R"} +{"name":"HTPI.Exercises.Exercise_4_4_18a","declaration":"theorem HTPI.Exercises.Exercise_4_4_18a {A : Type} (R : HTPI.BinRel A) (B1 : Set A) (B2 : Set A) (h1 : HTPI.partial_order R) (h2 : ∀ x ∈ B1, ∃ y ∈ B2, R x y) (h3 : ∀ x ∈ B2, ∃ y ∈ B1, R x y) (x : A) : HTPI.upperBd R x B1 ↔ HTPI.upperBd R x B2"} +{"name":"HTPI.Theorem_4_4_6_3","declaration":"theorem HTPI.Theorem_4_4_6_3 {A : Type} (R : HTPI.BinRel A) (B : Set A) (b : A) (h1 : HTPI.total_order R) (h2 : HTPI.minimalElt R b B) : HTPI.smallestElt R b B"} +{"name":"HTPI.Exercises.Lemma_4_5_7_trans","declaration":"theorem HTPI.Exercises.Lemma_4_5_7_trans {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : HTPI.transitive (HTPI.EqRelFromPart F)"} +{"name":"HTPI.extension","declaration":"def HTPI.extension {A : Type} {B : Type} (R : Rel A B) : Set (A × B)"} +{"name":"HTPI.reflexive","declaration":"def HTPI.reflexive {A : Type} (R : HTPI.BinRel A) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_3_12a","declaration":"theorem HTPI.Exercises.Exercise_4_3_12a {A : Type} (R : HTPI.BinRel A) (h1 : HTPI.reflexive R) : HTPI.reflexive (HTPI.RelFromExt (HTPI.inv (HTPI.extension R)))"} +{"name":"HTPI.Exercises.Lemma_4_5_8","declaration":"theorem HTPI.Exercises.Lemma_4_5_8 {A : Type} (F : Set (Set A)) (h : HTPI.partition F) (X : Set A) : X ∈ F → ∀ x ∈ X, HTPI.equivClass (HTPI.EqRelFromPart F) x = X"} +{"name":"HTPI.antisymmetric","declaration":"def HTPI.antisymmetric {A : Type} (R : HTPI.BinRel A) : Prop"} +{"name":"HTPI.Exercises.Exercise_4_4_8","declaration":"theorem HTPI.Exercises.Exercise_4_4_8 {A : Type} {B : Type} (R : HTPI.BinRel A) (S : HTPI.BinRel B) (T : HTPI.BinRel (A × B)) (h1 : HTPI.partial_order R) (h2 : HTPI.partial_order S) (h3 : ∀ (a a' : A) (b b' : B), T (a, b) (a', b') ↔ R a a' ∧ S b b') : HTPI.partial_order T"} +{"name":"HTPI.RelFromExt_def","declaration":"theorem HTPI.RelFromExt_def {A : Type} {B : Type} (R : Set (A × B)) (a : A) (b : B) : HTPI.RelFromExt R a b ↔ (a, b) ∈ R"} +{"name":"HTPI.Lemma_4_5_7_symm","declaration":"theorem HTPI.Lemma_4_5_7_symm {A : Type} (F : Set (Set A)) (h : HTPI.partition F) : HTPI.symmetric (HTPI.EqRelFromPart F)"} +{"name":"HTPI.Theorem_4_4_6_2","declaration":"theorem HTPI.Theorem_4_4_6_2 {A : Type} (R : HTPI.BinRel A) (B : Set A) (b : A) (h1 : HTPI.partial_order R) (h2 : HTPI.smallestElt R b B) : HTPI.minimalElt R b B ∧ ∀ (c : A), HTPI.minimalElt R c B → b = c"} +{"name":"HTPI.equivClass","declaration":"def HTPI.equivClass {A : Type} (R : HTPI.BinRel A) (x : A) : Set A"} +{"name":"HTPI.Exercises.Exercise_4_4_15b","declaration":"theorem HTPI.Exercises.Exercise_4_4_15b {A : Type} (R1 : HTPI.BinRel A) (R2 : HTPI.BinRel A) (B : Set A) (b : A) (h1 : HTPI.partial_order R1) (h2 : HTPI.partial_order R2) (h3 : HTPI.extension R1 ⊆ HTPI.extension R2) : HTPI.minimalElt R2 b B → HTPI.minimalElt R1 b B"} +{"name":"HTPI.total_order","declaration":"def HTPI.total_order {A : Type} (R : HTPI.BinRel A) : Prop"} +{"name":"HTPI.minimalElt","declaration":"def HTPI.minimalElt {A : Type} (R : HTPI.BinRel A) (b : A) (B : Set A) : Prop"} +{"name":"HTPI.Exercises.Theorem_4_4_6_1","declaration":"theorem HTPI.Exercises.Theorem_4_4_6_1 {A : Type} (R : HTPI.BinRel A) (B : Set A) (b : A) (h1 : HTPI.partial_order R) (h2 : HTPI.smallestElt R b B) (c : A) : HTPI.smallestElt R c B → b = c"} +{"name":"HTPI.partition","declaration":"def HTPI.partition {A : Type} (F : Set (Set A)) : Prop"} +{"name":"HTPI.Theorem_4_5_4_part_2","declaration":"theorem HTPI.Theorem_4_5_4_part_2 {A : Type} (R : HTPI.BinRel A) (h : HTPI.equiv_rel R) : HTPI.pairwise_disjoint (HTPI.mod A R)"} +{"name":"HTPI.Exercises.Exercise_4_4_24","declaration":"theorem HTPI.Exercises.Exercise_4_4_24 {A : Type} (R : Set (A × A)) : HTPI.smallestElt (HTPI.sub (A × A)) (R ∪ HTPI.inv R) {T | R ⊆ T ∧ HTPI.symmetric (HTPI.RelFromExt T)}"} +{"name":"HTPI.sub","declaration":"def HTPI.sub (A : Type) (X : Set A) (Y : Set A) : Prop"} +{"name":"HTPI.mod","declaration":"def HTPI.mod (A : Type) (R : HTPI.BinRel A) : Set (Set A)"} diff --git a/htpi-declarations/HTPILib.Chap5.jsonl b/htpi-declarations/HTPILib.Chap5.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c99ef98a8952079c3bd206c0412272fd0db9b1e2 --- /dev/null +++ b/htpi-declarations/HTPILib.Chap5.jsonl @@ -0,0 +1,72 @@ +{"name":"HTPI.Exercises.Theorem_5_3_3_2","declaration":"theorem HTPI.Exercises.Theorem_5_3_3_2 {A : Type} {B : Type} (f : A → B) (g : B → A) (h1 : f ∘ g = id) : HTPI.onto f"} +{"name":"HTPI.Exercises.Exercise_5_2_11b","declaration":"theorem HTPI.Exercises.Exercise_5_2_11b {A : Type} {B : Type} {C : Type} (f : A → B) (g : B → C) : ¬HTPI.onto f → HTPI.one_to_one g → ¬HTPI.onto (g ∘ f)"} +{"name":"HTPI.Exercises.Theorem_5_4_9","declaration":"theorem HTPI.Exercises.Theorem_5_4_9 {A : Type} (f : A → A → A) (B : Set A) : ∃ C, HTPI.closure2 f B C"} +{"name":"HTPI.plus''","declaration":"def HTPI.plus'' : ℤ → ℤ → ℤ"} +{"name":"HTPI.func_from_graph_rtl","declaration":"theorem HTPI.func_from_graph_rtl {A : Type} {B : Type} (F : Set (A × B)) : HTPI.is_func_graph F → ∃ f, HTPI.graph f = F"} +{"name":"HTPI.Theorem_5_3_1","declaration":"theorem HTPI.Theorem_5_3_1 {A : Type} {B : Type} (f : A → B) (h1 : HTPI.one_to_one f) (h2 : HTPI.onto f) : ∃ g, HTPI.graph g = HTPI.inv (HTPI.graph f)"} +{"name":"HTPI.graph_def","declaration":"theorem HTPI.graph_def {A : Type} {B : Type} (f : A → B) (a : A) (b : B) : (a, b) ∈ HTPI.graph f ↔ f a = b"} +{"name":"HTPI.image","declaration":"def HTPI.image {A : Type} {B : Type} (f : A → B) (X : Set A) : Set B"} +{"name":"HTPI.inverse_image_def","declaration":"theorem HTPI.inverse_image_def {A : Type} {B : Type} (f : A → B) (Y : Set B) (a : A) : a ∈ HTPI.inverse_image f Y ↔ f a ∈ Y"} +{"name":"HTPI.Exercises.Exercise_5_1_17a","declaration":"theorem HTPI.Exercises.Exercise_5_1_17a {A : Type} (f : A → A) (a : A) (h : ∀ (x : A), f x = a) (g : A → A) : f ∘ g = f"} +{"name":"HTPI.Exercises.Exercise_5_2_10b","declaration":"theorem HTPI.Exercises.Exercise_5_2_10b {A : Type} {B : Type} {C : Type} (f : A → B) (g : B → C) : HTPI.one_to_one (g ∘ f) → HTPI.one_to_one f"} +{"name":"HTPI.Exercises.Exercise_5_1_13a","declaration":"theorem HTPI.Exercises.Exercise_5_1_13a {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) (f : A → C) (h1 : ∀ (b : B), b ∈ HTPI.Ran R ∧ b ∈ HTPI.Dom S) (h2 : HTPI.graph f = HTPI.comp S R) : HTPI.is_func_graph S"} +{"name":"HTPI.Exercises.Exercise_5_3_14a","declaration":"theorem HTPI.Exercises.Exercise_5_3_14a {A : Type} {B : Type} (f : A → B) (g : B → A) (h : f ∘ g = id) (x : A) : x ∈ HTPI.Ran (HTPI.graph g) → g (f x) = x"} +{"name":"HTPI.graph","declaration":"def HTPI.graph {A : Type} {B : Type} (f : A → B) : Set (A × B)"} +{"name":"HTPI.Exercises.Exercise_5_2_12","declaration":"theorem HTPI.Exercises.Exercise_5_2_12 {A : Type} {B : Type} (f : A → B) (g : B → Set A) (h : ∀ (b : B), g b = {a | f a = b}) : HTPI.onto f → HTPI.one_to_one g"} +{"name":"HTPI.Theorem_5_3_2_1","declaration":"theorem HTPI.Theorem_5_3_2_1 {A : Type} {B : Type} (f : A → B) (g : B → A) (h1 : HTPI.graph g = HTPI.inv (HTPI.graph f)) : g ∘ f = id"} +{"name":"HTPI.Theorem_5_1_4","declaration":"theorem HTPI.Theorem_5_1_4 {A : Type} {B : Type} (f : A → B) (g : A → B) : (∀ (a : A), f a = g a) → f = g"} +{"name":"HTPI.Exercises.Exercise_5_4_7","declaration":"theorem HTPI.Exercises.Exercise_5_4_7 {A : Type} (f : A → A) (g : A → A) (C : Set A) (h1 : f ∘ g = id) (h2 : HTPI.closed f C) : HTPI.closed g (HTPI.Exercises.complement C)"} +{"name":"HTPI.closure_family","declaration":"def HTPI.closure_family {A : Type} (F : Set (A → A)) (B : Set A) (C : Set A) : Prop"} +{"name":"HTPI.Exercises.Exercise_5_3_11b","declaration":"theorem HTPI.Exercises.Exercise_5_3_11b {A : Type} {B : Type} (f : A → B) (g : B → A) : HTPI.onto f → g ∘ f = id → HTPI.graph g = HTPI.inv (HTPI.graph f)"} +{"name":"HTPI.Exercises.Exercise_5_3_18","declaration":"theorem HTPI.Exercises.Exercise_5_3_18 {A : Type} {B : Type} {C : Type} (f : A → C) (g : B → C) (h1 : HTPI.one_to_one g) (h2 : HTPI.onto g) : ∃ h, g ∘ h = f"} +{"name":"HTPI.Theorem_5_4_5","declaration":"theorem HTPI.Theorem_5_4_5 {A : Type} (f : A → A) (B : Set A) : ∃ C, HTPI.closure f B C"} +{"name":"HTPI.Exercises.Exercise_5_4_13a","declaration":"theorem HTPI.Exercises.Exercise_5_4_13a {A : Type} (F : Set (A → A)) (B : Set A) : ∃ C, HTPI.closure_family F B C"} +{"name":"HTPI.Exercises.Exercise_5_4_9a","declaration":"theorem HTPI.Exercises.Exercise_5_4_9a {A : Type} (f : A → A) (C1 : Set A) (C2 : Set A) (h1 : HTPI.closed f C1) (h2 : HTPI.closed f C2) : HTPI.closed f (C1 ∪ C2)"} +{"name":"HTPI.Exercises.Exercise_5_1_15a","declaration":"theorem HTPI.Exercises.Exercise_5_1_15a {A : Type} {B : Type} (f : A → B) (R : HTPI.BinRel A) (S : HTPI.BinRel B) (h : ∀ (x y : B), S x y ↔ ∃ u v, f u = x ∧ f v = y ∧ R u v) : HTPI.reflexive R → HTPI.reflexive S"} +{"name":"HTPI.plus'","declaration":"def HTPI.plus' : ℤ → ℤ → ℤ"} +{"name":"HTPI.Exercises.Exercise_5_2_17a","declaration":"theorem HTPI.Exercises.Exercise_5_2_17a {A : Type} {B : Type} (f : A → B) (R : HTPI.BinRel A) (S : HTPI.BinRel B) (h1 : ∀ (x y : B), S x y ↔ ∃ u v, f u = x ∧ f v = y ∧ R u v) (h2 : HTPI.onto f) : HTPI.reflexive R → HTPI.reflexive S"} +{"name":"HTPI.Theorem_5_5_2_2","declaration":"theorem HTPI.Theorem_5_5_2_2 {A : Type} {B : Type} (f : A → B) (W : Set A) (X : Set A) (h1 : HTPI.one_to_one f) : HTPI.image f (W ∩ X) = HTPI.image f W ∩ HTPI.image f X"} +{"name":"HTPI.is_func_graph","declaration":"def HTPI.is_func_graph {A : Type} {B : Type} (G : Set (A × B)) : Prop"} +{"name":"HTPI.square2","declaration":"def HTPI.square2 : ℕ → ℕ"} +{"name":"HTPI.Theorem_5_3_3_2","declaration":"theorem HTPI.Theorem_5_3_3_2 {A : Type} {B : Type} (f : A → B) (g : B → A) (h1 : f ∘ g = id) : HTPI.onto f"} +{"name":"HTPI.closed","declaration":"def HTPI.closed {A : Type} (f : A → A) (C : Set A) : Prop"} +{"name":"HTPI.square1","declaration":"def HTPI.square1 (n : ℕ) : ℕ"} +{"name":"HTPI.image_def","declaration":"theorem HTPI.image_def {A : Type} {B : Type} (f : A → B) (X : Set A) (b : B) : b ∈ HTPI.image f X ↔ ∃ x ∈ X, f x = b"} +{"name":"HTPI.closed_family","declaration":"def HTPI.closed_family {A : Type} (F : Set (A → A)) (C : Set A) : Prop"} +{"name":"HTPI.Exercises.Exercise_5_3_17a","declaration":"theorem HTPI.Exercises.Exercise_5_3_17a {A : Type} : HTPI.symmetric (HTPI.Exercises.conjugate A)"} +{"name":"HTPI.inverse_image","declaration":"def HTPI.inverse_image {A : Type} {B : Type} (f : A → B) (Y : Set B) : Set A"} +{"name":"HTPI.Exercises.Exercise_5_1_17b","declaration":"theorem HTPI.Exercises.Exercise_5_1_17b {A : Type} (f : A → A) (a : A) (h : ∀ (g : A → A), f ∘ g = f) : ∃ y, ∀ (x : A), f x = y"} +{"name":"HTPI.Exercises.Exercise_5_1_16b","declaration":"theorem HTPI.Exercises.Exercise_5_1_16b {A : Type} {B : Type} (R : HTPI.BinRel B) (S : HTPI.BinRel (A → B)) (h : ∀ (f g : A → B), S f g ↔ ∀ (x : A), R (f x) (g x)) : HTPI.symmetric R → HTPI.symmetric S"} +{"name":"HTPI.Theorem_5_2_5_1","declaration":"theorem HTPI.Theorem_5_2_5_1 {A : Type} {B : Type} {C : Type} (f : A → B) (g : B → C) : HTPI.one_to_one f → HTPI.one_to_one g → HTPI.one_to_one (g ∘ f)"} +{"name":"HTPI.Exercises.Exercise_5_4_10a","declaration":"theorem HTPI.Exercises.Exercise_5_4_10a {A : Type} (f : A → A) (B1 : Set A) (B2 : Set A) (C1 : Set A) (C2 : Set A) (h1 : HTPI.closure f B1 C1) (h2 : HTPI.closure f B2 C2) : B1 ⊆ B2 → C1 ⊆ C2"} +{"name":"HTPI.Exercises.Exercise_5_2_21a","declaration":"theorem HTPI.Exercises.Exercise_5_2_21a {A : Type} {B : Type} {C : Type} (f : B → C) (g : A → B) (h : A → B) (h1 : HTPI.one_to_one f) (h2 : f ∘ g = f ∘ h) : g = h"} +{"name":"HTPI.closed2","declaration":"def HTPI.closed2 {A : Type} (f : A → A → A) (C : Set A) : Prop"} +{"name":"HTPI.Theorem_5_1_5","declaration":"theorem HTPI.Theorem_5_1_5 {A : Type} {B : Type} {C : Type} (f : A → B) (g : B → C) : ∃ h, HTPI.graph h = HTPI.comp (HTPI.graph g) (HTPI.graph f)"} +{"name":"HTPI.Exercises.Exercise_5_2_16","declaration":"theorem HTPI.Exercises.Exercise_5_2_16 {A : Type} {B : Type} {C : Type} (R : Set (A × B)) (S : Set (B × C)) (f : A → C) (g : B → C) (h1 : HTPI.graph f = HTPI.comp S R) (h2 : HTPI.graph g = S) (h3 : HTPI.one_to_one g) : HTPI.is_func_graph R"} +{"name":"HTPI.plus","declaration":"def HTPI.plus (m : ℤ) (n : ℤ) : ℤ"} +{"name":"HTPI.Exercises.Exercise_5_2_17b","declaration":"theorem HTPI.Exercises.Exercise_5_2_17b {A : Type} {B : Type} (f : A → B) (R : HTPI.BinRel A) (S : HTPI.BinRel B) (h1 : ∀ (x y : B), S x y ↔ ∃ u v, f u = x ∧ f v = y ∧ R u v) (h2 : HTPI.one_to_one f) : HTPI.transitive R → HTPI.transitive S"} +{"name":"HTPI.Exercises.Exercise_5_2_10a","declaration":"theorem HTPI.Exercises.Exercise_5_2_10a {A : Type} {B : Type} {C : Type} (f : A → B) (g : B → C) : HTPI.onto (g ∘ f) → HTPI.onto g"} +{"name":"HTPI.Theorem_5_3_2_2","declaration":"theorem HTPI.Theorem_5_3_2_2 {A : Type} {B : Type} (f : A → B) (g : B → A) (h1 : HTPI.graph g = HTPI.inv (HTPI.graph f)) : f ∘ g = id"} +{"name":"HTPI.closure","declaration":"def HTPI.closure {A : Type} (f : A → A) (B : Set A) (C : Set A) : Prop"} +{"name":"HTPI.Exercises.Exercise_5_2_21b","declaration":"theorem HTPI.Exercises.Exercise_5_2_21b {A : Type} {B : Type} {C : Type} (f : B → C) (a : A) (h1 : ∀ (g h : A → B), f ∘ g = f ∘ h → g = h) : HTPI.one_to_one f"} +{"name":"HTPI.Exercises.complement","declaration":"def HTPI.Exercises.complement {A : Type} (B : Set A) : Set A"} +{"name":"HTPI.Exercises.Exercise_5_3_17b","declaration":"theorem HTPI.Exercises.Exercise_5_3_17b {A : Type} (f1 : A → A) (f2 : A → A) (h1 : HTPI.Exercises.conjugate A f1 f2) (h2 : ∃ a, f1 a = a) : ∃ a, f2 a = a"} +{"name":"HTPI.func_from_graph","declaration":"theorem HTPI.func_from_graph {A : Type} {B : Type} (F : Set (A × B)) : (∃ f, HTPI.graph f = F) ↔ HTPI.is_func_graph F"} +{"name":"HTPI.func_from_graph_ltr","declaration":"theorem HTPI.func_from_graph_ltr {A : Type} {B : Type} (F : Set (A × B)) : (∃ f, HTPI.graph f = F) → HTPI.is_func_graph F"} +{"name":"HTPI.Exercises.Exercise_5_2_11a","declaration":"theorem HTPI.Exercises.Exercise_5_2_11a {A : Type} {B : Type} {C : Type} (f : A → B) (g : B → C) : HTPI.onto f → ¬HTPI.one_to_one g → ¬HTPI.one_to_one (g ∘ f)"} +{"name":"HTPI.Theorem_5_2_5_2","declaration":"theorem HTPI.Theorem_5_2_5_2 {A : Type} {B : Type} {C : Type} (f : A → B) (g : B → C) : HTPI.onto f → HTPI.onto g → HTPI.onto (g ∘ f)"} +{"name":"HTPI.Exercises.Exercise_5_4_10b","declaration":"theorem HTPI.Exercises.Exercise_5_4_10b {A : Type} (f : A → A) (B1 : Set A) (B2 : Set A) (C1 : Set A) (C2 : Set A) (h1 : HTPI.closure f B1 C1) (h2 : HTPI.closure f B2 C2) : HTPI.closure f (B1 ∪ B2) (C1 ∪ C2)"} +{"name":"HTPI.Theorem_5_5_2_1","declaration":"theorem HTPI.Theorem_5_5_2_1 {A : Type} {B : Type} (f : A → B) (W : Set A) (X : Set A) : HTPI.image f (W ∩ X) ⊆ HTPI.image f W ∩ HTPI.image f X"} +{"name":"HTPI.one_to_one","declaration":"def HTPI.one_to_one {A : Type} {B : Type} (f : A → B) : Prop"} +{"name":"HTPI.comp_def","declaration":"theorem HTPI.comp_def {A : Type} {B : Type} {C : Type} (g : B → C) (f : A → B) (x : A) : (g ∘ f) x = g (f x)"} +{"name":"HTPI.Exercises.func_from_graph_ltr","declaration":"theorem HTPI.Exercises.func_from_graph_ltr {A : Type} {B : Type} (F : Set (A × B)) : (∃ f, HTPI.graph f = F) → HTPI.is_func_graph F"} +{"name":"HTPI.Exercises.Exercise_5_1_15c","declaration":"theorem HTPI.Exercises.Exercise_5_1_15c {A : Type} {B : Type} (f : A → B) (R : HTPI.BinRel A) (S : HTPI.BinRel B) (h : ∀ (x y : B), S x y ↔ ∃ u v, f u = x ∧ f v = y ∧ R u v) : HTPI.transitive R → HTPI.transitive S"} +{"name":"HTPI.Exercises.Exercise_5_1_14a","declaration":"theorem HTPI.Exercises.Exercise_5_1_14a {A : Type} {B : Type} (f : A → B) (R : HTPI.BinRel A) (S : HTPI.BinRel B) (h : ∀ (x y : A), R x y ↔ S (f x) (f y)) : HTPI.reflexive S → HTPI.reflexive R"} +{"name":"HTPI.Exercises.Theorem_5_3_2_2","declaration":"theorem HTPI.Exercises.Theorem_5_3_2_2 {A : Type} {B : Type} (f : A → B) (g : B → A) (h1 : HTPI.graph g = HTPI.inv (HTPI.graph f)) : f ∘ g = id"} +{"name":"HTPI.Theorem_5_3_3_1","declaration":"theorem HTPI.Theorem_5_3_3_1 {A : Type} {B : Type} (f : A → B) (g : B → A) (h1 : g ∘ f = id) : HTPI.one_to_one f"} +{"name":"HTPI.onto","declaration":"def HTPI.onto {A : Type} {B : Type} (f : A → B) : Prop"} +{"name":"HTPI.Exercises.conjugate","declaration":"def HTPI.Exercises.conjugate (A : Type) (f1 : A → A) (f2 : A → A) : Prop"} +{"name":"HTPI.closure2","declaration":"def HTPI.closure2 {A : Type} (f : A → A → A) (B : Set A) (C : Set A) : Prop"} +{"name":"HTPI.Theorem_5_3_5","declaration":"theorem HTPI.Theorem_5_3_5 {A : Type} {B : Type} (f : A → B) (g : B → A) (h1 : g ∘ f = id) (h2 : f ∘ g = id) : HTPI.graph g = HTPI.inv (HTPI.graph f)"} +{"name":"HTPI.Theorem_5_4_9","declaration":"theorem HTPI.Theorem_5_4_9 {A : Type} (f : A → A → A) (B : Set A) : ∃ C, HTPI.closure2 f B C"} +{"name":"HTPI.Exercises.Exercise_5_3_11a","declaration":"theorem HTPI.Exercises.Exercise_5_3_11a {A : Type} {B : Type} (f : A → B) (g : B → A) : HTPI.one_to_one f → f ∘ g = id → HTPI.graph g = HTPI.inv (HTPI.graph f)"} diff --git a/htpi-declarations/HTPILib.Chap6.jsonl b/htpi-declarations/HTPILib.Chap6.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..630ed01b635376fa5ce692bffdce01cb008e09ee --- /dev/null +++ b/htpi-declarations/HTPILib.Chap6.jsonl @@ -0,0 +1,102 @@ +{"name":"HTPI.extendPO_is_ref","declaration":"theorem HTPI.extendPO_is_ref {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) : HTPI.reflexive (HTPI.extendPO R b)"} +{"name":"HTPI.extendPO_extends","declaration":"theorem HTPI.extendPO_extends {A : Type} (R : HTPI.BinRel A) (b : A) (x : A) (y : A) : R x y → HTPI.extendPO R b x y"} +{"name":"HTPI.Example_6_1_1","declaration":"theorem HTPI.Example_6_1_1 (n : ℕ) : Sum i from 0 to n, 2 ^ i = 2 ^ (n + 1) - 1"} +{"name":"HTPI.cumul_image","declaration":"def HTPI.cumul_image {A : Type} (f : A → A) (B : Set A) : Set A"} +{"name":"HTPI.Exercises.Exercise_6_5_14","declaration":"theorem HTPI.Exercises.Exercise_6_5_14 {A : Type} (R : Set (A × A)) : HTPI.smallestElt (HTPI.sub (A × A)) (HTPI.Exercises.cumul_comp R) {S | R ⊆ S ∧ HTPI.transitive (HTPI.RelFromExt S)}"} +{"name":"HTPI.Exercises.Exercise_6_3_16","declaration":"theorem HTPI.Exercises.Exercise_6_3_16 {___ : ℕ} (n : ℕ) : HTPI.Exercises.seq_6_3_16 n = ___"} +{"name":"HTPI.Exercises.cumul_un_image2","declaration":"def HTPI.Exercises.cumul_un_image2 {A : Type} (f : A → A → A) (B : Set A) : Set A"} +{"name":"HTPI.Exercises.Exercise_6_4_7c","declaration":"theorem HTPI.Exercises.Exercise_6_4_7c (n : ℕ) : Sum i from 0 to n, HTPI.Fib (2 * i + 1) = HTPI.Fib (2 * n + 2)"} +{"name":"HTPI.Exercises.Exercise_6_4_4a","declaration":"theorem HTPI.Exercises.Exercise_6_4_4a : ¬∃ q p, p * p = 6 * (q * q) ∧ q ≠ 0"} +{"name":"HTPI.Exercises.Lemma_6_2_1_2","declaration":"theorem HTPI.Exercises.Lemma_6_2_1_2 {A : Type} {R : HTPI.BinRel A} {B : Set A} {b : A} {c : A} (h1 : HTPI.partial_order R) (h2 : b ∈ B) (h3 : HTPI.minimalElt R c (B \\ {b})) (h4 : ¬R b c) : HTPI.minimalElt R c B"} +{"name":"HTPI.Example_6_1_3","declaration":"theorem HTPI.Example_6_1_3 (n : ℕ) : n ≥ 5 → 2 ^ n > n ^ 2"} +{"name":"HTPI.Exercises.cumul_comp","declaration":"def HTPI.Exercises.cumul_comp {A : Type} (R : Set (A × A)) : Set (A × A)"} +{"name":"HTPI.Exercises.Exercise_6_1_15","declaration":"theorem HTPI.Exercises.Exercise_6_1_15 (n : ℕ) : n ≥ 10 → 2 ^ n > n ^ 3"} +{"name":"HTPI.Exercises.Exercise_6_4_5","declaration":"theorem HTPI.Exercises.Exercise_6_4_5 (n : ℕ) : n ≥ 12 → ∃ a b, 3 * a + 7 * b = n"} +{"name":"HTPI.Exercises.Exercise_6_4_8a","declaration":"theorem HTPI.Exercises.Exercise_6_4_8a (m : ℕ) (n : ℕ) : HTPI.Fib (m + n + 1) = HTPI.Fib m * HTPI.Fib n + HTPI.Fib (m + 1) * HTPI.Fib (n + 1)"} +{"name":"HTPI.Exercises.image2","declaration":"def HTPI.Exercises.image2 {A : Type} (f : A → A → A) (B : Set A) : Set A"} +{"name":"HTPI.Exercises.seq_6_3_15","declaration":"def HTPI.Exercises.seq_6_3_15 (k : ℕ) : ℤ"} +{"name":"HTPI.Exercises.rep_image_family_step","declaration":"theorem HTPI.Exercises.rep_image_family_step {A : Type} (F : Set (A → A)) (n : ℕ) (B : Set A) : HTPI.Exercises.rep_image_family F (n + 1) B = {x | ∃ f ∈ F, x ∈ HTPI.image f (HTPI.Exercises.rep_image_family F n B)}"} +{"name":"HTPI.Lemma_6_2_1_1","declaration":"theorem HTPI.Lemma_6_2_1_1 {A : Type} {R : HTPI.BinRel A} {B : Set A} {b : A} {c : A} (h1 : HTPI.partial_order R) (h2 : b ∈ B) (h3 : HTPI.minimalElt R c (B \\ {b})) (h4 : R b c) : HTPI.minimalElt R b B"} +{"name":"HTPI.Exercises.Exercise_6_2_4a","declaration":"theorem HTPI.Exercises.Exercise_6_2_4a {A : Type} (R : HTPI.BinRel A) (h : ∀ (x y : A), R x y ∨ R y x) (n : ℕ) : n ≥ 1 → ∀ (B : Set A), HTPI.numElts B n → ∃ x ∈ B, ∀ y ∈ B, ∃ z, R x z ∧ R z y"} +{"name":"HTPI.rep_image_sub_closed","declaration":"theorem HTPI.rep_image_sub_closed {A : Type} {f : A → A} {B : Set A} {D : Set A} (h1 : B ⊆ D) (h2 : HTPI.closed f D) (n : ℕ) : HTPI.rep_image f n B ⊆ D"} +{"name":"HTPI.extendPO_all_comp","declaration":"theorem HTPI.extendPO_all_comp {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) (x : A) : HTPI.extendPO R b b x ∨ HTPI.extendPO R b x b"} +{"name":"HTPI.rep_image","declaration":"def HTPI.rep_image {A : Type} (f : A → A) (n : ℕ) (B : Set A) : Set A"} +{"name":"HTPI.Exercises.Exercise_6_5_3","declaration":"theorem HTPI.Exercises.Exercise_6_5_3 {A : Type} (F : Set (A → A)) (B : Set A) : HTPI.closure_family F B (HTPI.Exercises.cumul_image_family F B)"} +{"name":"HTPI.Exercises.fact_pos","declaration":"theorem HTPI.Exercises.fact_pos (n : ℕ) : HTPI.fact n ≥ 1"} +{"name":"HTPI.Exercises.rep_image2","declaration":"def HTPI.Exercises.rep_image2 {A : Type} (f : A → A → A) (n : ℕ) (B : Set A) : Set A"} +{"name":"HTPI.fact","declaration":"def HTPI.fact (k : ℕ) : ℕ"} +{"name":"HTPI.Exercises.Exercise_6_3_13b","declaration":"theorem HTPI.Exercises.Exercise_6_3_13b (k : ℕ) (n : ℕ) : n ≥ 2 * k ^ 2 → HTPI.fact n ≥ k ^ n"} +{"name":"HTPI.Exercises.Exercise_6_2_3","declaration":"theorem HTPI.Exercises.Exercise_6_2_3 (A : Type) (R : HTPI.BinRel A) (h : HTPI.total_order R) (n : ℕ) : n ≥ 1 → ∀ (B : Set A), HTPI.numElts B n → ∃ b, HTPI.smallestElt R b B"} +{"name":"HTPI.Exercises.sq_even_iff_even","declaration":"theorem HTPI.Exercises.sq_even_iff_even (n : ℕ) : HTPI.nat_even (n * n) ↔ HTPI.nat_even n"} +{"name":"HTPI.Fib","declaration":"def HTPI.Fib (n : ℕ) : ℕ"} +{"name":"HTPI.Exercises.Example_6_2_2","declaration":"theorem HTPI.Exercises.Example_6_2_2 {A : Type} (R : HTPI.BinRel A) (h1 : ∃ n, HTPI.numElts {x | x = x} n) (h2 : HTPI.partial_order R) : ∃ T, HTPI.total_order T ∧ ∀ (x y : A), R x y → T x y"} +{"name":"HTPI.Exercises.Exercise_6_5_8a","declaration":"theorem HTPI.Exercises.Exercise_6_5_8a {A : Type} (f : A → A → A) (B : Set A) (m : ℕ) (n : ℕ) : m ≤ n → HTPI.Exercises.rep_un_image2 f m B ⊆ HTPI.Exercises.rep_un_image2 f n B"} +{"name":"HTPI.Exercises.Exercise_6_4_7a","declaration":"theorem HTPI.Exercises.Exercise_6_4_7a (n : ℕ) : (Sum i from 0 to n, HTPI.Fib i) + 1 = HTPI.Fib (n + 2)"} +{"name":"HTPI.Exercises.Exercise_6_3_7b","declaration":"theorem HTPI.Exercises.Exercise_6_3_7b (f : ℕ → ℝ) (c : ℝ) (n : ℕ) : Sum i from 0 to n, c * f i = c * Sum i from 0 to n, f i"} +{"name":"HTPI.Exercises.rep_image_family","declaration":"def HTPI.Exercises.rep_image_family {A : Type} (F : Set (A → A)) (n : ℕ) (B : Set A) : Set A"} +{"name":"HTPI.Exercises.Exercise_6_3_13a","declaration":"theorem HTPI.Exercises.Exercise_6_3_13a (k : ℕ) (n : ℕ) : HTPI.fact (k ^ 2 + n) ≥ k ^ (2 * n)"} +{"name":"HTPI.Exercises.div_mod_char","declaration":"theorem HTPI.Exercises.div_mod_char (m : ℕ) (n : ℕ) (q : ℕ) (r : ℕ) (h1 : n = m * q + r) (h2 : r < m) : q = n / m ∧ r = n % m"} +{"name":"HTPI.Example_6_4_1","declaration":"theorem HTPI.Example_6_4_1 (m : ℕ) : m > 0 → ∀ (n : ℕ), ∃ q r, n = m * q + r ∧ r < m"} +{"name":"HTPI.extendPO_is_antisymm","declaration":"theorem HTPI.extendPO_is_antisymm {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) : HTPI.antisymmetric (HTPI.extendPO R b)"} +{"name":"HTPI.Like_Example_6_1_1","declaration":"theorem HTPI.Like_Example_6_1_1 (n : ℕ) : (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1)"} +{"name":"HTPI.Exercises.extendPO_is_antisymm","declaration":"theorem HTPI.Exercises.extendPO_is_antisymm {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) : HTPI.antisymmetric (HTPI.extendPO R b)"} +{"name":"HTPI.Exercises.rep_un_image2_step","declaration":"theorem HTPI.Exercises.rep_un_image2_step {A : Type} (f : A → A → A) (n : ℕ) (B : Set A) : HTPI.Exercises.rep_un_image2 f (n + 1) B = HTPI.Exercises.un_image2 f (HTPI.Exercises.rep_un_image2 f n B)"} +{"name":"HTPI.Exercises.triple_rec_formula","declaration":"theorem HTPI.Exercises.triple_rec_formula (n : ℕ) : HTPI.Exercises.triple_rec n = 2 ^ n * HTPI.Fib n"} +{"name":"HTPI.Exercises.rep_un_image2_sub_closed","declaration":"theorem HTPI.Exercises.rep_un_image2_sub_closed {A : Type} {f : A → A → A} {B : Set A} {D : Set A} (h1 : B ⊆ D) (h2 : HTPI.closed2 f D) (n : ℕ) : HTPI.Exercises.rep_un_image2 f n B ⊆ D"} +{"name":"HTPI.Exercises.Exercise_6_5_11","declaration":"theorem HTPI.Exercises.Exercise_6_5_11 {A : Type} (R : Set (A × A)) (m : ℕ) (n : ℕ) : HTPI.Exercises.rep_comp R (m + n) = HTPI.comp (HTPI.Exercises.rep_comp R m) (HTPI.Exercises.rep_comp R n)"} +{"name":"HTPI.Exercises.Exercise_6_1_13","declaration":"theorem HTPI.Exercises.Exercise_6_1_13 (a : ℤ) (b : ℤ) (n : ℕ) : a - b ∣ a ^ n - b ^ n"} +{"name":"HTPI.Exercises.un_image2","declaration":"def HTPI.Exercises.un_image2 {A : Type} (f : A → A → A) (B : Set A) : Set A"} +{"name":"HTPI.Exercises.Like_Exercise_6_2_16","declaration":"theorem HTPI.Exercises.Like_Exercise_6_2_16 {A : Type} (f : A → A) (h : HTPI.one_to_one f) (n : ℕ) (B : Set A) : HTPI.numElts B n → HTPI.closed f B → ∀ y ∈ B, ∃ x ∈ B, f x = y"} +{"name":"HTPI.Exercises.rep_comp_sub_trans","declaration":"theorem HTPI.Exercises.rep_comp_sub_trans {A : Type} {R : Set (A × A)} {S : Set (A × A)} (h1 : R ⊆ S) (h2 : HTPI.transitive (HTPI.RelFromExt S)) (n : ℕ) : n ≥ 1 → HTPI.Exercises.rep_comp R n ⊆ S"} +{"name":"HTPI.Exercises.nonzero_is_successor","declaration":"theorem HTPI.Exercises.nonzero_is_successor (n : ℕ) : n ≠ 0 → ∃ m, n = m + 1"} +{"name":"HTPI.Exercises.extendPO_is_ref","declaration":"theorem HTPI.Exercises.extendPO_is_ref {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) : HTPI.reflexive (HTPI.extendPO R b)"} +{"name":"HTPI.Exercises.quot_rem_unique","declaration":"theorem HTPI.Exercises.quot_rem_unique (m : ℕ) (q : ℕ) (r : ℕ) (q' : ℕ) (r' : ℕ) (h1 : m * q + r = m * q' + r') (h2 : r < m) (h3 : r' < m) : q = q' ∧ r = r'"} +{"name":"HTPI.Exercises.seq_6_3_16","declaration":"def HTPI.Exercises.seq_6_3_16 (k : ℕ) : ℕ"} +{"name":"HTPI.extendPO_is_po","declaration":"theorem HTPI.extendPO_is_po {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) : HTPI.partial_order (HTPI.extendPO R b)"} +{"name":"HTPI.Example_6_3_2_cheating","declaration":"theorem HTPI.Example_6_3_2_cheating (a : ℝ) (m : ℕ) (n : ℕ) : a ^ (m + n) = a ^ m * a ^ n"} +{"name":"HTPI.Exercises.Fib_like_formula","declaration":"theorem HTPI.Exercises.Fib_like_formula (n : ℕ) : HTPI.Exercises.Fib_like n = 2 ^ n"} +{"name":"HTPI.Exercises.Exercise_6_1_16a2","declaration":"theorem HTPI.Exercises.Exercise_6_1_16a2 (n : ℕ) : ¬(HTPI.nat_even n ∧ HTPI.nat_odd n)"} +{"name":"HTPI.rep_image_base","declaration":"theorem HTPI.rep_image_base {A : Type} (f : A → A) (B : Set A) : HTPI.rep_image f 0 B = B"} +{"name":"HTPI.Exercises.rep_comp_one","declaration":"theorem HTPI.Exercises.rep_comp_one {A : Type} (R : Set (A × A)) : HTPI.Exercises.rep_comp R 1 = R"} +{"name":"HTPI.Exercises.Exercise_6_3_4","declaration":"theorem HTPI.Exercises.Exercise_6_3_4 (n : ℕ) : 3 * Sum i from 0 to n, (2 * i + 1) ^ 2 = (n + 1) * (2 * n + 1) * (2 * n + 3)"} +{"name":"HTPI.Exercises.rep_un_image2_base","declaration":"theorem HTPI.Exercises.rep_un_image2_base {A : Type} (f : A → A → A) (B : Set A) : HTPI.Exercises.rep_un_image2 f 0 B = B"} +{"name":"HTPI.nat_odd","declaration":"def HTPI.nat_odd (n : ℕ) : Prop"} +{"name":"HTPI.Exercises.quot_rem_unique_lemma","declaration":"theorem HTPI.Exercises.quot_rem_unique_lemma {m : ℕ} {q : ℕ} {r : ℕ} {q' : ℕ} {r' : ℕ} (h1 : m * q + r = m * q' + r') (h2 : r' < m) : q ≤ q'"} +{"name":"HTPI.Example_6_3_1","declaration":"theorem HTPI.Example_6_3_1 (n : ℕ) : n ≥ 4 → HTPI.fact n > 2 ^ n"} +{"name":"HTPI.Exercises.idExt","declaration":"def HTPI.Exercises.idExt (A : Type) : Set (A × A)"} +{"name":"HTPI.Exercises.rep_comp","declaration":"def HTPI.Exercises.rep_comp {A : Type} (R : Set (A × A)) (n : ℕ) : Set (A × A)"} +{"name":"HTPI.Exercises.Exercise_6_1_16a1","declaration":"theorem HTPI.Exercises.Exercise_6_1_16a1 (n : ℕ) : HTPI.nat_even n ∨ HTPI.nat_odd n"} +{"name":"HTPI.Exercises.Like_Exercise_6_1_4","declaration":"theorem HTPI.Exercises.Like_Exercise_6_1_4 (n : ℕ) : Sum i from 0 to n, 2 * i + 1 = (n + 1) ^ 2"} +{"name":"HTPI.Exercises.Exercise_6_5_8b","declaration":"theorem HTPI.Exercises.Exercise_6_5_8b {A : Type} (f : A → A → A) (B : Set A) : HTPI.closure2 f B (HTPI.Exercises.cumul_un_image2 f B)"} +{"name":"HTPI.Exercises.cumul_image2","declaration":"def HTPI.Exercises.cumul_image2 {A : Type} (f : A → A → A) (B : Set A) : Set A"} +{"name":"HTPI.Exercises.rep_image2_base","declaration":"theorem HTPI.Exercises.rep_image2_base {A : Type} (f : A → A → A) (B : Set A) : HTPI.Exercises.rep_image2 f 0 B = B"} +{"name":"HTPI.extendPO","declaration":"def HTPI.extendPO {A : Type} (R : HTPI.BinRel A) (b : A) (x : A) (y : A) : Prop"} +{"name":"HTPI.rep_image_step","declaration":"theorem HTPI.rep_image_step {A : Type} (f : A → A) (n : ℕ) (B : Set A) : HTPI.rep_image f (n + 1) B = HTPI.image f (HTPI.rep_image f n B)"} +{"name":"HTPI.extendPO_is_trans","declaration":"theorem HTPI.extendPO_is_trans {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) : HTPI.transitive (HTPI.extendPO R b)"} +{"name":"HTPI.Exercises.Exercise_6_5_6","declaration":"theorem HTPI.Exercises.Exercise_6_5_6 {A : Type} (f : A → A → A) (B : Set A) : HTPI.closed2 f (HTPI.Exercises.cumul_image2 f B)"} +{"name":"HTPI.exists_eq_add_one_of_ne_zero","declaration":"theorem HTPI.exists_eq_add_one_of_ne_zero {n : ℕ} (h1 : n ≠ 0) : ∃ k, n = k + 1"} +{"name":"HTPI.Like_Example_6_1_2","declaration":"theorem HTPI.Like_Example_6_1_2 (n : ℕ) : 3 ∣ n ^ 3 + 2 * n"} +{"name":"HTPI.Example_6_3_2","declaration":"theorem HTPI.Example_6_3_2 (a : ℝ) (m : ℕ) (n : ℕ) : a ^ (m + n) = a ^ m * a ^ n"} +{"name":"HTPI.well_ord_princ","declaration":"theorem HTPI.well_ord_princ (S : Set ℕ) : (∃ n, n ∈ S) → ∃ n ∈ S, ∀ m ∈ S, n ≤ m"} +{"name":"HTPI.Exercises.Like_Exercise_6_1_1","declaration":"theorem HTPI.Exercises.Like_Exercise_6_1_1 (n : ℕ) : 2 * Sum i from 0 to n, i = n * (n + 1)"} +{"name":"HTPI.Exercises.rep_un_image2","declaration":"def HTPI.Exercises.rep_un_image2 {A : Type} (f : A → A → A) (n : ℕ) (B : Set A) : Set A"} +{"name":"HTPI.Example_6_3_4","declaration":"theorem HTPI.Example_6_3_4 (x : ℝ) : x > -1 → ∀ (n : ℕ), (1 + x) ^ n ≥ 1 + ↑n * x"} +{"name":"HTPI.Exercises.Exercise_6_4_8d","declaration":"theorem HTPI.Exercises.Exercise_6_4_8d (m : ℕ) (k : ℕ) : HTPI.Fib m ∣ HTPI.Fib (m * k)"} +{"name":"HTPI.Theorem_6_5_1","declaration":"theorem HTPI.Theorem_6_5_1 {A : Type} (f : A → A) (B : Set A) : HTPI.closure f B (HTPI.cumul_image f B)"} +{"name":"HTPI.Exercises.Fib_like","declaration":"def HTPI.Exercises.Fib_like (n : ℕ) : ℕ"} +{"name":"HTPI.Exercises.extendPO_is_trans","declaration":"theorem HTPI.Exercises.extendPO_is_trans {A : Type} (R : HTPI.BinRel A) (b : A) (h : HTPI.partial_order R) : HTPI.transitive (HTPI.extendPO R b)"} +{"name":"HTPI.Exercises.rep_image_family_sub_closed","declaration":"theorem HTPI.Exercises.rep_image_family_sub_closed {A : Type} (F : Set (A → A)) (B : Set A) (D : Set A) (h1 : B ⊆ D) (h2 : HTPI.closed_family F D) (n : ℕ) : HTPI.Exercises.rep_image_family F n B ⊆ D"} +{"name":"HTPI.Exercises.rep_image2_step","declaration":"theorem HTPI.Exercises.rep_image2_step {A : Type} (f : A → A → A) (n : ℕ) (B : Set A) : HTPI.Exercises.rep_image2 f (n + 1) B = HTPI.Exercises.image2 f (HTPI.Exercises.rep_image2 f n B)"} +{"name":"HTPI.Exercises.triple_rec","declaration":"def HTPI.Exercises.triple_rec (n : ℕ) : ℕ"} +{"name":"HTPI.Exercises.closed_lemma","declaration":"theorem HTPI.Exercises.closed_lemma {A : Type} {f : A → A → A} {B : Set A} {x : A} {y : A} {nx : ℕ} {ny : ℕ} {n : ℕ} (h1 : x ∈ HTPI.Exercises.rep_un_image2 f nx B) (h2 : y ∈ HTPI.Exercises.rep_un_image2 f ny B) (h3 : nx ≤ n) (h4 : ny ≤ n) : f x y ∈ HTPI.Exercises.cumul_un_image2 f B"} +{"name":"HTPI.Exercises.Exercise_6_3_15","declaration":"theorem HTPI.Exercises.Exercise_6_3_15 (n : ℕ) : HTPI.Exercises.seq_6_3_15 n = 2 ^ n - ↑n - 1"} +{"name":"HTPI.exists_eq_add_two_of_ne_zero_one","declaration":"theorem HTPI.exists_eq_add_two_of_ne_zero_one {n : ℕ} (h1 : n ≠ 0) (h2 : n ≠ 1) : ∃ k, n = k + 2"} +{"name":"HTPI.sq_even_iff_even","declaration":"theorem HTPI.sq_even_iff_even (n : ℕ) : HTPI.nat_even (n * n) ↔ HTPI.nat_even n"} +{"name":"HTPI.Exercises.rep_image_family_base","declaration":"theorem HTPI.Exercises.rep_image_family_base {A : Type} (F : Set (A → A)) (B : Set A) : HTPI.Exercises.rep_image_family F 0 B = B"} +{"name":"HTPI.Exercises.Exercise_6_1_9a","declaration":"theorem HTPI.Exercises.Exercise_6_1_9a (n : ℕ) : 2 ∣ n ^ 2 + n"} +{"name":"HTPI.nat_even","declaration":"def HTPI.nat_even (n : ℕ) : Prop"} +{"name":"HTPI.Exercise_6_2_2","declaration":"theorem HTPI.Exercise_6_2_2 {A : Type} (R : HTPI.BinRel A) (h : HTPI.partial_order R) (n : ℕ) (B : Set A) : HTPI.numElts B n → ∃ T, HTPI.partial_order T ∧ (∀ (x y : A), R x y → T x y) ∧ ∀ x ∈ B, ∀ (y : A), T x y ∨ T y x"} +{"name":"HTPI.Exercises.cumul_image_family","declaration":"def HTPI.Exercises.cumul_image_family {A : Type} (F : Set (A → A)) (B : Set A) : Set A"} +{"name":"HTPI.Theorem_6_4_5","declaration":"theorem HTPI.Theorem_6_4_5 : ¬∃ q p, p * p = 2 * (q * q) ∧ q ≠ 0"} +{"name":"HTPI.Lemma_6_2_1_2","declaration":"theorem HTPI.Lemma_6_2_1_2 {A : Type} {R : HTPI.BinRel A} {B : Set A} {b : A} {c : A} (h1 : HTPI.partial_order R) (h2 : b ∈ B) (h3 : HTPI.minimalElt R c (B \\ {b})) (h4 : ¬R b c) : HTPI.minimalElt R c B"} +{"name":"HTPI.Example_6_2_1","declaration":"theorem HTPI.Example_6_2_1 {A : Type} (R : HTPI.BinRel A) (h : HTPI.partial_order R) (n : ℕ) : n ≥ 1 → ∀ (B : Set A), HTPI.numElts B n → ∃ x, HTPI.minimalElt R x B"} diff --git a/htpi-declarations/HTPILib.Chap7.jsonl b/htpi-declarations/HTPILib.Chap7.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..89f5c4eb926fd11e565512a80c2a5e19fb53916c --- /dev/null +++ b/htpi-declarations/HTPILib.Chap7.jsonl @@ -0,0 +1,213 @@ +{"name":"HTPI.nondec_cons","declaration":"theorem HTPI.nondec_cons (n : ℕ) (L : List ℕ) : HTPI.nondec (n :: L) ↔ (∀ m ∈ L, n ≤ m) ∧ HTPI.nondec L"} +{"name":"HTPI.gcd_c2_inv","declaration":"theorem HTPI.gcd_c2_inv {m : ℕ} {a : ℕ} (h1 : HTPI.rel_prime m a) : [↑a]_m * [HTPI.gcd_c2 m a]_m = [1]_m"} +{"name":"HTPI.Theorem_7_3_1","declaration":"theorem HTPI.Theorem_7_3_1 (m : ℕ) [NeZero m] (a : ℤ) : ∃! r, 0 ≤ r ∧ r < ↑m ∧ a ≡ r (MOD m)"} +{"name":"HTPI.num_rp_below_base","declaration":"theorem HTPI.num_rp_below_base {m : ℕ} : HTPI.num_rp_below m 0 = 0"} +{"name":"HTPI.Euler.F","declaration":"def HTPI.Euler.F (m : ℕ) (i : ℕ) : ZMod m"} +{"name":"HTPI.congr_symm","declaration":"theorem HTPI.congr_symm {m : ℕ} {a : ℤ} {b : ℤ} : a ≡ b (MOD m) → b ≡ a (MOD m)"} +{"name":"HTPI.le_nonzero_prod_left","declaration":"theorem HTPI.le_nonzero_prod_left {a : ℕ} {b : ℕ} (h : a * b ≠ 0) : a ≤ a * b"} +{"name":"HTPI.one_one_below","declaration":"def HTPI.one_one_below (n : ℕ) (g : ℕ → ℕ) : Prop"} +{"name":"HTPI.Theorem_7_2_3","declaration":"theorem HTPI.Theorem_7_2_3 {a : ℕ} {b : ℕ} {p : ℕ} (h1 : HTPI.prime p) (h2 : p ∣ a * b) : p ∣ a ∨ p ∣ b"} +{"name":"HTPI.cc_eq_iff_sub_zero","declaration":"theorem HTPI.cc_eq_iff_sub_zero (m : ℕ) (a : ℤ) (b : ℤ) : [a]_m = [b]_m ↔ [a - b]_m = [0]_m"} +{"name":"HTPI.all_prime_cons","declaration":"theorem HTPI.all_prime_cons (n : ℕ) (L : List ℕ) : HTPI.all_prime (n :: L) ↔ HTPI.prime n ∧ HTPI.all_prime L"} +{"name":"HTPI.swap_values","declaration":"theorem HTPI.swap_values (u : ℕ) (v : ℕ) (i : ℕ) : HTPI.swap u v i = v ∨ HTPI.swap u v i = u ∨ HTPI.swap u v i = i"} +{"name":"HTPI.swap_onto_below","declaration":"theorem HTPI.swap_onto_below {u : ℕ} {v : ℕ} {n : ℕ} (h1 : u < n) (h2 : v < n) : HTPI.onto_below n (HTPI.swap u v)"} +{"name":"HTPI.eq_one_of_dvd_one","declaration":"theorem HTPI.eq_one_of_dvd_one {n : ℕ} (h : n ∣ 1) : n = 1"} +{"name":"HTPI.swap_maps_below","declaration":"theorem HTPI.swap_maps_below {u : ℕ} {v : ℕ} {n : ℕ} (h1 : u < n) (h2 : v < n) : HTPI.maps_below n (HTPI.swap u v)"} +{"name":"HTPI.congr_mod_mod","declaration":"theorem HTPI.congr_mod_mod (m : ℕ) (a : ℤ) : a ≡ a % ↑m (MOD m)"} +{"name":"HTPI.mod_cmpl_res","declaration":"theorem HTPI.mod_cmpl_res (m : ℕ) [NeZero m] (a : ℤ) : 0 ≤ a % ↑m ∧ a % ↑m < ↑m ∧ a ≡ a % ↑m (MOD m)"} +{"name":"HTPI.Exercises.Lemma_7_4_6","declaration":"theorem HTPI.Exercises.Lemma_7_4_6 {a : ℕ} {b : ℕ} {c : ℕ} : HTPI.rel_prime (a * b) c ↔ HTPI.rel_prime a c ∧ HTPI.rel_prime b c"} +{"name":"HTPI.Exercises.Lemma_7_1_10c","declaration":"theorem HTPI.Exercises.Lemma_7_1_10c {a : ℕ} {b : ℕ} (h1 : a ∣ b) (h2 : b ∣ a) : a = b"} +{"name":"HTPI.gcd","declaration":"def HTPI.gcd (a : ℕ) (b : ℕ) : ℕ"} +{"name":"HTPI.Theorem_7_3_6_7","declaration":"theorem HTPI.Theorem_7_3_6_7 {m : ℕ} (X : ZMod m) : X * [1]_m = X"} +{"name":"HTPI.exists_cons_of_length_eq_succ","declaration":"theorem HTPI.exists_cons_of_length_eq_succ {A : Type} {l : List A} {n : ℕ} (h : List.length l = n + 1) : ∃ a L, l = a :: L ∧ List.length L = n"} +{"name":"HTPI.prime_factorization","declaration":"def HTPI.prime_factorization (n : ℕ) (l : List ℕ) : Prop"} +{"name":"HTPI.gcd_dvd_left","declaration":"theorem HTPI.gcd_dvd_left (a : ℕ) (b : ℕ) : HTPI.gcd a b ∣ a"} +{"name":"HTPI.fund_thm_arith","declaration":"theorem HTPI.fund_thm_arith (n : ℕ) (h : n ≥ 1) : ∃! l, HTPI.prime_factorization n l"} +{"name":"HTPI.num_rp_below","declaration":"def HTPI.num_rp_below (m : ℕ) (k : ℕ) : ℕ"} +{"name":"HTPI.break_prod","declaration":"theorem HTPI.break_prod {m : ℕ} (n : ℕ) (f : ℕ → ZMod m) (j : ℕ) : HTPI.prod_seq (n + j) 0 f = HTPI.prod_seq n 0 f * HTPI.prod_seq j n f"} +{"name":"HTPI.Exercises.gcd_greatest","declaration":"theorem HTPI.Exercises.gcd_greatest {a : ℕ} {b : ℕ} {d : ℕ} (h1 : HTPI.gcd a b ≠ 0) (h2 : d ∣ a) (h3 : d ∣ b) : d ≤ HTPI.gcd a b"} +{"name":"HTPI.FG_prod","declaration":"theorem HTPI.FG_prod {m : ℕ} {a : ℕ} (h1 : HTPI.rel_prime m a) (k : ℕ) : HTPI.prod_seq k 0 (HTPI.Euler.F m ∘ HTPI.Euler.G m a) =\n [↑a]_m ^ HTPI.num_rp_below m k * HTPI.prod_seq k 0 (HTPI.Euler.F m)"} +{"name":"HTPI.FG_not_rp","declaration":"theorem HTPI.FG_not_rp {m : ℕ} {a : ℕ} {i : ℕ} (h1 : HTPI.rel_prime m a) (h2 : ¬HTPI.rel_prime m i) : HTPI.Euler.F m (HTPI.Euler.G m a i) = [1]_m"} +{"name":"HTPI.mod_nonzero_lt","declaration":"theorem HTPI.mod_nonzero_lt (a : ℕ) {b : ℕ} (h : b ≠ 0) : a % b < b"} +{"name":"HTPI.«term[_]__»","declaration":"def HTPI.«term[_]__» : Lean.ParserDescr"} +{"name":"HTPI.Theorem_7_4_2","declaration":"theorem HTPI.Theorem_7_4_2 {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) : [↑a]_m ^ HTPI.phi m = [1]_m"} +{"name":"HTPI.Euler's_theorem","declaration":"theorem HTPI.Euler's_theorem {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) : ↑a ^ HTPI.phi m ≡ 1 (MOD m)"} +{"name":"HTPI.G_maps_below","declaration":"theorem HTPI.G_maps_below (m : ℕ) (a : ℕ) [NeZero m] : HTPI.maps_below m (HTPI.Euler.G m a)"} +{"name":"HTPI.cc_mul_inv_mod_eq_one","declaration":"theorem HTPI.cc_mul_inv_mod_eq_one {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) : [↑a]_m * [↑(HTPI.inv_mod m a)]_m = [1]_m"} +{"name":"HTPI.swap_prod_eq_prod_below","declaration":"theorem HTPI.swap_prod_eq_prod_below {m : ℕ} {u : ℕ} {n : ℕ} (f : ℕ → ZMod m) (h1 : u ≤ n) : HTPI.prod_seq u 0 (f ∘ HTPI.swap u n) = HTPI.prod_seq u 0 f"} +{"name":"HTPI.Exercises.prod_nonzero_nonzero","declaration":"theorem HTPI.Exercises.prod_nonzero_nonzero (l : List ℕ) : (∀ a ∈ l, a ≠ 0) → HTPI.prod l ≠ 0"} +{"name":"HTPI.prime_NeZero","declaration":"theorem HTPI.prime_NeZero {p : ℕ} (h : HTPI.prime p) : NeZero p"} +{"name":"HTPI.cc_mod_0","declaration":"theorem HTPI.cc_mod_0 (a : ℤ) : [a]_0 = a"} +{"name":"HTPI.Exercises.Exercise_7_3_4a","declaration":"theorem HTPI.Exercises.Exercise_7_3_4a {m : ℕ} (Z1 : ZMod m) (Z2 : ZMod m) (h1 : ∀ (X : ZMod m), X + Z1 = X) (h2 : ∀ (X : ZMod m), X + Z2 = X) : Z1 = Z2"} +{"name":"HTPI.Ginv_maps_below","declaration":"theorem HTPI.Ginv_maps_below (m : ℕ) (a : ℕ) [NeZero m] : HTPI.maps_below m (HTPI.Euler.Ginv m a)"} +{"name":"HTPI.prod_one","declaration":"theorem HTPI.prod_one {m : ℕ} (k : ℕ) (f : ℕ → ZMod m) : HTPI.prod_seq 1 k f = f k"} +{"name":"HTPI.right_inv_onto_below","declaration":"theorem HTPI.right_inv_onto_below {n : ℕ} {g : ℕ → ℕ} {g' : ℕ → ℕ} (h1 : ∀ i < n, g (g' i) = i) (h2 : HTPI.maps_below n g') : HTPI.onto_below n g"} +{"name":"HTPI.swap_one_one_below","declaration":"theorem HTPI.swap_one_one_below (u : ℕ) (v : ℕ) (n : ℕ) : HTPI.one_one_below n (HTPI.swap u v)"} +{"name":"HTPI.all_prime","declaration":"def HTPI.all_prime (l : List ℕ) : Prop"} +{"name":"HTPI.Theorem_7_2_5","declaration":"theorem HTPI.Theorem_7_2_5 (l1 : List ℕ) (l2 : List ℕ) : HTPI.nondec_prime_list l1 → HTPI.nondec_prime_list l2 → HTPI.prod l1 = HTPI.prod l2 → l1 = l2"} +{"name":"HTPI.mul_class","declaration":"theorem HTPI.mul_class (m : ℕ) (a : ℤ) (b : ℤ) : [a]_m * [b]_m = [a * b]_m"} +{"name":"HTPI.phi_def","declaration":"theorem HTPI.phi_def (m : ℕ) : HTPI.phi m = HTPI.num_rp_below m m"} +{"name":"HTPI.mod_lt","declaration":"theorem HTPI.mod_lt (m : ℕ) [NeZero m] (a : ℤ) : a % ↑m < ↑m"} +{"name":"HTPI.swap_swap","declaration":"theorem HTPI.swap_swap (u : ℕ) (v : ℕ) (n : ℕ) (i : ℕ) : i < n → HTPI.swap u v (HTPI.swap u v i) = i"} +{"name":"HTPI.Exercises.Exercise_7_2_17a","declaration":"theorem HTPI.Exercises.Exercise_7_2_17a (a : ℕ) (b : ℕ) (c : ℕ) : HTPI.gcd a (b * c) ∣ HTPI.gcd a b * HTPI.gcd a c"} +{"name":"HTPI.FG_rp","declaration":"theorem HTPI.FG_rp {m : ℕ} {a : ℕ} {i : ℕ} (h1 : HTPI.rel_prime m a) (h2 : HTPI.rel_prime m i) : HTPI.Euler.F m (HTPI.Euler.G m a i) = [↑a]_m * HTPI.Euler.F m i"} +{"name":"HTPI.phi","declaration":"def HTPI.phi (m : ℕ) : ℕ"} +{"name":"HTPI.break_prod_twice","declaration":"theorem HTPI.break_prod_twice {m : ℕ} {u : ℕ} {j : ℕ} {n : ℕ} (f : ℕ → ZMod m) (h1 : n = u + 1 + j) : HTPI.prod_seq (n + 1) 0 f = HTPI.prod_seq u 0 f * f u * HTPI.prod_seq j (u + 1) f * f n"} +{"name":"HTPI.prod_eq_fun","declaration":"theorem HTPI.prod_eq_fun {m : ℕ} (f : ℕ → ZMod m) (g : ℕ → ZMod m) (k : ℕ) (n : ℕ) : (∀ i < n, f (k + i) = g (k + i)) → HTPI.prod_seq n k f = HTPI.prod_seq n k g"} +{"name":"HTPI.comp_perm_below","declaration":"theorem HTPI.comp_perm_below {n : ℕ} {f : ℕ → ℕ} {g : ℕ → ℕ} (h1 : HTPI.perm_below n f) (h2 : HTPI.perm_below n g) : HTPI.perm_below n (f ∘ g)"} +{"name":"HTPI.Theorem_7_2_2_Int","declaration":"theorem HTPI.Theorem_7_2_2_Int {a : ℕ} {c : ℕ} {b : ℤ} (h1 : ↑c ∣ ↑a * b) (h2 : HTPI.rel_prime a c) : ↑c ∣ b"} +{"name":"HTPI.Lemma_7_4_5","declaration":"theorem HTPI.Lemma_7_4_5 {m : ℕ} {n : ℕ} (a : ℤ) (b : ℤ) (h1 : HTPI.rel_prime m n) : a ≡ b (MOD m * n) ↔ a ≡ b (MOD m) ∧ a ≡ b (MOD n)"} +{"name":"HTPI.F_rp_def","declaration":"theorem HTPI.F_rp_def {m : ℕ} {i : ℕ} (h : HTPI.rel_prime m i) : HTPI.Euler.F m i = [↑i]_m"} +{"name":"HTPI.congr_trans","declaration":"theorem HTPI.congr_trans {m : ℕ} {a : ℤ} {b : ℤ} {c : ℤ} : a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m)"} +{"name":"HTPI.Exercises.congr_iff_mod_eq_Nat","declaration":"theorem HTPI.Exercises.congr_iff_mod_eq_Nat (m : ℕ) (a : ℕ) (b : ℕ) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m"} +{"name":"HTPI.left_inv_one_one_below","declaration":"theorem HTPI.left_inv_one_one_below {n : ℕ} {g : ℕ → ℕ} {g' : ℕ → ℕ} (h1 : ∀ i < n, g' (g i) = i) : HTPI.one_one_below n g"} +{"name":"HTPI.rel_prime","declaration":"def HTPI.rel_prime (a : ℕ) (b : ℕ) : Prop"} +{"name":"HTPI.invertible","declaration":"def HTPI.invertible {m : ℕ} (X : ZMod m) : Prop"} +{"name":"HTPI.prod_nil","declaration":"theorem HTPI.prod_nil : HTPI.prod [] = 1"} +{"name":"HTPI.gcd_dvd","declaration":"theorem HTPI.gcd_dvd (b : ℕ) (a : ℕ) : HTPI.gcd a b ∣ a ∧ HTPI.gcd a b ∣ b"} +{"name":"HTPI.Exercises.gcd_comm_lt","declaration":"theorem HTPI.Exercises.gcd_comm_lt {a : ℕ} {b : ℕ} (h : a < b) : HTPI.gcd a b = HTPI.gcd b a"} +{"name":"HTPI.gcd_c1","declaration":"def HTPI.gcd_c1 (a : ℕ) (b : ℕ) : ℤ"} +{"name":"HTPI.swap_prod_eq_prod","declaration":"theorem HTPI.swap_prod_eq_prod {m : ℕ} {u : ℕ} {n : ℕ} (f : ℕ → ZMod m) (h1 : u ≤ n) : HTPI.prod_seq (n + 1) 0 (f ∘ HTPI.swap u n) = HTPI.prod_seq (n + 1) 0 f"} +{"name":"HTPI.F_not_rp_def","declaration":"theorem HTPI.F_not_rp_def {m : ℕ} {i : ℕ} (h : ¬HTPI.rel_prime m i) : HTPI.Euler.F m i = [1]_m"} +{"name":"HTPI.F_invertible","declaration":"theorem HTPI.F_invertible (m : ℕ) (i : ℕ) : HTPI.invertible (HTPI.Euler.F m i)"} +{"name":"HTPI.val_zero","declaration":"theorem HTPI.val_zero (n : ℕ) : ZMod.val [0]_(n + 1) = 0"} +{"name":"HTPI.mod_nonneg","declaration":"theorem HTPI.mod_nonneg (m : ℕ) [NeZero m] (a : ℤ) : 0 ≤ a % ↑m"} +{"name":"HTPI.Exercises.Exercise_7_3_4b","declaration":"theorem HTPI.Exercises.Exercise_7_3_4b {m : ℕ} (X : ZMod m) (Y1 : ZMod m) (Y2 : ZMod m) (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2"} +{"name":"HTPI.perm_below","declaration":"def HTPI.perm_below (n : ℕ) (g : ℕ → ℕ) : Prop"} +{"name":"HTPI.«term_≡_(MOD_)»","declaration":"def HTPI.«term_≡_(MOD_)» : Lean.TrailingParserDescr"} +{"name":"HTPI.trivial_swap","declaration":"theorem HTPI.trivial_swap (u : ℕ) : HTPI.swap u u = id"} +{"name":"HTPI.swap_snd","declaration":"theorem HTPI.swap_snd (u : ℕ) (v : ℕ) : HTPI.swap u v v = u"} +{"name":"HTPI.gcd_c2_base","declaration":"theorem HTPI.gcd_c2_base (a : ℕ) : HTPI.gcd_c2 a 0 = 0"} +{"name":"HTPI.phi_prime","declaration":"theorem HTPI.phi_prime {p : ℕ} (h1 : HTPI.prime p) : HTPI.phi p = p - 1"} +{"name":"HTPI.prod_seq","declaration":"def HTPI.prod_seq {m : ℕ} (j : ℕ) (k : ℕ) (f : ℕ → ZMod m) : ZMod m"} +{"name":"HTPI.inv_mod","declaration":"def HTPI.inv_mod (m : ℕ) (a : ℕ) : ℕ"} +{"name":"HTPI.Exercises.Exercise_7_5_13a","declaration":"theorem HTPI.Exercises.Exercise_7_5_13a (a : ℕ) (h1 : HTPI.rel_prime 561 a) : ↑a ^ 560 ≡ 1 (MOD 3)"} +{"name":"HTPI.Theorem_7_3_7","declaration":"theorem HTPI.Theorem_7_3_7 (m : ℕ) (a : ℕ) : HTPI.invertible [↑a]_m ↔ HTPI.rel_prime m a"} +{"name":"HTPI.exists_least_prime_factor","declaration":"theorem HTPI.exists_least_prime_factor {n : ℕ} (h : 2 ≤ n) : ∃ p, HTPI.prime_factor p n ∧ ∀ (q : ℕ), HTPI.prime_factor q n → p ≤ q"} +{"name":"HTPI.nondec","declaration":"def HTPI.nondec (l : List ℕ) : Prop"} +{"name":"HTPI.swap_other","declaration":"theorem HTPI.swap_other {u : ℕ} {v : ℕ} {i : ℕ} (h1 : i ≠ u) (h2 : i ≠ v) : HTPI.swap u v i = i"} +{"name":"HTPI.rel_prime_symm","declaration":"theorem HTPI.rel_prime_symm {a : ℕ} {b : ℕ} (h : HTPI.rel_prime a b) : HTPI.rel_prime b a"} +{"name":"HTPI.Exercises.rel_prime_iff_no_common_factor","declaration":"theorem HTPI.Exercises.rel_prime_iff_no_common_factor (a : ℕ) (b : ℕ) : HTPI.rel_prime a b ↔ ¬∃ p, HTPI.prime p ∧ p ∣ a ∧ p ∣ b"} +{"name":"HTPI.num_rp_below_step_not_rp","declaration":"theorem HTPI.num_rp_below_step_not_rp {m : ℕ} {j : ℕ} (h : ¬HTPI.rel_prime m j) : HTPI.num_rp_below m (j + 1) = HTPI.num_rp_below m j"} +{"name":"HTPI.Exercises.Exercise_7_5_14b","declaration":"theorem HTPI.Exercises.Exercise_7_5_14b (n : ℕ) (b : ℤ) (h1 : HTPI.prime n) (h2 : b ^ 2 ≡ 1 (MOD n)) : b ≡ 1 (MOD n) ∨ b ≡ -1 (MOD n)"} +{"name":"HTPI.G_one_one_below","declaration":"theorem HTPI.G_one_one_below {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) : HTPI.one_one_below m (HTPI.Euler.G m a)"} +{"name":"HTPI.Exercises.Theorem_7_3_6_3","declaration":"theorem HTPI.Exercises.Theorem_7_3_6_3 {m : ℕ} (X : ZMod m) : X + [0]_m = X"} +{"name":"HTPI.exists_prime_factorization","declaration":"theorem HTPI.exists_prime_factorization (n : ℕ) : n ≥ 1 → ∃ l, HTPI.prime_factorization n l"} +{"name":"HTPI.gcd_c1_base","declaration":"theorem HTPI.gcd_c1_base (a : ℕ) : HTPI.gcd_c1 a 0 = 1"} +{"name":"HTPI.cons_prod_not_one","declaration":"theorem HTPI.cons_prod_not_one {p : ℕ} {l : List ℕ} (h : HTPI.nondec_prime_list (p :: l)) : HTPI.prod (p :: l) ≠ 1"} +{"name":"HTPI.Exercises.congr_rel_prime","declaration":"theorem HTPI.Exercises.congr_rel_prime {m : ℕ} {a : ℕ} {b : ℕ} (h1 : ↑a ≡ ↑b (MOD m)) : HTPI.rel_prime m a ↔ HTPI.rel_prime m b"} +{"name":"HTPI.Exercises.rel_prime_mod","declaration":"theorem HTPI.Exercises.rel_prime_mod (m : ℕ) (a : ℕ) : HTPI.rel_prime m (a % m) ↔ HTPI.rel_prime m a"} +{"name":"HTPI.gcd_c2","declaration":"def HTPI.gcd_c2 (a : ℕ) (b : ℕ) : ℤ"} +{"name":"HTPI.G_onto_below","declaration":"theorem HTPI.G_onto_below {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) : HTPI.onto_below m (HTPI.Euler.G m a)"} +{"name":"HTPI.Exercises.three_prime","declaration":"theorem HTPI.Exercises.three_prime : HTPI.prime 3"} +{"name":"HTPI.G_def","declaration":"theorem HTPI.G_def (m : ℕ) (a : ℕ) (i : ℕ) : HTPI.Euler.G m a i = a * i % m"} +{"name":"HTPI.perm_prod","declaration":"theorem HTPI.perm_prod {m : ℕ} (f : ℕ → ZMod m) (n : ℕ) (g : ℕ → ℕ) : HTPI.perm_below n g → HTPI.prod_seq n 0 f = HTPI.prod_seq n 0 (f ∘ g)"} +{"name":"HTPI.Exercises.gcd_is_nonzero","declaration":"theorem HTPI.Exercises.gcd_is_nonzero {a : ℕ} {b : ℕ} (h : a ≠ 0 ∨ b ≠ 0) : HTPI.gcd a b ≠ 0"} +{"name":"HTPI.val_nat_eq_mod","declaration":"theorem HTPI.val_nat_eq_mod (n : ℕ) (k : ℕ) : ZMod.val [↑k]_(n + 1) = k % (n + 1)"} +{"name":"HTPI.mul_mod_mod_eq_mul_mod","declaration":"theorem HTPI.mul_mod_mod_eq_mul_mod (m : ℕ) (a : ℕ) (b : ℕ) : a * (b % m) % m = a * b % m"} +{"name":"HTPI.ge_one_of_prod_one","declaration":"theorem HTPI.ge_one_of_prod_one {a : ℕ} {b : ℕ} (h : a * b = 1) : a ≥ 1"} +{"name":"HTPI.swap_prod_eq_prod_between","declaration":"theorem HTPI.swap_prod_eq_prod_between {m : ℕ} {u : ℕ} {j : ℕ} {n : ℕ} (f : ℕ → ZMod m) (h1 : n = u + 1 + j) : HTPI.prod_seq j (u + 1) (f ∘ HTPI.swap u n) = HTPI.prod_seq j (u + 1) f"} +{"name":"HTPI.cc","declaration":"def HTPI.cc (m : ℕ) (a : ℤ) : ZMod m"} +{"name":"HTPI.Exercises.Lemma_7_1_10b","declaration":"theorem HTPI.Exercises.Lemma_7_1_10b {a : ℕ} {b : ℕ} {n : ℕ} (h1 : n ≠ 0) (h2 : n * a ∣ n * b) : a ∣ b"} +{"name":"HTPI.rel_prime_of_prime_not_dvd","declaration":"theorem HTPI.rel_prime_of_prime_not_dvd {a : ℕ} {p : ℕ} (h1 : HTPI.prime p) (h2 : ¬p ∣ a) : HTPI.rel_prime a p"} +{"name":"HTPI.first_le_first","declaration":"theorem HTPI.first_le_first {p : ℕ} {q : ℕ} {l : List ℕ} {m : List ℕ} (h1 : HTPI.nondec_prime_list (p :: l)) (h2 : HTPI.nondec_prime_list (q :: m)) (h3 : HTPI.prod (p :: l) = HTPI.prod (q :: m)) : p ≤ q"} +{"name":"HTPI.Exercises.num_rp_prime","declaration":"theorem HTPI.Exercises.num_rp_prime {p : ℕ} (h1 : HTPI.prime p) (k : ℕ) : k < p → HTPI.num_rp_below p (k + 1) = k"} +{"name":"HTPI.mod_succ_lt","declaration":"theorem HTPI.mod_succ_lt (a : ℕ) (n : ℕ) : a % (n + 1) < n + 1"} +{"name":"HTPI.Exercises.Lemma_7_1_10a","declaration":"theorem HTPI.Exercises.Lemma_7_1_10a {a : ℕ} {b : ℕ} (n : ℕ) (h : a ∣ b) : n * a ∣ n * b"} +{"name":"HTPI.G_perm_below","declaration":"theorem HTPI.G_perm_below {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) : HTPI.perm_below m (HTPI.Euler.G m a)"} +{"name":"HTPI.Exercises.Theorem_7_3_11","declaration":"theorem HTPI.Exercises.Theorem_7_3_11 (m : ℕ) (n : ℕ) (a : ℤ) (b : ℤ) (h1 : n ≠ 0) : ↑n * a ≡ ↑n * b (MOD n * m) ↔ a ≡ b (MOD m)"} +{"name":"HTPI.congr_mod","declaration":"def HTPI.congr_mod (m : ℕ) (a : ℤ) (b : ℤ) : Prop"} +{"name":"HTPI.Ginv_def","declaration":"theorem HTPI.Ginv_def {m : ℕ} {a : ℕ} {i : ℕ} : HTPI.Euler.Ginv m a i = HTPI.Euler.G m (HTPI.inv_mod m a) i"} +{"name":"HTPI.prod_cons","declaration":"theorem HTPI.prod_cons {n : ℕ} {L : List ℕ} : HTPI.prod (n :: L) = n * HTPI.prod L"} +{"name":"HTPI.Exercises.rel_prime_symm","declaration":"theorem HTPI.Exercises.rel_prime_symm {a : ℕ} {b : ℕ} (h : HTPI.rel_prime a b) : HTPI.rel_prime b a"} +{"name":"HTPI.dvd_prime","declaration":"theorem HTPI.dvd_prime {a : ℕ} {p : ℕ} (h1 : HTPI.prime p) (h2 : a ∣ p) : a = 1 ∨ a = p"} +{"name":"HTPI.Exercises.Exercise_7_2_6","declaration":"theorem HTPI.Exercises.Exercise_7_2_6 (a : ℕ) (b : ℕ) : HTPI.rel_prime a b ↔ ∃ s t, s * ↑a + t * ↑b = 1"} +{"name":"HTPI.prime","declaration":"def HTPI.prime (n : ℕ) : Prop"} +{"name":"HTPI.prod_seq_zero_step","declaration":"theorem HTPI.prod_seq_zero_step {m : ℕ} (n : ℕ) (f : ℕ → ZMod m) : HTPI.prod_seq (n + 1) 0 f = HTPI.prod_seq n 0 f * f n"} +{"name":"HTPI.exists_prime_factor","declaration":"theorem HTPI.exists_prime_factor (n : ℕ) : 2 ≤ n → ∃ p, HTPI.prime_factor p n"} +{"name":"HTPI.nondec_prime_list","declaration":"def HTPI.nondec_prime_list (l : List ℕ) : Prop"} +{"name":"HTPI.mod_mul_mod_eq_mul_mod","declaration":"theorem HTPI.mod_mul_mod_eq_mul_mod (m : ℕ) (a : ℕ) (b : ℕ) : a % m * b % m = a * b % m"} +{"name":"HTPI.prime_not_one","declaration":"theorem HTPI.prime_not_one {p : ℕ} (h : HTPI.prime p) : p ≠ 1"} +{"name":"HTPI.dvd_self","declaration":"theorem HTPI.dvd_self (n : ℕ) : n ∣ n"} +{"name":"HTPI.Exercises.Theorem_7_2_3_Int","declaration":"theorem HTPI.Exercises.Theorem_7_2_3_Int {p : ℕ} {a : ℤ} {b : ℤ} (h1 : HTPI.prime p) (h2 : ↑p ∣ a * b) : ↑p ∣ a ∨ ↑p ∣ b"} +{"name":"HTPI.congr_iff_mod_eq_Nat","declaration":"theorem HTPI.congr_iff_mod_eq_Nat (m : ℕ) (a : ℕ) (b : ℕ) [NeZero m] : ↑a ≡ ↑b (MOD m) ↔ a % m = b % m"} +{"name":"HTPI.dvd_trans","declaration":"theorem HTPI.dvd_trans {a : ℕ} {b : ℕ} {c : ℕ} (h1 : a ∣ b) (h2 : b ∣ c) : a ∣ c"} +{"name":"HTPI.nondec_nil","declaration":"theorem HTPI.nondec_nil : HTPI.nondec []"} +{"name":"HTPI.prod_seq_base","declaration":"theorem HTPI.prod_seq_base {m : ℕ} (k : ℕ) (f : ℕ → ZMod m) : HTPI.prod_seq 0 k f = [1]_m"} +{"name":"HTPI.Exercises.Exercise_7_3_16","declaration":"theorem HTPI.Exercises.Exercise_7_3_16 {m : ℕ} {a : ℤ} {b : ℤ} (h : a ≡ b (MOD m)) (n : ℕ) : a ^ n ≡ b ^ n (MOD m)"} +{"name":"HTPI.Exercises.dvd_a_of_dvd_b_mod","declaration":"theorem HTPI.Exercises.dvd_a_of_dvd_b_mod {a : ℕ} {b : ℕ} {d : ℕ} (h1 : d ∣ b) (h2 : d ∣ a % b) : d ∣ a"} +{"name":"HTPI.Exercises.Like_Exercise_7_4_12","declaration":"theorem HTPI.Exercises.Like_Exercise_7_4_12 {m : ℕ} {a : ℕ} {p : ℕ} {q : ℕ} {k : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) (h2 : p = q + HTPI.phi m * k) : ↑a ^ p ≡ ↑a ^ q (MOD m)"} +{"name":"HTPI.G_rp_iff","declaration":"theorem HTPI.G_rp_iff {m : ℕ} {a : ℕ} (h1 : HTPI.rel_prime m a) (i : ℕ) : HTPI.rel_prime m (HTPI.Euler.G m a i) ↔ HTPI.rel_prime m i"} +{"name":"HTPI.Exercises.left_inv_one_one_below","declaration":"theorem HTPI.Exercises.left_inv_one_one_below {n : ℕ} {g : ℕ → ℕ} {g' : ℕ → ℕ} (h1 : ∀ i < n, g' (g i) = i) : HTPI.one_one_below n g"} +{"name":"HTPI.Exercises.congr_trans","declaration":"theorem HTPI.Exercises.congr_trans {m : ℕ} {a : ℤ} {b : ℤ} {c : ℤ} : a ≡ b (MOD m) → b ≡ c (MOD m) → a ≡ c (MOD m)"} +{"name":"HTPI.Ginv_right_inv","declaration":"theorem HTPI.Ginv_right_inv {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) (i : ℕ) : i < m → HTPI.Euler.G m a (HTPI.Euler.Ginv m a i) = i"} +{"name":"HTPI.Exercises.gcd_comm","declaration":"theorem HTPI.Exercises.gcd_comm (a : ℕ) (b : ℕ) : HTPI.gcd a b = HTPI.gcd b a"} +{"name":"HTPI.Exercises.Exercise_7_1_5","declaration":"theorem HTPI.Exercises.Exercise_7_1_5 (a : ℕ) (b : ℕ) (n : ℤ) : (∃ s t, s * ↑a + t * ↑b = n) ↔ ↑(HTPI.gcd a b) ∣ n"} +{"name":"HTPI.cc_zero_iff_dvd","declaration":"theorem HTPI.cc_zero_iff_dvd (m : ℕ) (a : ℤ) : [a]_m = [0]_m ↔ ↑m ∣ a"} +{"name":"HTPI.Exercises.Theorem_7_3_6_4","declaration":"theorem HTPI.Exercises.Theorem_7_3_6_4 {m : ℕ} (X : ZMod m) : ∃ Y, X + Y = [0]_m"} +{"name":"HTPI.cc_neg_zero_of_cc_zero","declaration":"theorem HTPI.cc_neg_zero_of_cc_zero (m : ℕ) (a : ℤ) : [a]_m = [0]_m → [-a]_m = [0]_m"} +{"name":"HTPI.Exercise_7_4_5_Nat","declaration":"theorem HTPI.Exercise_7_4_5_Nat (m : ℕ) (a : ℕ) (n : ℕ) : [↑a]_m ^ n = [↑a ^ n]_m"} +{"name":"HTPI.list_elt_dvd_prod_by_length","declaration":"theorem HTPI.list_elt_dvd_prod_by_length (a : ℕ) (n : ℕ) (l : List ℕ) : List.length l = n → a ∈ l → a ∣ HTPI.prod l"} +{"name":"HTPI.Exercises.in_prime_factorization_iff_prime_factor","declaration":"theorem HTPI.Exercises.in_prime_factorization_iff_prime_factor {a : ℕ} {l : List ℕ} (h1 : HTPI.prime_factorization a l) (p : ℕ) : p ∈ l ↔ HTPI.prime_factor p a"} +{"name":"HTPI.Exercises.Exercise_7_2_5","declaration":"theorem HTPI.Exercises.Exercise_7_2_5 {a : ℕ} {b : ℕ} {l : List ℕ} {m : List ℕ} (h1 : HTPI.prime_factorization a l) (h2 : HTPI.prime_factorization b m) : HTPI.rel_prime a b ↔ ¬∃ p ∈ l, p ∈ m"} +{"name":"HTPI.prime_pos","declaration":"theorem HTPI.prime_pos {p : ℕ} (h : HTPI.prime p) : p > 0"} +{"name":"HTPI.Ginv_left_inv","declaration":"theorem HTPI.Ginv_left_inv {m : ℕ} {a : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) (i : ℕ) : i < m → HTPI.Euler.Ginv m a (HTPI.Euler.G m a i) = i"} +{"name":"HTPI.prime_factor","declaration":"def HTPI.prime_factor (p : ℕ) (n : ℕ) : Prop"} +{"name":"HTPI.congr_refl","declaration":"theorem HTPI.congr_refl (m : ℕ) (a : ℤ) : a ≡ a (MOD m)"} +{"name":"HTPI.cc_eq_iff_congr","declaration":"theorem HTPI.cc_eq_iff_congr (m : ℕ) (a : ℤ) (b : ℤ) : [a]_m = [b]_m ↔ a ≡ b (MOD m)"} +{"name":"HTPI.cc_eq_mod","declaration":"theorem HTPI.cc_eq_mod (m : ℕ) (a : ℤ) : [a]_m = [a % ↑m]_m"} +{"name":"HTPI.gcd_c2_nonzero","declaration":"theorem HTPI.gcd_c2_nonzero (a : ℕ) {b : ℕ} (h : b ≠ 0) : HTPI.gcd_c2 a b = HTPI.gcd_c1 b (a % b) - HTPI.gcd_c2 b (a % b) * ↑(a / b)"} +{"name":"HTPI.Theorem_7_1_6","declaration":"theorem HTPI.Theorem_7_1_6 {d : ℕ} {a : ℕ} {b : ℕ} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ HTPI.gcd a b"} +{"name":"HTPI.perm_below_fixed","declaration":"theorem HTPI.perm_below_fixed {n : ℕ} {g : ℕ → ℕ} (h1 : HTPI.perm_below (n + 1) g) (h2 : g n = n) : HTPI.perm_below n g"} +{"name":"HTPI.num_rp_below_step_rp","declaration":"theorem HTPI.num_rp_below_step_rp {m : ℕ} {j : ℕ} (h : HTPI.rel_prime m j) : HTPI.num_rp_below m (j + 1) = HTPI.num_rp_below m j + 1"} +{"name":"HTPI.add_class","declaration":"theorem HTPI.add_class (m : ℕ) (a : ℤ) (b : ℤ) : [a]_m + [b]_m = [a + b]_m"} +{"name":"HTPI.Exercises.perm_below_fixed","declaration":"theorem HTPI.Exercises.perm_below_fixed {n : ℕ} {g : ℕ → ℕ} (h1 : HTPI.perm_below (n + 1) g) (h2 : g n = n) : HTPI.perm_below n g"} +{"name":"HTPI.Theorem_7_5_1","declaration":"theorem HTPI.Theorem_7_5_1 (p : ℕ) (q : ℕ) (n : ℕ) (e : ℕ) (d : ℕ) (k : ℕ) (m : ℕ) (c : ℕ) (p_prime : HTPI.prime p) (q_prime : HTPI.prime q) (p_ne_q : p ≠ q) (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1) (h1 : [↑m]_n ^ e = [↑c]_n) : [↑c]_n ^ d = [↑m]_n"} +{"name":"HTPI.Fprod_invertible","declaration":"theorem HTPI.Fprod_invertible (m : ℕ) (k : ℕ) : HTPI.invertible (HTPI.prod_seq k 0 (HTPI.Euler.F m))"} +{"name":"HTPI.Theorem_7_2_2","declaration":"theorem HTPI.Theorem_7_2_2 {a : ℕ} {b : ℕ} {c : ℕ} (h1 : c ∣ a * b) (h2 : HTPI.rel_prime a c) : c ∣ b"} +{"name":"HTPI.Exercise_7_2_6","declaration":"theorem HTPI.Exercise_7_2_6 (a : ℕ) (b : ℕ) : HTPI.rel_prime a b ↔ ∃ s t, s * ↑a + t * ↑b = 1"} +{"name":"HTPI.swap","declaration":"def HTPI.swap (u : ℕ) (v : ℕ) (i : ℕ) : ℕ"} +{"name":"HTPI.eq_one_of_prod_one","declaration":"theorem HTPI.eq_one_of_prod_one {a : ℕ} {b : ℕ} (h : a * b = 1) : a = 1"} +{"name":"HTPI.Exercise_7_4_5_Int","declaration":"theorem HTPI.Exercise_7_4_5_Int (m : ℕ) (a : ℤ) (n : ℕ) : [a]_m ^ n = [a ^ n]_m"} +{"name":"HTPI.swap_fst","declaration":"theorem HTPI.swap_fst (u : ℕ) (v : ℕ) : HTPI.swap u v u = v"} +{"name":"HTPI.Exercises.comp_perm_below","declaration":"theorem HTPI.Exercises.comp_perm_below {n : ℕ} {f : ℕ → ℕ} {g : ℕ → ℕ} (h1 : HTPI.perm_below n f) (h2 : HTPI.perm_below n g) : HTPI.perm_below n (f ∘ g)"} +{"name":"HTPI.Exercises.Theorem_7_3_10","declaration":"theorem HTPI.Exercises.Theorem_7_3_10 (m : ℕ) (a : ℕ) (b : ℤ) : ¬↑(HTPI.gcd m a) ∣ b → ¬∃ x, ↑a * x ≡ b (MOD m)"} +{"name":"HTPI.Theorem_7_2_4","declaration":"theorem HTPI.Theorem_7_2_4 {p : ℕ} (h1 : HTPI.prime p) (l : List ℕ) : p ∣ HTPI.prod l → ∃ a ∈ l, p ∣ a"} +{"name":"HTPI.prod","declaration":"def HTPI.prod (l : List ℕ) : ℕ"} +{"name":"HTPI.prod_seq_step","declaration":"theorem HTPI.prod_seq_step {m : ℕ} (n : ℕ) (k : ℕ) (f : ℕ → ZMod m) : HTPI.prod_seq (n + 1) k f = HTPI.prod_seq n k f * f (k + n)"} +{"name":"HTPI.maps_below","declaration":"def HTPI.maps_below (n : ℕ) (g : ℕ → ℕ) : Prop"} +{"name":"HTPI.all_prime_nil","declaration":"theorem HTPI.all_prime_nil : HTPI.all_prime []"} +{"name":"HTPI.dvd_a_of_dvd_b_mod","declaration":"theorem HTPI.dvd_a_of_dvd_b_mod {a : ℕ} {b : ℕ} {d : ℕ} (h1 : d ∣ b) (h2 : d ∣ a % b) : d ∣ a"} +{"name":"HTPI.onto_below","declaration":"def HTPI.onto_below (n : ℕ) (g : ℕ → ℕ) : Prop"} +{"name":"HTPI.list_elt_dvd_prod","declaration":"theorem HTPI.list_elt_dvd_prod {a : ℕ} {l : List ℕ} (h : a ∈ l) : a ∣ HTPI.prod l"} +{"name":"HTPI.mul_inv_mod_cancel","declaration":"theorem HTPI.mul_inv_mod_cancel {m : ℕ} {a : ℕ} {i : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) (h2 : i < m) : a * HTPI.inv_mod m a * i % m = i"} +{"name":"HTPI.dvd_mod_of_dvd_a_b","declaration":"theorem HTPI.dvd_mod_of_dvd_a_b {a : ℕ} {b : ℕ} {d : ℕ} (h1 : d ∣ a) (h2 : d ∣ b) : d ∣ a % b"} +{"name":"HTPI.Exercises.Exercise_7_4_5_Int","declaration":"theorem HTPI.Exercises.Exercise_7_4_5_Int (m : ℕ) (a : ℤ) (n : ℕ) : [a]_m ^ n = [a ^ n]_m"} +{"name":"HTPI.Exercises.Like_Exercise_7_4_11","declaration":"theorem HTPI.Exercises.Like_Exercise_7_4_11 {m : ℕ} {a : ℕ} {p : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) (h2 : p + 1 = HTPI.phi m) : [↑a]_m * [↑a ^ p]_m = [1]_m"} +{"name":"HTPI.gcd_dvd_right","declaration":"theorem HTPI.gcd_dvd_right (a : ℕ) (b : ℕ) : HTPI.gcd a b ∣ b"} +{"name":"HTPI.cc_eq_iff_val_eq","declaration":"theorem HTPI.cc_eq_iff_val_eq {n : ℕ} (X : ZMod (n + 1)) (Y : ZMod (n + 1)) : X = Y ↔ ZMod.val X = ZMod.val Y"} +{"name":"HTPI.le_nonzero_prod_right","declaration":"theorem HTPI.le_nonzero_prod_right {a : ℕ} {b : ℕ} (h : a * b ≠ 0) : b ≤ a * b"} +{"name":"HTPI.gcd_nonzero","declaration":"theorem HTPI.gcd_nonzero (a : ℕ) {b : ℕ} (h : b ≠ 0) : HTPI.gcd a b = HTPI.gcd b (a % b)"} +{"name":"HTPI.Exercises.Exercise_7_1_6","declaration":"theorem HTPI.Exercises.Exercise_7_1_6 (a : ℕ) (b : ℕ) (c : ℕ) : HTPI.gcd a b = HTPI.gcd (a + b * c) b"} +{"name":"HTPI.Exercises.Exercise_7_2_9","declaration":"theorem HTPI.Exercises.Exercise_7_2_9 {a : ℕ} {b : ℕ} {j : ℕ} {k : ℕ} (h1 : HTPI.gcd a b ≠ 0) (h2 : a = j * HTPI.gcd a b) (h3 : b = k * HTPI.gcd a b) : HTPI.rel_prime j k"} +{"name":"HTPI.prod_inv_iff_inv","declaration":"theorem HTPI.prod_inv_iff_inv {m : ℕ} {X : ZMod m} (h1 : HTPI.invertible X) (Y : ZMod m) : HTPI.invertible (X * Y) ↔ HTPI.invertible Y"} +{"name":"HTPI.gcd_base","declaration":"theorem HTPI.gcd_base (a : ℕ) : HTPI.gcd a 0 = a"} +{"name":"HTPI.gcd_lin_comb","declaration":"theorem HTPI.gcd_lin_comb (b : ℕ) (a : ℕ) : HTPI.gcd_c1 a b * ↑a + HTPI.gcd_c2 a b * ↑b = ↑(HTPI.gcd a b)"} +{"name":"HTPI.num_rp_prime","declaration":"theorem HTPI.num_rp_prime {p : ℕ} (h1 : HTPI.prime p) (k : ℕ) : k < p → HTPI.num_rp_below p (k + 1) = k"} +{"name":"HTPI.Exercises.Exercise_7_2_7","declaration":"theorem HTPI.Exercises.Exercise_7_2_7 {a : ℕ} {b : ℕ} {a' : ℕ} {b' : ℕ} (h1 : HTPI.rel_prime a b) (h2 : a' ∣ a) (h3 : b' ∣ b) : HTPI.rel_prime a' b'"} +{"name":"HTPI.list_nil_iff_prod_one","declaration":"theorem HTPI.list_nil_iff_prod_one {l : List ℕ} (h : HTPI.nondec_prime_list l) : l = [] ↔ HTPI.prod l = 1"} +{"name":"HTPI.Euler.Ginv","declaration":"def HTPI.Euler.Ginv (m : ℕ) (a : ℕ) (i : ℕ) : ℕ"} +{"name":"HTPI.swap_perm_below","declaration":"theorem HTPI.swap_perm_below {u : ℕ} {v : ℕ} {n : ℕ} (h1 : u < n) (h2 : v < n) : HTPI.perm_below n (HTPI.swap u v)"} +{"name":"HTPI.gcd_c1_nonzero","declaration":"theorem HTPI.gcd_c1_nonzero (a : ℕ) {b : ℕ} (h : b ≠ 0) : HTPI.gcd_c1 a b = HTPI.gcd_c2 b (a % b)"} +{"name":"HTPI.cc_nat_zero_iff_dvd","declaration":"theorem HTPI.cc_nat_zero_iff_dvd (m : ℕ) (k : ℕ) : [↑k]_m = [0]_m ↔ m ∣ k"} +{"name":"HTPI.cc_G","declaration":"theorem HTPI.cc_G (m : ℕ) (a : ℕ) (i : ℕ) : [↑(HTPI.Euler.G m a i)]_m = [↑a]_m * [↑i]_m"} +{"name":"HTPI.Exercises.dvd_prime","declaration":"theorem HTPI.Exercises.dvd_prime {a : ℕ} {p : ℕ} (h1 : HTPI.prime p) (h2 : a ∣ p) : a = 1 ∨ a = p"} +{"name":"HTPI.cc_neg_zero_iff_cc_zero","declaration":"theorem HTPI.cc_neg_zero_iff_cc_zero (m : ℕ) (a : ℤ) : [-a]_m = [0]_m ↔ [a]_m = [0]_m"} +{"name":"HTPI.Theorem_7_3_6_1","declaration":"theorem HTPI.Theorem_7_3_6_1 {m : ℕ} (X : ZMod m) (Y : ZMod m) : X + Y = Y + X"} +{"name":"HTPI.Exercises.congr_iff_mod_eq_Int","declaration":"theorem HTPI.Exercises.congr_iff_mod_eq_Int (m : ℕ) (a : ℤ) (b : ℤ) [NeZero m] : a ≡ b (MOD m) ↔ a % ↑m = b % ↑m"} +{"name":"HTPI.Euler.G","declaration":"def HTPI.Euler.G (m : ℕ) (a : ℕ) (i : ℕ) : ℕ"} +{"name":"HTPI.Exercises.Exercise_7_1_10","declaration":"theorem HTPI.Exercises.Exercise_7_1_10 (a : ℕ) (b : ℕ) (n : ℕ) : HTPI.gcd (n * a) (n * b) = n * HTPI.gcd a b"} +{"name":"HTPI.prime_in_list","declaration":"theorem HTPI.prime_in_list {p : ℕ} {l : List ℕ} (h1 : HTPI.prime p) (h2 : HTPI.all_prime l) (h3 : p ∣ HTPI.prod l) : p ∈ l"} +{"name":"HTPI.nondec_prime_list_tail","declaration":"theorem HTPI.nondec_prime_list_tail {p : ℕ} {l : List ℕ} (h : HTPI.nondec_prime_list (p :: l)) : HTPI.nondec_prime_list l"} +{"name":"HTPI.Lemma_7_5_1","declaration":"theorem HTPI.Lemma_7_5_1 {p : ℕ} {e : ℕ} {d : ℕ} {m : ℕ} {c : ℕ} {s : ℕ} {t : ℤ} (h1 : HTPI.prime p) (h2 : e * d = (p - 1) * s + 1) (h3 : ↑m ^ e - ↑c = ↑p * t) : ↑c ^ d ≡ ↑m (MOD p)"} +{"name":"HTPI.cc_rep","declaration":"theorem HTPI.cc_rep {m : ℕ} (X : ZMod m) : ∃ a, X = [a]_m"} diff --git a/htpi-declarations/HTPILib.Chap8Part1.jsonl b/htpi-declarations/HTPILib.Chap8Part1.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..28f1b13e5b3d85d51be9d4430cc05da4398e466a --- /dev/null +++ b/htpi-declarations/HTPILib.Chap8Part1.jsonl @@ -0,0 +1,59 @@ +{"name":"HTPI.empty_match","declaration":"theorem HTPI.empty_match {U : Type} {V : Type} {A : Set U} {B : Set V} (h1 : HTPI.empty A) (h2 : HTPI.empty B) : HTPI.matching (HTPI.emptyRel U V) A B"} +{"name":"HTPI.finite_def","declaration":"theorem HTPI.finite_def {U : Type} (A : Set U) : HTPI.finite A ↔ ∃ n, HTPI.numElts A n"} +{"name":"HTPI.I_0_empty","declaration":"theorem HTPI.I_0_empty : HTPI.empty (HTPI.I 0)"} +{"name":"HTPI.Theorem_8_1_3_3","declaration":"theorem HTPI.Theorem_8_1_3_3 {U : Type} {V : Type} {W : Type} {A : Set U} {B : Set V} {C : Set W} (h1 : A ∼ B) (h2 : B ∼ C) : A ∼ C"} +{"name":"HTPI.remove_one_def","declaration":"theorem HTPI.remove_one_def {U : Type} {V : Type} (R : Rel U V) (u : U) (x : U) (v : V) (y : V) : HTPI.remove_one R u v x y ↔ x ≠ u ∧ y ≠ v ∧ (R x y ∨ R x v ∧ R u y)"} +{"name":"HTPI.denum","declaration":"def HTPI.denum {U : Type} (A : Set U) : Prop"} +{"name":"HTPI.remove_one_equinum","declaration":"theorem HTPI.remove_one_equinum {U : Type} {V : Type} {A : Set U} {B : Set V} {u : U} {v : V} (h1 : A ∼ B) (h2 : u ∈ A) (h3 : v ∈ B) : A \\ {u} ∼ B \\ {v}"} +{"name":"HTPI.I","declaration":"def HTPI.I (n : ℕ) : Set ℕ"} +{"name":"HTPI.id_one_one_on","declaration":"theorem HTPI.id_one_one_on {U : Type} (A : Set U) : HTPI.one_one_on id A"} +{"name":"HTPI.one_elt_iff_singleton","declaration":"theorem HTPI.one_elt_iff_singleton {U : Type} (A : Set U) : HTPI.numElts A 1 ↔ ∃ x, A = {x}"} +{"name":"HTPI.fcnl_on","declaration":"def HTPI.fcnl_on {U : Type} {V : Type} (R : Rel U V) (A : Set U) : Prop"} +{"name":"HTPI.I_def","declaration":"theorem HTPI.I_def (k : ℕ) (n : ℕ) : k ∈ HTPI.I n ↔ k < n"} +{"name":"HTPI.remove_one_iff","declaration":"theorem HTPI.remove_one_iff {U : Type} {V : Type} {A : Set U} {B : Set V} {R : Rel U V} (h1 : HTPI.matching R A B) {u : U} (h2 : u ∈ A) (v : V) {x : U} (h3 : x ∈ A \\ {u}) : ∃ w ∈ A, ∀ (y : V), HTPI.remove_one R u v x y ↔ R w y"} +{"name":"HTPI.I_diff","declaration":"theorem HTPI.I_diff (n : ℕ) : HTPI.I (n + 1) \\ {n} = HTPI.I n"} +{"name":"HTPI.fcnl_unique","declaration":"theorem HTPI.fcnl_unique {U : Type} {V : Type} {R : Rel U V} {A : Set U} {x : U} {y1 : V} {y2 : V} (h1 : HTPI.fcnl_on R A) (h2 : x ∈ A) (h3 : R x y1) (h4 : R x y2) : y1 = y2"} +{"name":"HTPI.denum_def","declaration":"theorem HTPI.denum_def {U : Type} (A : Set U) : HTPI.denum A ↔ HTPI.Univ ℕ ∼ A"} +{"name":"HTPI.I_max","declaration":"theorem HTPI.I_max (n : ℕ) : n ∈ HTPI.I (n + 1)"} +{"name":"HTPI.one_match_def","declaration":"theorem HTPI.one_match_def {U : Type} {V : Type} (a : U) (x : U) (b : V) (y : V) : HTPI.one_match a b x y ↔ x = a ∧ y = b"} +{"name":"HTPI.remove_one_fcnl","declaration":"theorem HTPI.remove_one_fcnl {U : Type} {V : Type} {R : Rel U V} {A : Set U} {B : Set V} {u : U} (h1 : HTPI.matching R A B) (h2 : u ∈ A) (v : V) : HTPI.fcnl_on (HTPI.remove_one R u v) (A \\ {u})"} +{"name":"HTPI.emptyRel","declaration":"def HTPI.emptyRel (U : Type) (V : Type) (x : U) (y : V) : Prop"} +{"name":"HTPI.I_1_singleton","declaration":"theorem HTPI.I_1_singleton : HTPI.I 1 = {0}"} +{"name":"HTPI.compRel_def","declaration":"theorem HTPI.compRel_def {U : Type} {V : Type} {W : Type} (S : Rel V W) (R : Rel U V) (u : U) (w : W) : HTPI.compRel S R u w ↔ ∃ x, R u x ∧ S x w"} +{"name":"HTPI.invRel_def","declaration":"theorem HTPI.invRel_def {U : Type} {V : Type} (R : Rel U V) (u : U) (v : V) : HTPI.invRel R v u ↔ R u v"} +{"name":"HTPI.«term_∼_»","declaration":"def HTPI.«term_∼_» : Lean.TrailingParserDescr"} +{"name":"HTPI.image_id","declaration":"theorem HTPI.image_id {U : Type} (A : Set U) : HTPI.image id A = A"} +{"name":"HTPI.fcnl_exists","declaration":"theorem HTPI.fcnl_exists {U : Type} {V : Type} {R : Rel U V} {A : Set U} {x : U} (h1 : HTPI.fcnl_on R A) (h2 : x ∈ A) : ∃ y, R x y"} +{"name":"HTPI.numElts","declaration":"def HTPI.numElts {U : Type} (A : Set U) (n : ℕ) : Prop"} +{"name":"HTPI.compRel","declaration":"def HTPI.compRel {U : Type} {V : Type} {W : Type} (S : Rel V W) (R : Rel U V) : Rel U W"} +{"name":"HTPI.singleton_of_diff_empty","declaration":"theorem HTPI.singleton_of_diff_empty {U : Type} {A : Set U} {a : U} (h1 : a ∈ A) (h2 : HTPI.empty (A \\ {a})) : A = {a}"} +{"name":"HTPI.rel_within","declaration":"def HTPI.rel_within {U : Type} {V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop"} +{"name":"HTPI.one_match","declaration":"def HTPI.one_match {U : Type} {V : Type} (a : U) (b : V) (x : U) (y : V) : Prop"} +{"name":"HTPI.remove_one_inv","declaration":"theorem HTPI.remove_one_inv {U : Type} {V : Type} (R : Rel U V) (u : U) (v : V) : HTPI.invRel (HTPI.remove_one R u v) = HTPI.remove_one (HTPI.invRel R) v u"} +{"name":"HTPI.one_match_match","declaration":"theorem HTPI.one_match_match {U : Type} {V : Type} (a : U) (b : V) : HTPI.matching (HTPI.one_match a b) {a} {b}"} +{"name":"HTPI.Theorem_8_1_3_1","declaration":"theorem HTPI.Theorem_8_1_3_1 {U : Type} (A : Set U) : A ∼ A"} +{"name":"HTPI.numElts_def","declaration":"theorem HTPI.numElts_def {U : Type} (A : Set U) (n : ℕ) : HTPI.numElts A n ↔ HTPI.I n ∼ A"} +{"name":"HTPI.zero_elts_iff_empty","declaration":"theorem HTPI.zero_elts_iff_empty {U : Type} (A : Set U) : HTPI.numElts A 0 ↔ HTPI.empty A"} +{"name":"HTPI.ctble","declaration":"def HTPI.ctble {U : Type} (A : Set U) : Prop"} +{"name":"HTPI.equinum_image","declaration":"theorem HTPI.equinum_image {U : Type} {V : Type} {A : Set U} {B : Set V} {f : U → V} (h1 : HTPI.one_one_on f A) (h2 : HTPI.image f A = B) : A ∼ B"} +{"name":"HTPI.remove_one_numElts","declaration":"theorem HTPI.remove_one_numElts {U : Type} {A : Set U} {n : ℕ} {a : U} (h1 : HTPI.numElts A (n + 1)) (h2 : a ∈ A) : HTPI.numElts (A \\ {a}) n"} +{"name":"HTPI.invRel","declaration":"def HTPI.invRel {U : Type} {V : Type} (R : Rel U V) : Rel V U"} +{"name":"HTPI.equinum","declaration":"def HTPI.equinum {U : Type} {V : Type} (A : Set U) (B : Set V) : Prop"} +{"name":"HTPI.remove_one_match","declaration":"theorem HTPI.remove_one_match {U : Type} {V : Type} {R : Rel U V} {A : Set U} {B : Set V} {u : U} {v : V} (h1 : HTPI.matching R A B) (h2 : u ∈ A) (h3 : v ∈ B) : HTPI.matching (HTPI.remove_one R u v) (A \\ {u}) (B \\ {v})"} +{"name":"HTPI.nonempty_of_pos_numElts","declaration":"theorem HTPI.nonempty_of_pos_numElts {U : Type} {A : Set U} {n : ℕ} (h1 : HTPI.numElts A n) (h2 : n > 0) : ∃ x, x ∈ A"} +{"name":"HTPI.relext","declaration":"theorem HTPI.relext {U : Type} {V : Type} {R : Rel U V} {S : Rel U V} (h : ∀ (u : U) (v : V), R u v ↔ S u v) : R = S"} +{"name":"HTPI.comp_fcnl","declaration":"theorem HTPI.comp_fcnl {U : Type} {V : Type} {W : Type} {R : Rel U V} {S : Rel V W} {A : Set U} {B : Set V} {C : Set W} (h1 : HTPI.matching R A B) (h2 : HTPI.matching S B C) : HTPI.fcnl_on (HTPI.compRel S R) A"} +{"name":"HTPI.Univ","declaration":"def HTPI.Univ (U : Type) : Set U"} +{"name":"HTPI.remove_one","declaration":"def HTPI.remove_one {U : Type} {V : Type} (R : Rel U V) (u : U) (v : V) (x : U) (y : V) : Prop"} +{"name":"HTPI.inv_comp","declaration":"theorem HTPI.inv_comp {U : Type} {V : Type} {W : Type} (R : Rel U V) (S : Rel V W) : HTPI.invRel (HTPI.compRel S R) = HTPI.compRel (HTPI.invRel R) (HTPI.invRel S)"} +{"name":"HTPI.matching","declaration":"def HTPI.matching {U : Type} {V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop"} +{"name":"HTPI.singleton_one_elt","declaration":"theorem HTPI.singleton_one_elt {U : Type} (u : U) : HTPI.numElts {u} 1"} +{"name":"HTPI.finite","declaration":"def HTPI.finite {U : Type} (A : Set U) : Prop"} +{"name":"HTPI.comp_match","declaration":"theorem HTPI.comp_match {U : Type} {V : Type} {W : Type} {R : Rel U V} {S : Rel V W} {A : Set U} {B : Set V} {C : Set W} (h1 : HTPI.matching R A B) (h2 : HTPI.matching S B C) : HTPI.matching (HTPI.compRel S R) A C"} +{"name":"HTPI.one_one_on","declaration":"def HTPI.one_one_on {U : Type} {V : Type} (f : U → V) (A : Set U) : Prop"} +{"name":"HTPI.fcnl_on_empty","declaration":"theorem HTPI.fcnl_on_empty {U : Type} {V : Type} (R : Rel U V) {A : Set U} (h1 : HTPI.empty A) : HTPI.fcnl_on R A"} +{"name":"HTPI.inv_inv","declaration":"theorem HTPI.inv_inv {U : Type} {V : Type} (R : Rel U V) : HTPI.invRel (HTPI.invRel R) = R"} +{"name":"HTPI.inv_match","declaration":"theorem HTPI.inv_match {U : Type} {V : Type} {R : Rel U V} {A : Set U} {B : Set V} (h : HTPI.matching R A B) : HTPI.matching (HTPI.invRel R) B A"} +{"name":"HTPI.remove_one_rel_within","declaration":"theorem HTPI.remove_one_rel_within {U : Type} {V : Type} {R : Rel U V} {A : Set U} {B : Set V} {x : U} {u : U} {y : V} {v : V} (h1 : HTPI.matching R A B) (h2 : HTPI.remove_one R u v x y) : x ∈ A \\ {u} ∧ y ∈ B \\ {v}"} +{"name":"HTPI.Theorem_8_1_3_2","declaration":"theorem HTPI.Theorem_8_1_3_2 {U : Type} {V : Type} {A : Set U} {B : Set V} (h : A ∼ B) : B ∼ A"} +{"name":"HTPI.RelWithinFromFunc","declaration":"def HTPI.RelWithinFromFunc {U : Type} {V : Type} (f : U → V) (A : Set U) (x : U) (y : V) : Prop"} diff --git a/htpi-declarations/HTPILib.Chap8Part2.jsonl b/htpi-declarations/HTPILib.Chap8Part2.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..04c13bea371cfa0ba7c3b105c823359910621819 --- /dev/null +++ b/htpi-declarations/HTPILib.Chap8Part2.jsonl @@ -0,0 +1,183 @@ +{"name":"HTPI.Theorem_8_1_5_1_to_2","declaration":"theorem HTPI.Theorem_8_1_5_1_to_2 {U : Type} {A : Set U} (h1 : HTPI.ctble A) : ∃ R, HTPI.fcnl_onto_from_nat R A"} +{"name":"HTPI.numElts_prod","declaration":"theorem HTPI.numElts_prod {U : Type} {V : Type} {A : Set U} {B : Set V} {m : ℕ} {n : ℕ} (h1 : HTPI.numElts A m) (h2 : HTPI.numElts B n) : HTPI.numElts (A ×ₛ B) (m * n)"} +{"name":"HTPI.seq_cons_image","declaration":"theorem HTPI.seq_cons_image {U : Type} (A : Set U) (n : ℕ) : HTPI.image (HTPI.seq_cons U) (A ×ₛ HTPI.seq_by_length A n) = HTPI.seq_by_length A (n + 1)"} +{"name":"HTPI.Set_prod","declaration":"def HTPI.Set_prod {U : Type} {V : Type} (A : Set U) (B : Set V) : Set (U × V)"} +{"name":"HTPI.neb_phi","declaration":"theorem HTPI.neb_phi (m : ℕ) : HTPI.num_elts_below (HTPI.Set_rp_below m) m (HTPI.phi m)"} +{"name":"HTPI.fnnn_one_one","declaration":"theorem HTPI.fnnn_one_one : HTPI.one_to_one HTPI.fnnn"} +{"name":"HTPI.left_NeZero_of_mul","declaration":"theorem HTPI.left_NeZero_of_mul {m : ℕ} {n : ℕ} (h : m * n ≠ 0) : NeZero m"} +{"name":"HTPI.phi_is_numElts","declaration":"theorem HTPI.phi_is_numElts (m : ℕ) : HTPI.numElts (HTPI.Set_rp_below m) (HTPI.phi m)"} +{"name":"HTPI.cum_rep_image","declaration":"def HTPI.cum_rep_image {U : Type} {V : Type} (R : Rel U V) (S : Rel U V) (X0 : Set U) : Set U"} +{"name":"HTPI.enum_not_skip","declaration":"theorem HTPI.enum_not_skip {A : Set ℕ} ⦃m : ℕ⦄ ⦃s : ℕ⦄ : HTPI.num_elts_below A m s → ∀ t < s, ∃ n, HTPI.enum A t n"} +{"name":"HTPI.Rel_prod","declaration":"def HTPI.Rel_prod {U : Type} {V : Type} {W : Type} {X : Type} (R : Rel U V) (S : Rel W X) (p : U × W) (q : V × X) : Prop"} +{"name":"HTPI.Exercises.Theorem_8_2_1_2","declaration":"theorem HTPI.Exercises.Theorem_8_2_1_2 {U : Type} {A : Set U} {B : Set U} (h1 : HTPI.ctble A) (h2 : HTPI.ctble B) : HTPI.ctble (A ∪ B)"} +{"name":"HTPI.fzn_nat","declaration":"theorem HTPI.fzn_nat (k : ℕ) : HTPI.fzn ↑k = 2 * k"} +{"name":"HTPI.Lemma_8_2_2_2","declaration":"theorem HTPI.Lemma_8_2_2_2 {U : Type} {F : Set (Set U)} (h : ∀ A ∈ F, HTPI.ctble A) : ∃ f, ∀ A ∈ F, HTPI.fcnl_onto_from_nat (f A) A"} +{"name":"HTPI.ctble_of_onto_func_from_N","declaration":"theorem HTPI.ctble_of_onto_func_from_N {U : Type} {A : Set U} {f : ℕ → U} (h1 : ∀ x ∈ A, ∃ n, f n = x) : HTPI.ctble A"} +{"name":"HTPI.fqn","declaration":"def HTPI.fqn (q : ℚ) : ℕ"} +{"name":"HTPI.csb_match_not_cri","declaration":"theorem HTPI.csb_match_not_cri {U : Type} {V : Type} {R : Rel U V} {S : Rel U V} {X0 : Set U} {x : U} {y : V} (h1 : HTPI.csb_match R S X0 x y) (h2 : x ∉ HTPI.cum_rep_image R S X0) : S x y"} +{"name":"HTPI.equinum_Univ","declaration":"theorem HTPI.equinum_Univ {U : Type} {V : Type} {f : U → V} (h1 : HTPI.one_to_one f) (h2 : HTPI.onto f) : HTPI.Univ U ∼ HTPI.Univ V"} +{"name":"HTPI.fnz_onto","declaration":"theorem HTPI.fnz_onto : HTPI.onto HTPI.fnz"} +{"name":"HTPI.Theorem_8_1_5_2","declaration":"theorem HTPI.Theorem_8_1_5_2 {U : Type} (A : Set U) : HTPI.ctble A ↔ ∃ R, HTPI.fcnl_onto_from_nat R A"} +{"name":"HTPI.Theorem_8_1_6","declaration":"theorem HTPI.Theorem_8_1_6 : HTPI.denum (HTPI.Univ ℚ)"} +{"name":"HTPI.fzn_neg_succ_nat","declaration":"theorem HTPI.fzn_neg_succ_nat (k : ℕ) : HTPI.fzn (-↑(k + 1)) = 2 * k + 1"} +{"name":"HTPI.qr_image","declaration":"theorem HTPI.qr_image (m : ℕ) (n : ℕ) : HTPI.image (HTPI.qr n) (HTPI.I (m * n)) = HTPI.I m ×ₛ HTPI.I n"} +{"name":"HTPI.Theorem_8_2_2","declaration":"theorem HTPI.Theorem_8_2_2 {U : Type} {F : Set (Set U)} (h1 : HTPI.ctble F) (h2 : ∀ A ∈ F, HTPI.ctble A) : HTPI.ctble (⋃₀ F)"} +{"name":"HTPI.bdd_subset_nat_match","declaration":"theorem HTPI.bdd_subset_nat_match {A : Set ℕ} {m : ℕ} {s : ℕ} (h1 : ∀ n ∈ A, n < m) (h2 : HTPI.num_elts_below A m s) : HTPI.matching (HTPI.enum A) (HTPI.I s) A"} +{"name":"HTPI.Exercises.Rel_image","declaration":"def HTPI.Exercises.Rel_image {U : Type} {V : Type} (R : Rel U V) (A : Set U) : Set V"} +{"name":"HTPI.fqn_one_one","declaration":"theorem HTPI.fqn_one_one : HTPI.one_to_one HTPI.fqn"} +{"name":"HTPI.Z_equinum_N","declaration":"theorem HTPI.Z_equinum_N : HTPI.Univ ℤ ∼ HTPI.Univ ℕ"} +{"name":"HTPI.restrict_to","declaration":"def HTPI.restrict_to {U : Type} {V : Type} (S : Rel U V) (A : Set U) (x : U) (y : V) : Prop"} +{"name":"HTPI.tri","declaration":"def HTPI.tri (k : ℕ) : ℕ"} +{"name":"HTPI.Exercises.Part","declaration":"def HTPI.Exercises.Part (A : Type) : Set (Set (Set A))"} +{"name":"HTPI.fcnl_onto_from_nat","declaration":"def HTPI.fcnl_onto_from_nat {U : Type} (R : Rel ℕ U) (A : Set U) : Prop"} +{"name":"HTPI.csb_match","declaration":"def HTPI.csb_match {U : Type} {V : Type} (R : Rel U V) (S : Rel U V) (X0 : Set U) (x : U) (y : V) : Prop"} +{"name":"HTPI.unique_val_on_N","declaration":"def HTPI.unique_val_on_N {U : Type} (R : Rel ℕ U) : Prop"} +{"name":"HTPI.neb_exists","declaration":"theorem HTPI.neb_exists (A : Set ℕ) (n : ℕ) : ∃ s, HTPI.num_elts_below A n s"} +{"name":"HTPI.Exercises.saz_pair_part","declaration":"theorem HTPI.Exercises.saz_pair_part (X : Set ℕ) : HTPI.partition (HTPI.Exercises.saz_pair X)"} +{"name":"HTPI.least_rel_to","declaration":"def HTPI.least_rel_to {U : Type} (S : Rel ℕ U) (x : U) (n : ℕ) : Prop"} +{"name":"HTPI.Exercises.pair_ctble","declaration":"theorem HTPI.Exercises.pair_ctble {U : Type} (a : U) (b : U) : HTPI.ctble {a, b}"} +{"name":"HTPI.neb_step_elt","declaration":"theorem HTPI.neb_step_elt {A : Set ℕ} {n : ℕ} (h1 : n ∈ A) (s : ℕ) : HTPI.num_elts_below A (n + 1) s ↔ 1 ≤ s ∧ HTPI.num_elts_below A n (s - 1)"} +{"name":"HTPI.Exercises.EqRelExt","declaration":"def HTPI.Exercises.EqRelExt (A : Type) : Set (Set (A × A))"} +{"name":"HTPI.ctble_iff_equinum_set_nat","declaration":"theorem HTPI.ctble_iff_equinum_set_nat {U : Type} (A : Set U) : HTPI.ctble A ↔ ∃ I, I ∼ A"} +{"name":"HTPI.neb_unique","declaration":"theorem HTPI.neb_unique (A : Set ℕ) ⦃n : ℕ⦄ ⦃s1 : ℕ⦄ ⦃s2 : ℕ⦄ : HTPI.num_elts_below A n s1 → HTPI.num_elts_below A n s2 → s1 = s2"} +{"name":"HTPI.Set_rp_below_def","declaration":"theorem HTPI.Set_rp_below_def (a : ℕ) (m : ℕ) : a ∈ HTPI.Set_rp_below m ↔ HTPI.rel_prime m a ∧ a < m"} +{"name":"HTPI.right_NeZero_of_mul","declaration":"theorem HTPI.right_NeZero_of_mul {m : ℕ} {n : ℕ} (h : m * n ≠ 0) : NeZero n"} +{"name":"HTPI.unbdd_subset_nat","declaration":"theorem HTPI.unbdd_subset_nat {A : Set ℕ} (h1 : ∀ (m : ℕ), ∃ n ∈ A, n ≥ m) : HTPI.denum A"} +{"name":"HTPI.Exercises.Exercise_8_1_5","declaration":"theorem HTPI.Exercises.Exercise_8_1_5 {U : Type} {V : Type} {A : Set U} {B : Set V} (h1 : A ∼ B) : 𝒫 A ∼ 𝒫 B"} +{"name":"HTPI.Rel_prod_def","declaration":"theorem HTPI.Rel_prod_def {U : Type} {V : Type} {W : Type} {X : Type} (R : Rel U V) (S : Rel W X) (u : U) (v : V) (w : W) (x : X) : (R ×ᵣ S) (u, w) (v, x) ↔ R u v ∧ S w x"} +{"name":"HTPI.Exercises.CSB_func","declaration":"theorem HTPI.Exercises.CSB_func {U : Type} {V : Type} {f : U → V} {g : V → U} (h1 : HTPI.one_to_one f) (h2 : HTPI.one_to_one g) : HTPI.Univ U ∼ HTPI.Univ V"} +{"name":"HTPI.Exercises.Like_Exercise_8_2_7","declaration":"theorem HTPI.Exercises.Like_Exercise_8_2_7 : ∃ P, HTPI.partition P ∧ HTPI.denum P ∧ ∀ X ∈ P, HTPI.denum X"} +{"name":"HTPI.ctble_of_equinum_ctble","declaration":"theorem HTPI.ctble_of_equinum_ctble {U : Type} {V : Type} {A : Set U} {B : Set V} (h1 : A ∼ B) (h2 : HTPI.ctble A) : HTPI.ctble B"} +{"name":"HTPI.Exercises.remove_one_iff","declaration":"theorem HTPI.Exercises.remove_one_iff {U : Type} {V : Type} {A : Set U} {B : Set V} {R : Rel U V} (h1 : HTPI.matching R A B) {u : U} (h2 : u ∈ A) (v : V) {x : U} (h3 : x ∈ A \\ {u}) : ∃ w ∈ A, ∀ (y : V), HTPI.remove_one R u v x y ↔ R w y"} +{"name":"HTPI.Exercises.EqRel_equinum_EqRelExt","declaration":"theorem HTPI.Exercises.EqRel_equinum_EqRelExt (A : Type) : HTPI.Exercises.EqRel A ∼ HTPI.Exercises.EqRelExt A"} +{"name":"HTPI.Exercises.unctbly_many_inf_set_nat","declaration":"theorem HTPI.Exercises.unctbly_many_inf_set_nat : ¬HTPI.ctble {X | ¬HTPI.finite X}"} +{"name":"HTPI.set_elt_powerset_univ","declaration":"theorem HTPI.set_elt_powerset_univ {U : Type} (A : Set U) : A ∈ 𝒫 HTPI.Univ U"} +{"name":"HTPI.neb_step_not_elt","declaration":"theorem HTPI.neb_step_not_elt {A : Set ℕ} {n : ℕ} (h1 : n ∉ A) (s : ℕ) : HTPI.num_elts_below A (n + 1) s ↔ HTPI.num_elts_below A n s"} +{"name":"HTPI.«term_×ₛ_»","declaration":"def HTPI.«term_×ₛ_» : Lean.TrailingParserDescr"} +{"name":"HTPI.Exercises.shift_I_equinum","declaration":"theorem HTPI.Exercises.shift_I_equinum (n : ℕ) (m : ℕ) : HTPI.I m ∼ HTPI.I (n + m) \\ HTPI.I n"} +{"name":"HTPI.exists_least_rel_to","declaration":"theorem HTPI.exists_least_rel_to {U : Type} {S : Rel ℕ U} {x : U} (h1 : ∃ n, S n x) : ∃ n, HTPI.least_rel_to S x n"} +{"name":"HTPI.Exercises.qr_image","declaration":"theorem HTPI.Exercises.qr_image (m : ℕ) (n : ℕ) : HTPI.image (HTPI.qr n) (HTPI.I (m * n)) = HTPI.I m ×ₛ HTPI.I n"} +{"name":"HTPI.subset_nat_ctble","declaration":"theorem HTPI.subset_nat_ctble (A : Set ℕ) : HTPI.ctble A"} +{"name":"HTPI.Exercises.neb_nrpb","declaration":"theorem HTPI.Exercises.neb_nrpb (m : ℕ) ⦃k : ℕ⦄ : k ≤ m → HTPI.num_elts_below (HTPI.Set_rp_below m) k (HTPI.num_rp_below m k)"} +{"name":"HTPI.Lemma_8_2_4_2","declaration":"theorem HTPI.Lemma_8_2_4_2 {U : Type} {A : Set U} (h1 : HTPI.ctble A) (n : ℕ) : HTPI.ctble (HTPI.seq_by_length A n)"} +{"name":"HTPI.Exercise_6_1_16a1","declaration":"theorem HTPI.Exercise_6_1_16a1 (n : ℕ) : HTPI.nat_even n ∨ HTPI.nat_odd n"} +{"name":"HTPI.mod_mod","declaration":"def HTPI.mod_mod (m : ℕ) (n : ℕ) (a : ℕ) : ℕ × ℕ"} +{"name":"HTPI.fnnn","declaration":"def HTPI.fnnn (p : ℕ × ℕ) : ℕ"} +{"name":"HTPI.Exercises.Rel_image_on_power_set","declaration":"theorem HTPI.Exercises.Rel_image_on_power_set {U : Type} {V : Type} {R : Rel U V} {A : Set U} {C : Set U} {B : Set V} (h1 : HTPI.matching R A B) (h2 : C ∈ 𝒫 A) : HTPI.Exercises.Rel_image R C ∈ 𝒫 B"} +{"name":"HTPI.seq_cons_one_one","declaration":"theorem HTPI.seq_cons_one_one (U : Type) : HTPI.one_to_one (HTPI.seq_cons U)"} +{"name":"HTPI.le_of_fnnn_eq","declaration":"theorem HTPI.le_of_fnnn_eq {a1 : ℕ} {b1 : ℕ} {a2 : ℕ} {b2 : ℕ} (h1 : HTPI.fnnn (a1, b1) = HTPI.fnnn (a2, b2)) : a1 + b1 ≤ a2 + b2"} +{"name":"HTPI.Exercises.denum_not_finite","declaration":"theorem HTPI.Exercises.denum_not_finite {U : Type} {A : Set U} (h : HTPI.denum A) : ¬HTPI.finite A"} +{"name":"HTPI.Exercises.ctble_of_onto_func_from_N","declaration":"theorem HTPI.Exercises.ctble_of_onto_func_from_N {U : Type} {A : Set U} {f : ℕ → U} (h1 : ∀ x ∈ A, ∃ n, f n = x) : HTPI.ctble A"} +{"name":"HTPI.Exercises.EqRel_Nat_equinum_PN","declaration":"theorem HTPI.Exercises.EqRel_Nat_equinum_PN : HTPI.Exercises.EqRel ℕ ∼ 𝒫 HTPI.Univ ℕ"} +{"name":"HTPI.Exercises.one_match_fcnl","declaration":"theorem HTPI.Exercises.one_match_fcnl {U : Type} {V : Type} (a : U) (b : V) : HTPI.fcnl_on (HTPI.one_match a b) {a}"} +{"name":"HTPI.Theorem_8_2_4","declaration":"theorem HTPI.Theorem_8_2_4 {U : Type} {A : Set U} (h1 : HTPI.ctble A) : HTPI.ctble (HTPI.seq A)"} +{"name":"HTPI.Exercises.fnz_fzn","declaration":"theorem HTPI.Exercises.fnz_fzn : HTPI.fnz ∘ HTPI.fzn = id"} +{"name":"HTPI.Exercises.tri_step","declaration":"theorem HTPI.Exercises.tri_step (k : ℕ) : HTPI.tri (k + 1) = HTPI.tri k + k + 1"} +{"name":"HTPI.image_fqn_unbdd","declaration":"theorem HTPI.image_fqn_unbdd (m : ℕ) : ∃ n ∈ HTPI.image HTPI.fqn (HTPI.Univ ℚ), n ≥ m"} +{"name":"HTPI.seq_cons","declaration":"def HTPI.seq_cons (U : Type) (p : U × List U) : List U"} +{"name":"HTPI.neb_increase","declaration":"theorem HTPI.neb_increase {A : Set ℕ} {n : ℕ} {s : ℕ} (h1 : HTPI.enum A s n) ⦃m : ℕ⦄ : m ≥ n + 1 → ∀ ⦃t : ℕ⦄, HTPI.num_elts_below A m t → s < t"} +{"name":"HTPI.Cantor's_theorem","declaration":"theorem HTPI.Cantor's_theorem : ¬HTPI.ctble (𝒫 HTPI.Univ ℕ)"} +{"name":"HTPI.enum_le","declaration":"theorem HTPI.enum_le {A : Set ℕ} {t : ℕ} {n1 : ℕ} {n2 : ℕ} (h1 : HTPI.enum A t n1) (h2 : HTPI.enum A t n2) : n1 ≤ n2"} +{"name":"HTPI.mod_elt_Set_rp_below","declaration":"theorem HTPI.mod_elt_Set_rp_below {a : ℕ} {m : ℕ} [NeZero m] (h1 : HTPI.rel_prime m a) : a % m ∈ HTPI.Set_rp_below m"} +{"name":"HTPI.inv_enum_fcnl","declaration":"theorem HTPI.inv_enum_fcnl (A : Set ℕ) : HTPI.fcnl_on (HTPI.invRel (HTPI.enum A)) A"} +{"name":"HTPI.neb_nrpb","declaration":"theorem HTPI.neb_nrpb (m : ℕ) ⦃k : ℕ⦄ : k ≤ m → HTPI.num_elts_below (HTPI.Set_rp_below m) k (HTPI.num_rp_below m k)"} +{"name":"HTPI.Lemma_8_2_2_1","declaration":"theorem HTPI.Lemma_8_2_2_1 {U : Type} {F : Set (Set U)} {f : Set U → Rel ℕ U} (h1 : HTPI.ctble F) (h2 : ∀ A ∈ F, HTPI.fcnl_onto_from_nat (f A) A) : HTPI.ctble (⋃₀ F)"} +{"name":"HTPI.elt_Univ","declaration":"theorem HTPI.elt_Univ {U : Type} (u : U) : u ∈ HTPI.Univ U"} +{"name":"HTPI.fzn_one_one","declaration":"theorem HTPI.fzn_one_one : HTPI.one_to_one HTPI.fzn"} +{"name":"HTPI.Set_rp_below_prod","declaration":"theorem HTPI.Set_rp_below_prod {m : ℕ} {n : ℕ} (h1 : HTPI.rel_prime m n) : HTPI.Set_rp_below (m * n) ∼ HTPI.Set_rp_below m ×ₛ HTPI.Set_rp_below n"} +{"name":"HTPI.Exercises.Theorem_8_1_2_2","declaration":"theorem HTPI.Exercises.Theorem_8_1_2_2 {U : Type} {V : Type} {A : Set U} {C : Set U} {B : Set V} {D : Set V} (h1 : HTPI.empty (A ∩ C)) (h2 : HTPI.empty (B ∩ D)) (h3 : A ∼ B) (h4 : C ∼ D) : A ∪ C ∼ B ∪ D"} +{"name":"HTPI.eq_numElts_of_equinum","declaration":"theorem HTPI.eq_numElts_of_equinum {U : Type} {V : Type} {A : Set U} {B : Set V} {n : ℕ} (h1 : A ∼ B) (h2 : HTPI.numElts A n) : HTPI.numElts B n"} +{"name":"HTPI.sbl_base","declaration":"theorem HTPI.sbl_base {U : Type} (A : Set U) : HTPI.seq_by_length A 0 = {[]}"} +{"name":"HTPI.Lemma_7_4_7_aux","declaration":"theorem HTPI.Lemma_7_4_7_aux {m : ℕ} {n : ℕ} {s : ℤ} {t : ℤ} (h : s * ↑m + t * ↑n = 1) (a : ℕ) (b : ℕ) : t * ↑n * ↑a + s * ↑m * ↑b ≡ ↑a (MOD m)"} +{"name":"HTPI.Exercises.ctble_of_equinum_ctble","declaration":"theorem HTPI.Exercises.ctble_of_equinum_ctble {U : Type} {V : Type} {A : Set U} {B : Set V} (h1 : A ∼ B) (h2 : HTPI.ctble A) : HTPI.ctble B"} +{"name":"HTPI.fnnn_def","declaration":"theorem HTPI.fnnn_def (a : ℕ) (b : ℕ) : HTPI.fnnn (a, b) = HTPI.tri (a + b) + a"} +{"name":"HTPI.Exercises.set_to_list","declaration":"theorem HTPI.Exercises.set_to_list {U : Type} {A : Set U} (h : HTPI.finite A) : ∃ l, ∀ (x : U), x ∈ l ↔ x ∈ A"} +{"name":"HTPI.Exercises.Rel_image_one_one_on","declaration":"theorem HTPI.Exercises.Rel_image_one_one_on {U : Type} {V : Type} {R : Rel U V} {A : Set U} {B : Set V} (h1 : HTPI.matching R A B) : HTPI.one_one_on (HTPI.Exercises.Rel_image R) (𝒫 A)"} +{"name":"HTPI.csb_match_cri","declaration":"theorem HTPI.csb_match_cri {U : Type} {V : Type} {R : Rel U V} {S : Rel U V} {X0 : Set U} {x : U} {y : V} (h1 : HTPI.csb_match R S X0 x y) (h2 : x ∈ HTPI.cum_rep_image R S X0) : R x y"} +{"name":"HTPI.eq_of_I_equinum","declaration":"theorem HTPI.eq_of_I_equinum ⦃m : ℕ⦄ ⦃n : ℕ⦄ : HTPI.I m ∼ HTPI.I n → m = n"} +{"name":"HTPI.Exercises.equinum_sub","declaration":"theorem HTPI.Exercises.equinum_sub {U : Type} {V : Type} {A : Set U} {C : Set U} {B : Set V} (h1 : A ∼ B) (h2 : C ⊆ A) : ∃ D ⊆ B, C ∼ D"} +{"name":"HTPI.Exercises.EqRel","declaration":"def HTPI.Exercises.EqRel (A : Type) : Set (HTPI.BinRel A)"} +{"name":"HTPI.I_prod","declaration":"theorem HTPI.I_prod (m : ℕ) (n : ℕ) : HTPI.I (m * n) ∼ HTPI.I m ×ₛ HTPI.I n"} +{"name":"HTPI.Exercises.sub_EqRel_Nat_equinum_PN","declaration":"theorem HTPI.Exercises.sub_EqRel_Nat_equinum_PN : ∃ C ⊆ HTPI.Exercises.EqRel ℕ, C ∼ 𝒫 HTPI.Univ ℕ"} +{"name":"HTPI.Set_prod_def","declaration":"theorem HTPI.Set_prod_def {U : Type} {V : Type} (A : Set U) (B : Set V) (a : U) (b : V) : (a, b) ∈ A ×ₛ B ↔ a ∈ A ∧ b ∈ B"} +{"name":"HTPI.fzn_onto","declaration":"theorem HTPI.fzn_onto : HTPI.onto HTPI.fzn"} +{"name":"HTPI.Lemma_7_4_6","declaration":"theorem HTPI.Lemma_7_4_6 {a : ℕ} {b : ℕ} {c : ℕ} : HTPI.rel_prime (a * b) c ↔ HTPI.rel_prime a c ∧ HTPI.rel_prime b c"} +{"name":"HTPI.enum_fcnl_of_unbdd","declaration":"theorem HTPI.enum_fcnl_of_unbdd {A : Set ℕ} (h1 : ∀ (m : ℕ), ∃ n ∈ A, n ≥ m) : HTPI.fcnl_on (HTPI.enum A) (HTPI.Univ ℕ)"} +{"name":"HTPI.Theorem_8_2_1_1","declaration":"theorem HTPI.Theorem_8_2_1_1 {U : Type} {V : Type} {A : Set U} {B : Set V} (h1 : HTPI.ctble A) (h2 : HTPI.ctble B) : HTPI.ctble (A ×ₛ B)"} +{"name":"HTPI.Lemma_8_2_4_4","declaration":"theorem HTPI.Lemma_8_2_4_4 {U : Type} (A : Set U) : HTPI.ctble (HTPI.sbl_set A)"} +{"name":"HTPI.Theorem_8_1_5_3_to_1","declaration":"theorem HTPI.Theorem_8_1_5_3_to_1 {U : Type} {A : Set U} (h1 : ∃ R, HTPI.fcnl_one_one_to_nat R A) : HTPI.ctble A"} +{"name":"HTPI.qr_def","declaration":"theorem HTPI.qr_def (n : ℕ) (a : ℕ) : HTPI.qr n a = (a / n, a % n)"} +{"name":"HTPI.Exercises.one_match_match","declaration":"theorem HTPI.Exercises.one_match_match {U : Type} {V : Type} (a : U) (b : V) : HTPI.matching (HTPI.one_match a b) {a} {b}"} +{"name":"HTPI.nat_rel_onto","declaration":"def HTPI.nat_rel_onto {U : Type} (R : Rel ℕ U) (A : Set U) : Prop"} +{"name":"HTPI.Theorem_8_1_5_2_to_3","declaration":"theorem HTPI.Theorem_8_1_5_2_to_3 {U : Type} {A : Set U} (h1 : ∃ R, HTPI.fcnl_onto_from_nat R A) : ∃ R, HTPI.fcnl_one_one_to_nat R A"} +{"name":"HTPI.unbdd_subset_nat_match","declaration":"theorem HTPI.unbdd_subset_nat_match {A : Set ℕ} (h1 : ∀ (m : ℕ), ∃ n ∈ A, n ≥ m) : HTPI.matching (HTPI.enum A) (HTPI.Univ ℕ) A"} +{"name":"HTPI.Exercises.prod_match","declaration":"theorem HTPI.Exercises.prod_match {U : Type} {V : Type} {W : Type} {X : Type} {A : Set U} {B : Set V} {C : Set W} {D : Set X} {R : Rel U V} {S : Rel W X} (h1 : HTPI.matching R A B) (h2 : HTPI.matching S C D) : HTPI.matching (R ×ᵣ S) (A ×ₛ C) (B ×ₛ D)"} +{"name":"HTPI.num_elts_below","declaration":"def HTPI.num_elts_below (A : Set ℕ) (m : ℕ) (s : ℕ) : Prop"} +{"name":"HTPI.fnz","declaration":"def HTPI.fnz (n : ℕ) : ℤ"} +{"name":"HTPI.Exercises.shift_and_zero","declaration":"def HTPI.Exercises.shift_and_zero (X : Set ℕ) : Set ℕ"} +{"name":"HTPI.Lemma_8_2_4_3","declaration":"theorem HTPI.Lemma_8_2_4_3 {U : Type} (A : Set U) : ⋃₀ HTPI.sbl_set A = HTPI.seq A"} +{"name":"HTPI.csb_cri_of_cri","declaration":"theorem HTPI.csb_cri_of_cri {U : Type} {V : Type} {R : Rel U V} {S : Rel U V} {X0 : Set U} {x1 : U} {x2 : U} {y : V} (h1 : HTPI.csb_match R S X0 x1 y) (h2 : HTPI.csb_match R S X0 x2 y) (h3 : x1 ∈ HTPI.cum_rep_image R S X0) : x2 ∈ HTPI.cum_rep_image R S X0"} +{"name":"HTPI.Exercises.ctble_of_one_one_func_to_N","declaration":"theorem HTPI.Exercises.ctble_of_one_one_func_to_N {U : Type} {A : Set U} {f : U → ℕ} (h1 : HTPI.one_one_on f A) : HTPI.ctble A"} +{"name":"HTPI.fnz_odd","declaration":"theorem HTPI.fnz_odd (k : ℕ) : HTPI.fnz (2 * k + 1) = -↑(k + 1)"} +{"name":"HTPI.Exercises.EqRel_equinum_Part","declaration":"theorem HTPI.Exercises.EqRel_equinum_Part (A : Type) : HTPI.Exercises.EqRel A ∼ HTPI.Exercises.Part A"} +{"name":"HTPI.one_one_on_of_one_one","declaration":"theorem HTPI.one_one_on_of_one_one {U : Type} {V : Type} {f : U → V} (h : HTPI.one_to_one f) (A : Set U) : HTPI.one_one_on f A"} +{"name":"HTPI.fnz_fzn","declaration":"theorem HTPI.fnz_fzn : HTPI.fnz ∘ HTPI.fzn = id"} +{"name":"HTPI.tri_incr","declaration":"theorem HTPI.tri_incr {j : ℕ} {k : ℕ} (h1 : j ≤ k) : HTPI.tri j ≤ HTPI.tri k"} +{"name":"HTPI.seq_cons_def","declaration":"theorem HTPI.seq_cons_def {U : Type} (x : U) (l : List U) : HTPI.seq_cons U (x, l) = x :: l"} +{"name":"HTPI.mod_mod_image","declaration":"theorem HTPI.mod_mod_image {m : ℕ} {n : ℕ} (h1 : HTPI.rel_prime m n) : HTPI.image (HTPI.mod_mod m n) (HTPI.Set_rp_below (m * n)) = HTPI.Set_rp_below m ×ₛ HTPI.Set_rp_below n"} +{"name":"HTPI.Cantor_Schroeder_Bernstein_theorem","declaration":"theorem HTPI.Cantor_Schroeder_Bernstein_theorem {U : Type} {V : Type} {A : Set U} {C : Set U} {B : Set V} {D : Set V} (h1 : C ⊆ A) (h2 : D ⊆ B) (h3 : A ∼ D) (h4 : C ∼ B) : A ∼ B"} +{"name":"HTPI.Exercises.Exercise_8_2_8","declaration":"theorem HTPI.Exercises.Exercise_8_2_8 {U : Type} {A : Set U} {B : Set U} (h : HTPI.empty (A ∩ B)) : 𝒫(A ∪ B) ∼ 𝒫 A ×ₛ 𝒫 B"} +{"name":"HTPI.Exercises.tri_incr","declaration":"theorem HTPI.Exercises.tri_incr {j : ℕ} {k : ℕ} (h1 : j ≤ k) : HTPI.tri j ≤ HTPI.tri k"} +{"name":"HTPI.Exercises.intervals_equinum","declaration":"theorem HTPI.Exercises.intervals_equinum : {x | 0 < x ∧ x < 1} ∼ {x | 0 < x ∧ x ≤ 1}"} +{"name":"HTPI.fnz_one_one","declaration":"theorem HTPI.fnz_one_one : HTPI.one_to_one HTPI.fnz"} +{"name":"HTPI.prod_match","declaration":"theorem HTPI.prod_match {U : Type} {V : Type} {W : Type} {X : Type} {A : Set U} {B : Set V} {C : Set W} {D : Set X} {R : Rel U V} {S : Rel W X} (h1 : HTPI.matching R A B) (h2 : HTPI.matching S C D) : HTPI.matching (R ×ᵣ S) (A ×ₛ C) (B ×ₛ D)"} +{"name":"HTPI.rel_prime_mod","declaration":"theorem HTPI.rel_prime_mod (m : ℕ) (a : ℕ) : HTPI.rel_prime m (a % m) ↔ HTPI.rel_prime m a"} +{"name":"HTPI.rep_common_image","declaration":"def HTPI.rep_common_image {U : Type} {V : Type} (R : Rel U V) (S : Rel U V) (X0 : Set U) (n : ℕ) : Set U"} +{"name":"HTPI.Exercises.Theorem_8_1_7","declaration":"theorem HTPI.Exercises.Theorem_8_1_7 {U : Type} {A : Set U} {B : Set U} {n : ℕ} {m : ℕ} (h1 : HTPI.empty (A ∩ B)) (h2 : HTPI.numElts A n) (h3 : HTPI.numElts B m) : HTPI.numElts (A ∪ B) (n + m)"} +{"name":"HTPI.Exercises.Like_Exercise_8_2_4","declaration":"theorem HTPI.Exercises.Like_Exercise_8_2_4 {U : Type} {A : Set U} (h : HTPI.ctble A) : HTPI.ctble {X | X ⊆ A ∧ HTPI.finite X}"} +{"name":"HTPI.fzn_fnz","declaration":"theorem HTPI.fzn_fnz : HTPI.fzn ∘ HTPI.fnz = id"} +{"name":"HTPI.fcnl_one_one_to_nat","declaration":"def HTPI.fcnl_one_one_to_nat {U : Type} (R : Rel U ℕ) (A : Set U) : Prop"} +{"name":"HTPI.Exercises.inv_one_match","declaration":"theorem HTPI.Exercises.inv_one_match {U : Type} {V : Type} (a : U) (b : V) : HTPI.invRel (HTPI.one_match a b) = HTPI.one_match b a"} +{"name":"HTPI.Exercises.EqRel_Nat_equinum_sub_PN","declaration":"theorem HTPI.Exercises.EqRel_Nat_equinum_sub_PN : ∃ D ⊆ 𝒫 HTPI.Univ ℕ, HTPI.Exercises.EqRel ℕ ∼ D"} +{"name":"HTPI.bdd_subset_nat","declaration":"theorem HTPI.bdd_subset_nat {A : Set ℕ} {m : ℕ} {s : ℕ} (h1 : ∀ n ∈ A, n < m) (h2 : HTPI.num_elts_below A m s) : HTPI.I s ∼ A"} +{"name":"HTPI.Exercises.prod_fcnl","declaration":"theorem HTPI.Exercises.prod_fcnl {U : Type} {V : Type} {W : Type} {X : Type} {R : Rel U V} {S : Rel W X} {A : Set U} {C : Set W} (h1 : HTPI.fcnl_on R A) (h2 : HTPI.fcnl_on S C) : HTPI.fcnl_on (R ×ᵣ S) (A ×ₛ C)"} +{"name":"HTPI.Theorem_7_4_4","declaration":"theorem HTPI.Theorem_7_4_4 {m : ℕ} {n : ℕ} (h1 : HTPI.rel_prime m n) : HTPI.phi (m * n) = HTPI.phi m * HTPI.phi n"} +{"name":"HTPI.enum_unique","declaration":"theorem HTPI.enum_unique (A : Set ℕ) (t : ℕ) ⦃n1 : ℕ⦄ ⦃n2 : ℕ⦄ : HTPI.enum A t n1 → HTPI.enum A t n2 → n1 = n2"} +{"name":"HTPI.seq","declaration":"def HTPI.seq {U : Type} (A : Set U) : Set (List U)"} +{"name":"HTPI.mod_mod_one_one_on","declaration":"theorem HTPI.mod_mod_one_one_on {m : ℕ} {n : ℕ} (h1 : HTPI.rel_prime m n) : HTPI.one_one_on (HTPI.mod_mod m n) (HTPI.Set_rp_below (m * n))"} +{"name":"HTPI.Lemma_7_4_7","declaration":"theorem HTPI.Lemma_7_4_7 {m : ℕ} {n : ℕ} [NeZero m] [NeZero n] (h1 : HTPI.rel_prime m n) (a : ℕ) (b : ℕ) : ∃ r < m * n, ↑r ≡ ↑a (MOD m) ∧ ↑r ≡ ↑b (MOD n)"} +{"name":"HTPI.fnnn_onto","declaration":"theorem HTPI.fnnn_onto : HTPI.onto HTPI.fnnn"} +{"name":"HTPI.Exercises.Rel_image_image","declaration":"theorem HTPI.Exercises.Rel_image_image {U : Type} {V : Type} {R : Rel U V} {A : Set U} {B : Set V} (h1 : HTPI.matching R A B) : HTPI.image (HTPI.Exercises.Rel_image R) (𝒫 A) = 𝒫 B"} +{"name":"HTPI.tri_step","declaration":"theorem HTPI.tri_step (k : ℕ) : HTPI.tri (k + 1) = HTPI.tri k + k + 1"} +{"name":"HTPI.Exercises.saz_pair","declaration":"def HTPI.Exercises.saz_pair (X : Set ℕ) : Set (Set ℕ)"} +{"name":"HTPI.Theorem_8_1_5_3","declaration":"theorem HTPI.Theorem_8_1_5_3 {U : Type} (A : Set U) : HTPI.ctble A ↔ ∃ R, HTPI.fcnl_one_one_to_nat R A"} +{"name":"HTPI.Exercise_8_1_17","declaration":"theorem HTPI.Exercise_8_1_17 {U : Type} {A : Set U} {B : Set U} (h1 : B ⊆ A) (h2 : HTPI.ctble A) : HTPI.ctble B"} +{"name":"HTPI.«term_×ᵣ_»","declaration":"def HTPI.«term_×ᵣ_» : Lean.TrailingParserDescr"} +{"name":"HTPI.Exercises.fnz_odd","declaration":"theorem HTPI.Exercises.fnz_odd (k : ℕ) : HTPI.fnz (2 * k + 1) = -↑(k + 1)"} +{"name":"HTPI.rep_common_image_step","declaration":"theorem HTPI.rep_common_image_step {U : Type} {V : Type} (R : Rel U V) (S : Rel U V) (X0 : Set U) (m : ℕ) (a : U) : a ∈ HTPI.rep_common_image R S X0 (m + 1) ↔ ∃ x ∈ HTPI.rep_common_image R S X0 m, ∃ y, R x y ∧ S a y"} +{"name":"HTPI.sbl_set","declaration":"def HTPI.sbl_set {U : Type} (A : Set U) : Set (Set (List U))"} +{"name":"HTPI.Exercises.Exercise_8_1_8b","declaration":"theorem HTPI.Exercises.Exercise_8_1_8b {U : Type} {A : Set U} {B : Set U} (h1 : HTPI.finite A) (h2 : B ⊆ A) : HTPI.finite B"} +{"name":"HTPI.qr_one_one","declaration":"theorem HTPI.qr_one_one (n : ℕ) : HTPI.one_to_one (HTPI.qr n)"} +{"name":"HTPI.NxN_equinum_N","declaration":"theorem HTPI.NxN_equinum_N : HTPI.Univ (ℕ × ℕ) ∼ HTPI.Univ ℕ"} +{"name":"HTPI.qr","declaration":"def HTPI.qr (n : ℕ) (a : ℕ) : ℕ × ℕ"} +{"name":"HTPI.Lemma_8_2_4_1","declaration":"theorem HTPI.Lemma_8_2_4_1 {U : Type} (A : Set U) (n : ℕ) : A ×ₛ HTPI.seq_by_length A n ∼ HTPI.seq_by_length A (n + 1)"} +{"name":"HTPI.fzn","declaration":"def HTPI.fzn (a : ℤ) : ℕ"} +{"name":"HTPI.numElts_unique","declaration":"theorem HTPI.numElts_unique {U : Type} {A : Set U} {m : ℕ} {n : ℕ} (h1 : HTPI.numElts A m) (h2 : HTPI.numElts A n) : m = n"} +{"name":"HTPI.congr_rel_prime","declaration":"theorem HTPI.congr_rel_prime {m : ℕ} {a : ℕ} {b : ℕ} (h1 : ↑a ≡ ↑b (MOD m)) : HTPI.rel_prime m a ↔ HTPI.rel_prime m b"} +{"name":"HTPI.Set_rp_below","declaration":"def HTPI.Set_rp_below (m : ℕ) : Set ℕ"} +{"name":"HTPI.Theorem_8_1_2_1","declaration":"theorem HTPI.Theorem_8_1_2_1 {U : Type} {V : Type} {W : Type} {X : Type} {A : Set U} {B : Set V} {C : Set W} {D : Set X} (h1 : A ∼ B) (h2 : C ∼ D) : A ×ₛ C ∼ B ×ₛ D"} +{"name":"HTPI.Exercises.Rel_image_inv","declaration":"theorem HTPI.Exercises.Rel_image_inv {U : Type} {V : Type} {R : Rel U V} {A : Set U} {C : Set U} {B : Set V} (h1 : HTPI.matching R A B) (h2 : C ∈ 𝒫 A) : HTPI.Exercises.Rel_image (HTPI.invRel R) (HTPI.Exercises.Rel_image R C) = C"} +{"name":"HTPI.enum","declaration":"def HTPI.enum (A : Set ℕ) (s : ℕ) (n : ℕ) : Prop"} +{"name":"HTPI.Exercises.Exercise_8_2_6b","declaration":"theorem HTPI.Exercises.Exercise_8_2_6b (A : Type) (B : Type) (C : Type) : HTPI.Univ (A × B → C) ∼ HTPI.Univ (A → B → C)"} +{"name":"HTPI.fnz_even","declaration":"theorem HTPI.fnz_even (k : ℕ) : HTPI.fnz (2 * k) = ↑k"} +{"name":"HTPI.Exercises.seq_cons_image","declaration":"theorem HTPI.Exercises.seq_cons_image {U : Type} (A : Set U) (n : ℕ) : HTPI.image (HTPI.seq_cons U) (A ×ₛ HTPI.seq_by_length A n) = HTPI.seq_by_length A (n + 1)"} +{"name":"HTPI.seq_def","declaration":"theorem HTPI.seq_def {U : Type} (A : Set U) (l : List U) : l ∈ HTPI.seq A ↔ ∀ x ∈ l, x ∈ A"} +{"name":"HTPI.Exercises.Exercise_8_1_1_b","declaration":"theorem HTPI.Exercises.Exercise_8_1_1_b : HTPI.denum {n | HTPI.even n}"} +{"name":"HTPI.fqn_def","declaration":"theorem HTPI.fqn_def (q : ℚ) : HTPI.fqn q = HTPI.fnnn (HTPI.fzn q.num, q.den)"} +{"name":"HTPI.seq_by_length","declaration":"def HTPI.seq_by_length {U : Type} (A : Set U) (n : ℕ) : Set (List U)"} +{"name":"HTPI.mod_mod_def","declaration":"theorem HTPI.mod_mod_def (m : ℕ) (n : ℕ) (a : ℕ) : HTPI.mod_mod m n a = (a % m, a % n)"} +{"name":"HTPI.neb_step","declaration":"theorem HTPI.neb_step (A : Set ℕ) (n : ℕ) (s : ℕ) : HTPI.num_elts_below A (n + 1) s ↔ n ∈ A ∧ 1 ≤ s ∧ HTPI.num_elts_below A n (s - 1) ∨ n ∉ A ∧ HTPI.num_elts_below A n s"} +{"name":"HTPI.enum_union_fam","declaration":"def HTPI.enum_union_fam {U : Type} (F : Set (Set U)) (f : Set U → Rel ℕ U) (R : Rel ℕ (Set U)) (n : ℕ) (a : U) : Prop"} +{"name":"HTPI.Exercises.N_not_finite","declaration":"theorem HTPI.Exercises.N_not_finite : ¬HTPI.finite (HTPI.Univ ℕ)"} +{"name":"HTPI.eq_zero_of_I_zero_equinum","declaration":"theorem HTPI.eq_zero_of_I_zero_equinum {n : ℕ} (h1 : HTPI.I 0 ∼ HTPI.I n) : n = 0"} +{"name":"HTPI.Exercises.Exercise_8_1_17","declaration":"theorem HTPI.Exercises.Exercise_8_1_17 {U : Type} {A : Set U} {B : Set U} (h1 : B ⊆ A) (h2 : HTPI.ctble A) : HTPI.ctble B"} diff --git a/htpi-declarations/HTPILib.HTPIDefs.jsonl b/htpi-declarations/HTPILib.HTPIDefs.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a58a4e5f34fd0c8e486be41af3ce397be921ce6a --- /dev/null +++ b/htpi-declarations/HTPILib.HTPIDefs.jsonl @@ -0,0 +1,153 @@ +{"name":"HTPI.«tacticFix_:_»","declaration":"def HTPI.«tacticFix_:_» : Lean.ParserDescr"} +{"name":"HTPI.ExprToString","declaration":"def HTPI.ExprToString (e : Lean.Expr) : String"} +{"name":"HTPI.doSuffices","declaration":"def HTPI.doSuffices (form : Lean.Expr) (pfstx : Lean.Syntax) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.tacticDouble_neg__","declaration":"def HTPI.tacticDouble_neg__ : Lean.ParserDescr"} +{"name":"HTPI.parseIdOrTerm","declaration":"def HTPI.parseIdOrTerm (it : HTPI.IdOrTerm) : Lean.Term"} +{"name":"HTPI.tacticBy_strong_induc","declaration":"def HTPI.tacticBy_strong_induc : Lean.ParserDescr"} +{"name":"HTPI.idOrTerm?Type","declaration":"def HTPI.idOrTerm?Type : Lean.ParserDescr"} +{"name":"HTPI.doInduc","declaration":"def HTPI.doInduc (strong : Bool) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.doRewrite","declaration":"def HTPI.doRewrite (symm : Bool) (rule : Lean.Term) (l : Option HTPI.OneLoc) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.matchNeg","declaration":"def HTPI.matchNeg (e1 : Lean.Expr) (e2 : Lean.Expr) : Lean.Elab.Tactic.TacticM (Bool × Bool)"} +{"name":"HTPI.ColonTerm","declaration":"def HTPI.ColonTerm : Type"} +{"name":"HTPI.tacticFix_","declaration":"def HTPI.tacticFix_ : Lean.ParserDescr"} +{"name":"HTPI.doSwap","declaration":"def HTPI.doSwap : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.tacticContrapos__","declaration":"def HTPI.tacticContrapos__ : Lean.ParserDescr"} +{"name":"HTPI.PropForm.ex","declaration":"ctor HTPI.PropForm.ex : Lean.Level → Lean.Name → Lean.Expr → Lean.Expr → Lean.BinderInfo → HTPI.PropForm"} +{"name":"HTPI.sum_empty","declaration":"theorem HTPI.sum_empty {A : Type} [AddZeroClass A] {k : ℕ} {n : ℕ} {f : ℕ → A} (h : n < k) : Sum i from k to n, f i = 0"} +{"name":"HTPI.getHeadData","declaration":"def HTPI.getHeadData (e : Lean.Expr) : Lean.Expr × List Lean.Expr"} +{"name":"HTPI.doEquivTac","declaration":"def HTPI.doEquivTac (f : Option HTPI.ColonTerm) (l : Option HTPI.OneLoc) (tac : Lean.Name) (ruleFunc : Lean.Expr → Lean.Elab.Tactic.TacticM HTPI.ruleType) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.tacticShow_By_","declaration":"def HTPI.tacticShow_By_ : Lean.ParserDescr"} +{"name":"HTPI.PropForm.iff","declaration":"ctor HTPI.PropForm.iff : Lean.Expr → Lean.Expr → HTPI.PropForm"} +{"name":"HTPI.withId","declaration":"def HTPI.withId : Lean.ParserDescr"} +{"name":"HTPI.tacticAssume_","declaration":"def HTPI.tacticAssume_ : Lean.ParserDescr"} +{"name":"HTPI.With2Ids","declaration":"def HTPI.With2Ids : Type"} +{"name":"HTPI.with2Ids","declaration":"def HTPI.with2Ids : Lean.ParserDescr"} +{"name":"HTPI.PropForm.all","declaration":"ctor HTPI.PropForm.all : Lean.Name → Lean.Expr → Lean.Expr → Lean.BinderInfo → HTPI.PropForm"} +{"name":"HTPI.exun_elim","declaration":"theorem HTPI.exun_elim {α : Sort u} {p : α → Prop} {b : Prop} (h2 : ∃! x, p x) (h1 : ∀ (x : α), p x → (∀ (y z : α), p y → p z → y = z) → b) : b"} +{"name":"HTPI.not_imp_not_iff_and","declaration":"theorem HTPI.not_imp_not_iff_and {p : Prop} {q : Prop} : ¬(p → ¬q) ↔ p ∧ q"} +{"name":"HTPI.mkIff","declaration":"def HTPI.mkIff (l : Lean.Expr) (r : Lean.Expr) : Lean.Expr"} +{"name":"HTPI.doAssume","declaration":"def HTPI.doAssume (w : Lean.Term) (t : Option Lean.Term) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.whnfNotExUn","declaration":"def HTPI.whnfNotExUn (e : Lean.Expr) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.tacticDefinition___1","declaration":"def HTPI.tacticDefinition___1 : Lean.ParserDescr"} +{"name":"HTPI.addToName","declaration":"def HTPI.addToName (n : Lean.Name) (s : String) : Lean.Name"} +{"name":"HTPI.mkUn","declaration":"def HTPI.mkUn (lev : Lean.Level) (v : Lean.Name) (t : Lean.Expr) (b : Lean.Expr) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.myTrace","declaration":"def HTPI.myTrace (msg : String) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.tacticExists_unique","declaration":"def HTPI.tacticExists_unique : Lean.ParserDescr"} +{"name":"HTPI.sum_from_zero_step","declaration":"theorem HTPI.sum_from_zero_step {A : Type} [AddZeroClass A] {n : ℕ} {f : ℕ → A} : Sum i from 0 to n + 1, f i = (Sum i from 0 to n, f i) + f (n + 1)"} +{"name":"HTPI.dontUnfoldNum","declaration":"def HTPI.dontUnfoldNum : List Lean.Name"} +{"name":"HTPI.unfoldOrWhnf","declaration":"def HTPI.unfoldOrWhnf (tac : Lean.Name) (e : Lean.Expr) (w : Bool) (rep : Bool) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.«tacticAssume_:_»","declaration":"def HTPI.«tacticAssume_:_» : Lean.ParserDescr"} +{"name":"HTPI.tacticTraceExpr__","declaration":"def HTPI.tacticTraceExpr__ : Lean.ParserDescr"} +{"name":"HTPI.PropForm.or","declaration":"ctor HTPI.PropForm.or : Lean.Expr → Lean.Expr → HTPI.PropForm"} +{"name":"HTPI.not_or_not_distrib","declaration":"theorem HTPI.not_or_not_distrib {p : Prop} {q : Prop} : ¬(p ∨ ¬q) ↔ ¬p ∧ q"} +{"name":"HTPI.getPropForm","declaration":"def HTPI.getPropForm (e : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.PropForm"} +{"name":"HTPI.tacticOr_right_","declaration":"def HTPI.tacticOr_right_ : Lean.ParserDescr"} +{"name":"HTPI.induc_from","declaration":"theorem HTPI.induc_from (P : ℕ → Prop) (k : ℕ) (h1 : P k) (h2 : ∀ n ≥ k, P n → P (n + 1)) (n : ℕ) : n ≥ k → P n"} +{"name":"Iff.rtl","declaration":"def Iff.rtl {p : Prop} {q : Prop} (h : p ↔ q) : q → p"} +{"name":"HTPI.cpRule","declaration":"def HTPI.cpRule (form : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.binderString","declaration":"def HTPI.binderString (bi : Lean.BinderInfo) : String"} +{"name":"HTPI.tacticDefinition!__","declaration":"def HTPI.tacticDefinition!__ : Lean.ParserDescr"} +{"name":"HTPI.mkExists","declaration":"def HTPI.mkExists (l : Lean.Level) (x : Lean.Name) (bi : Lean.BinderInfo) (t : Lean.Expr) (b : Lean.Expr) : Lean.Expr"} +{"name":"HTPI.checkIdUsed","declaration":"opaque HTPI.checkIdUsed (tac : Lean.Name) (i : Lean.Syntax) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.sum_from_to","declaration":"def HTPI.sum_from_to {A : Type} [AddZeroClass A] (k : ℕ) (n : ℕ) (f : ℕ → A) : A"} +{"name":"HTPI.equivRuleFromForm","declaration":"def HTPI.equivRuleFromForm (p : Lean.Expr) (ruleFunc : Lean.Expr → Lean.Elab.Tactic.TacticM HTPI.ruleType) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.colonTerm","declaration":"def HTPI.colonTerm : Lean.ParserDescr"} +{"name":"HTPI.doDefinitionRep","declaration":"opaque HTPI.doDefinitionRep (label : Lean.Name) (e : Lean.Expr) (e1 : Lean.Expr) (prop : Bool) (rule : Lean.Ident) (firstNum : ℕ) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.qnRuleFromInfoNoNeg","declaration":"def HTPI.qnRuleFromInfoNoNeg (v : Lean.Name) (t : Lean.Expr) (b : Lean.Expr) (qf : Lean.Name → Lean.BinderInfo → Lean.Expr → Lean.Expr → Lean.Expr) (rs : Lean.Name × Lean.Name) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.getDefineFormLabel","declaration":"def HTPI.getDefineFormLabel (f : Option HTPI.ColonTerm) (l : Option HTPI.OneLoc) : Lean.Elab.Tactic.TacticM (Lean.Expr × Lean.Name)"} +{"name":"HTPI.tacticBy_casesOn__","declaration":"def HTPI.tacticBy_casesOn__ : Lean.ParserDescr"} +{"name":"HTPI.PropForm.t","declaration":"ctor HTPI.PropForm.t : HTPI.PropForm"} +{"name":"HTPI.setUpCases","declaration":"def HTPI.setUpCases (t : Lean.Term) (wids : Option HTPI.With2Ids) : Lean.Elab.Tactic.TacticM (Bool × Lean.Ident × Lean.Ident)"} +{"name":"HTPI.tacticDefine__","declaration":"def HTPI.tacticDefine__ : Lean.ParserDescr"} +{"name":"HTPI.tacticContradict__","declaration":"def HTPI.tacticContradict__ : Lean.ParserDescr"} +{"name":"HTPI.unfoldHeadCore","declaration":"opaque HTPI.unfoldHeadCore (e : Lean.Expr) (first : Bool) (rep : Bool) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.PropForm.exun","declaration":"ctor HTPI.PropForm.exun : Lean.Level → Lean.Name → Lean.Expr → Lean.Expr → Lean.BinderInfo → HTPI.PropForm"} +{"name":"HTPI.myFail","declaration":"def HTPI.myFail {α : Type} (tac : Lean.Name) (msg : String) : Lean.Elab.Tactic.TacticM α"} +{"name":"HTPI.not_not_iff","declaration":"theorem HTPI.not_not_iff {p : Prop} {q : Prop} : ¬(¬p ↔ q) ↔ (p ↔ q)"} +{"name":"HTPI.BinRel","declaration":"def HTPI.BinRel (t : Type u) : Type u"} +{"name":"HTPI.tacticDefinition!___1","declaration":"def HTPI.tacticDefinition!___1 : Lean.ParserDescr"} +{"name":"HTPI.not_not_or_distrib","declaration":"theorem HTPI.not_not_or_distrib {p : Prop} {q : Prop} : ¬(¬p ∨ q) ↔ p ∧ ¬q"} +{"name":"HTPI.not_and_not_distrib","declaration":"theorem HTPI.not_and_not_distrib {p : Prop} {q : Prop} : ¬(p ∧ ¬q) ↔ ¬p ∨ q"} +{"name":"HTPI.«tacticShow_:=_»","declaration":"def HTPI.«tacticShow_:=_» : Lean.ParserDescr"} +{"name":"HTPI.mkRel","declaration":"def HTPI.mkRel (e1 : Lean.Expr) (e2 : Lean.Expr) (prop : Bool) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.exprIsProp","declaration":"def HTPI.exprIsProp (e : Lean.Expr) : Lean.Elab.Tactic.TacticM Bool"} +{"name":"HTPI.doObtain","declaration":"def HTPI.doObtain (itw : HTPI.IdOrTerm?Type) (ith : HTPI.IdOrTerm?Type) (tm : Lean.Term) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.ensureContra","declaration":"def HTPI.ensureContra (w : Option HTPI.WithId) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.DisjSyllData","declaration":"def HTPI.DisjSyllData (disj : Lean.Expr) (neg : Lean.Expr) : Lean.Elab.Tactic.TacticM (Bool × Bool)"} +{"name":"HTPI.doFix","declaration":"def HTPI.doFix (w : Lean.Term) (t : Option Lean.Term) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.sum_seq","declaration":"def HTPI.sum_seq {A : Type} [AddZeroClass A] (m : ℕ) (k : ℕ) (f : ℕ → A) : A"} +{"name":"HTPI.PropForm.none","declaration":"ctor HTPI.PropForm.none : HTPI.PropForm"} +{"name":"HTPI.binegRule","declaration":"def HTPI.binegRule (form : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"«term_⊈_»","declaration":"def «term_⊈_» : Lean.TrailingParserDescr"} +{"name":"HTPI.formFromIdent","declaration":"def HTPI.formFromIdent (h : Lean.Syntax) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.OneLoc","declaration":"def HTPI.OneLoc : Type"} +{"name":"HTPI.SyntaxToString.SyntaxListToString","declaration":"opaque HTPI.SyntaxToString.SyntaxListToString (ss : List Lean.Syntax) : String"} +{"name":"HTPI.findNegPropAll","declaration":"def HTPI.findNegPropAll (e : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.PropForm"} +{"name":"HTPI.tacticDefinition__","declaration":"def HTPI.tacticDefinition__ : Lean.ParserDescr"} +{"name":"HTPI.tacticDemorgan__","declaration":"def HTPI.tacticDemorgan__ : Lean.ParserDescr"} +{"name":"HTPI.not_not_and_distrib","declaration":"theorem HTPI.not_not_and_distrib {p : Prop} {q : Prop} : ¬(¬p ∧ q) ↔ p ∨ ¬q"} +{"name":"HTPI.PropForm.implies","declaration":"ctor HTPI.PropForm.implies : Lean.Expr → Lean.Expr → HTPI.PropForm"} +{"name":"HTPI.ruleType","declaration":"def HTPI.ruleType : Type"} +{"name":"HTPI.PropForm.and","declaration":"ctor HTPI.PropForm.and : Lean.Expr → Lean.Expr → HTPI.PropForm"} +{"name":"HTPI.orstrat","declaration":"def HTPI.orstrat (tac : Lean.Name) (w : Option HTPI.WithId) (left : Bool) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.tacticOr_left_","declaration":"def HTPI.tacticOr_left_ : Lean.ParserDescr"} +{"name":"HTPI.PropForm.not","declaration":"ctor HTPI.PropForm.not : Lean.Expr → HTPI.PropForm"} +{"name":"HTPI.getLabel","declaration":"def HTPI.getLabel (tac : Lean.Name) (w : Option HTPI.WithId) (dflt : optParam Lean.Ident (Lean.mkIdent `this)) : Lean.Elab.Tactic.TacticM Lean.Ident"} +{"name":"HTPI.unfoldExUn","declaration":"def HTPI.unfoldExUn (lev : Lean.Level) (v : Lean.Name) (t : Lean.Expr) (b : Lean.Expr) : Lean.BinderInfo → Lean.Expr"} +{"name":"HTPI.doObtainExUn","declaration":"def HTPI.doObtainExUn (itw : HTPI.IdOrTerm?Type) (ith1 : HTPI.IdOrTerm?Type) (ith2 : HTPI.IdOrTerm?Type) (tm : Lean.Term) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.equivRule","declaration":"def HTPI.equivRule (f : Option HTPI.ColonTerm) (l : Option HTPI.OneLoc) (ruleFunc : Lean.Expr → Lean.Elab.Tactic.TacticM HTPI.ruleType) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.qnRuleFromInfo","declaration":"def HTPI.qnRuleFromInfo (v : Lean.Name) (t : Lean.Expr) (b : Lean.Expr) (qf : Lean.Name → Lean.BinderInfo → Lean.Expr → Lean.Expr → Lean.Expr) (n : Bool) (rs : Lean.Name × Lean.Name) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.oneLoc","declaration":"def HTPI.oneLoc : Lean.ParserDescr"} +{"name":"HTPI.doDefine","declaration":"def HTPI.doDefine (tac : Lean.Name) (f : Option HTPI.ColonTerm) (l : Option HTPI.OneLoc) (w : Bool) (rep : Bool) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.qnRule","declaration":"def HTPI.qnRule (form : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.isLocalVar","declaration":"def HTPI.isLocalVar (s : Lean.Syntax) : Lean.Elab.Tactic.TacticM Bool"} +{"name":"HTPI.unfoldHead","declaration":"def HTPI.unfoldHead (e : Lean.Expr) (tac : Lean.Name) (first : Bool) (rep : Bool) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.dmRuleFromInfoNoNeg","declaration":"def HTPI.dmRuleFromInfoNoNeg (l : Lean.Expr) (r : Lean.Expr) (conn : Lean.Expr → Lean.Expr → Lean.Expr) (rs : Array Lean.Name) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.IdOrTerm?Type","declaration":"def HTPI.IdOrTerm?Type : Type"} +{"name":"HTPI.doIntroOption","declaration":"def HTPI.doIntroOption (tac : Lean.Name) (i : Lean.Term) (t : Option Lean.Term) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.tacticDisj_syll___","declaration":"def HTPI.tacticDisj_syll___ : Lean.ParserDescr"} +{"name":"HTPI.tacticConditional__","declaration":"def HTPI.tacticConditional__ : Lean.ParserDescr"} +{"name":"HTPI.mkAppList","declaration":"def HTPI.mkAppList (h : Lean.Expr) (args : List Lean.Expr) : Lean.Expr"} +{"name":"HTPI.tacticDef_step__","declaration":"def HTPI.tacticDef_step__ : Lean.ParserDescr"} +{"name":"HTPI.tacticObtain__From_","declaration":"def HTPI.tacticObtain__From_ : Lean.ParserDescr"} +{"name":"HTPI.mkDeclaredTypeButIsExpectedMsg","declaration":"def HTPI.mkDeclaredTypeButIsExpectedMsg (decType : Lean.Expr) (expectedType : Lean.Expr) : Lean.MetaM Lean.MessageData"} +{"name":"HTPI.cdlRule","declaration":"def HTPI.cdlRule (form : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.dnRule","declaration":"def HTPI.dnRule (form : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.tacticWhnf__","declaration":"def HTPI.tacticWhnf__ : Lean.ParserDescr"} +{"name":"HTPI.idOrTerm","declaration":"def HTPI.idOrTerm : Lean.ParserDescr"} +{"name":"HTPI.sumFromTo","declaration":"def HTPI.sumFromTo : Lean.ParserDescr"} +{"name":"HTPI.exprFromPf","declaration":"def HTPI.exprFromPf (t : Lean.Term) (w : ℕ) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.applyToExData","declaration":"def HTPI.applyToExData {α : Type} (f : Lean.Level → Lean.Name → Lean.Expr → Lean.Expr → Lean.BinderInfo → α) (lev : Lean.Level) (l : Lean.Expr) (r : Lean.Expr) : α"} +{"name":"HTPI.traceThisSyntax","declaration":"def HTPI.traceThisSyntax (s : Lean.Syntax) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.fixCase","declaration":"def HTPI.fixCase (clear : Bool) (label : Lean.Ident) (g : Lean.Name) (c : String) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.dontUnfold","declaration":"def HTPI.dontUnfold : List Lean.Name"} +{"name":"HTPI.sum_step","declaration":"theorem HTPI.sum_step {A : Type} [AddZeroClass A] {k : ℕ} {n : ℕ} {f : ℕ → A} (h : k ≤ n) : Sum i from k to n + 1, f i = (Sum i from k to n, f i) + f (n + 1)"} +{"name":"HTPI.parseIdOrTerm?Type","declaration":"def HTPI.parseIdOrTerm?Type (tac : Lean.Name) (it : HTPI.IdOrTerm?Type) : Lean.Elab.Tactic.TacticM (Lean.Term × Option Lean.Term)"} +{"name":"HTPI.doDefinition","declaration":"def HTPI.doDefinition (all : Bool) (f : Option HTPI.ColonTerm) (l : Option HTPI.OneLoc) (wid : Option HTPI.WithId) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.dmRule","declaration":"def HTPI.dmRule (form : Lean.Expr) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.doReplace","declaration":"def HTPI.doReplace (tac : Lean.Name) (l : Option HTPI.OneLoc) (res : Lean.Expr) (pf : Lean.Syntax) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.Pred","declaration":"def HTPI.Pred (t : Type u) : Type u"} +{"name":"HTPI.IdOrTerm","declaration":"def HTPI.IdOrTerm : Type"} +{"name":"HTPI.tacticObtain___From_","declaration":"def HTPI.tacticObtain___From_ : Lean.ParserDescr"} +{"name":"«term_∆_»","declaration":"def «term_∆_» : Lean.TrailingParserDescr"} +{"name":"HTPI.dmRuleFromInfo","declaration":"def HTPI.dmRuleFromInfo (l : Lean.Expr) (r : Lean.Expr) (conn : Lean.Expr → Lean.Expr → Lean.Expr) (n : Bool) (rs : Array Lean.Name) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.formFromLoc","declaration":"def HTPI.formFromLoc (l : Option HTPI.OneLoc) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.SyntaxToString","declaration":"opaque HTPI.SyntaxToString (s : Lean.Syntax) : String"} +{"name":"HTPI.WithId","declaration":"def HTPI.WithId : Type"} +{"name":"HTPI.not_imp_iff_not_and","declaration":"theorem HTPI.not_imp_iff_not_and {p : Prop} {q : Prop} : ¬(q → p) ↔ ¬p ∧ q"} +{"name":"HTPI.equivMakeRule","declaration":"def HTPI.equivMakeRule (f : Lean.Expr) (ruleFunc : Lean.Expr → Lean.Elab.Tactic.TacticM HTPI.ruleType) : Lean.Elab.Tactic.TacticM HTPI.ruleType"} +{"name":"HTPI.doHave","declaration":"def HTPI.doHave (n : Lean.Name) (form : Lean.Expr) (pfstx : Lean.Syntax) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"HTPI.PropForm","declaration":"inductive HTPI.PropForm : Type"} +{"name":"HTPI.unexpandSumFromTo","declaration":"def HTPI.unexpandSumFromTo : Lean.PrettyPrinter.Unexpander"} +{"name":"HTPI.numNames","declaration":"def HTPI.numNames : List Lean.Name"} +{"name":"HTPI.PropForm.f","declaration":"ctor HTPI.PropForm.f : HTPI.PropForm"} +{"name":"HTPI.tacticBicond_neg__","declaration":"def HTPI.tacticBicond_neg__ : Lean.ParserDescr"} +{"name":"HTPI.tacticQuant_neg__","declaration":"def HTPI.tacticQuant_neg__ : Lean.ParserDescr"} +{"name":"HTPI.fixElt","declaration":"def HTPI.fixElt (e : Lean.Expr) (doFix : Bool) : Lean.Elab.Tactic.TacticM Lean.Expr"} +{"name":"HTPI.matchFirstNeg","declaration":"def HTPI.matchFirstNeg (e1 : Lean.Expr) (e2 : Lean.Expr) : Lean.Elab.Tactic.TacticM Bool"} +{"name":"HTPI.sum_base","declaration":"theorem HTPI.sum_base {A : Type} [AddZeroClass A] {k : ℕ} {f : ℕ → A} : Sum i from k to k, f i = f k"} +{"name":"HTPI.tacticShow_From_","declaration":"def HTPI.tacticShow_From_ : Lean.ParserDescr"} +{"name":"HTPI.negData","declaration":"def HTPI.negData (e : Lean.Expr) : Lean.Elab.Tactic.TacticM (Lean.Expr × Bool)"} +{"name":"HTPI.traceThisExpr","declaration":"def HTPI.traceThisExpr (e : Lean.Expr) : Lean.Elab.Tactic.TacticM Unit"} +{"name":"Iff.ltr","declaration":"def Iff.ltr {p : Prop} {q : Prop} (h : p ↔ q) : p → q"} +{"name":"HTPI.tacticBy_induc","declaration":"def HTPI.tacticBy_induc : Lean.ParserDescr"} diff --git a/htpi-declarations/HTPILib.IntroLean.jsonl b/htpi-declarations/HTPILib.IntroLean.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7041e0d8db745efaaefa0791022eed0872103f4d --- /dev/null +++ b/htpi-declarations/HTPILib.IntroLean.jsonl @@ -0,0 +1,7 @@ +{"name":"HTPI.two_imp","declaration":"theorem HTPI.two_imp (P : Prop) (Q : Prop) (R : Prop) (h1 : P → Q) (h2 : Q → ¬R) : R → ¬P"} +{"name":"HTPI.Example_3_2_4","declaration":"theorem HTPI.Example_3_2_4 (P : Prop) (Q : Prop) (R : Prop) (h : P → Q → R) : ¬R → P → ¬Q"} +{"name":"HTPI.extremely_easy","declaration":"theorem HTPI.extremely_easy (P : Prop) (h : P) : P"} +{"name":"HTPI.very_easy","declaration":"theorem HTPI.very_easy (P : Prop) (Q : Prop) (h1 : P → Q) (h2 : P) : Q"} +{"name":"HTPI.Example_3_2_5_simple","declaration":"theorem HTPI.Example_3_2_5_simple (B : Set ℕ) (C : Set ℕ) (a : ℕ) (h1 : a ∈ B) (h2 : a ∉ B \\ C) : a ∈ C"} +{"name":"HTPI.Example_3_2_5_simple_general","declaration":"theorem HTPI.Example_3_2_5_simple_general (U : Type) (B : Set U) (C : Set U) (a : U) (h1 : a ∈ B) (h2 : a ∉ B \\ C) : a ∈ C"} +{"name":"HTPI.easy","declaration":"theorem HTPI.easy (P : Prop) (Q : Prop) (R : Prop) (h1 : P → Q) (h2 : Q → R) (h3 : P) : R"} diff --git a/htpi-declarations/HTPILib.jsonl b/htpi-declarations/HTPILib.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/minictx/hep.jsonl b/minictx/hep.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ef4d5675b023c5f3110724817470993158d91154 --- /dev/null +++ b/minictx/hep.jsonl @@ -0,0 +1,61 @@ +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Analysis.Complex.Basic\n/-!\n# The Clifford Algebra\n\nThis file defines the Gamma matrices.\n\n-/\n/-! TODO: Prove algebra generated by gamma matrices is isomorphic to Clifford algebra. -/\n/-! TODO: Define relations between the gamma matrices. -/\nnamespace spaceTime\nopen Complex\n\nnoncomputable section diracRepresentation\n\n/-- The \u03b3\u2070 gamma matrix in the Dirac representation. -/\ndef \u03b30 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]\n\n/-- The \u03b3\u00b9 gamma matrix in the Dirac representation. -/\ndef \u03b31 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 0, 1], ![0, 0, 1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]]\n\n/-- The \u03b3\u00b2 gamma matrix in the Dirac representation. -/\ndef \u03b32 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 0, - I], ![0, 0, I, 0], ![0, I, 0, 0], ![-I, 0, 0, 0]]\n\n/-- The \u03b3\u00b3 gamma matrix in the Dirac representation. -/\ndef \u03b33 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, 1, 0, 0]]\n\n/-- The \u03b3\u2075 gamma matrix in the Dirac representation. -/\ndef \u03b35 : Matrix (Fin 4) (Fin 4) \u2102 := I \u2022 (\u03b30 * \u03b31 * \u03b32 * \u03b33)\n\n/-- The \u03b3 gamma matrices in the Dirac representation. -/\n@[simp]\ndef \u03b3 : Fin 4 \u2192 Matrix (Fin 4) (Fin 4) \u2102 := ![\u03b30, \u03b31, \u03b32, \u03b33]\n\nnamespace \u03b3\n\nopen spaceTime\n\nvariable (\u03bc \u03bd : Fin 4)\n\n/-- The subset of `Matrix (Fin 4) (Fin 4) \u2102` formed by the gamma matrices in the Dirac\nrepresentation. -/\n@[simp]\ndef \u03b3Set : Set (Matrix (Fin 4) (Fin 4) \u2102) := {\u03b3 i | i : Fin 4}\n\n", "theoremStatement": "lemma \u03b3_in_\u03b3Set (\u03bc : Fin 4) : \u03b3 \u03bc \u2208 \u03b3Set ", "theoremName": "spaceTime.\u03b3.\u03b3_in_\u03b3Set", "fileCreated": {"commit": "7020263", "date": "2024-05-14"}, "theoremCreated": {"commit": "b9c2086", "date": "2024-05-10"}, "file": "hep_lean/HepLean/SpaceTime/CliffordAlgebra.lean", "module": "HepLean.SpaceTime.CliffordAlgebra", "jsonFile": "HepLean.SpaceTime.CliffordAlgebra.jsonl", "positionMetadata": {"lineInFile": 54, "tokenPositionInFile": 1613, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 23, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [\u03b3Set]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 19}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Analysis.Complex.Basic\n/-!\n# The Clifford Algebra\n\nThis file defines the Gamma matrices.\n\n-/\n/-! TODO: Prove algebra generated by gamma matrices is isomorphic to Clifford algebra. -/\n/-! TODO: Define relations between the gamma matrices. -/\nnamespace spaceTime\nopen Complex\n\nnoncomputable section diracRepresentation\n\n/-- The \u03b3\u2070 gamma matrix in the Dirac representation. -/\ndef \u03b30 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]\n\n/-- The \u03b3\u00b9 gamma matrix in the Dirac representation. -/\ndef \u03b31 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 0, 1], ![0, 0, 1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]]\n\n/-- The \u03b3\u00b2 gamma matrix in the Dirac representation. -/\ndef \u03b32 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 0, - I], ![0, 0, I, 0], ![0, I, 0, 0], ![-I, 0, 0, 0]]\n\n/-- The \u03b3\u00b3 gamma matrix in the Dirac representation. -/\ndef \u03b33 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, 1, 0, 0]]\n\n/-- The \u03b3\u2075 gamma matrix in the Dirac representation. -/\ndef \u03b35 : Matrix (Fin 4) (Fin 4) \u2102 := I \u2022 (\u03b30 * \u03b31 * \u03b32 * \u03b33)\n\n/-- The \u03b3 gamma matrices in the Dirac representation. -/\n@[simp]\ndef \u03b3 : Fin 4 \u2192 Matrix (Fin 4) (Fin 4) \u2102 := ![\u03b30, \u03b31, \u03b32, \u03b33]\n\nnamespace \u03b3\n\nopen spaceTime\n\nvariable (\u03bc \u03bd : Fin 4)\n\n/-- The subset of `Matrix (Fin 4) (Fin 4) \u2102` formed by the gamma matrices in the Dirac\nrepresentation. -/\n@[simp]\ndef \u03b3Set : Set (Matrix (Fin 4) (Fin 4) \u2102) := {\u03b3 i | i : Fin 4}\n\nlemma \u03b3_in_\u03b3Set (\u03bc : Fin 4) : \u03b3 \u03bc \u2208 \u03b3Set := by\n simp [\u03b3Set]\n\n/-- The algebra generated by the gamma matrices in the Dirac representation. -/\ndef diracAlgebra : Subalgebra \u211d (Matrix (Fin 4) (Fin 4) \u2102) :=\n Algebra.adjoin \u211d \u03b3Set\n\n", "theoremStatement": "lemma \u03b3Set_subset_diracAlgebra : \u03b3Set \u2286 diracAlgebra ", "theoremName": "spaceTime.\u03b3.\u03b3Set_subset_diracAlgebra", "fileCreated": {"commit": "7020263", "date": "2024-05-14"}, "theoremCreated": {"commit": "b9c2086", "date": "2024-05-10"}, "file": "hep_lean/HepLean/SpaceTime/CliffordAlgebra.lean", "module": "HepLean.SpaceTime.CliffordAlgebra", "jsonFile": "HepLean.SpaceTime.CliffordAlgebra.jsonl", "positionMetadata": {"lineInFile": 61, "tokenPositionInFile": 1842, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 30, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n Algebra.subset_adjoin", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 26}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Analysis.Complex.Basic\n/-!\n# The Clifford Algebra\n\nThis file defines the Gamma matrices.\n\n-/\n/-! TODO: Prove algebra generated by gamma matrices is isomorphic to Clifford algebra. -/\n/-! TODO: Define relations between the gamma matrices. -/\nnamespace spaceTime\nopen Complex\n\nnoncomputable section diracRepresentation\n\n/-- The \u03b3\u2070 gamma matrix in the Dirac representation. -/\ndef \u03b30 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]\n\n/-- The \u03b3\u00b9 gamma matrix in the Dirac representation. -/\ndef \u03b31 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 0, 1], ![0, 0, 1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]]\n\n/-- The \u03b3\u00b2 gamma matrix in the Dirac representation. -/\ndef \u03b32 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 0, - I], ![0, 0, I, 0], ![0, I, 0, 0], ![-I, 0, 0, 0]]\n\n/-- The \u03b3\u00b3 gamma matrix in the Dirac representation. -/\ndef \u03b33 : Matrix (Fin 4) (Fin 4) \u2102 :=\n ![![0, 0, 1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, 1, 0, 0]]\n\n/-- The \u03b3\u2075 gamma matrix in the Dirac representation. -/\ndef \u03b35 : Matrix (Fin 4) (Fin 4) \u2102 := I \u2022 (\u03b30 * \u03b31 * \u03b32 * \u03b33)\n\n/-- The \u03b3 gamma matrices in the Dirac representation. -/\n@[simp]\ndef \u03b3 : Fin 4 \u2192 Matrix (Fin 4) (Fin 4) \u2102 := ![\u03b30, \u03b31, \u03b32, \u03b33]\n\nnamespace \u03b3\n\nopen spaceTime\n\nvariable (\u03bc \u03bd : Fin 4)\n\n/-- The subset of `Matrix (Fin 4) (Fin 4) \u2102` formed by the gamma matrices in the Dirac\nrepresentation. -/\n@[simp]\ndef \u03b3Set : Set (Matrix (Fin 4) (Fin 4) \u2102) := {\u03b3 i | i : Fin 4}\n\nlemma \u03b3_in_\u03b3Set (\u03bc : Fin 4) : \u03b3 \u03bc \u2208 \u03b3Set := by\n simp [\u03b3Set]\n\n/-- The algebra generated by the gamma matrices in the Dirac representation. -/\ndef diracAlgebra : Subalgebra \u211d (Matrix (Fin 4) (Fin 4) \u2102) :=\n Algebra.adjoin \u211d \u03b3Set\n\nlemma \u03b3Set_subset_diracAlgebra : \u03b3Set \u2286 diracAlgebra :=\n Algebra.subset_adjoin\n\n", "theoremStatement": "lemma \u03b3_in_diracAlgebra (\u03bc : Fin 4) : \u03b3 \u03bc \u2208 diracAlgebra ", "theoremName": "spaceTime.\u03b3.\u03b3_in_diracAlgebra", "fileCreated": {"commit": "7020263", "date": "2024-05-14"}, "theoremCreated": {"commit": "b9c2086", "date": "2024-05-10"}, "file": "hep_lean/HepLean/SpaceTime/CliffordAlgebra.lean", "module": "HepLean.SpaceTime.CliffordAlgebra", "jsonFile": "HepLean.SpaceTime.CliffordAlgebra.jsonl", "positionMetadata": {"lineInFile": 64, "tokenPositionInFile": 1923, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 29, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n \u03b3Set_subset_diracAlgebra (\u03b3_in_\u03b3Set \u03bc)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 43}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\n", "theoremStatement": "lemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 ", "theoremName": "LorentzGroup.mem_iff_norm", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 55, "tokenPositionInFile": 1355, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 208, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 328}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\n", "theoremStatement": "lemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 ", "theoremName": "LorentzGroup.mem_iff_on_right", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 66, "tokenPositionInFile": 1784, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 71, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 202}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\n", "theoremStatement": "lemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 ", "theoremName": "LorentzGroup.mem_iff_dual_mul_self", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 75, "tokenPositionInFile": 2099, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 68, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 67}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\n", "theoremStatement": "lemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 ", "theoremName": "LorentzGroup.mem_iff_self_mul_dual", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 79, "tokenPositionInFile": 2234, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 58}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\n", "theoremStatement": "lemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d ", "theoremName": "LorentzGroup.mem_iff_transpose", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 83, "tokenPositionInFile": 2360, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 65, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring", "proofType": "tactic", "proofLengthLines": 11, "proofLengthTokens": 579}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\n", "theoremStatement": "lemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d ", "theoremName": "LorentzGroup.mem_mul", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 96, "tokenPositionInFile": 3008, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 59, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 178}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\n", "theoremStatement": "lemma one_mem : 1 \u2208 LorentzGroup d ", "theoremName": "LorentzGroup.one_mem", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 103, "tokenPositionInFile": 3282, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 41, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mem_iff_dual_mul_self]\n simp", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 41}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\n", "theoremStatement": "lemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d ", "theoremName": "LorentzGroup.dual_mem", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 107, "tokenPositionInFile": 3360, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 80}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\n", "theoremStatement": "lemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9", "theoremName": "LorentzGroup.coe_inv", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 138, "tokenPositionInFile": 4254, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 40, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 56}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n", "theoremStatement": "@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 ", "theoremName": "LorentzGroup.subtype_inv_mul", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "a65fb06", "date": "2024-07-30"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 140, "tokenPositionInFile": 4343, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 50, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 94}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n", "theoremStatement": "@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 ", "theoremName": "LorentzGroup.subtype_mul_inv", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "a65fb06", "date": "2024-07-30"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 148, "tokenPositionInFile": 4511, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 50, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 95}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n", "theoremStatement": "@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix ", "theoremName": "LorentzGroup.mul_minkowskiMatrix_mul_transpose", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "7b0b979", "date": "2024-07-31"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 156, "tokenPositionInFile": 4680, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 71, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 185}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n", "theoremStatement": "@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix ", "theoremName": "LorentzGroup.transpose_mul_minkowskiMatrix_mul_self", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "7b0b979", "date": "2024-07-31"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 166, "tokenPositionInFile": 5001, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 71, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 178}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n", "theoremStatement": "@[simp]\nlemma transpose_one : @transpose d 1 = 1 ", "theoremName": "LorentzGroup.transpose_one", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "99f4e85", "date": "2024-07-29"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 180, "tokenPositionInFile": 5501, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 26, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= Subtype.eq Matrix.transpose_one", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 34}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n", "theoremStatement": "@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b ", "theoremName": "LorentzGroup.transpose_mul", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "99f4e85", "date": "2024-07-29"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 183, "tokenPositionInFile": 5586, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 28, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 47}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b :=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)\n\n/-!\n\n## Lorentz group as a topological group\n\nWe now show that the Lorentz group is a topological group.\nWe do this by showing that the natrual map from the Lorentz group to `GL (Fin 4) \u211d` is an\nembedding.\n\n-/\n\n/-- The homomorphism of the Lorentz group into `GL (Fin 4) \u211d`. -/\ndef toGL : LorentzGroup d \u2192* GL (Fin 1 \u2295 Fin d) \u211d where\n toFun A := \u27e8A.1, (A\u207b\u00b9).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,\n mem_iff_dual_mul_self.mp A.2\u27e9\n map_one' :=\n (GeneralLinearGroup.ext_iff _ 1).mpr fun _ => congrFun rfl\n map_mul' _ _ :=\n (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl\n\n", "theoremStatement": "lemma toGL_injective : Function.Injective (@toGL d) ", "theoremName": "LorentzGroup.toGL_injective", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 206, "tokenPositionInFile": 6319, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 41, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine fun A B h => Subtype.eq ?_\n rw [@Units.ext_iff] at h\n exact h", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 78}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b :=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)\n\n/-!\n\n## Lorentz group as a topological group\n\nWe now show that the Lorentz group is a topological group.\nWe do this by showing that the natrual map from the Lorentz group to `GL (Fin 4) \u211d` is an\nembedding.\n\n-/\n\n/-- The homomorphism of the Lorentz group into `GL (Fin 4) \u211d`. -/\ndef toGL : LorentzGroup d \u2192* GL (Fin 1 \u2295 Fin d) \u211d where\n toFun A := \u27e8A.1, (A\u207b\u00b9).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,\n mem_iff_dual_mul_self.mp A.2\u27e9\n map_one' :=\n (GeneralLinearGroup.ext_iff _ 1).mpr fun _ => congrFun rfl\n map_mul' _ _ :=\n (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl\n\nlemma toGL_injective : Function.Injective (@toGL d) := by\n refine fun A B h => Subtype.eq ?_\n rw [@Units.ext_iff] at h\n exact h\n\n/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\n@[simps!]\ndef toProd : LorentzGroup d \u2192* (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) \u00d7\n (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d)\u1d50\u1d52\u1d56 :=\n MonoidHom.comp (Units.embedProduct _) toGL\n\n", "theoremStatement": "lemma toProd_eq_transpose_\u03b7 : toProd \u039b = (\u039b.1, MulOpposite.op $ minkowskiMetric.dual \u039b.1) ", "theoremName": "LorentzGroup.toProd_eq_transpose_\u03b7", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 218, "tokenPositionInFile": 6759, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 39, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b :=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)\n\n/-!\n\n## Lorentz group as a topological group\n\nWe now show that the Lorentz group is a topological group.\nWe do this by showing that the natrual map from the Lorentz group to `GL (Fin 4) \u211d` is an\nembedding.\n\n-/\n\n/-- The homomorphism of the Lorentz group into `GL (Fin 4) \u211d`. -/\ndef toGL : LorentzGroup d \u2192* GL (Fin 1 \u2295 Fin d) \u211d where\n toFun A := \u27e8A.1, (A\u207b\u00b9).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,\n mem_iff_dual_mul_self.mp A.2\u27e9\n map_one' :=\n (GeneralLinearGroup.ext_iff _ 1).mpr fun _ => congrFun rfl\n map_mul' _ _ :=\n (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl\n\nlemma toGL_injective : Function.Injective (@toGL d) := by\n refine fun A B h => Subtype.eq ?_\n rw [@Units.ext_iff] at h\n exact h\n\n/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\n@[simps!]\ndef toProd : LorentzGroup d \u2192* (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) \u00d7\n (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d)\u1d50\u1d52\u1d56 :=\n MonoidHom.comp (Units.embedProduct _) toGL\n\nlemma toProd_eq_transpose_\u03b7 : toProd \u039b = (\u039b.1, MulOpposite.op $ minkowskiMetric.dual \u039b.1) := rfl\n\n", "theoremStatement": "lemma toProd_injective : Function.Injective (@toProd d) ", "theoremName": "LorentzGroup.toProd_injective", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 220, "tokenPositionInFile": 6857, "theoremPositionInFile": 21}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 47, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n intro A B h\n rw [toProd_eq_transpose_\u03b7, toProd_eq_transpose_\u03b7] at h\n rw [@Prod.mk.inj_iff] at h\n exact Subtype.eq h.1", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 128}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b :=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)\n\n/-!\n\n## Lorentz group as a topological group\n\nWe now show that the Lorentz group is a topological group.\nWe do this by showing that the natrual map from the Lorentz group to `GL (Fin 4) \u211d` is an\nembedding.\n\n-/\n\n/-- The homomorphism of the Lorentz group into `GL (Fin 4) \u211d`. -/\ndef toGL : LorentzGroup d \u2192* GL (Fin 1 \u2295 Fin d) \u211d where\n toFun A := \u27e8A.1, (A\u207b\u00b9).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,\n mem_iff_dual_mul_self.mp A.2\u27e9\n map_one' :=\n (GeneralLinearGroup.ext_iff _ 1).mpr fun _ => congrFun rfl\n map_mul' _ _ :=\n (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl\n\nlemma toGL_injective : Function.Injective (@toGL d) := by\n refine fun A B h => Subtype.eq ?_\n rw [@Units.ext_iff] at h\n exact h\n\n/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\n@[simps!]\ndef toProd : LorentzGroup d \u2192* (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) \u00d7\n (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d)\u1d50\u1d52\u1d56 :=\n MonoidHom.comp (Units.embedProduct _) toGL\n\nlemma toProd_eq_transpose_\u03b7 : toProd \u039b = (\u039b.1, MulOpposite.op $ minkowskiMetric.dual \u039b.1) := rfl\n\nlemma toProd_injective : Function.Injective (@toProd d) := by\n intro A B h\n rw [toProd_eq_transpose_\u03b7, toProd_eq_transpose_\u03b7] at h\n rw [@Prod.mk.inj_iff] at h\n exact Subtype.eq h.1\n\n", "theoremStatement": "lemma toProd_continuous : Continuous (@toProd d) ", "theoremName": "LorentzGroup.toProd_continuous", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 226, "tokenPositionInFile": 7043, "theoremPositionInFile": 22}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 82, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n change Continuous (fun A => (A.1, \u27e8dual A.1\u27e9))\n refine continuous_prod_mk.mpr \u27e8continuous_iff_le_induced.mpr fun U a \u21a6 a,\n MulOpposite.continuous_op.comp' ((continuous_const.matrix_mul (continuous_iff_le_induced.mpr\n fun U a => a).matrix_transpose).matrix_mul continuous_const)\u27e9", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 295}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b :=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)\n\n/-!\n\n## Lorentz group as a topological group\n\nWe now show that the Lorentz group is a topological group.\nWe do this by showing that the natrual map from the Lorentz group to `GL (Fin 4) \u211d` is an\nembedding.\n\n-/\n\n/-- The homomorphism of the Lorentz group into `GL (Fin 4) \u211d`. -/\ndef toGL : LorentzGroup d \u2192* GL (Fin 1 \u2295 Fin d) \u211d where\n toFun A := \u27e8A.1, (A\u207b\u00b9).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,\n mem_iff_dual_mul_self.mp A.2\u27e9\n map_one' :=\n (GeneralLinearGroup.ext_iff _ 1).mpr fun _ => congrFun rfl\n map_mul' _ _ :=\n (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl\n\nlemma toGL_injective : Function.Injective (@toGL d) := by\n refine fun A B h => Subtype.eq ?_\n rw [@Units.ext_iff] at h\n exact h\n\n/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\n@[simps!]\ndef toProd : LorentzGroup d \u2192* (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) \u00d7\n (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d)\u1d50\u1d52\u1d56 :=\n MonoidHom.comp (Units.embedProduct _) toGL\n\nlemma toProd_eq_transpose_\u03b7 : toProd \u039b = (\u039b.1, MulOpposite.op $ minkowskiMetric.dual \u039b.1) := rfl\n\nlemma toProd_injective : Function.Injective (@toProd d) := by\n intro A B h\n rw [toProd_eq_transpose_\u03b7, toProd_eq_transpose_\u03b7] at h\n rw [@Prod.mk.inj_iff] at h\n exact Subtype.eq h.1\n\nlemma toProd_continuous : Continuous (@toProd d) := by\n change Continuous (fun A => (A.1, \u27e8dual A.1\u27e9))\n refine continuous_prod_mk.mpr \u27e8continuous_iff_le_induced.mpr fun U a \u21a6 a,\n MulOpposite.continuous_op.comp' ((continuous_const.matrix_mul (continuous_iff_le_induced.mpr\n fun U a => a).matrix_transpose).matrix_mul continuous_const)\u27e9\n\n/-- The embedding from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\nlemma toProd_embedding : Embedding (@toProd d) where\n inj := toProd_injective\n induced :=\n (inducing_iff \u21d1toProd).mp (inducing_of_inducing_compose toProd_continuous continuous_fst\n ((inducing_iff (Prod.fst \u2218 \u21d1toProd)).mpr rfl))\n\n/-- The embedding from the Lorentz Group into `GL (Fin 4) \u211d`. -/\nlemma toGL_embedding : Embedding (@toGL d).toFun where\n inj := toGL_injective\n induced := by\n refine ((fun {X} {t t'} => TopologicalSpace.ext_iff.mpr) fun _ \u21a6 ?_).symm\n rw [TopologicalSpace.ext_iff.mp toProd_embedding.induced _, isOpen_induced_iff,\n isOpen_induced_iff]\n exact exists_exists_and_eq_and\n\ninstance : TopologicalGroup (LorentzGroup d) :=\nInducing.topologicalGroup toGL toGL_embedding.toInducing\n\nsection\nopen LorentzVector\n/-!\n\n# To a norm one Lorentz vector\n\n-/\n\n/-- The first column of a Lorentz matrix as a `NormOneLorentzVector`. -/\n@[simps!]\ndef toNormOneLorentzVector (\u039b : LorentzGroup d) : NormOneLorentzVector d :=\n \u27e8\u039b.1 *\u1d65 timeVec, by rw [NormOneLorentzVector.mem_iff, \u039b.2, minkowskiMetric.on_timeVec]\u27e9\n\n/-!\n\n# The time like element\n\n-/\n\n/-- The time like element of a Lorentz matrix. -/\n@[simp]\ndef timeComp (\u039b : LorentzGroup d) : \u211d := \u039b.1 (Sum.inl 0) (Sum.inl 0)\n\n", "theoremStatement": "theorem _root_.Matrix.mulVec_single_one [Fintype n] [DecidableEq n] [NonAssocSemiring R]\n (M : Matrix m n R) (j : n) :\n M *\u1d65 Pi.single j 1 = M\u1d40 j ", "theoremName": "Matrix.mulVec_single_one", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "1b9e77b", "date": "2024-09-23"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 275, "tokenPositionInFile": 8718, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 29, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by ext; simp", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 15}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b :=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)\n\n/-!\n\n## Lorentz group as a topological group\n\nWe now show that the Lorentz group is a topological group.\nWe do this by showing that the natrual map from the Lorentz group to `GL (Fin 4) \u211d` is an\nembedding.\n\n-/\n\n/-- The homomorphism of the Lorentz group into `GL (Fin 4) \u211d`. -/\ndef toGL : LorentzGroup d \u2192* GL (Fin 1 \u2295 Fin d) \u211d where\n toFun A := \u27e8A.1, (A\u207b\u00b9).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,\n mem_iff_dual_mul_self.mp A.2\u27e9\n map_one' :=\n (GeneralLinearGroup.ext_iff _ 1).mpr fun _ => congrFun rfl\n map_mul' _ _ :=\n (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl\n\nlemma toGL_injective : Function.Injective (@toGL d) := by\n refine fun A B h => Subtype.eq ?_\n rw [@Units.ext_iff] at h\n exact h\n\n/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\n@[simps!]\ndef toProd : LorentzGroup d \u2192* (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) \u00d7\n (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d)\u1d50\u1d52\u1d56 :=\n MonoidHom.comp (Units.embedProduct _) toGL\n\nlemma toProd_eq_transpose_\u03b7 : toProd \u039b = (\u039b.1, MulOpposite.op $ minkowskiMetric.dual \u039b.1) := rfl\n\nlemma toProd_injective : Function.Injective (@toProd d) := by\n intro A B h\n rw [toProd_eq_transpose_\u03b7, toProd_eq_transpose_\u03b7] at h\n rw [@Prod.mk.inj_iff] at h\n exact Subtype.eq h.1\n\nlemma toProd_continuous : Continuous (@toProd d) := by\n change Continuous (fun A => (A.1, \u27e8dual A.1\u27e9))\n refine continuous_prod_mk.mpr \u27e8continuous_iff_le_induced.mpr fun U a \u21a6 a,\n MulOpposite.continuous_op.comp' ((continuous_const.matrix_mul (continuous_iff_le_induced.mpr\n fun U a => a).matrix_transpose).matrix_mul continuous_const)\u27e9\n\n/-- The embedding from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\nlemma toProd_embedding : Embedding (@toProd d) where\n inj := toProd_injective\n induced :=\n (inducing_iff \u21d1toProd).mp (inducing_of_inducing_compose toProd_continuous continuous_fst\n ((inducing_iff (Prod.fst \u2218 \u21d1toProd)).mpr rfl))\n\n/-- The embedding from the Lorentz Group into `GL (Fin 4) \u211d`. -/\nlemma toGL_embedding : Embedding (@toGL d).toFun where\n inj := toGL_injective\n induced := by\n refine ((fun {X} {t t'} => TopologicalSpace.ext_iff.mpr) fun _ \u21a6 ?_).symm\n rw [TopologicalSpace.ext_iff.mp toProd_embedding.induced _, isOpen_induced_iff,\n isOpen_induced_iff]\n exact exists_exists_and_eq_and\n\ninstance : TopologicalGroup (LorentzGroup d) :=\nInducing.topologicalGroup toGL toGL_embedding.toInducing\n\nsection\nopen LorentzVector\n/-!\n\n# To a norm one Lorentz vector\n\n-/\n\n/-- The first column of a Lorentz matrix as a `NormOneLorentzVector`. -/\n@[simps!]\ndef toNormOneLorentzVector (\u039b : LorentzGroup d) : NormOneLorentzVector d :=\n \u27e8\u039b.1 *\u1d65 timeVec, by rw [NormOneLorentzVector.mem_iff, \u039b.2, minkowskiMetric.on_timeVec]\u27e9\n\n/-!\n\n# The time like element\n\n-/\n\n/-- The time like element of a Lorentz matrix. -/\n@[simp]\ndef timeComp (\u039b : LorentzGroup d) : \u211d := \u039b.1 (Sum.inl 0) (Sum.inl 0)\n\ntheorem _root_.Matrix.mulVec_single_one [Fintype n] [DecidableEq n] [NonAssocSemiring R]\n (M : Matrix m n R) (j : n) :\n M *\u1d65 Pi.single j 1 = M\u1d40 j := by ext; simp\n\n", "theoremStatement": "lemma timeComp_eq_toNormOneLorentzVector : timeComp \u039b = (toNormOneLorentzVector \u039b).1.time ", "theoremName": "LorentzGroup.timeComp_eq_toNormOneLorentzVector", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 279, "tokenPositionInFile": 8887, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 10, "numPremises": 65, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue, timeComp]\n erw [Pi.basisFun_apply, Matrix.mulVec_single_one]\n rfl", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 138}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.MinkowskiMetric\nimport HepLean.SpaceTime.LorentzVector.NormOne\n/-!\n# The Lorentz Group\n\nWe define the Lorentz group.\n\n## References\n\n- http://home.ku.edu.tr/~amostafazadeh/phys517_518/phys517_2016f/Handouts/A_Jaffi_Lorentz_Group.pdf\n\n-/\n/-! TODO: Show that the Lorentz is a Lie group. -/\n\nnoncomputable section\n\nopen Matrix\nopen Complex\nopen ComplexConjugate\n\n/-!\n## Matrices which preserves the Minkowski metric\n\nWe start studying the properties of matrices which preserve `\u03b7Lin`.\nThese matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.\n\n-/\nvariable {d : \u2115}\n\nopen minkowskiMetric in\n/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/\ndef LorentzGroup (d : \u2115) : Set (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) :=\n {\u039b : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d |\n \u2200 (x y : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098}\n\nnamespace LorentzGroup\n/-- Notation for the Lorentz group. -/\nscoped[LorentzGroup] notation (name := lorentzGroup_notation) \"\ud835\udcdb\" => LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d}\n\n/-!\n\n# Membership conditions\n\n-/\n\nlemma mem_iff_norm : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x : LorentzVector d), \u27ea\u039b *\u1d65 x, \u039b *\u1d65 x\u27eb\u2098 = \u27eax, x\u27eb\u2098 := by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp := h (x + y)\n have hn := h (x - y)\n rw [mulVec_add] at hp\n rw [mulVec_sub] at hn\n simp only [map_add, LinearMap.add_apply, map_sub, LinearMap.sub_apply] at hp hn\n rw [symm (\u039b *\u1d65 y) (\u039b *\u1d65 x), symm y x] at hp hn\n linear_combination hp / 4 + -1 * hn / 4\n\nlemma mem_iff_on_right : \u039b \u2208 LorentzGroup d \u2194\n \u2200 (x y : LorentzVector d), \u27eax, (dual \u039b * \u039b) *\u1d65 y\u27eb\u2098 = \u27eax, y\u27eb\u2098 := by\n refine Iff.intro (fun h x y \u21a6 ?_) (fun h x y \u21a6 ?_)\n \u00b7 have h1 := h x y\n rw [\u2190 dual_mulVec_right, mulVec_mulVec] at h1\n exact h1\n \u00b7 rw [\u2190 dual_mulVec_right, mulVec_mulVec]\n exact h x y\n\nlemma mem_iff_dual_mul_self : \u039b \u2208 LorentzGroup d \u2194 dual \u039b * \u039b = 1 := by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm\n\nlemma mem_iff_self_mul_dual : \u039b \u2208 LorentzGroup d \u2194 \u039b * dual \u039b = 1 := by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm\n\nlemma mem_iff_transpose : \u039b \u2208 LorentzGroup d \u2194 \u039b\u1d40 \u2208 LorentzGroup d := by\n refine Iff.intro (fun h \u21a6 ?_) (fun h \u21a6 ?_)\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n \u00b7 have h1 := congrArg transpose ((mem_iff_dual_mul_self).mp h)\n rw [dual, transpose_mul, transpose_mul, transpose_mul, minkowskiMatrix.eq_transpose,\n \u2190 mul_assoc, transpose_one, transpose_transpose] at h1\n rw [mem_iff_self_mul_dual, \u2190 h1, dual]\n noncomm_ring\n\nlemma mem_mul (h\u039b : \u039b \u2208 LorentzGroup d) (h\u039b' : \u039b' \u2208 LorentzGroup d) : \u039b * \u039b' \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual \u039b' * (dual \u039b * \u039b) * \u039b'\n \u00b7 noncomm_ring\n \u00b7 rw [(mem_iff_dual_mul_self).mp h\u039b]\n simp [(mem_iff_dual_mul_self).mp h\u039b']\n\nlemma one_mem : 1 \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self]\n simp\n\nlemma dual_mem (h : \u039b \u2208 LorentzGroup d) : dual \u039b \u2208 LorentzGroup d := by\n rw [mem_iff_dual_mul_self, dual_dual]\n exact mem_iff_self_mul_dual.mp h\n\nend LorentzGroup\n\n/-!\n\n# The Lorentz group as a group\n\n-/\n\n@[simps! mul_coe one_coe inv div]\ninstance lorentzGroupIsGroup : Group (LorentzGroup d) where\n mul A B := \u27e8A.1 * B.1, LorentzGroup.mem_mul A.2 B.2\u27e9\n mul_assoc A B C := Subtype.eq (Matrix.mul_assoc A.1 B.1 C.1)\n one := \u27e81, LorentzGroup.one_mem\u27e9\n one_mul A := Subtype.eq (Matrix.one_mul A.1)\n mul_one A := Subtype.eq (Matrix.mul_one A.1)\n inv A := \u27e8minkowskiMetric.dual A.1, LorentzGroup.dual_mem A.2\u27e9\n mul_left_inv A := Subtype.eq (LorentzGroup.mem_iff_dual_mul_self.mp A.2)\n\n/-- `LorentzGroup` has the subtype topology. -/\ninstance : TopologicalSpace (LorentzGroup d) := instTopologicalSpaceSubtype\n\nnamespace LorentzGroup\n\nopen minkowskiMetric\n\nvariable {\u039b \u039b' : LorentzGroup d}\n\nlemma coe_inv : (\u039b\u207b\u00b9).1 = \u039b.1\u207b\u00b9:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp \u039b.2)).symm\n\n@[simp]\nlemma subtype_inv_mul : (Subtype.val \u039b)\u207b\u00b9 * (Subtype.val \u039b) = 1 := by\n trans Subtype.val (\u039b\u207b\u00b9 * \u039b)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_left_inv \u039b]\n rfl\n\n@[simp]\nlemma subtype_mul_inv : (Subtype.val \u039b) * (Subtype.val \u039b)\u207b\u00b9 = 1 := by\n trans Subtype.val (\u039b * \u039b\u207b\u00b9)\n \u00b7 rw [\u2190 coe_inv]\n rfl\n \u00b7 rw [mul_right_inv \u039b]\n rfl\n\n@[simp]\nlemma mul_minkowskiMatrix_mul_transpose :\n (Subtype.val \u039b) * minkowskiMatrix * (Subtype.val \u039b).transpose = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_self_mul_dual] at h2\n simp only [dual] at h2\n refine (right_inv_eq_left_inv minkowskiMatrix.sq ?_).symm\n rw [\u2190 h2]\n noncomm_ring\n\n@[simp]\nlemma transpose_mul_minkowskiMatrix_mul_self :\n (Subtype.val \u039b).transpose * minkowskiMatrix * (Subtype.val \u039b) = minkowskiMatrix := by\n have h2 := \u039b.prop\n rw [LorentzGroup.mem_iff_dual_mul_self] at h2\n simp only [dual] at h2\n refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq\n rw [\u2190 h2]\n noncomm_ring\n\n/-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/\ndef transpose (\u039b : LorentzGroup d) : LorentzGroup d :=\n \u27e8\u039b.1\u1d40, mem_iff_transpose.mp \u039b.2\u27e9\n\n@[simp]\nlemma transpose_one : @transpose d 1 = 1 := Subtype.eq Matrix.transpose_one\n\n@[simp]\nlemma transpose_mul : transpose (\u039b * \u039b') = transpose \u039b' * transpose \u039b :=\n Subtype.eq (Matrix.transpose_mul \u039b.1 \u039b'.1)\n\n/-!\n\n## Lorentz group as a topological group\n\nWe now show that the Lorentz group is a topological group.\nWe do this by showing that the natrual map from the Lorentz group to `GL (Fin 4) \u211d` is an\nembedding.\n\n-/\n\n/-- The homomorphism of the Lorentz group into `GL (Fin 4) \u211d`. -/\ndef toGL : LorentzGroup d \u2192* GL (Fin 1 \u2295 Fin d) \u211d where\n toFun A := \u27e8A.1, (A\u207b\u00b9).1, mul_eq_one_comm.mpr $ mem_iff_dual_mul_self.mp A.2,\n mem_iff_dual_mul_self.mp A.2\u27e9\n map_one' :=\n (GeneralLinearGroup.ext_iff _ 1).mpr fun _ => congrFun rfl\n map_mul' _ _ :=\n (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl\n\nlemma toGL_injective : Function.Injective (@toGL d) := by\n refine fun A B h => Subtype.eq ?_\n rw [@Units.ext_iff] at h\n exact h\n\n/-- The homomorphism from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\n@[simps!]\ndef toProd : LorentzGroup d \u2192* (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d) \u00d7\n (Matrix (Fin 1 \u2295 Fin d) (Fin 1 \u2295 Fin d) \u211d)\u1d50\u1d52\u1d56 :=\n MonoidHom.comp (Units.embedProduct _) toGL\n\nlemma toProd_eq_transpose_\u03b7 : toProd \u039b = (\u039b.1, MulOpposite.op $ minkowskiMetric.dual \u039b.1) := rfl\n\nlemma toProd_injective : Function.Injective (@toProd d) := by\n intro A B h\n rw [toProd_eq_transpose_\u03b7, toProd_eq_transpose_\u03b7] at h\n rw [@Prod.mk.inj_iff] at h\n exact Subtype.eq h.1\n\nlemma toProd_continuous : Continuous (@toProd d) := by\n change Continuous (fun A => (A.1, \u27e8dual A.1\u27e9))\n refine continuous_prod_mk.mpr \u27e8continuous_iff_le_induced.mpr fun U a \u21a6 a,\n MulOpposite.continuous_op.comp' ((continuous_const.matrix_mul (continuous_iff_le_induced.mpr\n fun U a => a).matrix_transpose).matrix_mul continuous_const)\u27e9\n\n/-- The embedding from the Lorentz Group into the monoid of matrices times the opposite of\n the monoid of matrices. -/\nlemma toProd_embedding : Embedding (@toProd d) where\n inj := toProd_injective\n induced :=\n (inducing_iff \u21d1toProd).mp (inducing_of_inducing_compose toProd_continuous continuous_fst\n ((inducing_iff (Prod.fst \u2218 \u21d1toProd)).mpr rfl))\n\n/-- The embedding from the Lorentz Group into `GL (Fin 4) \u211d`. -/\nlemma toGL_embedding : Embedding (@toGL d).toFun where\n inj := toGL_injective\n induced := by\n refine ((fun {X} {t t'} => TopologicalSpace.ext_iff.mpr) fun _ \u21a6 ?_).symm\n rw [TopologicalSpace.ext_iff.mp toProd_embedding.induced _, isOpen_induced_iff,\n isOpen_induced_iff]\n exact exists_exists_and_eq_and\n\ninstance : TopologicalGroup (LorentzGroup d) :=\nInducing.topologicalGroup toGL toGL_embedding.toInducing\n\nsection\nopen LorentzVector\n/-!\n\n# To a norm one Lorentz vector\n\n-/\n\n/-- The first column of a Lorentz matrix as a `NormOneLorentzVector`. -/\n@[simps!]\ndef toNormOneLorentzVector (\u039b : LorentzGroup d) : NormOneLorentzVector d :=\n \u27e8\u039b.1 *\u1d65 timeVec, by rw [NormOneLorentzVector.mem_iff, \u039b.2, minkowskiMetric.on_timeVec]\u27e9\n\n/-!\n\n# The time like element\n\n-/\n\n/-- The time like element of a Lorentz matrix. -/\n@[simp]\ndef timeComp (\u039b : LorentzGroup d) : \u211d := \u039b.1 (Sum.inl 0) (Sum.inl 0)\n\ntheorem _root_.Matrix.mulVec_single_one [Fintype n] [DecidableEq n] [NonAssocSemiring R]\n (M : Matrix m n R) (j : n) :\n M *\u1d65 Pi.single j 1 = M\u1d40 j := by ext; simp\n\nlemma timeComp_eq_toNormOneLorentzVector : timeComp \u039b = (toNormOneLorentzVector \u039b).1.time := by\n simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue, timeComp]\n erw [Pi.basisFun_apply, Matrix.mulVec_single_one]\n rfl\n\n", "theoremStatement": "lemma timeComp_mul (\u039b \u039b' : LorentzGroup d) : timeComp (\u039b * \u039b') =\n \u27eatoNormOneLorentzVector (transpose \u039b), (toNormOneLorentzVector \u039b').1.spaceReflection\u27eb\u2098 ", "theoremName": "LorentzGroup.timeComp_mul", "fileCreated": {"commit": "89e940a", "date": "2024-05-17"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean", "module": "HepLean.SpaceTime.LorentzGroup.Basic", "jsonFile": "HepLean.SpaceTime.LorentzGroup.Basic.jsonl", "positionMetadata": {"lineInFile": 284, "tokenPositionInFile": 9117, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 16, "numPremises": 151, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp only [timeComp, Fin.isValue, lorentzGroupIsGroup_mul_coe, mul_apply, Fintype.sum_sum_type,\n Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton, toNormOneLorentzVector,\n transpose, timeVec, right_spaceReflection, time, space, PiLp.inner_apply, Function.comp_apply,\n RCLike.inner_apply, conj_trivial]\n erw [Pi.basisFun_apply, Matrix.mulVec_single_one]\n simp", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 390}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.Tensors.IndexNotation.TensorIndex\nimport HepLean.Tensors.IndexNotation.IndexString\nimport HepLean.SpaceTime.LorentzTensor.Real.Basic\n/-!\n\n# Index notation for real Lorentz tensors\n\nThis uses the general concepts of index notation in `HepLean.SpaceTime.LorentzTensor.IndexNotation`\nto define the index notation for real Lorentz tensors.\n\n-/\n\ninstance : IndexNotation realTensorColor.Color where\n charList := {'\u1d58', '\u1d64'}\n notaEquiv :=\n \u27e8fun c =>\n match c with\n | realTensorColor.ColorType.up => \u27e8'\u1d58', Finset.mem_insert_self '\u1d58' {'\u1d64'}\u27e9\n | realTensorColor.ColorType.down => \u27e8'\u1d64', Finset.insert_eq_self.mp (by rfl)\u27e9,\n fun c =>\n if c = '\u1d58' then realTensorColor.ColorType.up\n else realTensorColor.ColorType.down,\n by\n intro c\n match c with\n | realTensorColor.ColorType.up => rfl\n | realTensorColor.ColorType.down => rfl,\n by\n intro c\n by_cases hc : c = '\u1d58'\n \u00b7 simp only [\u2193Char.isValue, hc, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc))\n \u00b7 have hc' : c = '\u1d64' := by\n have hc2 := c.2\n simp only [\u2193Char.isValue, Finset.mem_insert, Finset.mem_singleton] at hc2\n simp_all\n simp only [\u2193Char.isValue, hc', Char.reduceEq, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc'))\u27e9\n\nnamespace realLorentzTensor\n\nopen realTensorColor\nopen IndexNotation IndexString\nopen TensorStructure TensorIndex\n\nvariable {d : \u2115}\n\ninstance instIndexNotationColor : IndexNotation (realLorentzTensor d).Color := instIndexNotationColorRealTensorColor\ninstance instDecidableEqColor : DecidableEq (realLorentzTensor d).Color := instDecidableEqColorRealTensorColor\n\n", "theoremStatement": "@[simp]\nlemma indexNotation_eq_color : @realLorentzTensor.instIndexNotationColor d =\n instIndexNotationColorRealTensorColor ", "theoremName": "realLorentzTensor.indexNotation_eq_color", "fileCreated": {"commit": "cef7e57", "date": "2024-08-06"}, "theoremCreated": {"commit": "9123431", "date": "2024-08-06"}, "file": "hep_lean/HepLean/SpaceTime/LorentzTensor/Real/IndexNotation.lean", "module": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation", "jsonFile": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation.jsonl", "positionMetadata": {"lineInFile": 56, "tokenPositionInFile": 1824, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Propext", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Tactic.FinCases", "HepLean.Tensors.IndexNotation.Basic", "HepLean.Tensors.IndexNotation.IndexList.Basic", "HepLean.Tensors.IndexNotation.IndexList.Duals", "Mathlib.Data.List.FinRange", "HepLean.Tensors.IndexNotation.IndexList.CountId", "HepLean.Tensors.IndexNotation.IndexList.Equivs", "HepLean.Tensors.IndexNotation.IndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.OnlyUniqueDuals", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Fintype.Sort", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.Multilinear.TensorProduct", "Mathlib.LinearAlgebra.PiTensorProduct", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "HepLean.Tensors.Basic", "HepLean.Tensors.IndexNotation.IndexList.Color", "HepLean.Tensors.IndexNotation.ColorIndexList.Basic", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.GroupTheory.Finiteness", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.ZMod.Basic", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.List", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.FieldTheory.Subfield", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.Contraction", "Mathlib.RepresentationTheory.Basic", "HepLean.Tensors.MulActionTensor", "HepLean.Tensors.Contraction", "HepLean.Tensors.IndexNotation.ColorIndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.Subperm", "HepLean.Tensors.IndexNotation.ColorIndexList.ContrPerm", "HepLean.Tensors.IndexNotation.ColorIndexList.Append", "HepLean.Tensors.RisingLowering", "HepLean.Tensors.IndexNotation.TensorIndex", "HepLean.Tensors.IndexNotation.IndexString", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne", "HepLean.SpaceTime.LorentzGroup.Basic", "HepLean.SpaceTime.LorentzVector.LorentzAction", "HepLean.SpaceTime.LorentzVector.Covariant", "HepLean.SpaceTime.LorentzVector.Contraction", "HepLean.SpaceTime.LorentzTensor.Real.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rfl", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 11}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.Tensors.IndexNotation.TensorIndex\nimport HepLean.Tensors.IndexNotation.IndexString\nimport HepLean.SpaceTime.LorentzTensor.Real.Basic\n/-!\n\n# Index notation for real Lorentz tensors\n\nThis uses the general concepts of index notation in `HepLean.SpaceTime.LorentzTensor.IndexNotation`\nto define the index notation for real Lorentz tensors.\n\n-/\n\ninstance : IndexNotation realTensorColor.Color where\n charList := {'\u1d58', '\u1d64'}\n notaEquiv :=\n \u27e8fun c =>\n match c with\n | realTensorColor.ColorType.up => \u27e8'\u1d58', Finset.mem_insert_self '\u1d58' {'\u1d64'}\u27e9\n | realTensorColor.ColorType.down => \u27e8'\u1d64', Finset.insert_eq_self.mp (by rfl)\u27e9,\n fun c =>\n if c = '\u1d58' then realTensorColor.ColorType.up\n else realTensorColor.ColorType.down,\n by\n intro c\n match c with\n | realTensorColor.ColorType.up => rfl\n | realTensorColor.ColorType.down => rfl,\n by\n intro c\n by_cases hc : c = '\u1d58'\n \u00b7 simp only [\u2193Char.isValue, hc, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc))\n \u00b7 have hc' : c = '\u1d64' := by\n have hc2 := c.2\n simp only [\u2193Char.isValue, Finset.mem_insert, Finset.mem_singleton] at hc2\n simp_all\n simp only [\u2193Char.isValue, hc', Char.reduceEq, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc'))\u27e9\n\nnamespace realLorentzTensor\n\nopen realTensorColor\nopen IndexNotation IndexString\nopen TensorStructure TensorIndex\n\nvariable {d : \u2115}\n\ninstance instIndexNotationColor : IndexNotation (realLorentzTensor d).Color := instIndexNotationColorRealTensorColor\ninstance instDecidableEqColor : DecidableEq (realLorentzTensor d).Color := instDecidableEqColorRealTensorColor\n\n@[simp]\nlemma indexNotation_eq_color : @realLorentzTensor.instIndexNotationColor d =\n instIndexNotationColorRealTensorColor := by\n rfl\n\n", "theoremStatement": "@[simp]\nlemma decidableEq_eq_color : @realLorentzTensor.instDecidableEqColor d =\n instDecidableEqColorRealTensorColor ", "theoremName": "realLorentzTensor.decidableEq_eq_color", "fileCreated": {"commit": "cef7e57", "date": "2024-08-06"}, "theoremCreated": {"commit": "d419a17", "date": "2024-08-14"}, "file": "hep_lean/HepLean/SpaceTime/LorentzTensor/Real/IndexNotation.lean", "module": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation", "jsonFile": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation.jsonl", "positionMetadata": {"lineInFile": 61, "tokenPositionInFile": 1964, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Propext", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Tactic.FinCases", "HepLean.Tensors.IndexNotation.Basic", "HepLean.Tensors.IndexNotation.IndexList.Basic", "HepLean.Tensors.IndexNotation.IndexList.Duals", "Mathlib.Data.List.FinRange", "HepLean.Tensors.IndexNotation.IndexList.CountId", "HepLean.Tensors.IndexNotation.IndexList.Equivs", "HepLean.Tensors.IndexNotation.IndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.OnlyUniqueDuals", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Fintype.Sort", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.Multilinear.TensorProduct", "Mathlib.LinearAlgebra.PiTensorProduct", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "HepLean.Tensors.Basic", "HepLean.Tensors.IndexNotation.IndexList.Color", "HepLean.Tensors.IndexNotation.ColorIndexList.Basic", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.GroupTheory.Finiteness", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.ZMod.Basic", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.List", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.FieldTheory.Subfield", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.Contraction", "Mathlib.RepresentationTheory.Basic", "HepLean.Tensors.MulActionTensor", "HepLean.Tensors.Contraction", "HepLean.Tensors.IndexNotation.ColorIndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.Subperm", "HepLean.Tensors.IndexNotation.ColorIndexList.ContrPerm", "HepLean.Tensors.IndexNotation.ColorIndexList.Append", "HepLean.Tensors.RisingLowering", "HepLean.Tensors.IndexNotation.TensorIndex", "HepLean.Tensors.IndexNotation.IndexString", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne", "HepLean.SpaceTime.LorentzGroup.Basic", "HepLean.SpaceTime.LorentzVector.LorentzAction", "HepLean.SpaceTime.LorentzVector.Covariant", "HepLean.SpaceTime.LorentzVector.Contraction", "HepLean.SpaceTime.LorentzTensor.Real.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rfl", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 11}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.Tensors.IndexNotation.TensorIndex\nimport HepLean.Tensors.IndexNotation.IndexString\nimport HepLean.SpaceTime.LorentzTensor.Real.Basic\n/-!\n\n# Index notation for real Lorentz tensors\n\nThis uses the general concepts of index notation in `HepLean.SpaceTime.LorentzTensor.IndexNotation`\nto define the index notation for real Lorentz tensors.\n\n-/\n\ninstance : IndexNotation realTensorColor.Color where\n charList := {'\u1d58', '\u1d64'}\n notaEquiv :=\n \u27e8fun c =>\n match c with\n | realTensorColor.ColorType.up => \u27e8'\u1d58', Finset.mem_insert_self '\u1d58' {'\u1d64'}\u27e9\n | realTensorColor.ColorType.down => \u27e8'\u1d64', Finset.insert_eq_self.mp (by rfl)\u27e9,\n fun c =>\n if c = '\u1d58' then realTensorColor.ColorType.up\n else realTensorColor.ColorType.down,\n by\n intro c\n match c with\n | realTensorColor.ColorType.up => rfl\n | realTensorColor.ColorType.down => rfl,\n by\n intro c\n by_cases hc : c = '\u1d58'\n \u00b7 simp only [\u2193Char.isValue, hc, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc))\n \u00b7 have hc' : c = '\u1d64' := by\n have hc2 := c.2\n simp only [\u2193Char.isValue, Finset.mem_insert, Finset.mem_singleton] at hc2\n simp_all\n simp only [\u2193Char.isValue, hc', Char.reduceEq, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc'))\u27e9\n\nnamespace realLorentzTensor\n\nopen realTensorColor\nopen IndexNotation IndexString\nopen TensorStructure TensorIndex\n\nvariable {d : \u2115}\n\ninstance instIndexNotationColor : IndexNotation (realLorentzTensor d).Color := instIndexNotationColorRealTensorColor\ninstance instDecidableEqColor : DecidableEq (realLorentzTensor d).Color := instDecidableEqColorRealTensorColor\n\n@[simp]\nlemma indexNotation_eq_color : @realLorentzTensor.instIndexNotationColor d =\n instIndexNotationColorRealTensorColor := by\n rfl\n\n@[simp]\nlemma decidableEq_eq_color : @realLorentzTensor.instDecidableEqColor d =\n instDecidableEqColorRealTensorColor := by\n rfl\n\n", "theoremStatement": "@[simp]\nlemma realLorentzTensor_color : (realLorentzTensor d).Color = realTensorColor.Color ", "theoremName": "realLorentzTensor.realLorentzTensor_color", "fileCreated": {"commit": "cef7e57", "date": "2024-08-06"}, "theoremCreated": {"commit": "cef7e57", "date": "2024-08-06"}, "file": "hep_lean/HepLean/SpaceTime/LorentzTensor/Real/IndexNotation.lean", "module": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation", "jsonFile": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation.jsonl", "positionMetadata": {"lineInFile": 66, "tokenPositionInFile": 2098, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Propext", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Tactic.FinCases", "HepLean.Tensors.IndexNotation.Basic", "HepLean.Tensors.IndexNotation.IndexList.Basic", "HepLean.Tensors.IndexNotation.IndexList.Duals", "Mathlib.Data.List.FinRange", "HepLean.Tensors.IndexNotation.IndexList.CountId", "HepLean.Tensors.IndexNotation.IndexList.Equivs", "HepLean.Tensors.IndexNotation.IndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.OnlyUniqueDuals", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Fintype.Sort", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.Multilinear.TensorProduct", "Mathlib.LinearAlgebra.PiTensorProduct", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "HepLean.Tensors.Basic", "HepLean.Tensors.IndexNotation.IndexList.Color", "HepLean.Tensors.IndexNotation.ColorIndexList.Basic", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.GroupTheory.Finiteness", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.ZMod.Basic", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.List", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.FieldTheory.Subfield", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.Contraction", "Mathlib.RepresentationTheory.Basic", "HepLean.Tensors.MulActionTensor", "HepLean.Tensors.Contraction", "HepLean.Tensors.IndexNotation.ColorIndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.Subperm", "HepLean.Tensors.IndexNotation.ColorIndexList.ContrPerm", "HepLean.Tensors.IndexNotation.ColorIndexList.Append", "HepLean.Tensors.RisingLowering", "HepLean.Tensors.IndexNotation.TensorIndex", "HepLean.Tensors.IndexNotation.IndexString", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne", "HepLean.SpaceTime.LorentzGroup.Basic", "HepLean.SpaceTime.LorentzVector.LorentzAction", "HepLean.SpaceTime.LorentzVector.Covariant", "HepLean.SpaceTime.LorentzVector.Contraction", "HepLean.SpaceTime.LorentzTensor.Real.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rfl", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 11}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.Tensors.IndexNotation.TensorIndex\nimport HepLean.Tensors.IndexNotation.IndexString\nimport HepLean.SpaceTime.LorentzTensor.Real.Basic\n/-!\n\n# Index notation for real Lorentz tensors\n\nThis uses the general concepts of index notation in `HepLean.SpaceTime.LorentzTensor.IndexNotation`\nto define the index notation for real Lorentz tensors.\n\n-/\n\ninstance : IndexNotation realTensorColor.Color where\n charList := {'\u1d58', '\u1d64'}\n notaEquiv :=\n \u27e8fun c =>\n match c with\n | realTensorColor.ColorType.up => \u27e8'\u1d58', Finset.mem_insert_self '\u1d58' {'\u1d64'}\u27e9\n | realTensorColor.ColorType.down => \u27e8'\u1d64', Finset.insert_eq_self.mp (by rfl)\u27e9,\n fun c =>\n if c = '\u1d58' then realTensorColor.ColorType.up\n else realTensorColor.ColorType.down,\n by\n intro c\n match c with\n | realTensorColor.ColorType.up => rfl\n | realTensorColor.ColorType.down => rfl,\n by\n intro c\n by_cases hc : c = '\u1d58'\n \u00b7 simp only [\u2193Char.isValue, hc, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc))\n \u00b7 have hc' : c = '\u1d64' := by\n have hc2 := c.2\n simp only [\u2193Char.isValue, Finset.mem_insert, Finset.mem_singleton] at hc2\n simp_all\n simp only [\u2193Char.isValue, hc', Char.reduceEq, \u2193reduceIte]\n exact SetCoe.ext (id (Eq.symm hc'))\u27e9\n\nnamespace realLorentzTensor\n\nopen realTensorColor\nopen IndexNotation IndexString\nopen TensorStructure TensorIndex\n\nvariable {d : \u2115}\n\ninstance instIndexNotationColor : IndexNotation (realLorentzTensor d).Color := instIndexNotationColorRealTensorColor\ninstance instDecidableEqColor : DecidableEq (realLorentzTensor d).Color := instDecidableEqColorRealTensorColor\n\n@[simp]\nlemma indexNotation_eq_color : @realLorentzTensor.instIndexNotationColor d =\n instIndexNotationColorRealTensorColor := by\n rfl\n\n@[simp]\nlemma decidableEq_eq_color : @realLorentzTensor.instDecidableEqColor d =\n instDecidableEqColorRealTensorColor := by\n rfl\n\n@[simp]\nlemma realLorentzTensor_color : (realLorentzTensor d).Color = realTensorColor.Color := by\n rfl\n\n", "theoremStatement": "@[simp]\nlemma toTensorColor_eq : (realLorentzTensor d).toTensorColor = realTensorColor ", "theoremName": "realLorentzTensor.toTensorColor_eq", "fileCreated": {"commit": "cef7e57", "date": "2024-08-06"}, "theoremCreated": {"commit": "cef7e57", "date": "2024-08-06"}, "file": "hep_lean/HepLean/SpaceTime/LorentzTensor/Real/IndexNotation.lean", "module": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation", "jsonFile": "HepLean.SpaceTime.LorentzTensor.Real.IndexNotation.jsonl", "positionMetadata": {"lineInFile": 70, "tokenPositionInFile": 2203, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Propext", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Tactic.FinCases", "HepLean.Tensors.IndexNotation.Basic", "HepLean.Tensors.IndexNotation.IndexList.Basic", "HepLean.Tensors.IndexNotation.IndexList.Duals", "Mathlib.Data.List.FinRange", "HepLean.Tensors.IndexNotation.IndexList.CountId", "HepLean.Tensors.IndexNotation.IndexList.Equivs", "HepLean.Tensors.IndexNotation.IndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.OnlyUniqueDuals", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Fintype.Sort", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.Multilinear.TensorProduct", "Mathlib.LinearAlgebra.PiTensorProduct", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "HepLean.Tensors.Basic", "HepLean.Tensors.IndexNotation.IndexList.Color", "HepLean.Tensors.IndexNotation.ColorIndexList.Basic", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.GroupTheory.Finiteness", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.ZMod.Basic", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.List", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.FieldTheory.Subfield", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.Contraction", "Mathlib.RepresentationTheory.Basic", "HepLean.Tensors.MulActionTensor", "HepLean.Tensors.Contraction", "HepLean.Tensors.IndexNotation.ColorIndexList.Contraction", "HepLean.Tensors.IndexNotation.IndexList.Subperm", "HepLean.Tensors.IndexNotation.ColorIndexList.ContrPerm", "HepLean.Tensors.IndexNotation.ColorIndexList.Append", "HepLean.Tensors.RisingLowering", "HepLean.Tensors.IndexNotation.TensorIndex", "HepLean.Tensors.IndexNotation.IndexString", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric", "HepLean.SpaceTime.LorentzVector.NormOne", "HepLean.SpaceTime.LorentzGroup.Basic", "HepLean.SpaceTime.LorentzVector.LorentzAction", "HepLean.SpaceTime.LorentzVector.Covariant", "HepLean.SpaceTime.LorentzVector.Contraction", "HepLean.SpaceTime.LorentzTensor.Real.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rfl", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 11}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Data.Complex.Exponential\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n/-!\n\n# Lorentz vectors\n\n(aka 4-vectors)\n\nIn this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).\n\nOne of the most important example of a Lorentz vector is SpaceTime.\n\nWe will define the group action of the Lorentz group on Lorentz vectors in\n`HepLean.SpaceTime.LorentzVector.LorentzAction` in such a way that `LorentzVector`\ncorresponds to contravariant Lorentz tensors.\n\n-/\n\nopen BigOperators\n\n/- The number of space dimensions . -/\nvariable (d : \u2115)\n\n/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector : Type := (Fin 1 \u2295 Fin d) \u2192 \u211d\n\n/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid\n\n/-- An instance of a module on `LorentzVector`. -/\nnoncomputable instance instModuleRealLorentzVector : Module \u211d (LorentzVector d) := Pi.module _ _ _\n\ninstance : AddCommGroup (LorentzVector d) := Pi.addCommGroup\n\n/-- The structure of a topological space `LorentzVector d`. -/\ninstance : TopologicalSpace (LorentzVector d) :=\n haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup\n UniformSpace.toTopologicalSpace\n\nnamespace LorentzVector\n\nvariable {d : \u2115} (v : LorentzVector d)\n\n/-- The space components. -/\n@[simp]\ndef space : EuclideanSpace \u211d (Fin d) := v \u2218 Sum.inr\n\n/-- The time component. -/\n@[simp]\ndef time : \u211d := v (Sum.inl 0)\n\n/-!\n\n# The standard basis\n\n-/\n\n/-- The standard basis of `LorentzVector` indexed by `Fin 1 \u2295 Fin (d)`. -/\n@[simps!]\nnoncomputable def stdBasis : Basis (Fin 1 \u2295 Fin (d)) \u211d (LorentzVector d) := Pi.basisFun \u211d _\n\n/-- Notation for `stdBasis`. -/\nscoped[LorentzVector] notation \"e\" => stdBasis\n\n", "theoremStatement": "lemma stdBasis_apply (\u03bc \u03bd : Fin 1 \u2295 Fin d) : e \u03bc \u03bd = if \u03bc = \u03bd then 1 else 0 ", "theoremName": "LorentzVector.stdBasis_apply", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "c64d926", "date": "2024-07-02"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/Basic.lean", "module": "HepLean.SpaceTime.LorentzVector.Basic", "jsonFile": "HepLean.SpaceTime.LorentzVector.Basic.jsonl", "positionMetadata": {"lineInFile": 70, "tokenPositionInFile": 1936, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 44, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 68}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Data.Complex.Exponential\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n/-!\n\n# Lorentz vectors\n\n(aka 4-vectors)\n\nIn this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).\n\nOne of the most important example of a Lorentz vector is SpaceTime.\n\nWe will define the group action of the Lorentz group on Lorentz vectors in\n`HepLean.SpaceTime.LorentzVector.LorentzAction` in such a way that `LorentzVector`\ncorresponds to contravariant Lorentz tensors.\n\n-/\n\nopen BigOperators\n\n/- The number of space dimensions . -/\nvariable (d : \u2115)\n\n/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector : Type := (Fin 1 \u2295 Fin d) \u2192 \u211d\n\n/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid\n\n/-- An instance of a module on `LorentzVector`. -/\nnoncomputable instance instModuleRealLorentzVector : Module \u211d (LorentzVector d) := Pi.module _ _ _\n\ninstance : AddCommGroup (LorentzVector d) := Pi.addCommGroup\n\n/-- The structure of a topological space `LorentzVector d`. -/\ninstance : TopologicalSpace (LorentzVector d) :=\n haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup\n UniformSpace.toTopologicalSpace\n\nnamespace LorentzVector\n\nvariable {d : \u2115} (v : LorentzVector d)\n\n/-- The space components. -/\n@[simp]\ndef space : EuclideanSpace \u211d (Fin d) := v \u2218 Sum.inr\n\n/-- The time component. -/\n@[simp]\ndef time : \u211d := v (Sum.inl 0)\n\n/-!\n\n# The standard basis\n\n-/\n\n/-- The standard basis of `LorentzVector` indexed by `Fin 1 \u2295 Fin (d)`. -/\n@[simps!]\nnoncomputable def stdBasis : Basis (Fin 1 \u2295 Fin (d)) \u211d (LorentzVector d) := Pi.basisFun \u211d _\n\n/-- Notation for `stdBasis`. -/\nscoped[LorentzVector] notation \"e\" => stdBasis\n\nlemma stdBasis_apply (\u03bc \u03bd : Fin 1 \u2295 Fin d) : e \u03bc \u03bd = if \u03bc = \u03bd then 1 else 0 := by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']\n\n", "theoremStatement": "lemma decomp_stdBasis (v : LorentzVector d) : \u2211 i, v i \u2022 e i = v ", "theoremName": "LorentzVector.decomp_stdBasis", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "99f4e85", "date": "2024-07-29"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/Basic.lean", "module": "HepLean.SpaceTime.LorentzVector.Basic", "jsonFile": "HepLean.SpaceTime.LorentzVector.Basic.jsonl", "positionMetadata": {"lineInFile": 73, "tokenPositionInFile": 2082, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 115, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n funext \u03bd\n rw [Finset.sum_apply]\n rw [Finset.sum_eq_single_of_mem \u03bd]\n \u00b7 simp only [HSMul.hSMul, SMul.smul, stdBasis]\n erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', \u2193reduceIte, mul_one]\n \u00b7 exact Finset.mem_univ \u03bd\n \u00b7 intros b _ hbi\n simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]\n erw [Pi.basisFun_apply]\n simp only [Pi.single]\n apply Or.inr $ Function.update_noteq (id (Ne.symm hbi)) 1 0", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 444}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Data.Complex.Exponential\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n/-!\n\n# Lorentz vectors\n\n(aka 4-vectors)\n\nIn this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).\n\nOne of the most important example of a Lorentz vector is SpaceTime.\n\nWe will define the group action of the Lorentz group on Lorentz vectors in\n`HepLean.SpaceTime.LorentzVector.LorentzAction` in such a way that `LorentzVector`\ncorresponds to contravariant Lorentz tensors.\n\n-/\n\nopen BigOperators\n\n/- The number of space dimensions . -/\nvariable (d : \u2115)\n\n/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector : Type := (Fin 1 \u2295 Fin d) \u2192 \u211d\n\n/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid\n\n/-- An instance of a module on `LorentzVector`. -/\nnoncomputable instance instModuleRealLorentzVector : Module \u211d (LorentzVector d) := Pi.module _ _ _\n\ninstance : AddCommGroup (LorentzVector d) := Pi.addCommGroup\n\n/-- The structure of a topological space `LorentzVector d`. -/\ninstance : TopologicalSpace (LorentzVector d) :=\n haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup\n UniformSpace.toTopologicalSpace\n\nnamespace LorentzVector\n\nvariable {d : \u2115} (v : LorentzVector d)\n\n/-- The space components. -/\n@[simp]\ndef space : EuclideanSpace \u211d (Fin d) := v \u2218 Sum.inr\n\n/-- The time component. -/\n@[simp]\ndef time : \u211d := v (Sum.inl 0)\n\n/-!\n\n# The standard basis\n\n-/\n\n/-- The standard basis of `LorentzVector` indexed by `Fin 1 \u2295 Fin (d)`. -/\n@[simps!]\nnoncomputable def stdBasis : Basis (Fin 1 \u2295 Fin (d)) \u211d (LorentzVector d) := Pi.basisFun \u211d _\n\n/-- Notation for `stdBasis`. -/\nscoped[LorentzVector] notation \"e\" => stdBasis\n\nlemma stdBasis_apply (\u03bc \u03bd : Fin 1 \u2295 Fin d) : e \u03bc \u03bd = if \u03bc = \u03bd then 1 else 0 := by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']\n\nlemma decomp_stdBasis (v : LorentzVector d) : \u2211 i, v i \u2022 e i = v := by\n funext \u03bd\n rw [Finset.sum_apply]\n rw [Finset.sum_eq_single_of_mem \u03bd]\n \u00b7 simp only [HSMul.hSMul, SMul.smul, stdBasis]\n erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', \u2193reduceIte, mul_one]\n \u00b7 exact Finset.mem_univ \u03bd\n \u00b7 intros b _ hbi\n simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]\n erw [Pi.basisFun_apply]\n simp only [Pi.single]\n apply Or.inr $ Function.update_noteq (id (Ne.symm hbi)) 1 0\n\n", "theoremStatement": "@[simp]\nlemma decomp_stdBasis' (v : LorentzVector d) :\n v (Sum.inl 0) \u2022 e (Sum.inl 0) + \u2211 a\u2082 : Fin d, v (Sum.inr a\u2082) \u2022 e (Sum.inr a\u2082) = v ", "theoremName": "LorentzVector.decomp_stdBasis'", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "a65fb06", "date": "2024-07-30"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/Basic.lean", "module": "HepLean.SpaceTime.LorentzVector.Basic", "jsonFile": "HepLean.SpaceTime.LorentzVector.Basic.jsonl", "positionMetadata": {"lineInFile": 87, "tokenPositionInFile": 2593, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 65, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n trans \u2211 i, v i \u2022 e i\n \u00b7 simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,\n Finset.sum_singleton]\n \u00b7 exact decomp_stdBasis v", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 175}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Data.Complex.Exponential\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n/-!\n\n# Lorentz vectors\n\n(aka 4-vectors)\n\nIn this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).\n\nOne of the most important example of a Lorentz vector is SpaceTime.\n\nWe will define the group action of the Lorentz group on Lorentz vectors in\n`HepLean.SpaceTime.LorentzVector.LorentzAction` in such a way that `LorentzVector`\ncorresponds to contravariant Lorentz tensors.\n\n-/\n\nopen BigOperators\n\n/- The number of space dimensions . -/\nvariable (d : \u2115)\n\n/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector : Type := (Fin 1 \u2295 Fin d) \u2192 \u211d\n\n/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid\n\n/-- An instance of a module on `LorentzVector`. -/\nnoncomputable instance instModuleRealLorentzVector : Module \u211d (LorentzVector d) := Pi.module _ _ _\n\ninstance : AddCommGroup (LorentzVector d) := Pi.addCommGroup\n\n/-- The structure of a topological space `LorentzVector d`. -/\ninstance : TopologicalSpace (LorentzVector d) :=\n haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup\n UniformSpace.toTopologicalSpace\n\nnamespace LorentzVector\n\nvariable {d : \u2115} (v : LorentzVector d)\n\n/-- The space components. -/\n@[simp]\ndef space : EuclideanSpace \u211d (Fin d) := v \u2218 Sum.inr\n\n/-- The time component. -/\n@[simp]\ndef time : \u211d := v (Sum.inl 0)\n\n/-!\n\n# The standard basis\n\n-/\n\n/-- The standard basis of `LorentzVector` indexed by `Fin 1 \u2295 Fin (d)`. -/\n@[simps!]\nnoncomputable def stdBasis : Basis (Fin 1 \u2295 Fin (d)) \u211d (LorentzVector d) := Pi.basisFun \u211d _\n\n/-- Notation for `stdBasis`. -/\nscoped[LorentzVector] notation \"e\" => stdBasis\n\nlemma stdBasis_apply (\u03bc \u03bd : Fin 1 \u2295 Fin d) : e \u03bc \u03bd = if \u03bc = \u03bd then 1 else 0 := by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']\n\nlemma decomp_stdBasis (v : LorentzVector d) : \u2211 i, v i \u2022 e i = v := by\n funext \u03bd\n rw [Finset.sum_apply]\n rw [Finset.sum_eq_single_of_mem \u03bd]\n \u00b7 simp only [HSMul.hSMul, SMul.smul, stdBasis]\n erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', \u2193reduceIte, mul_one]\n \u00b7 exact Finset.mem_univ \u03bd\n \u00b7 intros b _ hbi\n simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]\n erw [Pi.basisFun_apply]\n simp only [Pi.single]\n apply Or.inr $ Function.update_noteq (id (Ne.symm hbi)) 1 0\n\n@[simp]\nlemma decomp_stdBasis' (v : LorentzVector d) :\n v (Sum.inl 0) \u2022 e (Sum.inl 0) + \u2211 a\u2082 : Fin d, v (Sum.inr a\u2082) \u2022 e (Sum.inr a\u2082) = v := by\n trans \u2211 i, v i \u2022 e i\n \u00b7 simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,\n Finset.sum_singleton]\n \u00b7 exact decomp_stdBasis v\n\n/-- The standard unit time vector. -/\nnoncomputable abbrev timeVec : (LorentzVector d) := e (Sum.inl 0)\n\n", "theoremStatement": "lemma timeVec_space : (@timeVec d).space = 0 ", "theoremName": "LorentzVector.timeVec_space", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/Basic.lean", "module": "HepLean.SpaceTime.LorentzVector.Basic", "jsonFile": "HepLean.SpaceTime.LorentzVector.Basic.jsonl", "positionMetadata": {"lineInFile": 98, "tokenPositionInFile": 3016, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 60, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n funext i\n simp only [space, Function.comp_apply, stdBasis_apply, Fin.isValue, \u2193reduceIte, PiLp.zero_apply]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 115}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Data.Complex.Exponential\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n/-!\n\n# Lorentz vectors\n\n(aka 4-vectors)\n\nIn this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).\n\nOne of the most important example of a Lorentz vector is SpaceTime.\n\nWe will define the group action of the Lorentz group on Lorentz vectors in\n`HepLean.SpaceTime.LorentzVector.LorentzAction` in such a way that `LorentzVector`\ncorresponds to contravariant Lorentz tensors.\n\n-/\n\nopen BigOperators\n\n/- The number of space dimensions . -/\nvariable (d : \u2115)\n\n/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector : Type := (Fin 1 \u2295 Fin d) \u2192 \u211d\n\n/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid\n\n/-- An instance of a module on `LorentzVector`. -/\nnoncomputable instance instModuleRealLorentzVector : Module \u211d (LorentzVector d) := Pi.module _ _ _\n\ninstance : AddCommGroup (LorentzVector d) := Pi.addCommGroup\n\n/-- The structure of a topological space `LorentzVector d`. -/\ninstance : TopologicalSpace (LorentzVector d) :=\n haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup\n UniformSpace.toTopologicalSpace\n\nnamespace LorentzVector\n\nvariable {d : \u2115} (v : LorentzVector d)\n\n/-- The space components. -/\n@[simp]\ndef space : EuclideanSpace \u211d (Fin d) := v \u2218 Sum.inr\n\n/-- The time component. -/\n@[simp]\ndef time : \u211d := v (Sum.inl 0)\n\n/-!\n\n# The standard basis\n\n-/\n\n/-- The standard basis of `LorentzVector` indexed by `Fin 1 \u2295 Fin (d)`. -/\n@[simps!]\nnoncomputable def stdBasis : Basis (Fin 1 \u2295 Fin (d)) \u211d (LorentzVector d) := Pi.basisFun \u211d _\n\n/-- Notation for `stdBasis`. -/\nscoped[LorentzVector] notation \"e\" => stdBasis\n\nlemma stdBasis_apply (\u03bc \u03bd : Fin 1 \u2295 Fin d) : e \u03bc \u03bd = if \u03bc = \u03bd then 1 else 0 := by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']\n\nlemma decomp_stdBasis (v : LorentzVector d) : \u2211 i, v i \u2022 e i = v := by\n funext \u03bd\n rw [Finset.sum_apply]\n rw [Finset.sum_eq_single_of_mem \u03bd]\n \u00b7 simp only [HSMul.hSMul, SMul.smul, stdBasis]\n erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', \u2193reduceIte, mul_one]\n \u00b7 exact Finset.mem_univ \u03bd\n \u00b7 intros b _ hbi\n simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]\n erw [Pi.basisFun_apply]\n simp only [Pi.single]\n apply Or.inr $ Function.update_noteq (id (Ne.symm hbi)) 1 0\n\n@[simp]\nlemma decomp_stdBasis' (v : LorentzVector d) :\n v (Sum.inl 0) \u2022 e (Sum.inl 0) + \u2211 a\u2082 : Fin d, v (Sum.inr a\u2082) \u2022 e (Sum.inr a\u2082) = v := by\n trans \u2211 i, v i \u2022 e i\n \u00b7 simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,\n Finset.sum_singleton]\n \u00b7 exact decomp_stdBasis v\n\n/-- The standard unit time vector. -/\nnoncomputable abbrev timeVec : (LorentzVector d) := e (Sum.inl 0)\n\nlemma timeVec_space : (@timeVec d).space = 0 := by\n funext i\n simp only [space, Function.comp_apply, stdBasis_apply, Fin.isValue, \u2193reduceIte, PiLp.zero_apply]\n\n", "theoremStatement": "lemma timeVec_time: (@timeVec d).time = 1 ", "theoremName": "LorentzVector.timeVec_time", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/Basic.lean", "module": "HepLean.SpaceTime.LorentzVector.Basic", "jsonFile": "HepLean.SpaceTime.LorentzVector.Basic.jsonl", "positionMetadata": {"lineInFile": 102, "tokenPositionInFile": 3178, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 35, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp only [time, Fin.isValue, stdBasis_apply, \u2193reduceIte]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 65}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Data.Complex.Exponential\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n/-!\n\n# Lorentz vectors\n\n(aka 4-vectors)\n\nIn this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).\n\nOne of the most important example of a Lorentz vector is SpaceTime.\n\nWe will define the group action of the Lorentz group on Lorentz vectors in\n`HepLean.SpaceTime.LorentzVector.LorentzAction` in such a way that `LorentzVector`\ncorresponds to contravariant Lorentz tensors.\n\n-/\n\nopen BigOperators\n\n/- The number of space dimensions . -/\nvariable (d : \u2115)\n\n/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector : Type := (Fin 1 \u2295 Fin d) \u2192 \u211d\n\n/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid\n\n/-- An instance of a module on `LorentzVector`. -/\nnoncomputable instance instModuleRealLorentzVector : Module \u211d (LorentzVector d) := Pi.module _ _ _\n\ninstance : AddCommGroup (LorentzVector d) := Pi.addCommGroup\n\n/-- The structure of a topological space `LorentzVector d`. -/\ninstance : TopologicalSpace (LorentzVector d) :=\n haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup\n UniformSpace.toTopologicalSpace\n\nnamespace LorentzVector\n\nvariable {d : \u2115} (v : LorentzVector d)\n\n/-- The space components. -/\n@[simp]\ndef space : EuclideanSpace \u211d (Fin d) := v \u2218 Sum.inr\n\n/-- The time component. -/\n@[simp]\ndef time : \u211d := v (Sum.inl 0)\n\n/-!\n\n# The standard basis\n\n-/\n\n/-- The standard basis of `LorentzVector` indexed by `Fin 1 \u2295 Fin (d)`. -/\n@[simps!]\nnoncomputable def stdBasis : Basis (Fin 1 \u2295 Fin (d)) \u211d (LorentzVector d) := Pi.basisFun \u211d _\n\n/-- Notation for `stdBasis`. -/\nscoped[LorentzVector] notation \"e\" => stdBasis\n\nlemma stdBasis_apply (\u03bc \u03bd : Fin 1 \u2295 Fin d) : e \u03bc \u03bd = if \u03bc = \u03bd then 1 else 0 := by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']\n\nlemma decomp_stdBasis (v : LorentzVector d) : \u2211 i, v i \u2022 e i = v := by\n funext \u03bd\n rw [Finset.sum_apply]\n rw [Finset.sum_eq_single_of_mem \u03bd]\n \u00b7 simp only [HSMul.hSMul, SMul.smul, stdBasis]\n erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', \u2193reduceIte, mul_one]\n \u00b7 exact Finset.mem_univ \u03bd\n \u00b7 intros b _ hbi\n simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]\n erw [Pi.basisFun_apply]\n simp only [Pi.single]\n apply Or.inr $ Function.update_noteq (id (Ne.symm hbi)) 1 0\n\n@[simp]\nlemma decomp_stdBasis' (v : LorentzVector d) :\n v (Sum.inl 0) \u2022 e (Sum.inl 0) + \u2211 a\u2082 : Fin d, v (Sum.inr a\u2082) \u2022 e (Sum.inr a\u2082) = v := by\n trans \u2211 i, v i \u2022 e i\n \u00b7 simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,\n Finset.sum_singleton]\n \u00b7 exact decomp_stdBasis v\n\n/-- The standard unit time vector. -/\nnoncomputable abbrev timeVec : (LorentzVector d) := e (Sum.inl 0)\n\nlemma timeVec_space : (@timeVec d).space = 0 := by\n funext i\n simp only [space, Function.comp_apply, stdBasis_apply, Fin.isValue, \u2193reduceIte, PiLp.zero_apply]\n\nlemma timeVec_time: (@timeVec d).time = 1 := by\n simp only [time, Fin.isValue, stdBasis_apply, \u2193reduceIte]\n\n/-!\n\n# Reflection of space\n\n-/\n\n/-- The reflection of space as a linear map. -/\n@[simps!]\ndef spaceReflectionLin : LorentzVector d \u2192\u2097[\u211d] LorentzVector d where\n toFun x := Sum.elim (x \u2218 Sum.inl) (- x \u2218 Sum.inr)\n map_add' x y := by\n funext i\n rcases i with i | i\n \u00b7 rfl\n \u00b7 simp only [Sum.elim_inr, Pi.neg_apply]\n apply neg_add\n map_smul' c x := by\n funext i\n rcases i with i | i\n \u00b7 rfl\n \u00b7 simp [HSMul.hSMul, SMul.smul]\n\n/-- The reflection of space. -/\n@[simp]\ndef spaceReflection : LorentzVector d := spaceReflectionLin v\n\n", "theoremStatement": "lemma spaceReflection_space : v.spaceReflection.space = - v.space ", "theoremName": "LorentzVector.spaceReflection_space", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/Basic.lean", "module": "HepLean.SpaceTime.LorentzVector.Basic", "jsonFile": "HepLean.SpaceTime.LorentzVector.Basic.jsonl", "positionMetadata": {"lineInFile": 131, "tokenPositionInFile": 3840, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 24, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport Mathlib.Data.Complex.Exponential\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n/-!\n\n# Lorentz vectors\n\n(aka 4-vectors)\n\nIn this file we define a Lorentz vector (in 4d, this is more often called a 4-vector).\n\nOne of the most important example of a Lorentz vector is SpaceTime.\n\nWe will define the group action of the Lorentz group on Lorentz vectors in\n`HepLean.SpaceTime.LorentzVector.LorentzAction` in such a way that `LorentzVector`\ncorresponds to contravariant Lorentz tensors.\n\n-/\n\nopen BigOperators\n\n/- The number of space dimensions . -/\nvariable (d : \u2115)\n\n/-- The type of (contravariant) Lorentz Vectors in `d`-space dimensions. -/\ndef LorentzVector : Type := (Fin 1 \u2295 Fin d) \u2192 \u211d\n\n/-- An instance of an additive commutative monoid on `LorentzVector`. -/\ninstance : AddCommMonoid (LorentzVector d) := Pi.addCommMonoid\n\n/-- An instance of a module on `LorentzVector`. -/\nnoncomputable instance instModuleRealLorentzVector : Module \u211d (LorentzVector d) := Pi.module _ _ _\n\ninstance : AddCommGroup (LorentzVector d) := Pi.addCommGroup\n\n/-- The structure of a topological space `LorentzVector d`. -/\ninstance : TopologicalSpace (LorentzVector d) :=\n haveI : NormedAddCommGroup (LorentzVector d) := Pi.normedAddCommGroup\n UniformSpace.toTopologicalSpace\n\nnamespace LorentzVector\n\nvariable {d : \u2115} (v : LorentzVector d)\n\n/-- The space components. -/\n@[simp]\ndef space : EuclideanSpace \u211d (Fin d) := v \u2218 Sum.inr\n\n/-- The time component. -/\n@[simp]\ndef time : \u211d := v (Sum.inl 0)\n\n/-!\n\n# The standard basis\n\n-/\n\n/-- The standard basis of `LorentzVector` indexed by `Fin 1 \u2295 Fin (d)`. -/\n@[simps!]\nnoncomputable def stdBasis : Basis (Fin 1 \u2295 Fin (d)) \u211d (LorentzVector d) := Pi.basisFun \u211d _\n\n/-- Notation for `stdBasis`. -/\nscoped[LorentzVector] notation \"e\" => stdBasis\n\nlemma stdBasis_apply (\u03bc \u03bd : Fin 1 \u2295 Fin d) : e \u03bc \u03bd = if \u03bc = \u03bd then 1 else 0 := by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']\n\nlemma decomp_stdBasis (v : LorentzVector d) : \u2211 i, v i \u2022 e i = v := by\n funext \u03bd\n rw [Finset.sum_apply]\n rw [Finset.sum_eq_single_of_mem \u03bd]\n \u00b7 simp only [HSMul.hSMul, SMul.smul, stdBasis]\n erw [Pi.basisFun_apply]\n simp only [LinearMap.stdBasis_apply', \u2193reduceIte, mul_one]\n \u00b7 exact Finset.mem_univ \u03bd\n \u00b7 intros b _ hbi\n simp only [HSMul.hSMul, SMul.smul, stdBasis, mul_eq_zero]\n erw [Pi.basisFun_apply]\n simp only [Pi.single]\n apply Or.inr $ Function.update_noteq (id (Ne.symm hbi)) 1 0\n\n@[simp]\nlemma decomp_stdBasis' (v : LorentzVector d) :\n v (Sum.inl 0) \u2022 e (Sum.inl 0) + \u2211 a\u2082 : Fin d, v (Sum.inr a\u2082) \u2022 e (Sum.inr a\u2082) = v := by\n trans \u2211 i, v i \u2022 e i\n \u00b7 simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,\n Finset.sum_singleton]\n \u00b7 exact decomp_stdBasis v\n\n/-- The standard unit time vector. -/\nnoncomputable abbrev timeVec : (LorentzVector d) := e (Sum.inl 0)\n\nlemma timeVec_space : (@timeVec d).space = 0 := by\n funext i\n simp only [space, Function.comp_apply, stdBasis_apply, Fin.isValue, \u2193reduceIte, PiLp.zero_apply]\n\nlemma timeVec_time: (@timeVec d).time = 1 := by\n simp only [time, Fin.isValue, stdBasis_apply, \u2193reduceIte]\n\n/-!\n\n# Reflection of space\n\n-/\n\n/-- The reflection of space as a linear map. -/\n@[simps!]\ndef spaceReflectionLin : LorentzVector d \u2192\u2097[\u211d] LorentzVector d where\n toFun x := Sum.elim (x \u2218 Sum.inl) (- x \u2218 Sum.inr)\n map_add' x y := by\n funext i\n rcases i with i | i\n \u00b7 rfl\n \u00b7 simp only [Sum.elim_inr, Pi.neg_apply]\n apply neg_add\n map_smul' c x := by\n funext i\n rcases i with i | i\n \u00b7 rfl\n \u00b7 simp [HSMul.hSMul, SMul.smul]\n\n/-- The reflection of space. -/\n@[simp]\ndef spaceReflection : LorentzVector d := spaceReflectionLin v\n\nlemma spaceReflection_space : v.spaceReflection.space = - v.space := rfl\n\n", "theoremStatement": "lemma spaceReflection_time : v.spaceReflection.time = v.time ", "theoremName": "LorentzVector.spaceReflection_time", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/Basic.lean", "module": "HepLean.SpaceTime.LorentzVector.Basic", "jsonFile": "HepLean.SpaceTime.LorentzVector.Basic.jsonl", "positionMetadata": {"lineInFile": 133, "tokenPositionInFile": 3914, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 7, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\n", "theoremStatement": "lemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 ", "theoremName": "NormOneLorentzVector.mem_iff", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 30, "tokenPositionInFile": 635, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 43, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rfl", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 11}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\n", "theoremStatement": "lemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 ", "theoremName": "NormOneLorentzVector.time_sq", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 39, "tokenPositionInFile": 930, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 71, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [time_sq_eq_metric_add_space, v.2]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 45}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\n", "theoremStatement": "lemma abs_time_ge_one : 1 \u2264 |v.1.time| ", "theoremName": "NormOneLorentzVector.abs_time_ge_one", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 42, "tokenPositionInFile": 1028, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 78, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 94}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\n", "theoremStatement": "lemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| ", "theoremName": "NormOneLorentzVector.norm_space_le_abs_time", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 47, "tokenPositionInFile": 1163, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 86, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 91}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\n", "theoremStatement": "lemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| ", "theoremName": "NormOneLorentzVector.norm_space_leq_abs_time", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 51, "tokenPositionInFile": 1312, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 31, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n le_of_lt (norm_space_le_abs_time v)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 40}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\n", "theoremStatement": "lemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time ", "theoremName": "NormOneLorentzVector.time_le_minus_one_or_ge_one", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 54, "tokenPositionInFile": 1411, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 37, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n le_abs'.mp (abs_time_ge_one v)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 35}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\n", "theoremStatement": "lemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 ", "theoremName": "NormOneLorentzVector.time_nonpos_iff", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 57, "tokenPositionInFile": 1513, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 170, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 142}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\n", "theoremStatement": "lemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time ", "theoremName": "NormOneLorentzVector.time_nonneg_iff", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 66, "tokenPositionInFile": 1711, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 171, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 142}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\n", "theoremStatement": "lemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time ", "theoremName": "NormOneLorentzVector.time_pos_iff", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 75, "tokenPositionInFile": 1907, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 129, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 111}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\n", "theoremStatement": "lemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 ", "theoremName": "NormOneLorentzVector.time_abs_sub_space_norm", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 80, "tokenPositionInFile": 2069, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 73, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 170}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\n", "theoremStatement": "lemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time ", "theoremName": "NormOneLorentzVector.FuturePointing.mem_iff", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 104, "tokenPositionInFile": 2679, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rfl", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 11}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\n", "theoremStatement": "lemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time ", "theoremName": "NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 107, "tokenPositionInFile": 2744, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 134, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 114}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\n", "theoremStatement": "lemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 ", "theoremName": "NormOneLorentzVector.FuturePointing.not_mem_iff", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 113, "tokenPositionInFile": 2924, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 42, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 207}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\n", "theoremStatement": "lemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d ", "theoremName": "NormOneLorentzVector.FuturePointing.not_mem_iff_neg", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 120, "tokenPositionInFile": 3189, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 39, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 129}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\n", "theoremStatement": "lemma time_nonneg : 0 \u2264 f.1.1.time ", "theoremName": "NormOneLorentzVector.FuturePointing.time_nonneg", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 126, "tokenPositionInFile": 3392, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 19, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= le_of_lt f.2", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 15}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\n", "theoremStatement": "lemma abs_time : |f.1.1.time| = f.1.1.time ", "theoremName": "NormOneLorentzVector.FuturePointing.abs_time", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 128, "tokenPositionInFile": 3444, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 19, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= abs_of_nonneg (time_nonneg f)", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 32}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\n", "theoremStatement": "lemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt ", "theoremName": "NormOneLorentzVector.FuturePointing.time_eq_sqrt", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "1b9e77b", "date": "2024-09-23"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 130, "tokenPositionInFile": 3521, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 61, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 103}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\n", "theoremStatement": "lemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 ", "theoremName": "NormOneLorentzVector.FuturePointing.metric_nonneg", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "c64d926", "date": "2024-07-02"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 143, "tokenPositionInFile": 3741, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 76, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 119}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\nlemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1\n\n", "theoremStatement": "lemma one_add_metric_non_zero : 1 + \u27eaf, f'.1.1\u27eb\u2098 \u2260 0 ", "theoremName": "NormOneLorentzVector.FuturePointing.one_add_metric_non_zero", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "c64d926", "date": "2024-07-02"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 148, "tokenPositionInFile": 3901, "theoremPositionInFile": 21}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 141, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n linarith [metric_nonneg f f']", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 37}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\nlemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1\n\nlemma one_add_metric_non_zero : 1 + \u27eaf, f'.1.1\u27eb\u2098 \u2260 0 := by\n linarith [metric_nonneg f f']\n\n/-!\n\n# The sign of \u27eav, w.1.spaceReflection\u27eb\u2098 different v and w\n\n-/\n\nsection\nvariable {v w : NormOneLorentzVector d}\n\n", "theoremStatement": "lemma metric_reflect_mem_mem (h : v \u2208 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 ", "theoremName": "NormOneLorentzVector.FuturePointing.metric_reflect_mem_mem", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 160, "tokenPositionInFile": 4110, "theoremPositionInFile": 22}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 117, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply le_trans (time_abs_sub_space_norm v w)\n rw [abs_time \u27e8v, h\u27e9, abs_time \u27e8w, hw\u27e9, sub_eq_add_neg, right_spaceReflection]\n apply (add_le_add_iff_left _).mpr\n rw [neg_le]\n apply le_trans (neg_le_abs _ : _ \u2264 |\u27ea(v.1).space, (w.1).space\u27eb_\u211d|) ?_\n exact abs_real_inner_le_norm (v.1).space (w.1).space", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 309}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\nlemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1\n\nlemma one_add_metric_non_zero : 1 + \u27eaf, f'.1.1\u27eb\u2098 \u2260 0 := by\n linarith [metric_nonneg f f']\n\n/-!\n\n# The sign of \u27eav, w.1.spaceReflection\u27eb\u2098 different v and w\n\n-/\n\nsection\nvariable {v w : NormOneLorentzVector d}\n\nlemma metric_reflect_mem_mem (h : v \u2208 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm v w)\n rw [abs_time \u27e8v, h\u27e9, abs_time \u27e8w, hw\u27e9, sub_eq_add_neg, right_spaceReflection]\n apply (add_le_add_iff_left _).mpr\n rw [neg_le]\n apply le_trans (neg_le_abs _ : _ \u2264 |\u27ea(v.1).space, (w.1).space\u27eb_\u211d|) ?_\n exact abs_real_inner_le_norm (v.1).space (w.1).space\n\n", "theoremStatement": "lemma metric_reflect_not_mem_not_mem (h : v \u2209 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 ", "theoremName": "NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 169, "tokenPositionInFile": 4543, "theoremPositionInFile": 23}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 88, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 138}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\nlemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1\n\nlemma one_add_metric_non_zero : 1 + \u27eaf, f'.1.1\u27eb\u2098 \u2260 0 := by\n linarith [metric_nonneg f f']\n\n/-!\n\n# The sign of \u27eav, w.1.spaceReflection\u27eb\u2098 different v and w\n\n-/\n\nsection\nvariable {v w : NormOneLorentzVector d}\n\nlemma metric_reflect_mem_mem (h : v \u2208 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm v w)\n rw [abs_time \u27e8v, h\u27e9, abs_time \u27e8w, hw\u27e9, sub_eq_add_neg, right_spaceReflection]\n apply (add_le_add_iff_left _).mpr\n rw [neg_le]\n apply le_trans (neg_le_abs _ : _ \u2264 |\u27ea(v.1).space, (w.1).space\u27eb_\u211d|) ?_\n exact abs_real_inner_le_norm (v.1).space (w.1).space\n\nlemma metric_reflect_not_mem_not_mem (h : v \u2209 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\n", "theoremStatement": "lemma metric_reflect_mem_not_mem (h : v \u2208 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n \u27eav.1, w.1.spaceReflection\u27eb\u2098 \u2264 0 ", "theoremName": "NormOneLorentzVector.FuturePointing.metric_reflect_mem_not_mem", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 175, "tokenPositionInFile": 4813, "theoremPositionInFile": 24}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 94, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [show (0 : \u211d) = - 0 from zero_eq_neg.mpr rfl, le_neg]\n have h1 := metric_reflect_mem_mem h ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 172}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\nlemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1\n\nlemma one_add_metric_non_zero : 1 + \u27eaf, f'.1.1\u27eb\u2098 \u2260 0 := by\n linarith [metric_nonneg f f']\n\n/-!\n\n# The sign of \u27eav, w.1.spaceReflection\u27eb\u2098 different v and w\n\n-/\n\nsection\nvariable {v w : NormOneLorentzVector d}\n\nlemma metric_reflect_mem_mem (h : v \u2208 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm v w)\n rw [abs_time \u27e8v, h\u27e9, abs_time \u27e8w, hw\u27e9, sub_eq_add_neg, right_spaceReflection]\n apply (add_le_add_iff_left _).mpr\n rw [neg_le]\n apply le_trans (neg_le_abs _ : _ \u2264 |\u27ea(v.1).space, (w.1).space\u27eb_\u211d|) ?_\n exact abs_real_inner_le_norm (v.1).space (w.1).space\n\nlemma metric_reflect_not_mem_not_mem (h : v \u2209 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\nlemma metric_reflect_mem_not_mem (h : v \u2208 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n \u27eav.1, w.1.spaceReflection\u27eb\u2098 \u2264 0 := by\n rw [show (0 : \u211d) = - 0 from zero_eq_neg.mpr rfl, le_neg]\n have h1 := metric_reflect_mem_mem h ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\n", "theoremStatement": "lemma metric_reflect_not_mem_mem (h : v \u2209 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n \u27eav.1, w.1.spaceReflection\u27eb\u2098 \u2264 0 ", "theoremName": "NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_mem", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "1133b88", "date": "2024-07-12"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 182, "tokenPositionInFile": 5113, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 97, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [show (0 : \u211d) = - 0 from zero_eq_neg.mpr rfl, le_neg]\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) hw\n apply le_of_le_of_eq h1 ?_\n simp [neg]", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 172}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\nlemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1\n\nlemma one_add_metric_non_zero : 1 + \u27eaf, f'.1.1\u27eb\u2098 \u2260 0 := by\n linarith [metric_nonneg f f']\n\n/-!\n\n# The sign of \u27eav, w.1.spaceReflection\u27eb\u2098 different v and w\n\n-/\n\nsection\nvariable {v w : NormOneLorentzVector d}\n\nlemma metric_reflect_mem_mem (h : v \u2208 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm v w)\n rw [abs_time \u27e8v, h\u27e9, abs_time \u27e8w, hw\u27e9, sub_eq_add_neg, right_spaceReflection]\n apply (add_le_add_iff_left _).mpr\n rw [neg_le]\n apply le_trans (neg_le_abs _ : _ \u2264 |\u27ea(v.1).space, (w.1).space\u27eb_\u211d|) ?_\n exact abs_real_inner_le_norm (v.1).space (w.1).space\n\nlemma metric_reflect_not_mem_not_mem (h : v \u2209 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\nlemma metric_reflect_mem_not_mem (h : v \u2208 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n \u27eav.1, w.1.spaceReflection\u27eb\u2098 \u2264 0 := by\n rw [show (0 : \u211d) = - 0 from zero_eq_neg.mpr rfl, le_neg]\n have h1 := metric_reflect_mem_mem h ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\nlemma metric_reflect_not_mem_mem (h : v \u2209 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n \u27eav.1, w.1.spaceReflection\u27eb\u2098 \u2264 0 := by\n rw [show (0 : \u211d) = - 0 from zero_eq_neg.mpr rfl, le_neg]\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) hw\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\nend\nend\n\nend FuturePointing\nend\n\nnamespace FuturePointing\n/-!\n\n# Topology\n\n-/\nopen LorentzVector\n\n/-- The `FuturePointing d` which has all space components zero. -/\n@[simps!]\nnoncomputable def timeVecNormOneFuture : FuturePointing d := \u27e8\u27e8timeVec, by\n rw [NormOneLorentzVector.mem_iff, on_timeVec]\u27e9, by\n rw [mem_iff, timeVec_time]; exact Real.zero_lt_one\u27e9\n\n/-- A continuous path from `timeVecNormOneFuture` to any other. -/\nnoncomputable def pathFromTime (u : FuturePointing d) : Path timeVecNormOneFuture u where\n toFun t := \u27e8\n \u27e8fun i => match i with\n | Sum.inl 0 => (1 + t ^ 2 * \u2016u.1.1.space\u2016 ^ 2).sqrt\n | Sum.inr i => t * u.1.1.space i,\n by\n rw [NormOneLorentzVector.mem_iff, minkowskiMetric.eq_time_minus_inner_prod]\n simp only [time, space, Function.comp_apply, PiLp.inner_apply, RCLike.inner_apply, map_mul,\n conj_trivial]\n rw [Real.mul_self_sqrt, \u2190 @real_inner_self_eq_norm_sq, @PiLp.inner_apply]\n \u00b7 simp only [Function.comp_apply, RCLike.inner_apply, conj_trivial]\n refine Eq.symm (eq_sub_of_add_eq (congrArg (HAdd.hAdd 1) ?_))\n rw [Finset.mul_sum]\n apply Finset.sum_congr rfl\n intro i _\n ring\n \u00b7 exact Right.add_nonneg (zero_le_one' \u211d) $ mul_nonneg (sq_nonneg _) (sq_nonneg _)\u27e9,\n by\n simp only [space, Function.comp_apply, mem_iff_time_nonneg, time, Real.sqrt_pos]\n exact Real.sqrt_nonneg _\u27e9\n continuous_toFun := by\n refine Continuous.subtype_mk ?_ _\n refine Continuous.subtype_mk ?_ _\n apply (continuous_pi_iff).mpr\n intro i\n match i with\n | Sum.inl 0 =>\n continuity\n | Sum.inr i =>\n continuity\n source' := by\n ext\n funext i\n match i with\n | Sum.inl 0 =>\n simp only [Set.Icc.coe_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, space,\n zero_mul, add_zero, Real.sqrt_one, Fin.isValue, timeVecNormOneFuture_coe_coe]\n exact Eq.symm timeVec_time\n | Sum.inr i =>\n simp only [Set.Icc.coe_zero, space, Function.comp_apply, zero_mul,\n timeVecNormOneFuture_coe_coe]\n change _ = timeVec.space i\n rw [timeVec_space]\n rfl\n target' := by\n ext\n funext i\n match i with\n | Sum.inl 0 =>\n simp only [Set.Icc.coe_one, one_pow, space, one_mul, Fin.isValue]\n exact (time_eq_sqrt u).symm\n | Sum.inr i =>\n simp only [Set.Icc.coe_one, one_pow, space, one_mul, Fin.isValue]\n exact rfl\n\n", "theoremStatement": "lemma isPathConnected : IsPathConnected (@Set.univ (FuturePointing d)) ", "theoremName": "NormOneLorentzVector.FuturePointing.isPathConnected", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "c64d926", "date": "2024-07-02"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 265, "tokenPositionInFile": 7824, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 24, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n use timeVecNormOneFuture\n apply And.intro trivial ?_\n intro y a\n use pathFromTime y\n exact fun _ => a", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 113}} +{"srcContext": "/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Tooby-Smith\n-/\nimport HepLean.SpaceTime.LorentzVector.Basic\nimport HepLean.SpaceTime.MinkowskiMetric\n/-!\n\n# Lorentz vectors with norm one\n\n-/\n\nopen minkowskiMetric\n\n/-- The set of Lorentz vectors with norm 1. -/\n@[simp]\ndef NormOneLorentzVector (d : \u2115) : Set (LorentzVector d) :=\n fun x => \u27eax, x\u27eb\u2098 = 1\n\ninstance : TopologicalSpace (NormOneLorentzVector d) := instTopologicalSpaceSubtype\n\nnamespace NormOneLorentzVector\n\nvariable {d : \u2115}\n\nsection\nvariable (v w : NormOneLorentzVector d)\n\nlemma mem_iff {x : LorentzVector d} : x \u2208 NormOneLorentzVector d \u2194 \u27eax, x\u27eb\u2098 = 1 := by\n rfl\n\n/-- The negative of a `NormOneLorentzVector` as a `NormOneLorentzVector`. -/\ndef neg : NormOneLorentzVector d := \u27e8- v, by\n rw [mem_iff]\n simp only [map_neg, LinearMap.neg_apply, neg_neg]\n exact v.2\u27e9\n\nlemma time_sq : v.1.time ^ 2 = 1 + \u2016v.1.space\u2016 ^ 2 := by\n rw [time_sq_eq_metric_add_space, v.2]\n\nlemma abs_time_ge_one : 1 \u2264 |v.1.time| := by\n have h1 := leq_time_sq v.1\n rw [v.2] at h1\n exact (one_le_sq_iff_one_le_abs _).mp h1\n\nlemma norm_space_le_abs_time : \u2016v.1.space\u2016 < |v.1.time| := by\n rw [(abs_norm _).symm, \u2190 @sq_lt_sq, time_sq]\n exact lt_one_add (\u2016(v.1).space\u2016 ^ 2)\n\nlemma norm_space_leq_abs_time : \u2016v.1.space\u2016 \u2264 |v.1.time| :=\n le_of_lt (norm_space_le_abs_time v)\n\nlemma time_le_minus_one_or_ge_one : v.1.time \u2264 -1 \u2228 1 \u2264 v.1.time :=\n le_abs'.mp (abs_time_ge_one v)\n\nlemma time_nonpos_iff : v.1.time \u2264 0 \u2194 v.1.time \u2264 - 1 := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 exact h1\n \u00b7 linarith\n \u00b7 intro h\n linarith\n\nlemma time_nonneg_iff : 0 \u2264 v.1.time \u2194 1 \u2264 v.1.time := by\n apply Iff.intro\n \u00b7 intro h\n cases' time_le_minus_one_or_ge_one v with h1 h1\n \u00b7 linarith\n \u00b7 exact h1\n \u00b7 intro h\n linarith\n\nlemma time_pos_iff : 0 < v.1.time \u2194 1 \u2264 v.1.time := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact (time_nonneg_iff v).mp (le_of_lt h)\n \u00b7 linarith\n\nlemma time_abs_sub_space_norm :\n 0 \u2264 |v.1.time| * |w.1.time| - \u2016v.1.space\u2016 * \u2016w.1.space\u2016 := by\n apply sub_nonneg.mpr\n apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_\n \u00b7 exact norm_nonneg w.1.space\n \u00b7 exact abs_nonneg (v.1 _)\n\n/-!\n\n# Future pointing norm one Lorentz vectors\n\n-/\n\n/-- The future pointing Lorentz vectors with Norm one. -/\ndef FuturePointing (d : \u2115) : Set (NormOneLorentzVector d) :=\n fun x => 0 < x.1.time\n\ninstance : TopologicalSpace (FuturePointing d) := instTopologicalSpaceSubtype\n\nnamespace FuturePointing\n\nsection\nvariable (f f' : FuturePointing d)\n\nlemma mem_iff : v \u2208 FuturePointing d \u2194 0 < v.1.time := by\n rfl\n\nlemma mem_iff_time_nonneg : v \u2208 FuturePointing d \u2194 0 \u2264 v.1.time := by\n refine Iff.intro (fun h => le_of_lt h) (fun h => ?_)\n rw [time_nonneg_iff] at h\n rw [mem_iff]\n linarith\n\nlemma not_mem_iff : v \u2209 FuturePointing d \u2194 v.1.time \u2264 0 := by\n refine Iff.intro (fun h => ?_) (fun h => ?_)\n \u00b7 exact le_of_not_lt ((mem_iff v).mp.mt h)\n \u00b7 have h1 := (mem_iff v).mp.mt\n simp only [LorentzVector.time, Fin.isValue, not_lt] at h1\n exact h1 h\n\nlemma not_mem_iff_neg : v \u2209 FuturePointing d \u2194 neg v \u2208 FuturePointing d := by\n rw [not_mem_iff, mem_iff_time_nonneg]\n simp only [Fin.isValue, neg]\n change _ \u2194 0 \u2264 - v.1 _\n exact Iff.symm neg_nonneg\n\nlemma time_nonneg : 0 \u2264 f.1.1.time := le_of_lt f.2\n\nlemma abs_time : |f.1.1.time| = f.1.1.time := abs_of_nonneg (time_nonneg f)\n\nlemma time_eq_sqrt : f.1.1.time = (1 + \u2016f.1.1.space\u2016 ^ 2).sqrt := by\n symm\n rw [Real.sqrt_eq_cases]\n apply Or.inl\n rw [\u2190 time_sq, sq]\n exact \u27e8rfl, time_nonneg f\u27e9\n\n/-!\n\n# The sign of \u27eav, w.1\u27eb\u2098 different v and w\n\n-/\n\nlemma metric_nonneg : 0 \u2264 \u27eaf, f'.1.1\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm f f'.1)\n rw [abs_time f, abs_time f']\n exact ge_sub_norm f.1.1 f'.1.1\n\nlemma one_add_metric_non_zero : 1 + \u27eaf, f'.1.1\u27eb\u2098 \u2260 0 := by\n linarith [metric_nonneg f f']\n\n/-!\n\n# The sign of \u27eav, w.1.spaceReflection\u27eb\u2098 different v and w\n\n-/\n\nsection\nvariable {v w : NormOneLorentzVector d}\n\nlemma metric_reflect_mem_mem (h : v \u2208 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n apply le_trans (time_abs_sub_space_norm v w)\n rw [abs_time \u27e8v, h\u27e9, abs_time \u27e8w, hw\u27e9, sub_eq_add_neg, right_spaceReflection]\n apply (add_le_add_iff_left _).mpr\n rw [neg_le]\n apply le_trans (neg_le_abs _ : _ \u2264 |\u27ea(v.1).space, (w.1).space\u27eb_\u211d|) ?_\n exact abs_real_inner_le_norm (v.1).space (w.1).space\n\nlemma metric_reflect_not_mem_not_mem (h : v \u2209 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n 0 \u2264 \u27eav.1, w.1.spaceReflection\u27eb\u2098 := by\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\nlemma metric_reflect_mem_not_mem (h : v \u2208 FuturePointing d) (hw : w \u2209 FuturePointing d) :\n \u27eav.1, w.1.spaceReflection\u27eb\u2098 \u2264 0 := by\n rw [show (0 : \u211d) = - 0 from zero_eq_neg.mpr rfl, le_neg]\n have h1 := metric_reflect_mem_mem h ((not_mem_iff_neg w).mp hw)\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\nlemma metric_reflect_not_mem_mem (h : v \u2209 FuturePointing d) (hw : w \u2208 FuturePointing d) :\n \u27eav.1, w.1.spaceReflection\u27eb\u2098 \u2264 0 := by\n rw [show (0 : \u211d) = - 0 from zero_eq_neg.mpr rfl, le_neg]\n have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) hw\n apply le_of_le_of_eq h1 ?_\n simp [neg]\n\nend\nend\n\nend FuturePointing\nend\n\nnamespace FuturePointing\n/-!\n\n# Topology\n\n-/\nopen LorentzVector\n\n/-- The `FuturePointing d` which has all space components zero. -/\n@[simps!]\nnoncomputable def timeVecNormOneFuture : FuturePointing d := \u27e8\u27e8timeVec, by\n rw [NormOneLorentzVector.mem_iff, on_timeVec]\u27e9, by\n rw [mem_iff, timeVec_time]; exact Real.zero_lt_one\u27e9\n\n/-- A continuous path from `timeVecNormOneFuture` to any other. -/\nnoncomputable def pathFromTime (u : FuturePointing d) : Path timeVecNormOneFuture u where\n toFun t := \u27e8\n \u27e8fun i => match i with\n | Sum.inl 0 => (1 + t ^ 2 * \u2016u.1.1.space\u2016 ^ 2).sqrt\n | Sum.inr i => t * u.1.1.space i,\n by\n rw [NormOneLorentzVector.mem_iff, minkowskiMetric.eq_time_minus_inner_prod]\n simp only [time, space, Function.comp_apply, PiLp.inner_apply, RCLike.inner_apply, map_mul,\n conj_trivial]\n rw [Real.mul_self_sqrt, \u2190 @real_inner_self_eq_norm_sq, @PiLp.inner_apply]\n \u00b7 simp only [Function.comp_apply, RCLike.inner_apply, conj_trivial]\n refine Eq.symm (eq_sub_of_add_eq (congrArg (HAdd.hAdd 1) ?_))\n rw [Finset.mul_sum]\n apply Finset.sum_congr rfl\n intro i _\n ring\n \u00b7 exact Right.add_nonneg (zero_le_one' \u211d) $ mul_nonneg (sq_nonneg _) (sq_nonneg _)\u27e9,\n by\n simp only [space, Function.comp_apply, mem_iff_time_nonneg, time, Real.sqrt_pos]\n exact Real.sqrt_nonneg _\u27e9\n continuous_toFun := by\n refine Continuous.subtype_mk ?_ _\n refine Continuous.subtype_mk ?_ _\n apply (continuous_pi_iff).mpr\n intro i\n match i with\n | Sum.inl 0 =>\n continuity\n | Sum.inr i =>\n continuity\n source' := by\n ext\n funext i\n match i with\n | Sum.inl 0 =>\n simp only [Set.Icc.coe_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, space,\n zero_mul, add_zero, Real.sqrt_one, Fin.isValue, timeVecNormOneFuture_coe_coe]\n exact Eq.symm timeVec_time\n | Sum.inr i =>\n simp only [Set.Icc.coe_zero, space, Function.comp_apply, zero_mul,\n timeVecNormOneFuture_coe_coe]\n change _ = timeVec.space i\n rw [timeVec_space]\n rfl\n target' := by\n ext\n funext i\n match i with\n | Sum.inl 0 =>\n simp only [Set.Icc.coe_one, one_pow, space, one_mul, Fin.isValue]\n exact (time_eq_sqrt u).symm\n | Sum.inr i =>\n simp only [Set.Icc.coe_one, one_pow, space, one_mul, Fin.isValue]\n exact rfl\n\nlemma isPathConnected : IsPathConnected (@Set.univ (FuturePointing d)) := by\n use timeVecNormOneFuture\n apply And.intro trivial ?_\n intro y a\n use pathFromTime y\n exact fun _ => a\n\n", "theoremStatement": "lemma metric_continuous (u : LorentzVector d) :\n Continuous (fun (a : FuturePointing d) => \u27eau, a.1.1\u27eb\u2098) ", "theoremName": "NormOneLorentzVector.FuturePointing.metric_continuous", "fileCreated": {"commit": "675b9a9", "date": "2024-07-01"}, "theoremCreated": {"commit": "c64d926", "date": "2024-07-02"}, "file": "hep_lean/HepLean/SpaceTime/LorentzVector/NormOne.lean", "module": "HepLean.SpaceTime.LorentzVector.NormOne", "jsonFile": "HepLean.SpaceTime.LorentzVector.NormOne.jsonl", "positionMetadata": {"lineInFile": 272, "tokenPositionInFile": 8010, "theoremPositionInFile": 28}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 10, "numPremises": 149, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Cases", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Tactic.ComputeDegree", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "HepLean.SpaceTime.LorentzVector.Basic", "Mathlib.Algebra.Lie.Basic", "Mathlib.Algebra.Lie.Subalgebra", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.Divisibility.Prod", "Mathlib.RingTheory.Nakayama", "Mathlib.RingTheory.Artinian", "Mathlib.Algebra.Lie.Submodule", "Mathlib.Algebra.Lie.OfAssociative", "Mathlib.Algebra.Lie.IdealOperations", "Mathlib.Algebra.Lie.Abelian", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.RingTheory.Localization.Integer", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.Algebra.Lie.Matrix", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.Tactic.NoncommRing", "Mathlib.Algebra.Lie.SkewAdjoint", "Mathlib.LinearAlgebra.SymplecticGroup", "Mathlib.Algebra.Lie.Classical", "HepLean.SpaceTime.MinkowskiMetric"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp only [minkowskiMetric.eq_time_minus_inner_prod]\n refine Continuous.add ?_ ?_\n \u00b7 refine Continuous.comp' (continuous_mul_left _) $ Continuous.comp'\n (continuous_apply (Sum.inl 0))\n (Continuous.comp' continuous_subtype_val continuous_subtype_val)\n \u00b7 refine Continuous.comp' continuous_neg $ Continuous.inner\n (Continuous.comp' (Pi.continuous_precomp Sum.inr) continuous_const)\n (Continuous.comp' (Pi.continuous_precomp Sum.inr) (Continuous.comp'\n continuous_subtype_val continuous_subtype_val))", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 535}} diff --git a/minictx/htpi.jsonl b/minictx/htpi.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..5eca1d53e19b95e4eb572b35b6aba5f1c96901d3 --- /dev/null +++ b/minictx/htpi.jsonl @@ -0,0 +1,45 @@ +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\n", "theoremStatement": "theorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a ", "theoremName": "HTPI.Exercises.dvd_a_of_dvd_b_mod", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1905, "tokenPositionInFile": 62474, "theoremPositionInFile": 166}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\n", "theoremStatement": "lemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a ", "theoremName": "HTPI.Exercises.gcd_comm_lt", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1909, "tokenPositionInFile": 62575, "theoremPositionInFile": 167}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 7, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\n", "theoremStatement": "theorem gcd_comm (a b : Nat) : gcd a b = gcd b a ", "theoremName": "HTPI.Exercises.gcd_comm", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1911, "tokenPositionInFile": 62647, "theoremPositionInFile": 168}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 5, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\n", "theoremStatement": "theorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n ", "theoremName": "HTPI.Exercises.Exercise_7_1_5", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "7e40747", "date": "2023-09-13"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1914, "tokenPositionInFile": 62712, "theoremPositionInFile": 169}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\n", "theoremStatement": "theorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b ", "theoremName": "HTPI.Exercises.Exercise_7_1_6", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "6b2542f", "date": "2023-04-08"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1918, "tokenPositionInFile": 62839, "theoremPositionInFile": 170}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\n", "theoremStatement": "theorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 ", "theoremName": "HTPI.Exercises.gcd_is_nonzero", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "68d6f9e", "date": "2023-04-28"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1922, "tokenPositionInFile": 62926, "theoremPositionInFile": 171}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\n", "theoremStatement": "theorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b ", "theoremName": "HTPI.Exercises.gcd_greatest", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "6b2542f", "date": "2023-04-08"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1926, "tokenPositionInFile": 63015, "theoremPositionInFile": 172}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\n", "theoremStatement": "lemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) ", "theoremName": "HTPI.Exercises.Lemma_7_1_10a", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "59c90ca", "date": "2023-09-12"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1930, "tokenPositionInFile": 63129, "theoremPositionInFile": 173}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\n", "theoremStatement": "lemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b ", "theoremName": "HTPI.Exercises.Lemma_7_1_10b", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "59c90ca", "date": "2023-09-12"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1933, "tokenPositionInFile": 63217, "theoremPositionInFile": 174}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\n", "theoremStatement": "lemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b ", "theoremName": "HTPI.Exercises.Lemma_7_1_10c", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "59c90ca", "date": "2023-09-12"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1936, "tokenPositionInFile": 63311, "theoremPositionInFile": 175}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 6, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\n", "theoremStatement": "theorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b ", "theoremName": "HTPI.Exercises.Exercise_7_1_10", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "6b2542f", "date": "2023-04-08"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1939, "tokenPositionInFile": 63391, "theoremPositionInFile": 176}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\n", "theoremStatement": "lemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p ", "theoremName": "HTPI.Exercises.dvd_prime", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1944, "tokenPositionInFile": 63503, "theoremPositionInFile": 177}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 10, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\n", "theoremStatement": "theorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 ", "theoremName": "HTPI.Exercises.prod_nonzero_nonzero", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "25681a1", "date": "2023-11-10"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1949, "tokenPositionInFile": 63666, "theoremPositionInFile": 178}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 10, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\n", "theoremStatement": "theorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b ", "theoremName": "HTPI.Exercises.rel_prime_iff_no_common_factor", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "68d6f9e", "date": "2023-04-28"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1953, "tokenPositionInFile": 63765, "theoremPositionInFile": 179}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\n", "theoremStatement": "theorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a ", "theoremName": "HTPI.Exercises.rel_prime_symm", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "fe2721d", "date": "2023-05-10"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1957, "tokenPositionInFile": 63892, "theoremPositionInFile": 180}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 4, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\n", "theoremStatement": "lemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a ", "theoremName": "HTPI.Exercises.in_prime_factorization_iff_prime_factor", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "ed81107", "date": "2023-09-14"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1961, "tokenPositionInFile": 63983, "theoremPositionInFile": 181}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\n", "theoremStatement": "theorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) ", "theoremName": "HTPI.Exercises.Exercise_7_2_5", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "68d6f9e", "date": "2023-04-28"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1966, "tokenPositionInFile": 64146, "theoremPositionInFile": 182}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 12, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\n", "theoremStatement": "theorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 ", "theoremName": "HTPI.Exercises.Exercise_7_2_6", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "6b2542f", "date": "2023-04-08"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1971, "tokenPositionInFile": 64332, "theoremPositionInFile": 183}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\n", "theoremStatement": "theorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' ", "theoremName": "HTPI.Exercises.Exercise_7_2_7", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "6b2542f", "date": "2023-04-08"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1975, "tokenPositionInFile": 64438, "theoremPositionInFile": 184}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 6, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\n", "theoremStatement": "theorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k ", "theoremName": "HTPI.Exercises.Exercise_7_2_9", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "ed6a4d7", "date": "2023-09-15"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1980, "tokenPositionInFile": 64570, "theoremPositionInFile": 185}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 12, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\n", "theoremStatement": "theorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c ", "theoremName": "HTPI.Exercises.Exercise_7_2_17a", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "6b2542f", "date": "2023-04-08"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1985, "tokenPositionInFile": 64715, "theoremPositionInFile": 186}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\n", "theoremStatement": "theorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) ", "theoremName": "HTPI.Exercises.congr_trans", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "fe2721d", "date": "2023-05-10"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1990, "tokenPositionInFile": 64828, "theoremPositionInFile": 187}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 5, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\n", "theoremStatement": "theorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X ", "theoremName": "HTPI.Exercises.Theorem_7_3_6_3", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1994, "tokenPositionInFile": 64943, "theoremPositionInFile": 188}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\n", "theoremStatement": "theorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m ", "theoremName": "HTPI.Exercises.Theorem_7_3_6_4", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1997, "tokenPositionInFile": 65022, "theoremPositionInFile": 189}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 19, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\n", "theoremStatement": "theorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 ", "theoremName": "HTPI.Exercises.Exercise_7_3_4a", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "515de92", "date": "2023-04-13"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2001, "tokenPositionInFile": 65121, "theoremPositionInFile": 190}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 14, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\n", "theoremStatement": "theorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 ", "theoremName": "HTPI.Exercises.Exercise_7_3_4b", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2006, "tokenPositionInFile": 65274, "theoremPositionInFile": 191}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\n", "theoremStatement": "theorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) ", "theoremName": "HTPI.Exercises.Theorem_7_3_10", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "a2a51e2", "date": "2023-09-25"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2010, "tokenPositionInFile": 65401, "theoremPositionInFile": 192}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 15, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\n", "theoremStatement": "theorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) ", "theoremName": "HTPI.Exercises.Theorem_7_3_11", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2014, "tokenPositionInFile": 65526, "theoremPositionInFile": 193}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 15, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\n", "theoremStatement": "theorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) ", "theoremName": "HTPI.Exercises.Exercise_7_3_16", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2018, "tokenPositionInFile": 65650, "theoremPositionInFile": 194}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 7, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\n", "theoremStatement": "theorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b ", "theoremName": "HTPI.Exercises.congr_rel_prime", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "54b1fe6", "date": "2023-06-29"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2026, "tokenPositionInFile": 65878, "theoremPositionInFile": 195}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\n", "theoremStatement": "theorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a ", "theoremName": "HTPI.Exercises.rel_prime_mod", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "54b1fe6", "date": "2023-06-29"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2031, "tokenPositionInFile": 66051, "theoremPositionInFile": 196}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\n", "theoremStatement": "lemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m ", "theoremName": "HTPI.Exercises.congr_iff_mod_eq_Int", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "28aefa3", "date": "2023-08-06"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2035, "tokenPositionInFile": 66144, "theoremPositionInFile": 197}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 15, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\n", "theoremStatement": "theorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m ", "theoremName": "HTPI.Exercises.congr_iff_mod_eq_Nat", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "28aefa3", "date": "2023-08-06"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2040, "tokenPositionInFile": 66365, "theoremPositionInFile": 198}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 15, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\n", "theoremStatement": "lemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m ", "theoremName": "HTPI.Exercises.Exercise_7_4_5_Int", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "fe2721d", "date": "2023-05-10"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2049, "tokenPositionInFile": 66663, "theoremPositionInFile": 199}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 16, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\n", "theoremStatement": "lemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g ", "theoremName": "HTPI.Exercises.left_inv_one_one_below", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "a690ad4", "date": "2023-04-16"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2053, "tokenPositionInFile": 66765, "theoremPositionInFile": 200}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 7, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\n", "theoremStatement": "lemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) ", "theoremName": "HTPI.Exercises.comp_perm_below", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "68d6f9e", "date": "2023-04-28"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2057, "tokenPositionInFile": 66892, "theoremPositionInFile": 201}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 5, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\n", "theoremStatement": "lemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g ", "theoremName": "HTPI.Exercises.perm_below_fixed", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "a690ad4", "date": "2023-04-16"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2062, "tokenPositionInFile": 67033, "theoremPositionInFile": 202}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 10, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\n", "theoremStatement": "lemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c ", "theoremName": "HTPI.Exercises.Lemma_7_4_6", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "68d6f9e", "date": "2023-04-28"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2066, "tokenPositionInFile": 67162, "theoremPositionInFile": 203}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\n-- 6.\nexample {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n a ^ (phi m + 1) \u2261 a (MOD m) := sorry\n\n-- 7.\n", "theoremStatement": "theorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p + 1 = phi m) :\n [a]_m * [a ^ p]_m = [1]_m ", "theoremName": "HTPI.Exercises.Like_Exercise_7_4_11", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2074, "tokenPositionInFile": 67370, "theoremPositionInFile": 204}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 29, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\n-- 6.\nexample {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n a ^ (phi m + 1) \u2261 a (MOD m) := sorry\n\n-- 7.\ntheorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p + 1 = phi m) :\n [a]_m * [a ^ p]_m = [1]_m := sorry\n\n-- 8.\n", "theoremStatement": "theorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) :\n a ^ p \u2261 a ^ q (MOD m) ", "theoremName": "HTPI.Exercises.Like_Exercise_7_4_12", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2079, "tokenPositionInFile": 67518, "theoremPositionInFile": 205}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\n-- 6.\nexample {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n a ^ (phi m + 1) \u2261 a (MOD m) := sorry\n\n-- 7.\ntheorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p + 1 = phi m) :\n [a]_m * [a ^ p]_m = [1]_m := sorry\n\n-- 8.\ntheorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) :\n a ^ p \u2261 a ^ q (MOD m) := sorry\n\n/- Section 7.5 -/\n-- 1.\n--Hint: Use induction.\n", "theoremStatement": "lemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k ", "theoremName": "HTPI.Exercises.num_rp_prime", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "25681a1", "date": "2023-11-10"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2086, "tokenPositionInFile": 67714, "theoremPositionInFile": 206}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\n-- 6.\nexample {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n a ^ (phi m + 1) \u2261 a (MOD m) := sorry\n\n-- 7.\ntheorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p + 1 = phi m) :\n [a]_m * [a ^ p]_m = [1]_m := sorry\n\n-- 8.\ntheorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) :\n a ^ p \u2261 a ^ q (MOD m) := sorry\n\n/- Section 7.5 -/\n-- 1.\n--Hint: Use induction.\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\n-- 2.\n", "theoremStatement": "lemma three_prime : prime 3 ", "theoremName": "HTPI.Exercises.three_prime", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "fe2721d", "date": "2023-05-10"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2090, "tokenPositionInFile": 67816, "theoremPositionInFile": 207}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 6, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\n-- 6.\nexample {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n a ^ (phi m + 1) \u2261 a (MOD m) := sorry\n\n-- 7.\ntheorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p + 1 = phi m) :\n [a]_m * [a ^ p]_m = [1]_m := sorry\n\n-- 8.\ntheorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) :\n a ^ p \u2261 a ^ q (MOD m) := sorry\n\n/- Section 7.5 -/\n-- 1.\n--Hint: Use induction.\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\n-- 2.\nlemma three_prime : prime 3 := sorry\n\n-- 3.\n--Hint: Use the previous exercise, Exercise_7_2_7, and Theorem_7_4_2.\n", "theoremStatement": "theorem Exercise_7_5_13a (a : Nat) (h1 : rel_prime 561 a) :\n a ^ 560 \u2261 1 (MOD 3) ", "theoremName": "HTPI.Exercises.Exercise_7_5_13a", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "fc48d15", "date": "2023-12-02"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2094, "tokenPositionInFile": 67931, "theoremPositionInFile": 208}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 15, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\n-- 6.\nexample {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n a ^ (phi m + 1) \u2261 a (MOD m) := sorry\n\n-- 7.\ntheorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p + 1 = phi m) :\n [a]_m * [a ^ p]_m = [1]_m := sorry\n\n-- 8.\ntheorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) :\n a ^ p \u2261 a ^ q (MOD m) := sorry\n\n/- Section 7.5 -/\n-- 1.\n--Hint: Use induction.\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\n-- 2.\nlemma three_prime : prime 3 := sorry\n\n-- 3.\n--Hint: Use the previous exercise, Exercise_7_2_7, and Theorem_7_4_2.\ntheorem Exercise_7_5_13a (a : Nat) (h1 : rel_prime 561 a) :\n a ^ 560 \u2261 1 (MOD 3) := sorry\n\n-- 4.\n--Hint: Imitate the way Theorem_7_2_2_Int was proven from Theorem_7_2_2.\n", "theoremStatement": "lemma Theorem_7_2_3_Int {p : Nat} {a b : Int}\n (h1 : prime p) (h2 : \u2191p \u2223 a * b) : \u2191p \u2223 a \u2228 \u2191p \u2223 b ", "theoremName": "HTPI.Exercises.Theorem_7_2_3_Int", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "fe2721d", "date": "2023-05-10"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2099, "tokenPositionInFile": 68105, "theoremPositionInFile": 209}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\nlemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m := by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done\n\ntheorem Euler's_theorem {m a : Nat} [NeZero m]\n (h1 : rel_prime m a) : a ^ (phi m) \u2261 1 (MOD m) := by\n have h2 : [a]_m ^ (phi m) = [1]_m := Theorem_7_4_2 h1\n rewrite [Exercise_7_4_5_Nat m a (phi m)] at h2\n --h2 : [a ^ phi m]_m = [1]_m\n show a ^ (phi m) \u2261 1 (MOD m) from (cc_eq_iff_congr _ _ _).ltr h2\n done\n\n#eval gcd 10 7 --Answer: 1. So 10 and 7 are relatively prime\n\n#eval 7 ^ phi 10 --Answer: 2401, which is congruent to 1 mod 10.\n\nend Euler\n\n/- Section 7.5 -/\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\nlemma phi_prime {p : Nat} (h1 : prime p) : phi p = p - 1 := by\n have h2 : 1 \u2264 p := prime_pos h1\n have h3 : p - 1 + 1 = p := Nat.sub_add_cancel h2\n have h4 : p - 1 < p := by linarith\n have h5 : num_rp_below p (p - 1 + 1) = p - 1 :=\n num_rp_prime h1 (p - 1) h4\n rewrite [h3] at h5\n show phi p = p - 1 from h5\n done\n\ntheorem Theorem_7_2_2_Int {a c : Nat} {b : Int}\n (h1 : \u2191c \u2223 \u2191a * b) (h2 : rel_prime a c) : \u2191c \u2223 b := by\n rewrite [Int.natCast_dvd, Int.natAbs_mul,\n Int.natAbs_ofNat] at h1 --h1 : c \u2223 a * Int.natAbs b\n rewrite [Int.natCast_dvd] --Goal : c \u2223 Int.natAbs b\n show c \u2223 Int.natAbs b from Theorem_7_2_2 h1 h2\n done\n\nlemma Lemma_7_4_5 {m n : Nat} (a b : Int) (h1 : rel_prime m n) :\n a \u2261 b (MOD m * n) \u2194 a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : a \u2261 b (MOD m * n)\n obtain (j : Int) (h3 : a - b = (m * n) * j) from h2\n apply And.intro\n \u00b7 -- Proof of a \u2261 b (MOD m)\n apply Exists.intro (n * j)\n show a - b = m * (n * j) from\n calc a - b\n _ = m * n * j := h3\n _ = m * (n * j) := by ring\n done\n \u00b7 -- Proof of a \u2261 b (MOD n)\n apply Exists.intro (m * j)\n show a - b = n * (m * j) from\n calc a - b\n _ = m * n * j := h3\n _ = n * (m * j) := by ring\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : a \u2261 b (MOD m) \u2227 a \u2261 b (MOD n)\n obtain (j : Int) (h3 : a - b = m * j) from h2.left\n have h4 : (\u2191n : Int) \u2223 a - b := h2.right\n rewrite [h3] at h4 --h4 : \u2191n \u2223 \u2191m * j\n have h5 : \u2191n \u2223 j := Theorem_7_2_2_Int h4 h1\n obtain (k : Int) (h6 : j = n * k) from h5\n apply Exists.intro k --Goal : a - b = \u2191(m * n) * k\n rewrite [Nat.cast_mul] --Goal : a - b = \u2191m * \u2191n * k\n show a - b = (m * n) * k from\n calc a - b\n _ = m * j := h3\n _ = m * (n * k) := by rw [h6]\n _ = (m * n) * k := by ring\n done\n done\n\n--From exercises of Section 7.2\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\nlemma prime_NeZero {p : Nat} (h : prime p) : NeZero p := by\n rewrite [neZero_iff] --Goal : p \u2260 0\n define at h\n linarith\n done\n\nlemma Lemma_7_5_1 {p e d m c s : Nat} {t : Int}\n (h1 : prime p) (h2 : e * d = (p - 1) * s + 1)\n (h3 : m ^ e - c = p * t) :\n c ^ d \u2261 m (MOD p) := by\n have h4 : m ^ e \u2261 c (MOD p) := Exists.intro t h3\n have h5 : [m ^ e]_p = [c]_p := (cc_eq_iff_congr _ _ _).rtl h4\n rewrite [\u2190Exercise_7_4_5_Nat] at h5 --h5 : [m]_p ^ e = [c]_p\n by_cases h6 : p \u2223 m\n \u00b7 -- Case 1. h6 : p \u2223 m\n have h7 : m \u2261 0 (MOD p) := by\n obtain (j : Nat) (h8 : m = p * j) from h6\n apply Exists.intro (\u2191j : Int) --Goal : \u2191m - 0 = \u2191p * \u2191j\n rewrite [h8, Nat.cast_mul]\n ring\n done\n have h8 : [m]_p = [0]_p := (cc_eq_iff_congr _ _ _).rtl h7\n have h9 : e * d \u2260 0 := by\n rewrite [h2]\n show (p - 1) * s + 1 \u2260 0 from Nat.add_one_ne_zero _\n done\n have h10 : (0 : Int) ^ (e * d) = 0 := zero_pow h9\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [0]_p ^ (e * d) := by rw [h8]\n _ = [0 ^ (e * d)]_p := Exercise_7_4_5_Int _ _ _\n _ = [0]_p := by rw [h10]\n _ = [m]_p := by rw [h8]\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n \u00b7 -- Case 2. h6 : \u00acp \u2223 m\n have h7 : rel_prime m p := rel_prime_of_prime_not_dvd h1 h6\n have h8 : rel_prime p m := rel_prime_symm h7\n have h9 : NeZero p := prime_NeZero h1\n have h10 : (1 : Int) ^ s = 1 := by ring\n have h11 : [c ^ d]_p = [m]_p :=\n calc [c ^ d]_p\n _ = [c]_p ^ d := by rw [Exercise_7_4_5_Nat]\n _ = ([m]_p ^ e) ^ d := by rw [h5]\n _ = [m]_p ^ (e * d) := by ring\n _ = [m]_p ^ ((p - 1) * s + 1) := by rw [h2]\n _ = ([m]_p ^ (p - 1)) ^ s * [m]_p := by ring\n _ = ([m]_p ^ (phi p)) ^ s * [m]_p := by rw [phi_prime h1]\n _ = [1]_p ^ s * [m]_p := by rw [Theorem_7_4_2 h8]\n _ = [1 ^ s]_p * [m]_p := by rw [Exercise_7_4_5_Int]\n _ = [1]_p * [m]_p := by rw [h10]\n _ = [m]_p * [1]_p := by ring\n _ = [m]_p := Theorem_7_3_6_7 _\n show c ^ d \u2261 m (MOD p) from (cc_eq_iff_congr _ _ _).ltr h11\n done\n done\n\ntheorem Theorem_7_5_1 (p q n e d k m c : Nat)\n (p_prime : prime p) (q_prime : prime q) (p_ne_q : p \u2260 q)\n (n_pq : n = p * q) (ed_congr_1 : e * d = k * (p - 1) * (q - 1) + 1)\n (h1 : [m]_n ^ e = [c]_n) : [c]_n ^ d = [m]_n := by\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr] at h1\n --h1 : m ^ e \u2261 c (MOD n)\n rewrite [Exercise_7_4_5_Nat, cc_eq_iff_congr]\n --Goal : c ^ d \u2261 m (MOD n)\n obtain (j : Int) (h2 : m ^ e - c = n * j) from h1\n rewrite [n_pq, Nat.cast_mul] at h2\n --h2 : m ^ e - c = p * q * j\n have h3 : e * d = (p - 1) * (k * (q - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h4 : m ^ e - c = p * (q * j) := by\n rewrite [h2]\n ring\n done\n have congr_p : c ^ d \u2261 m (MOD p) := Lemma_7_5_1 p_prime h3 h4\n have h5 : e * d = (q - 1) * (k * (p - 1)) + 1 := by\n rewrite [ed_congr_1]\n ring\n done\n have h6 : m ^ e - c = q * (p * j) := by\n rewrite [h2]\n ring\n done\n have congr_q : c ^ d \u2261 m (MOD q) := Lemma_7_5_1 q_prime h5 h6\n have h7 : \u00acq \u2223 p := by\n by_contra h8\n have h9 : q = 1 \u2228 q = p := dvd_prime p_prime h8\n disj_syll h9 (prime_not_one q_prime)\n show False from p_ne_q h9.symm\n done\n have h8 : rel_prime p q := rel_prime_of_prime_not_dvd q_prime h7\n rewrite [n_pq, Lemma_7_4_5 _ _ h8]\n show c ^ d \u2261 m (MOD p) \u2227 c ^ d \u2261 m (MOD q) from\n And.intro congr_p congr_q\n done\n\n/- BEGIN EXERCISES -/\n\nnamespace Exercises\n\n/- Section 7.1 -/\n-- 1.\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n-- 2.\nlemma gcd_comm_lt {a b : Nat} (h : a < b) : gcd a b = gcd b a := sorry\n\ntheorem gcd_comm (a b : Nat) : gcd a b = gcd b a := sorry\n\n-- 3.\ntheorem Exercise_7_1_5 (a b : Nat) (n : Int) :\n (\u2203 (s t : Int), s * a + t * b = n) \u2194 (\u2191(gcd a b) : Int) \u2223 n := sorry\n\n-- 4.\ntheorem Exercise_7_1_6 (a b c : Nat) :\n gcd a b = gcd (a + b * c) b := sorry\n\n-- 5.\ntheorem gcd_is_nonzero {a b : Nat} (h : a \u2260 0 \u2228 b \u2260 0) :\n gcd a b \u2260 0 := sorry\n\n-- 6.\ntheorem gcd_greatest {a b d : Nat} (h1 : gcd a b \u2260 0)\n (h2 : d \u2223 a) (h3 : d \u2223 b) : d \u2264 gcd a b := sorry\n\n-- 7.\nlemma Lemma_7_1_10a {a b : Nat}\n (n : Nat) (h : a \u2223 b) : (n * a) \u2223 (n * b) := sorry\n\nlemma Lemma_7_1_10b {a b n : Nat}\n (h1 : n \u2260 0) (h2 : (n * a) \u2223 (n * b)) : a \u2223 b := sorry\n\nlemma Lemma_7_1_10c {a b : Nat}\n (h1 : a \u2223 b) (h2 : b \u2223 a) : a = b := sorry\n\ntheorem Exercise_7_1_10 (a b n : Nat) :\n gcd (n * a) (n * b) = n * gcd a b := sorry\n\n/- Section 7.2 -/\n-- 1.\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n-- 2.\n-- Hints: Start with apply List.rec. You may find mul_ne_zero useful\ntheorem prod_nonzero_nonzero : \u2200 (l : List Nat),\n (\u2200 a \u2208 l, a \u2260 0) \u2192 prod l \u2260 0 := sorry\n\n-- 3.\ntheorem rel_prime_iff_no_common_factor (a b : Nat) :\n rel_prime a b \u2194 \u00ac\u2203 (p : Nat), prime p \u2227 p \u2223 a \u2227 p \u2223 b := sorry\n\n-- 4.\ntheorem rel_prime_symm {a b : Nat} (h : rel_prime a b) :\n rel_prime b a := sorry\n\n-- 5.\nlemma in_prime_factorization_iff_prime_factor {a : Nat} {l : List Nat}\n (h1 : prime_factorization a l) (p : Nat) :\n p \u2208 l \u2194 prime_factor p a := sorry\n\n-- 6.\ntheorem Exercise_7_2_5 {a b : Nat} {l m : List Nat}\n (h1 : prime_factorization a l) (h2 : prime_factorization b m) :\n rel_prime a b \u2194 (\u00ac\u2203 (p : Nat), p \u2208 l \u2227 p \u2208 m) := sorry\n\n-- 7.\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\n-- 8.\ntheorem Exercise_7_2_7 {a b a' b' : Nat}\n (h1 : rel_prime a b) (h2 : a' \u2223 a) (h3 : b' \u2223 b) :\n rel_prime a' b' := sorry\n\n-- 9.\ntheorem Exercise_7_2_9 {a b j k : Nat}\n (h1 : gcd a b \u2260 0) (h2 : a = j * gcd a b) (h3 : b = k * gcd a b) :\n rel_prime j k := sorry\n\n-- 10.\ntheorem Exercise_7_2_17a (a b c : Nat) :\n gcd a (b * c) \u2223 gcd a b * gcd a c := sorry\n\n/- Section 7.3 -/\n-- 1.\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n-- 2.\ntheorem Theorem_7_3_6_3 {m : Nat} (X : ZMod m) : X + [0]_m = X := sorry\n\n-- 3.\ntheorem Theorem_7_3_6_4 {m : Nat} (X : ZMod m) :\n \u2203 (Y : ZMod m), X + Y = [0]_m := sorry\n\n-- 4.\ntheorem Exercise_7_3_4a {m : Nat} (Z1 Z2 : ZMod m)\n (h1 : \u2200 (X : ZMod m), X + Z1 = X)\n (h2 : \u2200 (X : ZMod m), X + Z2 = X) : Z1 = Z2 := sorry\n\n-- 5.\ntheorem Exercise_7_3_4b {m : Nat} (X Y1 Y2 : ZMod m)\n (h1 : X + Y1 = [0]_m) (h2 : X + Y2 = [0]_m) : Y1 = Y2 := sorry\n\n-- 6.\ntheorem Theorem_7_3_10 (m a : Nat) (b : Int) :\n \u00ac(\u2191(gcd m a) : Int) \u2223 b \u2192 \u00ac\u2203 (x : Int), a * x \u2261 b (MOD m) := sorry\n\n-- 7.\ntheorem Theorem_7_3_11 (m n : Nat) (a b : Int) (h1 : n \u2260 0) :\n n * a \u2261 n * b (MOD n * m) \u2194 a \u2261 b (MOD m) := sorry\n\n-- 8.\ntheorem Exercise_7_3_16 {m : Nat} {a b : Int} (h : a \u2261 b (MOD m)) :\n \u2200 (n : Nat), a ^ n \u2261 b ^ n (MOD m) := sorry\n\n-- 9.\nexample {m : Nat} [NeZero m] (X : ZMod m) :\n \u2203! (a : Int), 0 \u2264 a \u2227 a < m \u2227 X = [a]_m := sorry\n\n-- 10.\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\n-- 11.\n--Hint: You may find the theorem Int.ofNat_mod_ofNat useful.\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n-- 12.\nlemma congr_iff_mod_eq_Int (m : Nat) (a b : Int) [NeZero m] :\n a \u2261 b (MOD m) \u2194 a % \u2191m = b % \u2191m := sorry\n\n--Hint for next theorem: Use the lemma above,\n--together with the theorems Int.ofNat_mod_ofNat and Nat.cast_inj.\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\n/- Section 7.4 -/\n-- 1.\n--Hint: Use induction.\n--For the base case, compute [a]_m ^ 0 * [1]_m in two ways:\n--by Theorem_7_3_6_7, [a] ^ 0 * [1]_m = [a]_m ^ 0\n--by ring, [a]_m ^ 0 * [1]_m = [1]_m.\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n-- 2.\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\n-- 3.\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\n-- 4.\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n-- 5.\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\n-- 6.\nexample {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n a ^ (phi m + 1) \u2261 a (MOD m) := sorry\n\n-- 7.\ntheorem Like_Exercise_7_4_11 {m a p : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p + 1 = phi m) :\n [a]_m * [a ^ p]_m = [1]_m := sorry\n\n-- 8.\ntheorem Like_Exercise_7_4_12 {m a p q k : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : p = q + (phi m) * k) :\n a ^ p \u2261 a ^ q (MOD m) := sorry\n\n/- Section 7.5 -/\n-- 1.\n--Hint: Use induction.\nlemma num_rp_prime {p : Nat} (h1 : prime p) :\n \u2200 k < p, num_rp_below p (k + 1) = k := sorry\n\n-- 2.\nlemma three_prime : prime 3 := sorry\n\n-- 3.\n--Hint: Use the previous exercise, Exercise_7_2_7, and Theorem_7_4_2.\ntheorem Exercise_7_5_13a (a : Nat) (h1 : rel_prime 561 a) :\n a ^ 560 \u2261 1 (MOD 3) := sorry\n\n-- 4.\n--Hint: Imitate the way Theorem_7_2_2_Int was proven from Theorem_7_2_2.\nlemma Theorem_7_2_3_Int {p : Nat} {a b : Int}\n (h1 : prime p) (h2 : \u2191p \u2223 a * b) : \u2191p \u2223 a \u2228 \u2191p \u2223 b := sorry\n\n-- 5.\n--Hint: Use the previous exercise.\n", "theoremStatement": "theorem Exercise_7_5_14b (n : Nat) (b : Int)\n (h1 : prime n) (h2 : b ^ 2 \u2261 1 (MOD n)) :\n b \u2261 1 (MOD n) \u2228 b \u2261 -1 (MOD n) ", "theoremName": "HTPI.Exercises.Exercise_7_5_14b", "fileCreated": {"commit": "c83e1d1", "date": "2023-04-04"}, "theoremCreated": {"commit": "3199c75", "date": "2023-05-17"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 2104, "tokenPositionInFile": 68258, "theoremPositionInFile": 210}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 16, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.LOption", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.FindMVar", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Util.ForEachExpr", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Rename", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.Tactic.ElabTerm", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.Offset", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Meta.KAbstract", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Meta.Tactic.Assumption", "Lean.Elab.Tactic.Injection", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.CollectFVars", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.ForEachExpr", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Elab.Tactic.Match", "Lean.Meta.Tactic.Rewrite", "Lean.Meta.Eval", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Init.Data.Int.Bitwise", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.BigOperators.Order", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} diff --git a/minictx/mathlib.jsonl b/minictx/mathlib.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..11623355735077a428aa39f16a5f89474880e097 --- /dev/null +++ b/minictx/mathlib.jsonl @@ -0,0 +1,50 @@ +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\nimport Mathlib.Algebra.Polynomial.AlgebraMap\nimport Mathlib.Algebra.Polynomial.Basic\nimport Mathlib.RingTheory.MvPowerSeries.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-!\n# Formal power series (in one variable)\n\nThis file defines (univariate) formal power series\nand develops the basic properties of these objects.\n\nA formal power series is to a polynomial like an infinite sum is to a finite sum.\n\nFormal power series in one variable are defined from multivariate\npower series as `PowerSeries R := MvPowerSeries Unit R`.\n\nThe file sets up the (semi)ring structure on univariate power series.\n\nWe provide the natural inclusion from polynomials to formal power series.\n\nAdditional results can be found in:\n* `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series;\n* `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series,\nand the fact that power series over a local ring form a local ring;\n* `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0,\nand application to the fact that power series over an integral domain\nform an integral domain.\n\n## Implementation notes\n\nBecause of its definition,\n `PowerSeries R := MvPowerSeries Unit R`.\na lot of proofs and properties from the multivariate case\ncan be ported to the single variable case.\nHowever, it means that formal power series are indexed by `Unit \u2192\u2080 \u2115`,\nwhich is of course canonically isomorphic to `\u2115`.\nWe then build some glue to treat formal power series as if they were indexed by `\u2115`.\nOccasionally this leads to proofs that are uglier than expected.\n\n-/\n\nnoncomputable section\n\nopen BigOperators\n\nopen Finset (antidiagonal mem_antidiagonal)\n\n/-- Formal power series over a coefficient type `R` -/\ndef PowerSeries (R : Type*) :=\n MvPowerSeries Unit R\n#align power_series PowerSeries\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection\n\n-- Porting note: not available in Lean 4\n-- local reducible PowerSeries\n\n\n/--\n`R\u27e6X\u27e7` is notation for `PowerSeries R`,\nthe semiring of formal power series in one variable over a semiring `R`.\n-/\nscoped notation:9000 R \"\u27e6X\u27e7\" => PowerSeries R\n\ninstance [Inhabited R] : Inhabited R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Zero R] : Zero R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddMonoid R] : AddMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddGroup R] : AddGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommMonoid R] : AddCommMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommGroup R] : AddCommGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Semiring R] : Semiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommSemiring R] : CommSemiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Ring R] : Ring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommRing R] : CommRing R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Nontrivial R] : Nontrivial R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]\n [IsScalarTower R S A] : IsScalarTower R S A\u27e6X\u27e7 :=\n Pi.isScalarTower\n\ninstance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\nend\n\nsection Semiring\n\nvariable (R) [Semiring R]\n\n/-- The `n`th coefficient of a formal power series. -/\ndef coeff (n : \u2115) : R\u27e6X\u27e7 \u2192\u2097[R] R :=\n MvPowerSeries.coeff R (single () n)\n#align power_series.coeff PowerSeries.coeff\n\n/-- The `n`th monomial with coefficient `a` as formal power series. -/\ndef monomial (n : \u2115) : R \u2192\u2097[R] R\u27e6X\u27e7 :=\n MvPowerSeries.monomial R (single () n)\n#align power_series.monomial PowerSeries.monomial\n\nvariable {R}\n\ntheorem coeff_def {s : Unit \u2192\u2080 \u2115} {n : \u2115} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by\n erw [coeff, \u2190 h, \u2190 Finsupp.unique_single s]\n#align power_series.coeff_def PowerSeries.coeff_def\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\n@[ext]\ntheorem ext {\u03c6 \u03c8 : R\u27e6X\u27e7} (h : \u2200 n, coeff R n \u03c6 = coeff R n \u03c8) : \u03c6 = \u03c8 :=\n MvPowerSeries.ext fun n => by\n rw [\u2190 coeff_def]\n \u00b7 apply h\n rfl\n#align power_series.ext PowerSeries.ext\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\ntheorem ext_iff {\u03c6 \u03c8 : R\u27e6X\u27e7} : \u03c6 = \u03c8 \u2194 \u2200 n, coeff R n \u03c6 = coeff R n \u03c8 :=\n \u27e8fun h n => congr_arg (coeff R n) h, ext\u27e9\n#align power_series.ext_iff PowerSeries.ext_iff\n\ninstance [Subsingleton R] : Subsingleton R\u27e6X\u27e7 := by\n simp only [subsingleton_iff, ext_iff]\n exact fun _ _ _ \u21a6 (subsingleton_iff).mp (by infer_instance) _ _\n\n/-- Constructor for formal power series. -/\ndef mk {R} (f : \u2115 \u2192 R) : R\u27e6X\u27e7 := fun s => f (s ())\n#align power_series.mk PowerSeries.mk\n\n@[simp]\ntheorem coeff_mk (n : \u2115) (f : \u2115 \u2192 R) : coeff R n (mk f) = f n :=\n congr_arg f Finsupp.single_eq_same\n#align power_series.coeff_mk PowerSeries.coeff_mk\n\ntheorem coeff_monomial (m n : \u2115) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=\n calc\n coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _\n _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff]\n\n#align power_series.coeff_monomial PowerSeries.coeff_monomial\n\ntheorem monomial_eq_mk (n : \u2115) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 :=\n ext fun m => by rw [coeff_monomial, coeff_mk]\n#align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk\n\n@[simp]\ntheorem coeff_monomial_same (n : \u2115) (a : R) : coeff R n (monomial R n a) = a :=\n MvPowerSeries.coeff_monomial_same _ _\n#align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same\n\n@[simp]\ntheorem coeff_comp_monomial (n : \u2115) : (coeff R n).comp (monomial R n) = LinearMap.id :=\n LinearMap.ext <| coeff_monomial_same n\n#align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial\n\nvariable (R)\n\n/-- The constant coefficient of a formal power series. -/\ndef constantCoeff : R\u27e6X\u27e7 \u2192+* R :=\n MvPowerSeries.constantCoeff Unit R\n#align power_series.constant_coeff PowerSeries.constantCoeff\n\n/-- The constant formal power series. -/\ndef C : R \u2192+* R\u27e6X\u27e7 :=\n MvPowerSeries.C Unit R\nset_option linter.uppercaseLean3 false in\n#align power_series.C PowerSeries.C\n\nvariable {R}\n\n/-- The variable of the formal power series ring. -/\ndef X : R\u27e6X\u27e7 :=\n MvPowerSeries.X ()\nset_option linter.uppercaseLean3 false in\n#align power_series.X PowerSeries.X\n\ntheorem commute_X (\u03c6 : R\u27e6X\u27e7) : Commute \u03c6 X :=\n MvPowerSeries.commute_X _ _\nset_option linter.uppercaseLean3 false in\n#align power_series.commute_X PowerSeries.commute_X\n\n@[simp]\ntheorem coeff_zero_eq_constantCoeff : \u21d1(coeff R 0) = constantCoeff R := by\n rw [coeff, Finsupp.single_zero]\n rfl\n#align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff\n\ntheorem coeff_zero_eq_constantCoeff_apply (\u03c6 : R\u27e6X\u27e7) : coeff R 0 \u03c6 = constantCoeff R \u03c6 :=\n by rw [coeff_zero_eq_constantCoeff]\n#align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply\n\n@[simp]\ntheorem monomial_zero_eq_C : \u21d1(monomial R 0) = C R := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C]\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C\n\ntheorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply\n\ntheorem coeff_C (n : \u2115) (a : R) : coeff R n (C R a : R\u27e6X\u27e7) = if n = 0 then a else 0 := by\n rw [\u2190 monomial_zero_eq_C_apply, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C PowerSeries.coeff_C\n\n@[simp]\ntheorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by\n rw [coeff_C, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_C PowerSeries.coeff_zero_C\n\ntheorem coeff_ne_zero_C {a : R} {n : \u2115} (h : n \u2260 0) : coeff R n (C R a) = 0 := by\n rw [coeff_C, if_neg h]\n\n@[simp]\ntheorem coeff_succ_C {a : R} {n : \u2115} : coeff R (n + 1) (C R a) = 0 :=\n coeff_ne_zero_C n.succ_ne_zero\n\n", "theoremStatement": "theorem C_injective : Function.Injective (C R)", "theoremName": "C_injective", "fileCreated": {"commit": "3d924717cd", "date": "2023-05-22"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean", "positionMetadata": {"lineInFile": 268, "tokenPositionInFile": 8671, "theoremPositionInFile": 39}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n intro a b H\n have := (ext_iff (\u03c6 := C R a) (\u03c8 := C R b)).mp H 0\n rwa [coeff_zero_C, coeff_zero_C] at this", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 112}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\nimport Mathlib.Algebra.Polynomial.AlgebraMap\nimport Mathlib.Algebra.Polynomial.Basic\nimport Mathlib.RingTheory.MvPowerSeries.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-!\n# Formal power series (in one variable)\n\nThis file defines (univariate) formal power series\nand develops the basic properties of these objects.\n\nA formal power series is to a polynomial like an infinite sum is to a finite sum.\n\nFormal power series in one variable are defined from multivariate\npower series as `PowerSeries R := MvPowerSeries Unit R`.\n\nThe file sets up the (semi)ring structure on univariate power series.\n\nWe provide the natural inclusion from polynomials to formal power series.\n\nAdditional results can be found in:\n* `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series;\n* `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series,\nand the fact that power series over a local ring form a local ring;\n* `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0,\nand application to the fact that power series over an integral domain\nform an integral domain.\n\n## Implementation notes\n\nBecause of its definition,\n `PowerSeries R := MvPowerSeries Unit R`.\na lot of proofs and properties from the multivariate case\ncan be ported to the single variable case.\nHowever, it means that formal power series are indexed by `Unit \u2192\u2080 \u2115`,\nwhich is of course canonically isomorphic to `\u2115`.\nWe then build some glue to treat formal power series as if they were indexed by `\u2115`.\nOccasionally this leads to proofs that are uglier than expected.\n\n-/\n\nnoncomputable section\n\nopen BigOperators\n\nopen Finset (antidiagonal mem_antidiagonal)\n\n/-- Formal power series over a coefficient type `R` -/\ndef PowerSeries (R : Type*) :=\n MvPowerSeries Unit R\n#align power_series PowerSeries\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection\n\n-- Porting note: not available in Lean 4\n-- local reducible PowerSeries\n\n\n/--\n`R\u27e6X\u27e7` is notation for `PowerSeries R`,\nthe semiring of formal power series in one variable over a semiring `R`.\n-/\nscoped notation:9000 R \"\u27e6X\u27e7\" => PowerSeries R\n\ninstance [Inhabited R] : Inhabited R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Zero R] : Zero R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddMonoid R] : AddMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddGroup R] : AddGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommMonoid R] : AddCommMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommGroup R] : AddCommGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Semiring R] : Semiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommSemiring R] : CommSemiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Ring R] : Ring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommRing R] : CommRing R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Nontrivial R] : Nontrivial R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]\n [IsScalarTower R S A] : IsScalarTower R S A\u27e6X\u27e7 :=\n Pi.isScalarTower\n\ninstance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\nend\n\nsection Semiring\n\nvariable (R) [Semiring R]\n\n/-- The `n`th coefficient of a formal power series. -/\ndef coeff (n : \u2115) : R\u27e6X\u27e7 \u2192\u2097[R] R :=\n MvPowerSeries.coeff R (single () n)\n#align power_series.coeff PowerSeries.coeff\n\n/-- The `n`th monomial with coefficient `a` as formal power series. -/\ndef monomial (n : \u2115) : R \u2192\u2097[R] R\u27e6X\u27e7 :=\n MvPowerSeries.monomial R (single () n)\n#align power_series.monomial PowerSeries.monomial\n\nvariable {R}\n\ntheorem coeff_def {s : Unit \u2192\u2080 \u2115} {n : \u2115} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by\n erw [coeff, \u2190 h, \u2190 Finsupp.unique_single s]\n#align power_series.coeff_def PowerSeries.coeff_def\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\n@[ext]\ntheorem ext {\u03c6 \u03c8 : R\u27e6X\u27e7} (h : \u2200 n, coeff R n \u03c6 = coeff R n \u03c8) : \u03c6 = \u03c8 :=\n MvPowerSeries.ext fun n => by\n rw [\u2190 coeff_def]\n \u00b7 apply h\n rfl\n#align power_series.ext PowerSeries.ext\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\ntheorem ext_iff {\u03c6 \u03c8 : R\u27e6X\u27e7} : \u03c6 = \u03c8 \u2194 \u2200 n, coeff R n \u03c6 = coeff R n \u03c8 :=\n \u27e8fun h n => congr_arg (coeff R n) h, ext\u27e9\n#align power_series.ext_iff PowerSeries.ext_iff\n\ninstance [Subsingleton R] : Subsingleton R\u27e6X\u27e7 := by\n simp only [subsingleton_iff, ext_iff]\n exact fun _ _ _ \u21a6 (subsingleton_iff).mp (by infer_instance) _ _\n\n/-- Constructor for formal power series. -/\ndef mk {R} (f : \u2115 \u2192 R) : R\u27e6X\u27e7 := fun s => f (s ())\n#align power_series.mk PowerSeries.mk\n\n@[simp]\ntheorem coeff_mk (n : \u2115) (f : \u2115 \u2192 R) : coeff R n (mk f) = f n :=\n congr_arg f Finsupp.single_eq_same\n#align power_series.coeff_mk PowerSeries.coeff_mk\n\ntheorem coeff_monomial (m n : \u2115) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=\n calc\n coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _\n _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff]\n\n#align power_series.coeff_monomial PowerSeries.coeff_monomial\n\ntheorem monomial_eq_mk (n : \u2115) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 :=\n ext fun m => by rw [coeff_monomial, coeff_mk]\n#align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk\n\n@[simp]\ntheorem coeff_monomial_same (n : \u2115) (a : R) : coeff R n (monomial R n a) = a :=\n MvPowerSeries.coeff_monomial_same _ _\n#align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same\n\n@[simp]\ntheorem coeff_comp_monomial (n : \u2115) : (coeff R n).comp (monomial R n) = LinearMap.id :=\n LinearMap.ext <| coeff_monomial_same n\n#align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial\n\nvariable (R)\n\n/-- The constant coefficient of a formal power series. -/\ndef constantCoeff : R\u27e6X\u27e7 \u2192+* R :=\n MvPowerSeries.constantCoeff Unit R\n#align power_series.constant_coeff PowerSeries.constantCoeff\n\n/-- The constant formal power series. -/\ndef C : R \u2192+* R\u27e6X\u27e7 :=\n MvPowerSeries.C Unit R\nset_option linter.uppercaseLean3 false in\n#align power_series.C PowerSeries.C\n\nvariable {R}\n\n/-- The variable of the formal power series ring. -/\ndef X : R\u27e6X\u27e7 :=\n MvPowerSeries.X ()\nset_option linter.uppercaseLean3 false in\n#align power_series.X PowerSeries.X\n\ntheorem commute_X (\u03c6 : R\u27e6X\u27e7) : Commute \u03c6 X :=\n MvPowerSeries.commute_X _ _\nset_option linter.uppercaseLean3 false in\n#align power_series.commute_X PowerSeries.commute_X\n\n@[simp]\ntheorem coeff_zero_eq_constantCoeff : \u21d1(coeff R 0) = constantCoeff R := by\n rw [coeff, Finsupp.single_zero]\n rfl\n#align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff\n\ntheorem coeff_zero_eq_constantCoeff_apply (\u03c6 : R\u27e6X\u27e7) : coeff R 0 \u03c6 = constantCoeff R \u03c6 :=\n by rw [coeff_zero_eq_constantCoeff]\n#align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply\n\n@[simp]\ntheorem monomial_zero_eq_C : \u21d1(monomial R 0) = C R := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C]\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C\n\ntheorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply\n\ntheorem coeff_C (n : \u2115) (a : R) : coeff R n (C R a : R\u27e6X\u27e7) = if n = 0 then a else 0 := by\n rw [\u2190 monomial_zero_eq_C_apply, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C PowerSeries.coeff_C\n\n@[simp]\ntheorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by\n rw [coeff_C, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_C PowerSeries.coeff_zero_C\n\ntheorem coeff_ne_zero_C {a : R} {n : \u2115} (h : n \u2260 0) : coeff R n (C R a) = 0 := by\n rw [coeff_C, if_neg h]\n\n@[simp]\ntheorem coeff_succ_C {a : R} {n : \u2115} : coeff R (n + 1) (C R a) = 0 :=\n coeff_ne_zero_C n.succ_ne_zero\n\ntheorem C_injective : Function.Injective (C R) := by\n intro a b H\n have := (ext_iff (\u03c6 := C R a) (\u03c8 := C R b)).mp H 0\n rwa [coeff_zero_C, coeff_zero_C] at this\n\n", "theoremStatement": "protected theorem subsingleton_iff : Subsingleton R\u27e6X\u27e7 \u2194 Subsingleton R", "theoremName": "subsingleton_iff", "fileCreated": {"commit": "3d924717cd", "date": "2023-05-22"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean", "positionMetadata": {"lineInFile": 273, "tokenPositionInFile": 8835, "theoremPositionInFile": 40}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n refine \u27e8fun h \u21a6 ?_, fun _ \u21a6 inferInstance\u27e9\n rw [subsingleton_iff] at h \u22a2\n exact fun a b \u21a6 C_injective (h (C R a) (C R b))", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 128}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\nimport Mathlib.Algebra.Polynomial.AlgebraMap\nimport Mathlib.Algebra.Polynomial.Basic\nimport Mathlib.RingTheory.MvPowerSeries.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-!\n# Formal power series (in one variable)\n\nThis file defines (univariate) formal power series\nand develops the basic properties of these objects.\n\nA formal power series is to a polynomial like an infinite sum is to a finite sum.\n\nFormal power series in one variable are defined from multivariate\npower series as `PowerSeries R := MvPowerSeries Unit R`.\n\nThe file sets up the (semi)ring structure on univariate power series.\n\nWe provide the natural inclusion from polynomials to formal power series.\n\nAdditional results can be found in:\n* `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series;\n* `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series,\nand the fact that power series over a local ring form a local ring;\n* `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0,\nand application to the fact that power series over an integral domain\nform an integral domain.\n\n## Implementation notes\n\nBecause of its definition,\n `PowerSeries R := MvPowerSeries Unit R`.\na lot of proofs and properties from the multivariate case\ncan be ported to the single variable case.\nHowever, it means that formal power series are indexed by `Unit \u2192\u2080 \u2115`,\nwhich is of course canonically isomorphic to `\u2115`.\nWe then build some glue to treat formal power series as if they were indexed by `\u2115`.\nOccasionally this leads to proofs that are uglier than expected.\n\n-/\n\nnoncomputable section\n\nopen BigOperators\n\nopen Finset (antidiagonal mem_antidiagonal)\n\n/-- Formal power series over a coefficient type `R` -/\ndef PowerSeries (R : Type*) :=\n MvPowerSeries Unit R\n#align power_series PowerSeries\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection\n\n-- Porting note: not available in Lean 4\n-- local reducible PowerSeries\n\n\n/--\n`R\u27e6X\u27e7` is notation for `PowerSeries R`,\nthe semiring of formal power series in one variable over a semiring `R`.\n-/\nscoped notation:9000 R \"\u27e6X\u27e7\" => PowerSeries R\n\ninstance [Inhabited R] : Inhabited R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Zero R] : Zero R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddMonoid R] : AddMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddGroup R] : AddGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommMonoid R] : AddCommMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommGroup R] : AddCommGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Semiring R] : Semiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommSemiring R] : CommSemiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Ring R] : Ring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommRing R] : CommRing R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Nontrivial R] : Nontrivial R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]\n [IsScalarTower R S A] : IsScalarTower R S A\u27e6X\u27e7 :=\n Pi.isScalarTower\n\ninstance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\nend\n\nsection Semiring\n\nvariable (R) [Semiring R]\n\n/-- The `n`th coefficient of a formal power series. -/\ndef coeff (n : \u2115) : R\u27e6X\u27e7 \u2192\u2097[R] R :=\n MvPowerSeries.coeff R (single () n)\n#align power_series.coeff PowerSeries.coeff\n\n/-- The `n`th monomial with coefficient `a` as formal power series. -/\ndef monomial (n : \u2115) : R \u2192\u2097[R] R\u27e6X\u27e7 :=\n MvPowerSeries.monomial R (single () n)\n#align power_series.monomial PowerSeries.monomial\n\nvariable {R}\n\ntheorem coeff_def {s : Unit \u2192\u2080 \u2115} {n : \u2115} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by\n erw [coeff, \u2190 h, \u2190 Finsupp.unique_single s]\n#align power_series.coeff_def PowerSeries.coeff_def\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\n@[ext]\ntheorem ext {\u03c6 \u03c8 : R\u27e6X\u27e7} (h : \u2200 n, coeff R n \u03c6 = coeff R n \u03c8) : \u03c6 = \u03c8 :=\n MvPowerSeries.ext fun n => by\n rw [\u2190 coeff_def]\n \u00b7 apply h\n rfl\n#align power_series.ext PowerSeries.ext\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\ntheorem ext_iff {\u03c6 \u03c8 : R\u27e6X\u27e7} : \u03c6 = \u03c8 \u2194 \u2200 n, coeff R n \u03c6 = coeff R n \u03c8 :=\n \u27e8fun h n => congr_arg (coeff R n) h, ext\u27e9\n#align power_series.ext_iff PowerSeries.ext_iff\n\ninstance [Subsingleton R] : Subsingleton R\u27e6X\u27e7 := by\n simp only [subsingleton_iff, ext_iff]\n exact fun _ _ _ \u21a6 (subsingleton_iff).mp (by infer_instance) _ _\n\n/-- Constructor for formal power series. -/\ndef mk {R} (f : \u2115 \u2192 R) : R\u27e6X\u27e7 := fun s => f (s ())\n#align power_series.mk PowerSeries.mk\n\n@[simp]\ntheorem coeff_mk (n : \u2115) (f : \u2115 \u2192 R) : coeff R n (mk f) = f n :=\n congr_arg f Finsupp.single_eq_same\n#align power_series.coeff_mk PowerSeries.coeff_mk\n\ntheorem coeff_monomial (m n : \u2115) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=\n calc\n coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _\n _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff]\n\n#align power_series.coeff_monomial PowerSeries.coeff_monomial\n\ntheorem monomial_eq_mk (n : \u2115) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 :=\n ext fun m => by rw [coeff_monomial, coeff_mk]\n#align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk\n\n@[simp]\ntheorem coeff_monomial_same (n : \u2115) (a : R) : coeff R n (monomial R n a) = a :=\n MvPowerSeries.coeff_monomial_same _ _\n#align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same\n\n@[simp]\ntheorem coeff_comp_monomial (n : \u2115) : (coeff R n).comp (monomial R n) = LinearMap.id :=\n LinearMap.ext <| coeff_monomial_same n\n#align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial\n\nvariable (R)\n\n/-- The constant coefficient of a formal power series. -/\ndef constantCoeff : R\u27e6X\u27e7 \u2192+* R :=\n MvPowerSeries.constantCoeff Unit R\n#align power_series.constant_coeff PowerSeries.constantCoeff\n\n/-- The constant formal power series. -/\ndef C : R \u2192+* R\u27e6X\u27e7 :=\n MvPowerSeries.C Unit R\nset_option linter.uppercaseLean3 false in\n#align power_series.C PowerSeries.C\n\nvariable {R}\n\n/-- The variable of the formal power series ring. -/\ndef X : R\u27e6X\u27e7 :=\n MvPowerSeries.X ()\nset_option linter.uppercaseLean3 false in\n#align power_series.X PowerSeries.X\n\ntheorem commute_X (\u03c6 : R\u27e6X\u27e7) : Commute \u03c6 X :=\n MvPowerSeries.commute_X _ _\nset_option linter.uppercaseLean3 false in\n#align power_series.commute_X PowerSeries.commute_X\n\n@[simp]\ntheorem coeff_zero_eq_constantCoeff : \u21d1(coeff R 0) = constantCoeff R := by\n rw [coeff, Finsupp.single_zero]\n rfl\n#align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff\n\ntheorem coeff_zero_eq_constantCoeff_apply (\u03c6 : R\u27e6X\u27e7) : coeff R 0 \u03c6 = constantCoeff R \u03c6 :=\n by rw [coeff_zero_eq_constantCoeff]\n#align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply\n\n@[simp]\ntheorem monomial_zero_eq_C : \u21d1(monomial R 0) = C R := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C]\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C\n\ntheorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply\n\ntheorem coeff_C (n : \u2115) (a : R) : coeff R n (C R a : R\u27e6X\u27e7) = if n = 0 then a else 0 := by\n rw [\u2190 monomial_zero_eq_C_apply, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C PowerSeries.coeff_C\n\n@[simp]\ntheorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by\n rw [coeff_C, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_C PowerSeries.coeff_zero_C\n\ntheorem coeff_ne_zero_C {a : R} {n : \u2115} (h : n \u2260 0) : coeff R n (C R a) = 0 := by\n rw [coeff_C, if_neg h]\n\n@[simp]\ntheorem coeff_succ_C {a : R} {n : \u2115} : coeff R (n + 1) (C R a) = 0 :=\n coeff_ne_zero_C n.succ_ne_zero\n\ntheorem C_injective : Function.Injective (C R) := by\n intro a b H\n have := (ext_iff (\u03c6 := C R a) (\u03c8 := C R b)).mp H 0\n rwa [coeff_zero_C, coeff_zero_C] at this\n\nprotected theorem subsingleton_iff : Subsingleton R\u27e6X\u27e7 \u2194 Subsingleton R := by\n refine \u27e8fun h \u21a6 ?_, fun _ \u21a6 inferInstance\u27e9\n rw [subsingleton_iff] at h \u22a2\n exact fun a b \u21a6 C_injective (h (C R a) (C R b))\n\ntheorem X_eq : (X : R\u27e6X\u27e7) = monomial R 1 1 :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.X_eq PowerSeries.X_eq\n\ntheorem coeff_X (n : \u2115) : coeff R n (X : R\u27e6X\u27e7) = if n = 1 then 1 else 0 := by\n rw [X_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X PowerSeries.coeff_X\n\n@[simp]\ntheorem coeff_zero_X : coeff R 0 (X : R\u27e6X\u27e7) = 0 := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_X PowerSeries.coeff_zero_X\n\n@[simp]\ntheorem coeff_one_X : coeff R 1 (X : R\u27e6X\u27e7) = 1 := by rw [coeff_X, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_one_X PowerSeries.coeff_one_X\n\n@[simp]\ntheorem X_ne_zero [Nontrivial R] : (X : R\u27e6X\u27e7) \u2260 0 := fun H => by\n simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H\nset_option linter.uppercaseLean3 false in\n#align power_series.X_ne_zero PowerSeries.X_ne_zero\n\ntheorem X_pow_eq (n : \u2115) : (X : R\u27e6X\u27e7) ^ n = monomial R n 1 :=\n MvPowerSeries.X_pow_eq _ n\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_eq PowerSeries.X_pow_eq\n\ntheorem coeff_X_pow (m n : \u2115) : coeff R m ((X : R\u27e6X\u27e7) ^ n) = if m = n then 1 else 0 := by\n rw [X_pow_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow PowerSeries.coeff_X_pow\n\n@[simp]\ntheorem coeff_X_pow_self (n : \u2115) : coeff R n ((X : R\u27e6X\u27e7) ^ n) = 1 := by\n rw [coeff_X_pow, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow_self PowerSeries.coeff_X_pow_self\n\n@[simp]\ntheorem coeff_one (n : \u2115) : coeff R n (1 : R\u27e6X\u27e7) = if n = 0 then 1 else 0 :=\n coeff_C n 1\n#align power_series.coeff_one PowerSeries.coeff_one\n\ntheorem coeff_zero_one : coeff R 0 (1 : R\u27e6X\u27e7) = 1 :=\n coeff_zero_C 1\n#align power_series.coeff_zero_one PowerSeries.coeff_zero_one\n\ntheorem coeff_mul (n : \u2115) (\u03c6 \u03c8 : R\u27e6X\u27e7) :\n coeff R n (\u03c6 * \u03c8) = \u2211 p in antidiagonal n, coeff R p.1 \u03c6 * coeff R p.2 \u03c8 := by\n -- `rw` can't see that `PowerSeries = MvPowerSeries Unit`, so use `.trans`\n refine (MvPowerSeries.coeff_mul _ \u03c6 \u03c8).trans ?_\n rw [Finsupp.antidiagonal_single, Finset.sum_map]\n rfl\n#align power_series.coeff_mul PowerSeries.coeff_mul\n\n@[simp]\ntheorem coeff_mul_C (n : \u2115) (\u03c6 : R\u27e6X\u27e7) (a : R) : coeff R n (\u03c6 * C R a) = coeff R n \u03c6 * a :=\n MvPowerSeries.coeff_mul_C _ \u03c6 a\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_mul_C PowerSeries.coeff_mul_C\n\n@[simp]\ntheorem coeff_C_mul (n : \u2115) (\u03c6 : R\u27e6X\u27e7) (a : R) : coeff R n (C R a * \u03c6) = a * coeff R n \u03c6 :=\n MvPowerSeries.coeff_C_mul _ \u03c6 a\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C_mul PowerSeries.coeff_C_mul\n\n@[simp]\ntheorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : \u2115) (\u03c6 : PowerSeries S) (a : R) :\n coeff S n (a \u2022 \u03c6) = a \u2022 coeff S n \u03c6 :=\n rfl\n#align power_series.coeff_smul PowerSeries.coeff_smul\n\ntheorem smul_eq_C_mul (f : R\u27e6X\u27e7) (a : R) : a \u2022 f = C R a * f := by\n ext\n simp\nset_option linter.uppercaseLean3 false in\n#align power_series.smul_eq_C_mul PowerSeries.smul_eq_C_mul\n\n@[simp]\ntheorem coeff_succ_mul_X (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff R (n + 1) (\u03c6 * X) = coeff R n \u03c6 := by\n simp only [coeff, Finsupp.single_add]\n convert \u03c6.coeff_add_mul_monomial (single () n) (single () 1) _\n rw [mul_one]; rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_succ_mul_X PowerSeries.coeff_succ_mul_X\n\n@[simp]\ntheorem coeff_succ_X_mul (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff R (n + 1) (X * \u03c6) = coeff R n \u03c6 := by\n simp only [coeff, Finsupp.single_add, add_comm n 1]\n convert \u03c6.coeff_add_monomial_mul (single () 1) (single () n) _\n rw [one_mul]; rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_succ_X_mul PowerSeries.coeff_succ_X_mul\n\n@[simp]\ntheorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_C PowerSeries.constantCoeff_C\n\n@[simp]\ntheorem constantCoeff_comp_C : (constantCoeff R).comp (C R) = RingHom.id R :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_comp_C PowerSeries.constantCoeff_comp_C\n\n-- Porting note (#10618): simp can prove this.\n-- @[simp]\ntheorem constantCoeff_zero : constantCoeff R 0 = 0 :=\n rfl\n#align power_series.constant_coeff_zero PowerSeries.constantCoeff_zero\n\n-- Porting note (#10618): simp can prove this.\n-- @[simp]\ntheorem constantCoeff_one : constantCoeff R 1 = 1 :=\n rfl\n#align power_series.constant_coeff_one PowerSeries.constantCoeff_one\n\n@[simp]\ntheorem constantCoeff_X : constantCoeff R X = 0 :=\n MvPowerSeries.coeff_zero_X _\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_X PowerSeries.constantCoeff_X\n\ntheorem coeff_zero_mul_X (\u03c6 : R\u27e6X\u27e7) : coeff R 0 (\u03c6 * X) = 0 := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_mul_X PowerSeries.coeff_zero_mul_X\n\ntheorem coeff_zero_X_mul (\u03c6 : R\u27e6X\u27e7) : coeff R 0 (X * \u03c6) = 0 := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_X_mul PowerSeries.coeff_zero_X_mul\n\n", "theoremStatement": "theorem constantCoeff_surj : Function.Surjective (constantCoeff R)", "theoremName": "constantCoeff_surj", "fileCreated": {"commit": "3d924717cd", "date": "2023-05-22"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean", "positionMetadata": {"lineInFile": 417, "tokenPositionInFile": 14246, "theoremPositionInFile": 65}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "fun r => \u27e8(C R) r, constantCoeff_C r\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 37}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\nimport Mathlib.Algebra.Polynomial.AlgebraMap\nimport Mathlib.Algebra.Polynomial.Basic\nimport Mathlib.RingTheory.MvPowerSeries.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-!\n# Formal power series (in one variable)\n\nThis file defines (univariate) formal power series\nand develops the basic properties of these objects.\n\nA formal power series is to a polynomial like an infinite sum is to a finite sum.\n\nFormal power series in one variable are defined from multivariate\npower series as `PowerSeries R := MvPowerSeries Unit R`.\n\nThe file sets up the (semi)ring structure on univariate power series.\n\nWe provide the natural inclusion from polynomials to formal power series.\n\nAdditional results can be found in:\n* `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series;\n* `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series,\nand the fact that power series over a local ring form a local ring;\n* `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0,\nand application to the fact that power series over an integral domain\nform an integral domain.\n\n## Implementation notes\n\nBecause of its definition,\n `PowerSeries R := MvPowerSeries Unit R`.\na lot of proofs and properties from the multivariate case\ncan be ported to the single variable case.\nHowever, it means that formal power series are indexed by `Unit \u2192\u2080 \u2115`,\nwhich is of course canonically isomorphic to `\u2115`.\nWe then build some glue to treat formal power series as if they were indexed by `\u2115`.\nOccasionally this leads to proofs that are uglier than expected.\n\n-/\n\nnoncomputable section\n\nopen BigOperators\n\nopen Finset (antidiagonal mem_antidiagonal)\n\n/-- Formal power series over a coefficient type `R` -/\ndef PowerSeries (R : Type*) :=\n MvPowerSeries Unit R\n#align power_series PowerSeries\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection\n\n-- Porting note: not available in Lean 4\n-- local reducible PowerSeries\n\n\n/--\n`R\u27e6X\u27e7` is notation for `PowerSeries R`,\nthe semiring of formal power series in one variable over a semiring `R`.\n-/\nscoped notation:9000 R \"\u27e6X\u27e7\" => PowerSeries R\n\ninstance [Inhabited R] : Inhabited R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Zero R] : Zero R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddMonoid R] : AddMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddGroup R] : AddGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommMonoid R] : AddCommMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommGroup R] : AddCommGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Semiring R] : Semiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommSemiring R] : CommSemiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Ring R] : Ring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommRing R] : CommRing R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Nontrivial R] : Nontrivial R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]\n [IsScalarTower R S A] : IsScalarTower R S A\u27e6X\u27e7 :=\n Pi.isScalarTower\n\ninstance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\nend\n\nsection Semiring\n\nvariable (R) [Semiring R]\n\n/-- The `n`th coefficient of a formal power series. -/\ndef coeff (n : \u2115) : R\u27e6X\u27e7 \u2192\u2097[R] R :=\n MvPowerSeries.coeff R (single () n)\n#align power_series.coeff PowerSeries.coeff\n\n/-- The `n`th monomial with coefficient `a` as formal power series. -/\ndef monomial (n : \u2115) : R \u2192\u2097[R] R\u27e6X\u27e7 :=\n MvPowerSeries.monomial R (single () n)\n#align power_series.monomial PowerSeries.monomial\n\nvariable {R}\n\ntheorem coeff_def {s : Unit \u2192\u2080 \u2115} {n : \u2115} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by\n erw [coeff, \u2190 h, \u2190 Finsupp.unique_single s]\n#align power_series.coeff_def PowerSeries.coeff_def\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\n@[ext]\ntheorem ext {\u03c6 \u03c8 : R\u27e6X\u27e7} (h : \u2200 n, coeff R n \u03c6 = coeff R n \u03c8) : \u03c6 = \u03c8 :=\n MvPowerSeries.ext fun n => by\n rw [\u2190 coeff_def]\n \u00b7 apply h\n rfl\n#align power_series.ext PowerSeries.ext\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\ntheorem ext_iff {\u03c6 \u03c8 : R\u27e6X\u27e7} : \u03c6 = \u03c8 \u2194 \u2200 n, coeff R n \u03c6 = coeff R n \u03c8 :=\n \u27e8fun h n => congr_arg (coeff R n) h, ext\u27e9\n#align power_series.ext_iff PowerSeries.ext_iff\n\ninstance [Subsingleton R] : Subsingleton R\u27e6X\u27e7 := by\n simp only [subsingleton_iff, ext_iff]\n exact fun _ _ _ \u21a6 (subsingleton_iff).mp (by infer_instance) _ _\n\n/-- Constructor for formal power series. -/\ndef mk {R} (f : \u2115 \u2192 R) : R\u27e6X\u27e7 := fun s => f (s ())\n#align power_series.mk PowerSeries.mk\n\n@[simp]\ntheorem coeff_mk (n : \u2115) (f : \u2115 \u2192 R) : coeff R n (mk f) = f n :=\n congr_arg f Finsupp.single_eq_same\n#align power_series.coeff_mk PowerSeries.coeff_mk\n\ntheorem coeff_monomial (m n : \u2115) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=\n calc\n coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _\n _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff]\n\n#align power_series.coeff_monomial PowerSeries.coeff_monomial\n\ntheorem monomial_eq_mk (n : \u2115) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 :=\n ext fun m => by rw [coeff_monomial, coeff_mk]\n#align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk\n\n@[simp]\ntheorem coeff_monomial_same (n : \u2115) (a : R) : coeff R n (monomial R n a) = a :=\n MvPowerSeries.coeff_monomial_same _ _\n#align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same\n\n@[simp]\ntheorem coeff_comp_monomial (n : \u2115) : (coeff R n).comp (monomial R n) = LinearMap.id :=\n LinearMap.ext <| coeff_monomial_same n\n#align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial\n\nvariable (R)\n\n/-- The constant coefficient of a formal power series. -/\ndef constantCoeff : R\u27e6X\u27e7 \u2192+* R :=\n MvPowerSeries.constantCoeff Unit R\n#align power_series.constant_coeff PowerSeries.constantCoeff\n\n/-- The constant formal power series. -/\ndef C : R \u2192+* R\u27e6X\u27e7 :=\n MvPowerSeries.C Unit R\nset_option linter.uppercaseLean3 false in\n#align power_series.C PowerSeries.C\n\nvariable {R}\n\n/-- The variable of the formal power series ring. -/\ndef X : R\u27e6X\u27e7 :=\n MvPowerSeries.X ()\nset_option linter.uppercaseLean3 false in\n#align power_series.X PowerSeries.X\n\ntheorem commute_X (\u03c6 : R\u27e6X\u27e7) : Commute \u03c6 X :=\n MvPowerSeries.commute_X _ _\nset_option linter.uppercaseLean3 false in\n#align power_series.commute_X PowerSeries.commute_X\n\n@[simp]\ntheorem coeff_zero_eq_constantCoeff : \u21d1(coeff R 0) = constantCoeff R := by\n rw [coeff, Finsupp.single_zero]\n rfl\n#align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff\n\ntheorem coeff_zero_eq_constantCoeff_apply (\u03c6 : R\u27e6X\u27e7) : coeff R 0 \u03c6 = constantCoeff R \u03c6 :=\n by rw [coeff_zero_eq_constantCoeff]\n#align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply\n\n@[simp]\ntheorem monomial_zero_eq_C : \u21d1(monomial R 0) = C R := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C]\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C\n\ntheorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply\n\ntheorem coeff_C (n : \u2115) (a : R) : coeff R n (C R a : R\u27e6X\u27e7) = if n = 0 then a else 0 := by\n rw [\u2190 monomial_zero_eq_C_apply, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C PowerSeries.coeff_C\n\n@[simp]\ntheorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by\n rw [coeff_C, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_C PowerSeries.coeff_zero_C\n\ntheorem coeff_ne_zero_C {a : R} {n : \u2115} (h : n \u2260 0) : coeff R n (C R a) = 0 := by\n rw [coeff_C, if_neg h]\n\n@[simp]\ntheorem coeff_succ_C {a : R} {n : \u2115} : coeff R (n + 1) (C R a) = 0 :=\n coeff_ne_zero_C n.succ_ne_zero\n\ntheorem C_injective : Function.Injective (C R) := by\n intro a b H\n have := (ext_iff (\u03c6 := C R a) (\u03c8 := C R b)).mp H 0\n rwa [coeff_zero_C, coeff_zero_C] at this\n\nprotected theorem subsingleton_iff : Subsingleton R\u27e6X\u27e7 \u2194 Subsingleton R := by\n refine \u27e8fun h \u21a6 ?_, fun _ \u21a6 inferInstance\u27e9\n rw [subsingleton_iff] at h \u22a2\n exact fun a b \u21a6 C_injective (h (C R a) (C R b))\n\ntheorem X_eq : (X : R\u27e6X\u27e7) = monomial R 1 1 :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.X_eq PowerSeries.X_eq\n\ntheorem coeff_X (n : \u2115) : coeff R n (X : R\u27e6X\u27e7) = if n = 1 then 1 else 0 := by\n rw [X_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X PowerSeries.coeff_X\n\n@[simp]\ntheorem coeff_zero_X : coeff R 0 (X : R\u27e6X\u27e7) = 0 := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_X PowerSeries.coeff_zero_X\n\n@[simp]\ntheorem coeff_one_X : coeff R 1 (X : R\u27e6X\u27e7) = 1 := by rw [coeff_X, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_one_X PowerSeries.coeff_one_X\n\n@[simp]\ntheorem X_ne_zero [Nontrivial R] : (X : R\u27e6X\u27e7) \u2260 0 := fun H => by\n simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H\nset_option linter.uppercaseLean3 false in\n#align power_series.X_ne_zero PowerSeries.X_ne_zero\n\ntheorem X_pow_eq (n : \u2115) : (X : R\u27e6X\u27e7) ^ n = monomial R n 1 :=\n MvPowerSeries.X_pow_eq _ n\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_eq PowerSeries.X_pow_eq\n\ntheorem coeff_X_pow (m n : \u2115) : coeff R m ((X : R\u27e6X\u27e7) ^ n) = if m = n then 1 else 0 := by\n rw [X_pow_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow PowerSeries.coeff_X_pow\n\n@[simp]\ntheorem coeff_X_pow_self (n : \u2115) : coeff R n ((X : R\u27e6X\u27e7) ^ n) = 1 := by\n rw [coeff_X_pow, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow_self PowerSeries.coeff_X_pow_self\n\n@[simp]\ntheorem coeff_one (n : \u2115) : coeff R n (1 : R\u27e6X\u27e7) = if n = 0 then 1 else 0 :=\n coeff_C n 1\n#align power_series.coeff_one PowerSeries.coeff_one\n\ntheorem coeff_zero_one : coeff R 0 (1 : R\u27e6X\u27e7) = 1 :=\n coeff_zero_C 1\n#align power_series.coeff_zero_one PowerSeries.coeff_zero_one\n\ntheorem coeff_mul (n : \u2115) (\u03c6 \u03c8 : R\u27e6X\u27e7) :\n coeff R n (\u03c6 * \u03c8) = \u2211 p in antidiagonal n, coeff R p.1 \u03c6 * coeff R p.2 \u03c8 := by\n -- `rw` can't see that `PowerSeries = MvPowerSeries Unit`, so use `.trans`\n refine (MvPowerSeries.coeff_mul _ \u03c6 \u03c8).trans ?_\n rw [Finsupp.antidiagonal_single, Finset.sum_map]\n rfl\n#align power_series.coeff_mul PowerSeries.coeff_mul\n\n@[simp]\ntheorem coeff_mul_C (n : \u2115) (\u03c6 : R\u27e6X\u27e7) (a : R) : coeff R n (\u03c6 * C R a) = coeff R n \u03c6 * a :=\n MvPowerSeries.coeff_mul_C _ \u03c6 a\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_mul_C PowerSeries.coeff_mul_C\n\n@[simp]\ntheorem coeff_C_mul (n : \u2115) (\u03c6 : R\u27e6X\u27e7) (a : R) : coeff R n (C R a * \u03c6) = a * coeff R n \u03c6 :=\n MvPowerSeries.coeff_C_mul _ \u03c6 a\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C_mul PowerSeries.coeff_C_mul\n\n@[simp]\ntheorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : \u2115) (\u03c6 : PowerSeries S) (a : R) :\n coeff S n (a \u2022 \u03c6) = a \u2022 coeff S n \u03c6 :=\n rfl\n#align power_series.coeff_smul PowerSeries.coeff_smul\n\ntheorem smul_eq_C_mul (f : R\u27e6X\u27e7) (a : R) : a \u2022 f = C R a * f := by\n ext\n simp\nset_option linter.uppercaseLean3 false in\n#align power_series.smul_eq_C_mul PowerSeries.smul_eq_C_mul\n\n@[simp]\ntheorem coeff_succ_mul_X (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff R (n + 1) (\u03c6 * X) = coeff R n \u03c6 := by\n simp only [coeff, Finsupp.single_add]\n convert \u03c6.coeff_add_mul_monomial (single () n) (single () 1) _\n rw [mul_one]; rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_succ_mul_X PowerSeries.coeff_succ_mul_X\n\n@[simp]\ntheorem coeff_succ_X_mul (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff R (n + 1) (X * \u03c6) = coeff R n \u03c6 := by\n simp only [coeff, Finsupp.single_add, add_comm n 1]\n convert \u03c6.coeff_add_monomial_mul (single () 1) (single () n) _\n rw [one_mul]; rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_succ_X_mul PowerSeries.coeff_succ_X_mul\n\n@[simp]\ntheorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_C PowerSeries.constantCoeff_C\n\n@[simp]\ntheorem constantCoeff_comp_C : (constantCoeff R).comp (C R) = RingHom.id R :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_comp_C PowerSeries.constantCoeff_comp_C\n\n-- Porting note (#10618): simp can prove this.\n-- @[simp]\ntheorem constantCoeff_zero : constantCoeff R 0 = 0 :=\n rfl\n#align power_series.constant_coeff_zero PowerSeries.constantCoeff_zero\n\n-- Porting note (#10618): simp can prove this.\n-- @[simp]\ntheorem constantCoeff_one : constantCoeff R 1 = 1 :=\n rfl\n#align power_series.constant_coeff_one PowerSeries.constantCoeff_one\n\n@[simp]\ntheorem constantCoeff_X : constantCoeff R X = 0 :=\n MvPowerSeries.coeff_zero_X _\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_X PowerSeries.constantCoeff_X\n\ntheorem coeff_zero_mul_X (\u03c6 : R\u27e6X\u27e7) : coeff R 0 (\u03c6 * X) = 0 := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_mul_X PowerSeries.coeff_zero_mul_X\n\ntheorem coeff_zero_X_mul (\u03c6 : R\u27e6X\u27e7) : coeff R 0 (X * \u03c6) = 0 := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_X_mul PowerSeries.coeff_zero_X_mul\n\ntheorem constantCoeff_surj : Function.Surjective (constantCoeff R) :=\n fun r => \u27e8(C R) r, constantCoeff_C r\u27e9\n\n-- The following section duplicates the API of `Data.Polynomial.Coeff` and should attempt to keep\n-- up to date with that\nsection\n\ntheorem coeff_C_mul_X_pow (x : R) (k n : \u2115) :\n coeff R n (C R x * X ^ k : R\u27e6X\u27e7) = if n = k then x else 0 := by\n simp [X_pow_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C_mul_X_pow PowerSeries.coeff_C_mul_X_pow\n\n@[simp]\ntheorem coeff_mul_X_pow (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R (d + n) (p * X ^ n) = coeff R d p := by\n rw [coeff_mul, Finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one]\n \u00b7 rintro \u27e8i, j\u27e9 h1 h2\n rw [coeff_X_pow, if_neg, mul_zero]\n rintro rfl\n apply h2\n rw [mem_antidiagonal, add_right_cancel_iff] at h1\n subst h1\n rfl\n \u00b7 exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_mul_X_pow PowerSeries.coeff_mul_X_pow\n\n@[simp]\ntheorem coeff_X_pow_mul (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R (d + n) (X ^ n * p) = coeff R d p := by\n rw [coeff_mul, Finset.sum_eq_single (n, d), coeff_X_pow, if_pos rfl, one_mul]\n \u00b7 rintro \u27e8i, j\u27e9 h1 h2\n rw [coeff_X_pow, if_neg, zero_mul]\n rintro rfl\n apply h2\n rw [mem_antidiagonal, add_comm, add_right_cancel_iff] at h1\n subst h1\n rfl\n \u00b7 rw [add_comm]\n exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow_mul PowerSeries.coeff_X_pow_mul\n\ntheorem coeff_mul_X_pow' (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R d (p * X ^ n) = ite (n \u2264 d) (coeff R (d - n) p) 0 := by\n split_ifs with h\n \u00b7 rw [\u2190 tsub_add_cancel_of_le h, coeff_mul_X_pow, add_tsub_cancel_right]\n \u00b7 refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _)\n rw [coeff_X_pow, if_neg, mul_zero]\n exact ((le_of_add_le_right (mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_mul_X_pow' PowerSeries.coeff_mul_X_pow'\n\ntheorem coeff_X_pow_mul' (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R d (X ^ n * p) = ite (n \u2264 d) (coeff R (d - n) p) 0 := by\n split_ifs with h\n \u00b7 rw [\u2190 tsub_add_cancel_of_le h, coeff_X_pow_mul]\n simp\n \u00b7 refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _)\n rw [coeff_X_pow, if_neg, zero_mul]\n have := mem_antidiagonal.mp hx\n rw [add_comm] at this\n exact ((le_of_add_le_right this.le).trans_lt <| not_le.mp h).ne\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow_mul' PowerSeries.coeff_X_pow_mul'\n\nend\n\n/-- If a formal power series is invertible, then so is its constant coefficient. -/\ntheorem isUnit_constantCoeff (\u03c6 : R\u27e6X\u27e7) (h : IsUnit \u03c6) : IsUnit (constantCoeff R \u03c6) :=\n MvPowerSeries.isUnit_constantCoeff \u03c6 h\n#align power_series.is_unit_constant_coeff PowerSeries.isUnit_constantCoeff\n\n/-- Split off the constant coefficient. -/\ntheorem eq_shift_mul_X_add_const (\u03c6 : R\u27e6X\u27e7) :\n \u03c6 = (mk fun p => coeff R (p + 1) \u03c6) * X + C R (constantCoeff R \u03c6) := by\n ext (_ | n)\n \u00b7 simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X,\n mul_zero, coeff_zero_C, zero_add]\n \u00b7 simp only [coeff_succ_mul_X, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero,\n if_false, add_zero]\nset_option linter.uppercaseLean3 false in\n#align power_series.eq_shift_mul_X_add_const PowerSeries.eq_shift_mul_X_add_const\n\n/-- Split off the constant coefficient. -/\ntheorem eq_X_mul_shift_add_const (\u03c6 : R\u27e6X\u27e7) :\n \u03c6 = (X * mk fun p => coeff R (p + 1) \u03c6) + C R (constantCoeff R \u03c6) := by\n ext (_ | n)\n \u00b7 simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X,\n zero_mul, coeff_zero_C, zero_add]\n \u00b7 simp only [coeff_succ_X_mul, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero,\n if_false, add_zero]\nset_option linter.uppercaseLean3 false in\n#align power_series.eq_X_mul_shift_add_const PowerSeries.eq_X_mul_shift_add_const\n\nsection Map\n\nvariable {S : Type*} {T : Type*} [Semiring S] [Semiring T]\nvariable (f : R \u2192+* S) (g : S \u2192+* T)\n\n/-- The map between formal power series induced by a map on the coefficients. -/\ndef map : R\u27e6X\u27e7 \u2192+* S\u27e6X\u27e7 :=\n MvPowerSeries.map _ f\n#align power_series.map PowerSeries.map\n\n@[simp]\ntheorem map_id : (map (RingHom.id R) : R\u27e6X\u27e7 \u2192 R\u27e6X\u27e7) = id :=\n rfl\n#align power_series.map_id PowerSeries.map_id\n\ntheorem map_comp : map (g.comp f) = (map g).comp (map f) :=\n rfl\n#align power_series.map_comp PowerSeries.map_comp\n\n@[simp]\ntheorem coeff_map (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff S n (map f \u03c6) = f (coeff R n \u03c6) :=\n rfl\n#align power_series.coeff_map PowerSeries.coeff_map\n\n@[simp]\ntheorem map_C (r : R) : map f (C _ r) = C _ (f r) := by\n ext\n simp [coeff_C, apply_ite f]\nset_option linter.uppercaseLean3 false in\n#align power_series.map_C PowerSeries.map_C\n\n@[simp]\ntheorem map_X : map f X = X := by\n ext\n simp [coeff_X, apply_ite f]\nset_option linter.uppercaseLean3 false in\n#align power_series.map_X PowerSeries.map_X\n\nend Map\n\ntheorem X_pow_dvd_iff {n : \u2115} {\u03c6 : R\u27e6X\u27e7} :\n (X : R\u27e6X\u27e7) ^ n \u2223 \u03c6 \u2194 \u2200 m, m < n \u2192 coeff R m \u03c6 = 0 := by\n convert@MvPowerSeries.X_pow_dvd_iff Unit R _ () n \u03c6\n constructor <;> intro h m hm\n \u00b7 rw [Finsupp.unique_single m]\n convert h _ hm\n \u00b7 apply h\n simpa only [Finsupp.single_eq_same] using hm\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_dvd_iff PowerSeries.X_pow_dvd_iff\n\ntheorem X_dvd_iff {\u03c6 : R\u27e6X\u27e7} : (X : R\u27e6X\u27e7) \u2223 \u03c6 \u2194 constantCoeff R \u03c6 = 0 := by\n rw [\u2190 pow_one (X : R\u27e6X\u27e7), X_pow_dvd_iff, \u2190 coeff_zero_eq_constantCoeff_apply]\n constructor <;> intro h\n \u00b7 exact h 0 zero_lt_one\n \u00b7 intro m hm\n rwa [Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ hm)]\nset_option linter.uppercaseLean3 false in\n#align power_series.X_dvd_iff PowerSeries.X_dvd_iff\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R]\n\nopen Finset Nat\n\n/-- The ring homomorphism taking a power series `f(X)` to `f(aX)`. -/\nnoncomputable def rescale (a : R) : R\u27e6X\u27e7 \u2192+* R\u27e6X\u27e7 where\n toFun f := PowerSeries.mk fun n => a ^ n * PowerSeries.coeff R n f\n map_zero' := by\n ext\n simp only [LinearMap.map_zero, PowerSeries.coeff_mk, mul_zero]\n map_one' := by\n ext1\n simp only [mul_boole, PowerSeries.coeff_mk, PowerSeries.coeff_one]\n split_ifs with h\n \u00b7 rw [h, pow_zero a]\n rfl\n map_add' := by\n intros\n ext\n dsimp only\n exact mul_add _ _ _\n map_mul' f g := by\n ext\n rw [PowerSeries.coeff_mul, PowerSeries.coeff_mk, PowerSeries.coeff_mul, Finset.mul_sum]\n apply sum_congr rfl\n simp only [coeff_mk, Prod.forall, mem_antidiagonal]\n intro b c H\n rw [\u2190 H, pow_add, mul_mul_mul_comm]\n#align power_series.rescale PowerSeries.rescale\n\n@[simp]\ntheorem coeff_rescale (f : R\u27e6X\u27e7) (a : R) (n : \u2115) :\n coeff R n (rescale a f) = a ^ n * coeff R n f :=\n coeff_mk n (fun n \u21a6 a ^ n * (coeff R n) f)\n#align power_series.coeff_rescale PowerSeries.coeff_rescale\n\n@[simp]\ntheorem rescale_zero : rescale 0 = (C R).comp (constantCoeff R) := by\n ext x n\n simp only [Function.comp_apply, RingHom.coe_comp, rescale, RingHom.coe_mk,\n PowerSeries.coeff_mk _ _, coeff_C]\n split_ifs with h <;> simp [h]\n#align power_series.rescale_zero PowerSeries.rescale_zero\n\ntheorem rescale_zero_apply : rescale 0 X = C R (constantCoeff R X) := by simp\n#align power_series.rescale_zero_apply PowerSeries.rescale_zero_apply\n\n@[simp]\ntheorem rescale_one : rescale 1 = RingHom.id R\u27e6X\u27e7 := by\n ext\n simp only [coeff_rescale, one_pow, one_mul, RingHom.id_apply]\n#align power_series.rescale_one PowerSeries.rescale_one\n\ntheorem rescale_mk (f : \u2115 \u2192 R) (a : R) : rescale a (mk f) = mk fun n : \u2115 => a ^ n * f n := by\n ext\n rw [coeff_rescale, coeff_mk, coeff_mk]\n#align power_series.rescale_mk PowerSeries.rescale_mk\n\ntheorem rescale_rescale (f : R\u27e6X\u27e7) (a b : R) :\n rescale b (rescale a f) = rescale (a * b) f := by\n ext n\n simp_rw [coeff_rescale]\n rw [mul_pow, mul_comm _ (b ^ n), mul_assoc]\n#align power_series.rescale_rescale PowerSeries.rescale_rescale\n\ntheorem rescale_mul (a b : R) : rescale (a * b) = (rescale b).comp (rescale a) := by\n ext\n simp [\u2190 rescale_rescale]\n#align power_series.rescale_mul PowerSeries.rescale_mul\n\nend CommSemiring\n\nsection CommSemiring\n\nopen Finset.HasAntidiagonal Finset\n\nvariable {R : Type*} [CommSemiring R] {\u03b9 : Type*} [DecidableEq \u03b9]\n\n/-- Coefficients of a product of power series -/\ntheorem coeff_prod (f : \u03b9 \u2192 PowerSeries R) (d : \u2115) (s : Finset \u03b9) :\n coeff R d (\u220f j in s, f j) = \u2211 l in piAntidiagonal s d, \u220f i in s, coeff R (l i) (f i) := by\n simp only [coeff]\n convert MvPowerSeries.coeff_prod _ _ _\n rw [\u2190 AddEquiv.finsuppUnique_symm d, \u2190 mapRange_piAntidiagonal_eq, sum_map, sum_congr rfl]\n intro x _\n apply prod_congr rfl\n intro i _\n congr 2\n simp only [AddEquiv.toEquiv_eq_coe, Finsupp.mapRange.addEquiv_toEquiv, AddEquiv.toEquiv_symm,\n Equiv.coe_toEmbedding, Finsupp.mapRange.equiv_apply, AddEquiv.coe_toEquiv_symm,\n Finsupp.mapRange_apply, AddEquiv.finsuppUnique_symm]\n\nend CommSemiring\n\nsection CommRing\n\nvariable {A : Type*} [CommRing A]\n\n", "theoremStatement": "theorem not_isField : \u00acIsField A\u27e6X\u27e7", "theoremName": "not_isField", "fileCreated": {"commit": "3d924717cd", "date": "2023-05-22"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean", "positionMetadata": {"lineInFile": 675, "tokenPositionInFile": 23335, "theoremPositionInFile": 91}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n by_cases hA : Subsingleton A\n \u00b7 exact not_isField_of_subsingleton _\n \u00b7 nontriviality A\n rw [Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top]\n use Ideal.span {X}\n constructor\n \u00b7 rw [bot_lt_iff_ne_bot, Ne.def, Ideal.span_singleton_eq_bot]\n exact X_ne_zero\n \u00b7 rw [lt_top_iff_ne_top, Ne.def, Ideal.eq_top_iff_one, Ideal.mem_span_singleton,\n X_dvd_iff, constantCoeff_one]\n exact one_ne_zero", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 428}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\nimport Mathlib.Algebra.Polynomial.AlgebraMap\nimport Mathlib.Algebra.Polynomial.Basic\nimport Mathlib.RingTheory.MvPowerSeries.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-!\n# Formal power series (in one variable)\n\nThis file defines (univariate) formal power series\nand develops the basic properties of these objects.\n\nA formal power series is to a polynomial like an infinite sum is to a finite sum.\n\nFormal power series in one variable are defined from multivariate\npower series as `PowerSeries R := MvPowerSeries Unit R`.\n\nThe file sets up the (semi)ring structure on univariate power series.\n\nWe provide the natural inclusion from polynomials to formal power series.\n\nAdditional results can be found in:\n* `Mathlib.RingTheory.PowerSeries.Trunc`, truncation of power series;\n* `Mathlib.RingTheory.PowerSeries.Inverse`, about inverses of power series,\nand the fact that power series over a local ring form a local ring;\n* `Mathlib.RingTheory.PowerSeries.Order`, the order of a power series at 0,\nand application to the fact that power series over an integral domain\nform an integral domain.\n\n## Implementation notes\n\nBecause of its definition,\n `PowerSeries R := MvPowerSeries Unit R`.\na lot of proofs and properties from the multivariate case\ncan be ported to the single variable case.\nHowever, it means that formal power series are indexed by `Unit \u2192\u2080 \u2115`,\nwhich is of course canonically isomorphic to `\u2115`.\nWe then build some glue to treat formal power series as if they were indexed by `\u2115`.\nOccasionally this leads to proofs that are uglier than expected.\n\n-/\n\nnoncomputable section\n\nopen BigOperators\n\nopen Finset (antidiagonal mem_antidiagonal)\n\n/-- Formal power series over a coefficient type `R` -/\ndef PowerSeries (R : Type*) :=\n MvPowerSeries Unit R\n#align power_series PowerSeries\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection\n\n-- Porting note: not available in Lean 4\n-- local reducible PowerSeries\n\n\n/--\n`R\u27e6X\u27e7` is notation for `PowerSeries R`,\nthe semiring of formal power series in one variable over a semiring `R`.\n-/\nscoped notation:9000 R \"\u27e6X\u27e7\" => PowerSeries R\n\ninstance [Inhabited R] : Inhabited R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Zero R] : Zero R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddMonoid R] : AddMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddGroup R] : AddGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommMonoid R] : AddCommMonoid R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [AddCommGroup R] : AddCommGroup R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Semiring R] : Semiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommSemiring R] : CommSemiring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Ring R] : Ring R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [CommRing R] : CommRing R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance [Nontrivial R] : Nontrivial R\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\ninstance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]\n [IsScalarTower R S A] : IsScalarTower R S A\u27e6X\u27e7 :=\n Pi.isScalarTower\n\ninstance {A} [Semiring A] [CommSemiring R] [Algebra R A] : Algebra R A\u27e6X\u27e7 := by\n dsimp only [PowerSeries]\n infer_instance\n\nend\n\nsection Semiring\n\nvariable (R) [Semiring R]\n\n/-- The `n`th coefficient of a formal power series. -/\ndef coeff (n : \u2115) : R\u27e6X\u27e7 \u2192\u2097[R] R :=\n MvPowerSeries.coeff R (single () n)\n#align power_series.coeff PowerSeries.coeff\n\n/-- The `n`th monomial with coefficient `a` as formal power series. -/\ndef monomial (n : \u2115) : R \u2192\u2097[R] R\u27e6X\u27e7 :=\n MvPowerSeries.monomial R (single () n)\n#align power_series.monomial PowerSeries.monomial\n\nvariable {R}\n\ntheorem coeff_def {s : Unit \u2192\u2080 \u2115} {n : \u2115} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by\n erw [coeff, \u2190 h, \u2190 Finsupp.unique_single s]\n#align power_series.coeff_def PowerSeries.coeff_def\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\n@[ext]\ntheorem ext {\u03c6 \u03c8 : R\u27e6X\u27e7} (h : \u2200 n, coeff R n \u03c6 = coeff R n \u03c8) : \u03c6 = \u03c8 :=\n MvPowerSeries.ext fun n => by\n rw [\u2190 coeff_def]\n \u00b7 apply h\n rfl\n#align power_series.ext PowerSeries.ext\n\n/-- Two formal power series are equal if all their coefficients are equal. -/\ntheorem ext_iff {\u03c6 \u03c8 : R\u27e6X\u27e7} : \u03c6 = \u03c8 \u2194 \u2200 n, coeff R n \u03c6 = coeff R n \u03c8 :=\n \u27e8fun h n => congr_arg (coeff R n) h, ext\u27e9\n#align power_series.ext_iff PowerSeries.ext_iff\n\ninstance [Subsingleton R] : Subsingleton R\u27e6X\u27e7 := by\n simp only [subsingleton_iff, ext_iff]\n exact fun _ _ _ \u21a6 (subsingleton_iff).mp (by infer_instance) _ _\n\n/-- Constructor for formal power series. -/\ndef mk {R} (f : \u2115 \u2192 R) : R\u27e6X\u27e7 := fun s => f (s ())\n#align power_series.mk PowerSeries.mk\n\n@[simp]\ntheorem coeff_mk (n : \u2115) (f : \u2115 \u2192 R) : coeff R n (mk f) = f n :=\n congr_arg f Finsupp.single_eq_same\n#align power_series.coeff_mk PowerSeries.coeff_mk\n\ntheorem coeff_monomial (m n : \u2115) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=\n calc\n coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _\n _ = if m = n then a else 0 := by simp only [Finsupp.unique_single_eq_iff]\n\n#align power_series.coeff_monomial PowerSeries.coeff_monomial\n\ntheorem monomial_eq_mk (n : \u2115) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 :=\n ext fun m => by rw [coeff_monomial, coeff_mk]\n#align power_series.monomial_eq_mk PowerSeries.monomial_eq_mk\n\n@[simp]\ntheorem coeff_monomial_same (n : \u2115) (a : R) : coeff R n (monomial R n a) = a :=\n MvPowerSeries.coeff_monomial_same _ _\n#align power_series.coeff_monomial_same PowerSeries.coeff_monomial_same\n\n@[simp]\ntheorem coeff_comp_monomial (n : \u2115) : (coeff R n).comp (monomial R n) = LinearMap.id :=\n LinearMap.ext <| coeff_monomial_same n\n#align power_series.coeff_comp_monomial PowerSeries.coeff_comp_monomial\n\nvariable (R)\n\n/-- The constant coefficient of a formal power series. -/\ndef constantCoeff : R\u27e6X\u27e7 \u2192+* R :=\n MvPowerSeries.constantCoeff Unit R\n#align power_series.constant_coeff PowerSeries.constantCoeff\n\n/-- The constant formal power series. -/\ndef C : R \u2192+* R\u27e6X\u27e7 :=\n MvPowerSeries.C Unit R\nset_option linter.uppercaseLean3 false in\n#align power_series.C PowerSeries.C\n\nvariable {R}\n\n/-- The variable of the formal power series ring. -/\ndef X : R\u27e6X\u27e7 :=\n MvPowerSeries.X ()\nset_option linter.uppercaseLean3 false in\n#align power_series.X PowerSeries.X\n\ntheorem commute_X (\u03c6 : R\u27e6X\u27e7) : Commute \u03c6 X :=\n MvPowerSeries.commute_X _ _\nset_option linter.uppercaseLean3 false in\n#align power_series.commute_X PowerSeries.commute_X\n\n@[simp]\ntheorem coeff_zero_eq_constantCoeff : \u21d1(coeff R 0) = constantCoeff R := by\n rw [coeff, Finsupp.single_zero]\n rfl\n#align power_series.coeff_zero_eq_constant_coeff PowerSeries.coeff_zero_eq_constantCoeff\n\ntheorem coeff_zero_eq_constantCoeff_apply (\u03c6 : R\u27e6X\u27e7) : coeff R 0 \u03c6 = constantCoeff R \u03c6 :=\n by rw [coeff_zero_eq_constantCoeff]\n#align power_series.coeff_zero_eq_constant_coeff_apply PowerSeries.coeff_zero_eq_constantCoeff_apply\n\n@[simp]\ntheorem monomial_zero_eq_C : \u21d1(monomial R 0) = C R := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C]\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C PowerSeries.monomial_zero_eq_C\n\ntheorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.monomial_zero_eq_C_apply PowerSeries.monomial_zero_eq_C_apply\n\ntheorem coeff_C (n : \u2115) (a : R) : coeff R n (C R a : R\u27e6X\u27e7) = if n = 0 then a else 0 := by\n rw [\u2190 monomial_zero_eq_C_apply, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C PowerSeries.coeff_C\n\n@[simp]\ntheorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by\n rw [coeff_C, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_C PowerSeries.coeff_zero_C\n\ntheorem coeff_ne_zero_C {a : R} {n : \u2115} (h : n \u2260 0) : coeff R n (C R a) = 0 := by\n rw [coeff_C, if_neg h]\n\n@[simp]\ntheorem coeff_succ_C {a : R} {n : \u2115} : coeff R (n + 1) (C R a) = 0 :=\n coeff_ne_zero_C n.succ_ne_zero\n\ntheorem C_injective : Function.Injective (C R) := by\n intro a b H\n have := (ext_iff (\u03c6 := C R a) (\u03c8 := C R b)).mp H 0\n rwa [coeff_zero_C, coeff_zero_C] at this\n\nprotected theorem subsingleton_iff : Subsingleton R\u27e6X\u27e7 \u2194 Subsingleton R := by\n refine \u27e8fun h \u21a6 ?_, fun _ \u21a6 inferInstance\u27e9\n rw [subsingleton_iff] at h \u22a2\n exact fun a b \u21a6 C_injective (h (C R a) (C R b))\n\ntheorem X_eq : (X : R\u27e6X\u27e7) = monomial R 1 1 :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.X_eq PowerSeries.X_eq\n\ntheorem coeff_X (n : \u2115) : coeff R n (X : R\u27e6X\u27e7) = if n = 1 then 1 else 0 := by\n rw [X_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X PowerSeries.coeff_X\n\n@[simp]\ntheorem coeff_zero_X : coeff R 0 (X : R\u27e6X\u27e7) = 0 := by\n -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644\n erw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_X PowerSeries.coeff_zero_X\n\n@[simp]\ntheorem coeff_one_X : coeff R 1 (X : R\u27e6X\u27e7) = 1 := by rw [coeff_X, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_one_X PowerSeries.coeff_one_X\n\n@[simp]\ntheorem X_ne_zero [Nontrivial R] : (X : R\u27e6X\u27e7) \u2260 0 := fun H => by\n simpa only [coeff_one_X, one_ne_zero, map_zero] using congr_arg (coeff R 1) H\nset_option linter.uppercaseLean3 false in\n#align power_series.X_ne_zero PowerSeries.X_ne_zero\n\ntheorem X_pow_eq (n : \u2115) : (X : R\u27e6X\u27e7) ^ n = monomial R n 1 :=\n MvPowerSeries.X_pow_eq _ n\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_eq PowerSeries.X_pow_eq\n\ntheorem coeff_X_pow (m n : \u2115) : coeff R m ((X : R\u27e6X\u27e7) ^ n) = if m = n then 1 else 0 := by\n rw [X_pow_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow PowerSeries.coeff_X_pow\n\n@[simp]\ntheorem coeff_X_pow_self (n : \u2115) : coeff R n ((X : R\u27e6X\u27e7) ^ n) = 1 := by\n rw [coeff_X_pow, if_pos rfl]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow_self PowerSeries.coeff_X_pow_self\n\n@[simp]\ntheorem coeff_one (n : \u2115) : coeff R n (1 : R\u27e6X\u27e7) = if n = 0 then 1 else 0 :=\n coeff_C n 1\n#align power_series.coeff_one PowerSeries.coeff_one\n\ntheorem coeff_zero_one : coeff R 0 (1 : R\u27e6X\u27e7) = 1 :=\n coeff_zero_C 1\n#align power_series.coeff_zero_one PowerSeries.coeff_zero_one\n\ntheorem coeff_mul (n : \u2115) (\u03c6 \u03c8 : R\u27e6X\u27e7) :\n coeff R n (\u03c6 * \u03c8) = \u2211 p in antidiagonal n, coeff R p.1 \u03c6 * coeff R p.2 \u03c8 := by\n -- `rw` can't see that `PowerSeries = MvPowerSeries Unit`, so use `.trans`\n refine (MvPowerSeries.coeff_mul _ \u03c6 \u03c8).trans ?_\n rw [Finsupp.antidiagonal_single, Finset.sum_map]\n rfl\n#align power_series.coeff_mul PowerSeries.coeff_mul\n\n@[simp]\ntheorem coeff_mul_C (n : \u2115) (\u03c6 : R\u27e6X\u27e7) (a : R) : coeff R n (\u03c6 * C R a) = coeff R n \u03c6 * a :=\n MvPowerSeries.coeff_mul_C _ \u03c6 a\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_mul_C PowerSeries.coeff_mul_C\n\n@[simp]\ntheorem coeff_C_mul (n : \u2115) (\u03c6 : R\u27e6X\u27e7) (a : R) : coeff R n (C R a * \u03c6) = a * coeff R n \u03c6 :=\n MvPowerSeries.coeff_C_mul _ \u03c6 a\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C_mul PowerSeries.coeff_C_mul\n\n@[simp]\ntheorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : \u2115) (\u03c6 : PowerSeries S) (a : R) :\n coeff S n (a \u2022 \u03c6) = a \u2022 coeff S n \u03c6 :=\n rfl\n#align power_series.coeff_smul PowerSeries.coeff_smul\n\ntheorem smul_eq_C_mul (f : R\u27e6X\u27e7) (a : R) : a \u2022 f = C R a * f := by\n ext\n simp\nset_option linter.uppercaseLean3 false in\n#align power_series.smul_eq_C_mul PowerSeries.smul_eq_C_mul\n\n@[simp]\ntheorem coeff_succ_mul_X (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff R (n + 1) (\u03c6 * X) = coeff R n \u03c6 := by\n simp only [coeff, Finsupp.single_add]\n convert \u03c6.coeff_add_mul_monomial (single () n) (single () 1) _\n rw [mul_one]; rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_succ_mul_X PowerSeries.coeff_succ_mul_X\n\n@[simp]\ntheorem coeff_succ_X_mul (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff R (n + 1) (X * \u03c6) = coeff R n \u03c6 := by\n simp only [coeff, Finsupp.single_add, add_comm n 1]\n convert \u03c6.coeff_add_monomial_mul (single () 1) (single () n) _\n rw [one_mul]; rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_succ_X_mul PowerSeries.coeff_succ_X_mul\n\n@[simp]\ntheorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_C PowerSeries.constantCoeff_C\n\n@[simp]\ntheorem constantCoeff_comp_C : (constantCoeff R).comp (C R) = RingHom.id R :=\n rfl\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_comp_C PowerSeries.constantCoeff_comp_C\n\n-- Porting note (#10618): simp can prove this.\n-- @[simp]\ntheorem constantCoeff_zero : constantCoeff R 0 = 0 :=\n rfl\n#align power_series.constant_coeff_zero PowerSeries.constantCoeff_zero\n\n-- Porting note (#10618): simp can prove this.\n-- @[simp]\ntheorem constantCoeff_one : constantCoeff R 1 = 1 :=\n rfl\n#align power_series.constant_coeff_one PowerSeries.constantCoeff_one\n\n@[simp]\ntheorem constantCoeff_X : constantCoeff R X = 0 :=\n MvPowerSeries.coeff_zero_X _\nset_option linter.uppercaseLean3 false in\n#align power_series.constant_coeff_X PowerSeries.constantCoeff_X\n\ntheorem coeff_zero_mul_X (\u03c6 : R\u27e6X\u27e7) : coeff R 0 (\u03c6 * X) = 0 := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_mul_X PowerSeries.coeff_zero_mul_X\n\ntheorem coeff_zero_X_mul (\u03c6 : R\u27e6X\u27e7) : coeff R 0 (X * \u03c6) = 0 := by simp\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_zero_X_mul PowerSeries.coeff_zero_X_mul\n\ntheorem constantCoeff_surj : Function.Surjective (constantCoeff R) :=\n fun r => \u27e8(C R) r, constantCoeff_C r\u27e9\n\n-- The following section duplicates the API of `Data.Polynomial.Coeff` and should attempt to keep\n-- up to date with that\nsection\n\ntheorem coeff_C_mul_X_pow (x : R) (k n : \u2115) :\n coeff R n (C R x * X ^ k : R\u27e6X\u27e7) = if n = k then x else 0 := by\n simp [X_pow_eq, coeff_monomial]\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_C_mul_X_pow PowerSeries.coeff_C_mul_X_pow\n\n@[simp]\ntheorem coeff_mul_X_pow (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R (d + n) (p * X ^ n) = coeff R d p := by\n rw [coeff_mul, Finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one]\n \u00b7 rintro \u27e8i, j\u27e9 h1 h2\n rw [coeff_X_pow, if_neg, mul_zero]\n rintro rfl\n apply h2\n rw [mem_antidiagonal, add_right_cancel_iff] at h1\n subst h1\n rfl\n \u00b7 exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_mul_X_pow PowerSeries.coeff_mul_X_pow\n\n@[simp]\ntheorem coeff_X_pow_mul (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R (d + n) (X ^ n * p) = coeff R d p := by\n rw [coeff_mul, Finset.sum_eq_single (n, d), coeff_X_pow, if_pos rfl, one_mul]\n \u00b7 rintro \u27e8i, j\u27e9 h1 h2\n rw [coeff_X_pow, if_neg, zero_mul]\n rintro rfl\n apply h2\n rw [mem_antidiagonal, add_comm, add_right_cancel_iff] at h1\n subst h1\n rfl\n \u00b7 rw [add_comm]\n exact fun h1 => (h1 (mem_antidiagonal.2 rfl)).elim\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow_mul PowerSeries.coeff_X_pow_mul\n\ntheorem coeff_mul_X_pow' (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R d (p * X ^ n) = ite (n \u2264 d) (coeff R (d - n) p) 0 := by\n split_ifs with h\n \u00b7 rw [\u2190 tsub_add_cancel_of_le h, coeff_mul_X_pow, add_tsub_cancel_right]\n \u00b7 refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _)\n rw [coeff_X_pow, if_neg, mul_zero]\n exact ((le_of_add_le_right (mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_mul_X_pow' PowerSeries.coeff_mul_X_pow'\n\ntheorem coeff_X_pow_mul' (p : R\u27e6X\u27e7) (n d : \u2115) :\n coeff R d (X ^ n * p) = ite (n \u2264 d) (coeff R (d - n) p) 0 := by\n split_ifs with h\n \u00b7 rw [\u2190 tsub_add_cancel_of_le h, coeff_X_pow_mul]\n simp\n \u00b7 refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _)\n rw [coeff_X_pow, if_neg, zero_mul]\n have := mem_antidiagonal.mp hx\n rw [add_comm] at this\n exact ((le_of_add_le_right this.le).trans_lt <| not_le.mp h).ne\nset_option linter.uppercaseLean3 false in\n#align power_series.coeff_X_pow_mul' PowerSeries.coeff_X_pow_mul'\n\nend\n\n/-- If a formal power series is invertible, then so is its constant coefficient. -/\ntheorem isUnit_constantCoeff (\u03c6 : R\u27e6X\u27e7) (h : IsUnit \u03c6) : IsUnit (constantCoeff R \u03c6) :=\n MvPowerSeries.isUnit_constantCoeff \u03c6 h\n#align power_series.is_unit_constant_coeff PowerSeries.isUnit_constantCoeff\n\n/-- Split off the constant coefficient. -/\ntheorem eq_shift_mul_X_add_const (\u03c6 : R\u27e6X\u27e7) :\n \u03c6 = (mk fun p => coeff R (p + 1) \u03c6) * X + C R (constantCoeff R \u03c6) := by\n ext (_ | n)\n \u00b7 simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X,\n mul_zero, coeff_zero_C, zero_add]\n \u00b7 simp only [coeff_succ_mul_X, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero,\n if_false, add_zero]\nset_option linter.uppercaseLean3 false in\n#align power_series.eq_shift_mul_X_add_const PowerSeries.eq_shift_mul_X_add_const\n\n/-- Split off the constant coefficient. -/\ntheorem eq_X_mul_shift_add_const (\u03c6 : R\u27e6X\u27e7) :\n \u03c6 = (X * mk fun p => coeff R (p + 1) \u03c6) + C R (constantCoeff R \u03c6) := by\n ext (_ | n)\n \u00b7 simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X,\n zero_mul, coeff_zero_C, zero_add]\n \u00b7 simp only [coeff_succ_X_mul, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero,\n if_false, add_zero]\nset_option linter.uppercaseLean3 false in\n#align power_series.eq_X_mul_shift_add_const PowerSeries.eq_X_mul_shift_add_const\n\nsection Map\n\nvariable {S : Type*} {T : Type*} [Semiring S] [Semiring T]\nvariable (f : R \u2192+* S) (g : S \u2192+* T)\n\n/-- The map between formal power series induced by a map on the coefficients. -/\ndef map : R\u27e6X\u27e7 \u2192+* S\u27e6X\u27e7 :=\n MvPowerSeries.map _ f\n#align power_series.map PowerSeries.map\n\n@[simp]\ntheorem map_id : (map (RingHom.id R) : R\u27e6X\u27e7 \u2192 R\u27e6X\u27e7) = id :=\n rfl\n#align power_series.map_id PowerSeries.map_id\n\ntheorem map_comp : map (g.comp f) = (map g).comp (map f) :=\n rfl\n#align power_series.map_comp PowerSeries.map_comp\n\n@[simp]\ntheorem coeff_map (n : \u2115) (\u03c6 : R\u27e6X\u27e7) : coeff S n (map f \u03c6) = f (coeff R n \u03c6) :=\n rfl\n#align power_series.coeff_map PowerSeries.coeff_map\n\n@[simp]\ntheorem map_C (r : R) : map f (C _ r) = C _ (f r) := by\n ext\n simp [coeff_C, apply_ite f]\nset_option linter.uppercaseLean3 false in\n#align power_series.map_C PowerSeries.map_C\n\n@[simp]\ntheorem map_X : map f X = X := by\n ext\n simp [coeff_X, apply_ite f]\nset_option linter.uppercaseLean3 false in\n#align power_series.map_X PowerSeries.map_X\n\nend Map\n\ntheorem X_pow_dvd_iff {n : \u2115} {\u03c6 : R\u27e6X\u27e7} :\n (X : R\u27e6X\u27e7) ^ n \u2223 \u03c6 \u2194 \u2200 m, m < n \u2192 coeff R m \u03c6 = 0 := by\n convert@MvPowerSeries.X_pow_dvd_iff Unit R _ () n \u03c6\n constructor <;> intro h m hm\n \u00b7 rw [Finsupp.unique_single m]\n convert h _ hm\n \u00b7 apply h\n simpa only [Finsupp.single_eq_same] using hm\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_dvd_iff PowerSeries.X_pow_dvd_iff\n\ntheorem X_dvd_iff {\u03c6 : R\u27e6X\u27e7} : (X : R\u27e6X\u27e7) \u2223 \u03c6 \u2194 constantCoeff R \u03c6 = 0 := by\n rw [\u2190 pow_one (X : R\u27e6X\u27e7), X_pow_dvd_iff, \u2190 coeff_zero_eq_constantCoeff_apply]\n constructor <;> intro h\n \u00b7 exact h 0 zero_lt_one\n \u00b7 intro m hm\n rwa [Nat.eq_zero_of_le_zero (Nat.le_of_succ_le_succ hm)]\nset_option linter.uppercaseLean3 false in\n#align power_series.X_dvd_iff PowerSeries.X_dvd_iff\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R]\n\nopen Finset Nat\n\n/-- The ring homomorphism taking a power series `f(X)` to `f(aX)`. -/\nnoncomputable def rescale (a : R) : R\u27e6X\u27e7 \u2192+* R\u27e6X\u27e7 where\n toFun f := PowerSeries.mk fun n => a ^ n * PowerSeries.coeff R n f\n map_zero' := by\n ext\n simp only [LinearMap.map_zero, PowerSeries.coeff_mk, mul_zero]\n map_one' := by\n ext1\n simp only [mul_boole, PowerSeries.coeff_mk, PowerSeries.coeff_one]\n split_ifs with h\n \u00b7 rw [h, pow_zero a]\n rfl\n map_add' := by\n intros\n ext\n dsimp only\n exact mul_add _ _ _\n map_mul' f g := by\n ext\n rw [PowerSeries.coeff_mul, PowerSeries.coeff_mk, PowerSeries.coeff_mul, Finset.mul_sum]\n apply sum_congr rfl\n simp only [coeff_mk, Prod.forall, mem_antidiagonal]\n intro b c H\n rw [\u2190 H, pow_add, mul_mul_mul_comm]\n#align power_series.rescale PowerSeries.rescale\n\n@[simp]\ntheorem coeff_rescale (f : R\u27e6X\u27e7) (a : R) (n : \u2115) :\n coeff R n (rescale a f) = a ^ n * coeff R n f :=\n coeff_mk n (fun n \u21a6 a ^ n * (coeff R n) f)\n#align power_series.coeff_rescale PowerSeries.coeff_rescale\n\n@[simp]\ntheorem rescale_zero : rescale 0 = (C R).comp (constantCoeff R) := by\n ext x n\n simp only [Function.comp_apply, RingHom.coe_comp, rescale, RingHom.coe_mk,\n PowerSeries.coeff_mk _ _, coeff_C]\n split_ifs with h <;> simp [h]\n#align power_series.rescale_zero PowerSeries.rescale_zero\n\ntheorem rescale_zero_apply : rescale 0 X = C R (constantCoeff R X) := by simp\n#align power_series.rescale_zero_apply PowerSeries.rescale_zero_apply\n\n@[simp]\ntheorem rescale_one : rescale 1 = RingHom.id R\u27e6X\u27e7 := by\n ext\n simp only [coeff_rescale, one_pow, one_mul, RingHom.id_apply]\n#align power_series.rescale_one PowerSeries.rescale_one\n\ntheorem rescale_mk (f : \u2115 \u2192 R) (a : R) : rescale a (mk f) = mk fun n : \u2115 => a ^ n * f n := by\n ext\n rw [coeff_rescale, coeff_mk, coeff_mk]\n#align power_series.rescale_mk PowerSeries.rescale_mk\n\ntheorem rescale_rescale (f : R\u27e6X\u27e7) (a b : R) :\n rescale b (rescale a f) = rescale (a * b) f := by\n ext n\n simp_rw [coeff_rescale]\n rw [mul_pow, mul_comm _ (b ^ n), mul_assoc]\n#align power_series.rescale_rescale PowerSeries.rescale_rescale\n\ntheorem rescale_mul (a b : R) : rescale (a * b) = (rescale b).comp (rescale a) := by\n ext\n simp [\u2190 rescale_rescale]\n#align power_series.rescale_mul PowerSeries.rescale_mul\n\nend CommSemiring\n\nsection CommSemiring\n\nopen Finset.HasAntidiagonal Finset\n\nvariable {R : Type*} [CommSemiring R] {\u03b9 : Type*} [DecidableEq \u03b9]\n\n/-- Coefficients of a product of power series -/\ntheorem coeff_prod (f : \u03b9 \u2192 PowerSeries R) (d : \u2115) (s : Finset \u03b9) :\n coeff R d (\u220f j in s, f j) = \u2211 l in piAntidiagonal s d, \u220f i in s, coeff R (l i) (f i) := by\n simp only [coeff]\n convert MvPowerSeries.coeff_prod _ _ _\n rw [\u2190 AddEquiv.finsuppUnique_symm d, \u2190 mapRange_piAntidiagonal_eq, sum_map, sum_congr rfl]\n intro x _\n apply prod_congr rfl\n intro i _\n congr 2\n simp only [AddEquiv.toEquiv_eq_coe, Finsupp.mapRange.addEquiv_toEquiv, AddEquiv.toEquiv_symm,\n Equiv.coe_toEmbedding, Finsupp.mapRange.equiv_apply, AddEquiv.coe_toEquiv_symm,\n Finsupp.mapRange_apply, AddEquiv.finsuppUnique_symm]\n\nend CommSemiring\n\nsection CommRing\n\nvariable {A : Type*} [CommRing A]\n\ntheorem not_isField : \u00acIsField A\u27e6X\u27e7 := by\n by_cases hA : Subsingleton A\n \u00b7 exact not_isField_of_subsingleton _\n \u00b7 nontriviality A\n rw [Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top]\n use Ideal.span {X}\n constructor\n \u00b7 rw [bot_lt_iff_ne_bot, Ne.def, Ideal.span_singleton_eq_bot]\n exact X_ne_zero\n \u00b7 rw [lt_top_iff_ne_top, Ne.def, Ideal.eq_top_iff_one, Ideal.mem_span_singleton,\n X_dvd_iff, constantCoeff_one]\n exact one_ne_zero\n\n@[simp]\ntheorem rescale_X (a : A) : rescale a X = C A a * X := by\n ext\n simp only [coeff_rescale, coeff_C_mul, coeff_X]\n split_ifs with h <;> simp [h]\nset_option linter.uppercaseLean3 false in\n#align power_series.rescale_X PowerSeries.rescale_X\n\ntheorem rescale_neg_one_X : rescale (-1 : A) X = -X := by\n rw [rescale_X, map_neg, map_one, neg_one_mul]\nset_option linter.uppercaseLean3 false in\n#align power_series.rescale_neg_one_X PowerSeries.rescale_neg_one_X\n\n/-- The ring homomorphism taking a power series `f(X)` to `f(-X)`. -/\nnoncomputable def evalNegHom : A\u27e6X\u27e7 \u2192+* A\u27e6X\u27e7 :=\n rescale (-1 : A)\n#align power_series.eval_neg_hom PowerSeries.evalNegHom\n\n@[simp]\ntheorem evalNegHom_X : evalNegHom (X : A\u27e6X\u27e7) = -X :=\n rescale_neg_one_X\nset_option linter.uppercaseLean3 false in\n#align power_series.eval_neg_hom_X PowerSeries.evalNegHom_X\n\nend CommRing\n\nsection Domain\n\nvariable [Ring R]\n\ntheorem eq_zero_or_eq_zero_of_mul_eq_zero [NoZeroDivisors R] (\u03c6 \u03c8 : R\u27e6X\u27e7) (h : \u03c6 * \u03c8 = 0) :\n \u03c6 = 0 \u2228 \u03c8 = 0 := by\n classical\n rw [or_iff_not_imp_left]\n intro H\n have ex : \u2203 m, coeff R m \u03c6 \u2260 0 := by\n contrapose! H\n exact ext H\n let m := Nat.find ex\n have hm\u2081 : coeff R m \u03c6 \u2260 0 := Nat.find_spec ex\n have hm\u2082 : \u2200 k < m, \u00accoeff R k \u03c6 \u2260 0 := fun k => Nat.find_min ex\n ext n\n rw [(coeff R n).map_zero]\n induction' n using Nat.strong_induction_on with n ih\n replace h := congr_arg (coeff R (m + n)) h\n rw [LinearMap.map_zero, coeff_mul, Finset.sum_eq_single (m, n)] at h\n \u00b7 replace h := NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero h\n rw [or_iff_not_imp_left] at h\n exact h hm\u2081\n \u00b7 rintro \u27e8i, j\u27e9 hij hne\n by_cases hj : j < n\n \u00b7 rw [ih j hj, mul_zero]\n by_cases hi : i < m\n \u00b7 specialize hm\u2082 _ hi\n push_neg at hm\u2082\n rw [hm\u2082, zero_mul]\n rw [mem_antidiagonal] at hij\n push_neg at hi hj\n suffices m < i by\n have : m + n < i + j := add_lt_add_of_lt_of_le this hj\n exfalso\n exact ne_of_lt this hij.symm\n contrapose! hne\n obtain rfl := le_antisymm hi hne\n simpa [Ne, Prod.mk.inj_iff] using (add_right_inj m).mp hij\n \u00b7 contrapose!\n intro\n rw [mem_antidiagonal]\n#align power_series.eq_zero_or_eq_zero_of_mul_eq_zero PowerSeries.eq_zero_or_eq_zero_of_mul_eq_zero\n\ninstance [NoZeroDivisors R] : NoZeroDivisors R\u27e6X\u27e7 where\n eq_zero_or_eq_zero_of_mul_eq_zero := eq_zero_or_eq_zero_of_mul_eq_zero _ _\n\ninstance [IsDomain R] : IsDomain R\u27e6X\u27e7 :=\n NoZeroDivisors.to_isDomain _\n\nend Domain\n\nsection IsDomain\n\nvariable [CommRing R] [IsDomain R]\n\n/-- The ideal spanned by the variable in the power series ring\n over an integral domain is a prime ideal. -/\ntheorem span_X_isPrime : (Ideal.span ({X} : Set R\u27e6X\u27e7)).IsPrime := by\n suffices Ideal.span ({X} : Set R\u27e6X\u27e7) = RingHom.ker (constantCoeff R) by\n rw [this]\n exact RingHom.ker_isPrime _\n apply Ideal.ext\n intro \u03c6\n rw [RingHom.mem_ker, Ideal.mem_span_singleton, X_dvd_iff]\nset_option linter.uppercaseLean3 false in\n#align power_series.span_X_is_prime PowerSeries.span_X_isPrime\n\n/-- The variable of the power series ring over an integral domain is prime. -/\ntheorem X_prime : Prime (X : R\u27e6X\u27e7) := by\n rw [\u2190 Ideal.span_singleton_prime]\n \u00b7 exact span_X_isPrime\n \u00b7 intro h\n simpa [map_zero (coeff R 1)] using congr_arg (coeff R 1) h\nset_option linter.uppercaseLean3 false in\n#align power_series.X_prime PowerSeries.X_prime\n\n/-- The variable of the power series ring over an integral domain is irreducible. -/\n", "theoremStatement": "theorem X_irreducible : Irreducible (X : R\u27e6X\u27e7)", "theoremName": "X_irreducible", "fileCreated": {"commit": "3d924717cd", "date": "2023-05-22"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Basic.lean", "positionMetadata": {"lineInFile": 792, "tokenPositionInFile": 27232, "theoremPositionInFile": 101}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "X_prime.irreducible", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 19}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\n\nimport Mathlib.RingTheory.PowerSeries.Basic\nimport Mathlib.Algebra.CharP.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-! # Formal power series (in one variable) - Order\n\nThe `PowerSeries.order` of a formal power series `\u03c6` is the multiplicity of the variable `X` in `\u03c6`.\n\nIf the coefficients form an integral domain, then `PowerSeries.order` is an\nadditive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`).\n\nWe prove that if the commutative ring `R` of coefficients is an integral domain,\nthen the ring `R\u27e6X\u27e7` of formal power series in one variable over `R`\nis an integral domain.\n\nGiven a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\ndividing out the largest power of X that divides `f`, that is its order. This is useful when\nproving that `R\u27e6X\u27e7` is a normalization monoid, which is done in `PowerSeries.Inverse`.\n\n-/\nnoncomputable section\n\nopen BigOperators Polynomial\n\nopen Finset (antidiagonal mem_antidiagonal)\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection OrderBasic\n\nopen multiplicity\n\nvariable [Semiring R] {\u03c6 : R\u27e6X\u27e7}\n\ntheorem exists_coeff_ne_zero_iff_ne_zero : (\u2203 n : \u2115, coeff R n \u03c6 \u2260 0) \u2194 \u03c6 \u2260 0 := by\n refine' not_iff_not.mp _\n push_neg\n -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?\n simp [PowerSeries.ext_iff, (coeff R _).map_zero]\n#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero\n\n/-- The order of a formal power series `\u03c6` is the greatest `n : PartENat`\nsuch that `X^n` divides `\u03c6`. The order is `\u22a4` if and only if `\u03c6 = 0`. -/\ndef order (\u03c6 : R\u27e6X\u27e7) : PartENat :=\n letI := Classical.decEq R\n letI := Classical.decEq R\u27e6X\u27e7\n if h : \u03c6 = 0 then \u22a4 else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)\n#align power_series.order PowerSeries.order\n\n/-- The order of the `0` power series is infinite. -/\n@[simp]\ntheorem order_zero : order (0 : R\u27e6X\u27e7) = \u22a4 :=\n dif_pos rfl\n#align power_series.order_zero PowerSeries.order_zero\n\ntheorem order_finite_iff_ne_zero : (order \u03c6).Dom \u2194 \u03c6 \u2260 0 := by\n simp only [order]\n constructor\n \u00b7 split_ifs with h <;> intro H\n \u00b7 simp only [PartENat.top_eq_none, Part.not_none_dom] at H\n \u00b7 exact h\n \u00b7 intro h\n simp [h]\n#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero\n\n/-- If the order of a formal power series is finite,\nthen the coefficient indexed by the order is nonzero. -/\ntheorem coeff_order (h : (order \u03c6).Dom) : coeff R (\u03c6.order.get h) \u03c6 \u2260 0 := by\n classical\n simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']\n generalize_proofs h\n exact Nat.find_spec h\n#align power_series.coeff_order PowerSeries.coeff_order\n\n/-- If the `n`th coefficient of a formal power series is nonzero,\nthen the order of the power series is less than or equal to `n`. -/\ntheorem order_le (n : \u2115) (h : coeff R n \u03c6 \u2260 0) : order \u03c6 \u2264 n := by\n classical\n rw [order, dif_neg]\n \u00b7 simp only [PartENat.coe_le_coe]\n exact Nat.find_le h\n \u00b7 exact exists_coeff_ne_zero_iff_ne_zero.mp \u27e8n, h\u27e9\n#align power_series.order_le PowerSeries.order_le\n\n/-- The `n`th coefficient of a formal power series is `0` if `n` is strictly\nsmaller than the order of the power series. -/\ntheorem coeff_of_lt_order (n : \u2115) (h : \u2191n < order \u03c6) : coeff R n \u03c6 = 0 := by\n contrapose! h\n exact order_le _ h\n#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order\n\n/-- The `0` power series is the unique power series with infinite order. -/\n@[simp]\ntheorem order_eq_top {\u03c6 : R\u27e6X\u27e7} : \u03c6.order = \u22a4 \u2194 \u03c6 = 0 :=\n PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left\n#align power_series.order_eq_top PowerSeries.order_eq_top\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem nat_le_order (\u03c6 : R\u27e6X\u27e7) (n : \u2115) (h : \u2200 i < n, coeff R i \u03c6 = 0) : \u2191n \u2264 order \u03c6 := by\n by_contra H; rw [not_le] at H\n have : (order \u03c6).Dom := PartENat.dom_of_le_natCast H.le\n rw [\u2190 PartENat.natCast_get this, PartENat.coe_lt_coe] at H\n exact coeff_order this (h _ H)\n#align power_series.nat_le_order PowerSeries.nat_le_order\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem le_order (\u03c6 : R\u27e6X\u27e7) (n : PartENat) (h : \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0) :\n n \u2264 order \u03c6 := by\n induction n using PartENat.casesOn\n \u00b7 show _ \u2264 _\n rw [top_le_iff, order_eq_top]\n ext i\n exact h _ (PartENat.natCast_lt_top i)\n \u00b7 apply nat_le_order\n simpa only [PartENat.coe_lt_coe] using h\n#align power_series.le_order PowerSeries.le_order\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq_nat {\u03c6 : R\u27e6X\u27e7} {n : \u2115} :\n order \u03c6 = n \u2194 coeff R n \u03c6 \u2260 0 \u2227 \u2200 i, i < n \u2192 coeff R i \u03c6 = 0 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm\n simp [order, dif_neg h\u03c6, Nat.find_eq_iff]\n#align power_series.order_eq_nat PowerSeries.order_eq_nat\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq {\u03c6 : R\u27e6X\u27e7} {n : PartENat} :\n order \u03c6 = n \u2194 (\u2200 i : \u2115, \u2191i = n \u2192 coeff R i \u03c6 \u2260 0) \u2227 \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0 := by\n induction n using PartENat.casesOn\n \u00b7 rw [order_eq_top]\n constructor\n \u00b7 rintro rfl\n constructor <;> intros\n \u00b7 exfalso\n exact PartENat.natCast_ne_top \u2039_\u203a \u2039_\u203a\n \u00b7 exact (coeff _ _).map_zero\n \u00b7 rintro \u27e8_h\u2081, h\u2082\u27e9\n ext i\n exact h\u2082 i (PartENat.natCast_lt_top i)\n \u00b7 simpa [PartENat.natCast_inj] using order_eq_nat\n#align power_series.order_eq PowerSeries.order_eq\n\n/-- The order of the sum of two formal power series\n is at least the minimum of their orders. -/\ntheorem le_order_add (\u03c6 \u03c8 : R\u27e6X\u27e7) : min (order \u03c6) (order \u03c8) \u2264 order (\u03c6 + \u03c8) := by\n refine' le_order _ _ _\n simp (config := { contextual := true }) [coeff_of_lt_order]\n#align power_series.le_order_add PowerSeries.le_order_add\n\nprivate theorem order_add_of_order_eq.aux (\u03c6 \u03c8 : R\u27e6X\u27e7) (_h : order \u03c6 \u2260 order \u03c8)\n (H : order \u03c6 < order \u03c8) : order (\u03c6 + \u03c8) \u2264 order \u03c6 \u2293 order \u03c8 := by\n suffices order (\u03c6 + \u03c8) = order \u03c6 by\n rw [le_inf_iff, this]\n exact \u27e8le_rfl, le_of_lt H\u27e9\n \u00b7 rw [order_eq]\n constructor\n \u00b7 intro i hi\n rw [\u2190 hi] at H\n rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero]\n exact (order_eq_nat.1 hi.symm).1\n \u00b7 intro i hi\n rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H),\n zero_add]\n-- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux\n\n/-- The order of the sum of two formal power series\n is the minimum of their orders if their orders differ. -/\ntheorem order_add_of_order_eq (\u03c6 \u03c8 : R\u27e6X\u27e7) (h : order \u03c6 \u2260 order \u03c8) :\n order (\u03c6 + \u03c8) = order \u03c6 \u2293 order \u03c8 := by\n refine' le_antisymm _ (le_order_add _ _)\n by_cases H\u2081 : order \u03c6 < order \u03c8\n \u00b7 apply order_add_of_order_eq.aux _ _ h H\u2081\n by_cases H\u2082 : order \u03c8 < order \u03c6\n \u00b7 simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H\u2082\n exfalso; exact h (le_antisymm (not_lt.1 H\u2082) (not_lt.1 H\u2081))\n#align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq\n\n/-- The order of the product of two formal power series\n is at least the sum of their orders. -/\ntheorem order_mul_ge (\u03c6 \u03c8 : R\u27e6X\u27e7) : order \u03c6 + order \u03c8 \u2264 order (\u03c6 * \u03c8) := by\n apply le_order\n intro n hn; rw [coeff_mul, Finset.sum_eq_zero]\n rintro \u27e8i, j\u27e9 hij\n by_cases hi : \u2191i < order \u03c6\n \u00b7 rw [coeff_of_lt_order i hi, zero_mul]\n by_cases hj : \u2191j < order \u03c8\n \u00b7 rw [coeff_of_lt_order j hj, mul_zero]\n rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij\n exfalso\n apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj)\n rw [\u2190 Nat.cast_add, hij]\n#align power_series.order_mul_ge PowerSeries.order_mul_ge\n\n/-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/\ntheorem order_monomial (n : \u2115) (a : R) [Decidable (a = 0)] :\n order (monomial R n a) = if a = 0 then (\u22a4 : PartENat) else n := by\n split_ifs with h\n \u00b7 rw [h, order_eq_top, LinearMap.map_zero]\n \u00b7 rw [order_eq]\n constructor <;> intro i hi\n \u00b7 rw [PartENat.natCast_inj] at hi\n rwa [hi, coeff_monomial_same]\n \u00b7 rw [PartENat.coe_lt_coe] at hi\n rw [coeff_monomial, if_neg]\n exact ne_of_lt hi\n#align power_series.order_monomial PowerSeries.order_monomial\n\n/-- The order of the monomial `a*X^n` is `n` if `a \u2260 0`. -/\ntheorem order_monomial_of_ne_zero (n : \u2115) (a : R) (h : a \u2260 0) : order (monomial R n a) = n := by\n classical\n rw [order_monomial, if_neg h]\n#align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero\n\n/-- If `n` is strictly smaller than the order of `\u03c8`, then the `n`th coefficient of its product\nwith any other power series is `0`. -/\ntheorem coeff_mul_of_lt_order {\u03c6 \u03c8 : R\u27e6X\u27e7} {n : \u2115} (h : \u2191n < \u03c8.order) :\n coeff R n (\u03c6 * \u03c8) = 0 := by\n suffices coeff R n (\u03c6 * \u03c8) = \u2211 p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero]\n rw [coeff_mul]\n apply Finset.sum_congr rfl\n intro x hx\n refine' mul_eq_zero_of_right (coeff R x.fst \u03c6) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h))\n rw [mem_antidiagonal] at hx\n norm_cast\n omega\n#align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order\n\ntheorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {\u03c6 \u03c8 : R\u27e6X\u27e7} (n : \u2115)\n (h : \u2191n < \u03c8.order) : coeff R n (\u03c6 * (1 - \u03c8)) = coeff R n \u03c6 := by\n simp [coeff_mul_of_lt_order h, mul_sub]\n#align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order\n\ntheorem coeff_mul_prod_one_sub_of_lt_order {R \u03b9 : Type*} [CommRing R] (k : \u2115) (s : Finset \u03b9)\n (\u03c6 : R\u27e6X\u27e7) (f : \u03b9 \u2192 R\u27e6X\u27e7) :\n (\u2200 i \u2208 s, \u2191k < (f i).order) \u2192 coeff R k (\u03c6 * \u220f i in s, (1 - f i)) = coeff R k \u03c6 := by\n classical\n induction' s using Finset.induction_on with a s ha ih t\n \u00b7 simp\n \u00b7 intro t\n simp only [Finset.mem_insert, forall_eq_or_imp] at t\n rw [Finset.prod_insert ha, \u2190 mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1]\n exact ih t.2\n#align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order\n\n-- TODO: link with `X_pow_dvd_iff`\ntheorem X_pow_order_dvd (h : (order \u03c6).Dom) : X ^ (order \u03c6).get h \u2223 \u03c6 := by\n refine' \u27e8PowerSeries.mk fun n => coeff R (n + (order \u03c6).get h) \u03c6, _\u27e9\n ext n\n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite,\n Finset.sum_const_zero, add_zero]\n rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order \u03c6) h)]\n split_ifs with hn\n \u00b7 simp [tsub_add_cancel_of_le hn]\n \u00b7 simp only [Finset.sum_empty]\n refine' coeff_of_lt_order _ _\n simpa [PartENat.coe_lt_iff] using fun _ => hn\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd\n\ntheorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R\u27e6X\u27e7 (\u00b7 \u2223 \u00b7)] (\u03c6 : R\u27e6X\u27e7) :\n order \u03c6 = multiplicity X \u03c6 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simp\n induction' ho : order \u03c6 using PartENat.casesOn with n\n \u00b7 simp [h\u03c6] at ho\n have hn : \u03c6.order.get (order_finite_iff_ne_zero.mpr h\u03c6) = n := by simp [ho]\n rw [\u2190 hn]\n refine'\n le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr h\u03c6))\n (PartENat.find_le _ _ _)\n rintro \u27e8\u03c8, H\u27e9\n have := congr_arg (coeff R n) H\n rw [\u2190 (\u03c8.commute_X.pow_right _).eq, coeff_mul_of_lt_order, \u2190 hn] at this\n \u00b7 exact coeff_order _ this\n \u00b7 rw [X_pow_eq, order_monomial]\n split_ifs\n \u00b7 exact PartENat.natCast_lt_top _\n \u00b7 rw [\u2190 hn, PartENat.coe_lt_coe]\n exact Nat.lt_succ_self _\nset_option linter.uppercaseLean3 false in\n#align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X\n\n/-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\n dividing out the largest power of X that divides `f`, that is its order-/\ndef divided_by_X_pow_order {f : PowerSeries R} (hf : f \u2260 0) : R\u27e6X\u27e7 :=\n (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose\n\n", "theoremStatement": "theorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R\u27e6X\u27e7} (hf : f \u2260 0) :\n X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f", "theoremName": "self_eq_X_pow_order_mul_divided_by_X_pow_order", "fileCreated": {"commit": "73d45f44c3", "date": "2024-02-29"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean", "positionMetadata": {"lineInFile": 305, "tokenPositionInFile": 12453, "theoremPositionInFile": 24}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "haveI dvd := X_pow_order_dvd (order_finite_iff_ne_zero.mpr hf)\n (exists_eq_mul_right_of_dvd dvd).choose_spec.symm", "proofType": "term", "proofLengthLines": 2, "proofLengthTokens": 114}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\n\nimport Mathlib.RingTheory.PowerSeries.Basic\nimport Mathlib.Algebra.CharP.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-! # Formal power series (in one variable) - Order\n\nThe `PowerSeries.order` of a formal power series `\u03c6` is the multiplicity of the variable `X` in `\u03c6`.\n\nIf the coefficients form an integral domain, then `PowerSeries.order` is an\nadditive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`).\n\nWe prove that if the commutative ring `R` of coefficients is an integral domain,\nthen the ring `R\u27e6X\u27e7` of formal power series in one variable over `R`\nis an integral domain.\n\nGiven a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\ndividing out the largest power of X that divides `f`, that is its order. This is useful when\nproving that `R\u27e6X\u27e7` is a normalization monoid, which is done in `PowerSeries.Inverse`.\n\n-/\nnoncomputable section\n\nopen BigOperators Polynomial\n\nopen Finset (antidiagonal mem_antidiagonal)\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection OrderBasic\n\nopen multiplicity\n\nvariable [Semiring R] {\u03c6 : R\u27e6X\u27e7}\n\ntheorem exists_coeff_ne_zero_iff_ne_zero : (\u2203 n : \u2115, coeff R n \u03c6 \u2260 0) \u2194 \u03c6 \u2260 0 := by\n refine' not_iff_not.mp _\n push_neg\n -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?\n simp [PowerSeries.ext_iff, (coeff R _).map_zero]\n#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero\n\n/-- The order of a formal power series `\u03c6` is the greatest `n : PartENat`\nsuch that `X^n` divides `\u03c6`. The order is `\u22a4` if and only if `\u03c6 = 0`. -/\ndef order (\u03c6 : R\u27e6X\u27e7) : PartENat :=\n letI := Classical.decEq R\n letI := Classical.decEq R\u27e6X\u27e7\n if h : \u03c6 = 0 then \u22a4 else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)\n#align power_series.order PowerSeries.order\n\n/-- The order of the `0` power series is infinite. -/\n@[simp]\ntheorem order_zero : order (0 : R\u27e6X\u27e7) = \u22a4 :=\n dif_pos rfl\n#align power_series.order_zero PowerSeries.order_zero\n\ntheorem order_finite_iff_ne_zero : (order \u03c6).Dom \u2194 \u03c6 \u2260 0 := by\n simp only [order]\n constructor\n \u00b7 split_ifs with h <;> intro H\n \u00b7 simp only [PartENat.top_eq_none, Part.not_none_dom] at H\n \u00b7 exact h\n \u00b7 intro h\n simp [h]\n#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero\n\n/-- If the order of a formal power series is finite,\nthen the coefficient indexed by the order is nonzero. -/\ntheorem coeff_order (h : (order \u03c6).Dom) : coeff R (\u03c6.order.get h) \u03c6 \u2260 0 := by\n classical\n simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']\n generalize_proofs h\n exact Nat.find_spec h\n#align power_series.coeff_order PowerSeries.coeff_order\n\n/-- If the `n`th coefficient of a formal power series is nonzero,\nthen the order of the power series is less than or equal to `n`. -/\ntheorem order_le (n : \u2115) (h : coeff R n \u03c6 \u2260 0) : order \u03c6 \u2264 n := by\n classical\n rw [order, dif_neg]\n \u00b7 simp only [PartENat.coe_le_coe]\n exact Nat.find_le h\n \u00b7 exact exists_coeff_ne_zero_iff_ne_zero.mp \u27e8n, h\u27e9\n#align power_series.order_le PowerSeries.order_le\n\n/-- The `n`th coefficient of a formal power series is `0` if `n` is strictly\nsmaller than the order of the power series. -/\ntheorem coeff_of_lt_order (n : \u2115) (h : \u2191n < order \u03c6) : coeff R n \u03c6 = 0 := by\n contrapose! h\n exact order_le _ h\n#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order\n\n/-- The `0` power series is the unique power series with infinite order. -/\n@[simp]\ntheorem order_eq_top {\u03c6 : R\u27e6X\u27e7} : \u03c6.order = \u22a4 \u2194 \u03c6 = 0 :=\n PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left\n#align power_series.order_eq_top PowerSeries.order_eq_top\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem nat_le_order (\u03c6 : R\u27e6X\u27e7) (n : \u2115) (h : \u2200 i < n, coeff R i \u03c6 = 0) : \u2191n \u2264 order \u03c6 := by\n by_contra H; rw [not_le] at H\n have : (order \u03c6).Dom := PartENat.dom_of_le_natCast H.le\n rw [\u2190 PartENat.natCast_get this, PartENat.coe_lt_coe] at H\n exact coeff_order this (h _ H)\n#align power_series.nat_le_order PowerSeries.nat_le_order\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem le_order (\u03c6 : R\u27e6X\u27e7) (n : PartENat) (h : \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0) :\n n \u2264 order \u03c6 := by\n induction n using PartENat.casesOn\n \u00b7 show _ \u2264 _\n rw [top_le_iff, order_eq_top]\n ext i\n exact h _ (PartENat.natCast_lt_top i)\n \u00b7 apply nat_le_order\n simpa only [PartENat.coe_lt_coe] using h\n#align power_series.le_order PowerSeries.le_order\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq_nat {\u03c6 : R\u27e6X\u27e7} {n : \u2115} :\n order \u03c6 = n \u2194 coeff R n \u03c6 \u2260 0 \u2227 \u2200 i, i < n \u2192 coeff R i \u03c6 = 0 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm\n simp [order, dif_neg h\u03c6, Nat.find_eq_iff]\n#align power_series.order_eq_nat PowerSeries.order_eq_nat\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq {\u03c6 : R\u27e6X\u27e7} {n : PartENat} :\n order \u03c6 = n \u2194 (\u2200 i : \u2115, \u2191i = n \u2192 coeff R i \u03c6 \u2260 0) \u2227 \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0 := by\n induction n using PartENat.casesOn\n \u00b7 rw [order_eq_top]\n constructor\n \u00b7 rintro rfl\n constructor <;> intros\n \u00b7 exfalso\n exact PartENat.natCast_ne_top \u2039_\u203a \u2039_\u203a\n \u00b7 exact (coeff _ _).map_zero\n \u00b7 rintro \u27e8_h\u2081, h\u2082\u27e9\n ext i\n exact h\u2082 i (PartENat.natCast_lt_top i)\n \u00b7 simpa [PartENat.natCast_inj] using order_eq_nat\n#align power_series.order_eq PowerSeries.order_eq\n\n/-- The order of the sum of two formal power series\n is at least the minimum of their orders. -/\ntheorem le_order_add (\u03c6 \u03c8 : R\u27e6X\u27e7) : min (order \u03c6) (order \u03c8) \u2264 order (\u03c6 + \u03c8) := by\n refine' le_order _ _ _\n simp (config := { contextual := true }) [coeff_of_lt_order]\n#align power_series.le_order_add PowerSeries.le_order_add\n\nprivate theorem order_add_of_order_eq.aux (\u03c6 \u03c8 : R\u27e6X\u27e7) (_h : order \u03c6 \u2260 order \u03c8)\n (H : order \u03c6 < order \u03c8) : order (\u03c6 + \u03c8) \u2264 order \u03c6 \u2293 order \u03c8 := by\n suffices order (\u03c6 + \u03c8) = order \u03c6 by\n rw [le_inf_iff, this]\n exact \u27e8le_rfl, le_of_lt H\u27e9\n \u00b7 rw [order_eq]\n constructor\n \u00b7 intro i hi\n rw [\u2190 hi] at H\n rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero]\n exact (order_eq_nat.1 hi.symm).1\n \u00b7 intro i hi\n rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H),\n zero_add]\n-- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux\n\n/-- The order of the sum of two formal power series\n is the minimum of their orders if their orders differ. -/\ntheorem order_add_of_order_eq (\u03c6 \u03c8 : R\u27e6X\u27e7) (h : order \u03c6 \u2260 order \u03c8) :\n order (\u03c6 + \u03c8) = order \u03c6 \u2293 order \u03c8 := by\n refine' le_antisymm _ (le_order_add _ _)\n by_cases H\u2081 : order \u03c6 < order \u03c8\n \u00b7 apply order_add_of_order_eq.aux _ _ h H\u2081\n by_cases H\u2082 : order \u03c8 < order \u03c6\n \u00b7 simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H\u2082\n exfalso; exact h (le_antisymm (not_lt.1 H\u2082) (not_lt.1 H\u2081))\n#align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq\n\n/-- The order of the product of two formal power series\n is at least the sum of their orders. -/\ntheorem order_mul_ge (\u03c6 \u03c8 : R\u27e6X\u27e7) : order \u03c6 + order \u03c8 \u2264 order (\u03c6 * \u03c8) := by\n apply le_order\n intro n hn; rw [coeff_mul, Finset.sum_eq_zero]\n rintro \u27e8i, j\u27e9 hij\n by_cases hi : \u2191i < order \u03c6\n \u00b7 rw [coeff_of_lt_order i hi, zero_mul]\n by_cases hj : \u2191j < order \u03c8\n \u00b7 rw [coeff_of_lt_order j hj, mul_zero]\n rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij\n exfalso\n apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj)\n rw [\u2190 Nat.cast_add, hij]\n#align power_series.order_mul_ge PowerSeries.order_mul_ge\n\n/-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/\ntheorem order_monomial (n : \u2115) (a : R) [Decidable (a = 0)] :\n order (monomial R n a) = if a = 0 then (\u22a4 : PartENat) else n := by\n split_ifs with h\n \u00b7 rw [h, order_eq_top, LinearMap.map_zero]\n \u00b7 rw [order_eq]\n constructor <;> intro i hi\n \u00b7 rw [PartENat.natCast_inj] at hi\n rwa [hi, coeff_monomial_same]\n \u00b7 rw [PartENat.coe_lt_coe] at hi\n rw [coeff_monomial, if_neg]\n exact ne_of_lt hi\n#align power_series.order_monomial PowerSeries.order_monomial\n\n/-- The order of the monomial `a*X^n` is `n` if `a \u2260 0`. -/\ntheorem order_monomial_of_ne_zero (n : \u2115) (a : R) (h : a \u2260 0) : order (monomial R n a) = n := by\n classical\n rw [order_monomial, if_neg h]\n#align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero\n\n/-- If `n` is strictly smaller than the order of `\u03c8`, then the `n`th coefficient of its product\nwith any other power series is `0`. -/\ntheorem coeff_mul_of_lt_order {\u03c6 \u03c8 : R\u27e6X\u27e7} {n : \u2115} (h : \u2191n < \u03c8.order) :\n coeff R n (\u03c6 * \u03c8) = 0 := by\n suffices coeff R n (\u03c6 * \u03c8) = \u2211 p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero]\n rw [coeff_mul]\n apply Finset.sum_congr rfl\n intro x hx\n refine' mul_eq_zero_of_right (coeff R x.fst \u03c6) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h))\n rw [mem_antidiagonal] at hx\n norm_cast\n omega\n#align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order\n\ntheorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {\u03c6 \u03c8 : R\u27e6X\u27e7} (n : \u2115)\n (h : \u2191n < \u03c8.order) : coeff R n (\u03c6 * (1 - \u03c8)) = coeff R n \u03c6 := by\n simp [coeff_mul_of_lt_order h, mul_sub]\n#align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order\n\ntheorem coeff_mul_prod_one_sub_of_lt_order {R \u03b9 : Type*} [CommRing R] (k : \u2115) (s : Finset \u03b9)\n (\u03c6 : R\u27e6X\u27e7) (f : \u03b9 \u2192 R\u27e6X\u27e7) :\n (\u2200 i \u2208 s, \u2191k < (f i).order) \u2192 coeff R k (\u03c6 * \u220f i in s, (1 - f i)) = coeff R k \u03c6 := by\n classical\n induction' s using Finset.induction_on with a s ha ih t\n \u00b7 simp\n \u00b7 intro t\n simp only [Finset.mem_insert, forall_eq_or_imp] at t\n rw [Finset.prod_insert ha, \u2190 mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1]\n exact ih t.2\n#align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order\n\n-- TODO: link with `X_pow_dvd_iff`\ntheorem X_pow_order_dvd (h : (order \u03c6).Dom) : X ^ (order \u03c6).get h \u2223 \u03c6 := by\n refine' \u27e8PowerSeries.mk fun n => coeff R (n + (order \u03c6).get h) \u03c6, _\u27e9\n ext n\n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite,\n Finset.sum_const_zero, add_zero]\n rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order \u03c6) h)]\n split_ifs with hn\n \u00b7 simp [tsub_add_cancel_of_le hn]\n \u00b7 simp only [Finset.sum_empty]\n refine' coeff_of_lt_order _ _\n simpa [PartENat.coe_lt_iff] using fun _ => hn\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd\n\ntheorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R\u27e6X\u27e7 (\u00b7 \u2223 \u00b7)] (\u03c6 : R\u27e6X\u27e7) :\n order \u03c6 = multiplicity X \u03c6 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simp\n induction' ho : order \u03c6 using PartENat.casesOn with n\n \u00b7 simp [h\u03c6] at ho\n have hn : \u03c6.order.get (order_finite_iff_ne_zero.mpr h\u03c6) = n := by simp [ho]\n rw [\u2190 hn]\n refine'\n le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr h\u03c6))\n (PartENat.find_le _ _ _)\n rintro \u27e8\u03c8, H\u27e9\n have := congr_arg (coeff R n) H\n rw [\u2190 (\u03c8.commute_X.pow_right _).eq, coeff_mul_of_lt_order, \u2190 hn] at this\n \u00b7 exact coeff_order _ this\n \u00b7 rw [X_pow_eq, order_monomial]\n split_ifs\n \u00b7 exact PartENat.natCast_lt_top _\n \u00b7 rw [\u2190 hn, PartENat.coe_lt_coe]\n exact Nat.lt_succ_self _\nset_option linter.uppercaseLean3 false in\n#align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X\n\n/-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\n dividing out the largest power of X that divides `f`, that is its order-/\ndef divided_by_X_pow_order {f : PowerSeries R} (hf : f \u2260 0) : R\u27e6X\u27e7 :=\n (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose\n\ntheorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R\u27e6X\u27e7} (hf : f \u2260 0) :\n X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f :=\n haveI dvd := X_pow_order_dvd (order_finite_iff_ne_zero.mpr hf)\n (exists_eq_mul_right_of_dvd dvd).choose_spec.symm\n\nend OrderBasic\n\nsection OrderZeroNeOne\n\nvariable [Semiring R] [Nontrivial R]\n\n/-- The order of the formal power series `1` is `0`. -/\n@[simp]\ntheorem order_one : order (1 : R\u27e6X\u27e7) = 0 := by\n simpa using order_monomial_of_ne_zero 0 (1 : R) one_ne_zero\n#align power_series.order_one PowerSeries.order_one\n\n/-- The order of an invertible power series is `0`. -/\n", "theoremStatement": "theorem order_zero_of_unit {f : PowerSeries R} : IsUnit f \u2192 f.order = 0", "theoremName": "order_zero_of_unit", "fileCreated": {"commit": "73d45f44c3", "date": "2024-02-29"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean", "positionMetadata": {"lineInFile": 323, "tokenPositionInFile": 13100, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rintro \u27e8\u27e8u, v, hu, hv\u27e9, hf\u27e9\n apply And.left\n rw [\u2190 add_eq_zero_iff, \u2190 hf, \u2190 nonpos_iff_eq_zero, \u2190 @order_one R _ _, \u2190 hu]\n exact order_mul_ge _ _", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 153}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\n\nimport Mathlib.RingTheory.PowerSeries.Basic\nimport Mathlib.Algebra.CharP.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-! # Formal power series (in one variable) - Order\n\nThe `PowerSeries.order` of a formal power series `\u03c6` is the multiplicity of the variable `X` in `\u03c6`.\n\nIf the coefficients form an integral domain, then `PowerSeries.order` is an\nadditive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`).\n\nWe prove that if the commutative ring `R` of coefficients is an integral domain,\nthen the ring `R\u27e6X\u27e7` of formal power series in one variable over `R`\nis an integral domain.\n\nGiven a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\ndividing out the largest power of X that divides `f`, that is its order. This is useful when\nproving that `R\u27e6X\u27e7` is a normalization monoid, which is done in `PowerSeries.Inverse`.\n\n-/\nnoncomputable section\n\nopen BigOperators Polynomial\n\nopen Finset (antidiagonal mem_antidiagonal)\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection OrderBasic\n\nopen multiplicity\n\nvariable [Semiring R] {\u03c6 : R\u27e6X\u27e7}\n\ntheorem exists_coeff_ne_zero_iff_ne_zero : (\u2203 n : \u2115, coeff R n \u03c6 \u2260 0) \u2194 \u03c6 \u2260 0 := by\n refine' not_iff_not.mp _\n push_neg\n -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?\n simp [PowerSeries.ext_iff, (coeff R _).map_zero]\n#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero\n\n/-- The order of a formal power series `\u03c6` is the greatest `n : PartENat`\nsuch that `X^n` divides `\u03c6`. The order is `\u22a4` if and only if `\u03c6 = 0`. -/\ndef order (\u03c6 : R\u27e6X\u27e7) : PartENat :=\n letI := Classical.decEq R\n letI := Classical.decEq R\u27e6X\u27e7\n if h : \u03c6 = 0 then \u22a4 else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)\n#align power_series.order PowerSeries.order\n\n/-- The order of the `0` power series is infinite. -/\n@[simp]\ntheorem order_zero : order (0 : R\u27e6X\u27e7) = \u22a4 :=\n dif_pos rfl\n#align power_series.order_zero PowerSeries.order_zero\n\ntheorem order_finite_iff_ne_zero : (order \u03c6).Dom \u2194 \u03c6 \u2260 0 := by\n simp only [order]\n constructor\n \u00b7 split_ifs with h <;> intro H\n \u00b7 simp only [PartENat.top_eq_none, Part.not_none_dom] at H\n \u00b7 exact h\n \u00b7 intro h\n simp [h]\n#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero\n\n/-- If the order of a formal power series is finite,\nthen the coefficient indexed by the order is nonzero. -/\ntheorem coeff_order (h : (order \u03c6).Dom) : coeff R (\u03c6.order.get h) \u03c6 \u2260 0 := by\n classical\n simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']\n generalize_proofs h\n exact Nat.find_spec h\n#align power_series.coeff_order PowerSeries.coeff_order\n\n/-- If the `n`th coefficient of a formal power series is nonzero,\nthen the order of the power series is less than or equal to `n`. -/\ntheorem order_le (n : \u2115) (h : coeff R n \u03c6 \u2260 0) : order \u03c6 \u2264 n := by\n classical\n rw [order, dif_neg]\n \u00b7 simp only [PartENat.coe_le_coe]\n exact Nat.find_le h\n \u00b7 exact exists_coeff_ne_zero_iff_ne_zero.mp \u27e8n, h\u27e9\n#align power_series.order_le PowerSeries.order_le\n\n/-- The `n`th coefficient of a formal power series is `0` if `n` is strictly\nsmaller than the order of the power series. -/\ntheorem coeff_of_lt_order (n : \u2115) (h : \u2191n < order \u03c6) : coeff R n \u03c6 = 0 := by\n contrapose! h\n exact order_le _ h\n#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order\n\n/-- The `0` power series is the unique power series with infinite order. -/\n@[simp]\ntheorem order_eq_top {\u03c6 : R\u27e6X\u27e7} : \u03c6.order = \u22a4 \u2194 \u03c6 = 0 :=\n PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left\n#align power_series.order_eq_top PowerSeries.order_eq_top\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem nat_le_order (\u03c6 : R\u27e6X\u27e7) (n : \u2115) (h : \u2200 i < n, coeff R i \u03c6 = 0) : \u2191n \u2264 order \u03c6 := by\n by_contra H; rw [not_le] at H\n have : (order \u03c6).Dom := PartENat.dom_of_le_natCast H.le\n rw [\u2190 PartENat.natCast_get this, PartENat.coe_lt_coe] at H\n exact coeff_order this (h _ H)\n#align power_series.nat_le_order PowerSeries.nat_le_order\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem le_order (\u03c6 : R\u27e6X\u27e7) (n : PartENat) (h : \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0) :\n n \u2264 order \u03c6 := by\n induction n using PartENat.casesOn\n \u00b7 show _ \u2264 _\n rw [top_le_iff, order_eq_top]\n ext i\n exact h _ (PartENat.natCast_lt_top i)\n \u00b7 apply nat_le_order\n simpa only [PartENat.coe_lt_coe] using h\n#align power_series.le_order PowerSeries.le_order\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq_nat {\u03c6 : R\u27e6X\u27e7} {n : \u2115} :\n order \u03c6 = n \u2194 coeff R n \u03c6 \u2260 0 \u2227 \u2200 i, i < n \u2192 coeff R i \u03c6 = 0 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm\n simp [order, dif_neg h\u03c6, Nat.find_eq_iff]\n#align power_series.order_eq_nat PowerSeries.order_eq_nat\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq {\u03c6 : R\u27e6X\u27e7} {n : PartENat} :\n order \u03c6 = n \u2194 (\u2200 i : \u2115, \u2191i = n \u2192 coeff R i \u03c6 \u2260 0) \u2227 \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0 := by\n induction n using PartENat.casesOn\n \u00b7 rw [order_eq_top]\n constructor\n \u00b7 rintro rfl\n constructor <;> intros\n \u00b7 exfalso\n exact PartENat.natCast_ne_top \u2039_\u203a \u2039_\u203a\n \u00b7 exact (coeff _ _).map_zero\n \u00b7 rintro \u27e8_h\u2081, h\u2082\u27e9\n ext i\n exact h\u2082 i (PartENat.natCast_lt_top i)\n \u00b7 simpa [PartENat.natCast_inj] using order_eq_nat\n#align power_series.order_eq PowerSeries.order_eq\n\n/-- The order of the sum of two formal power series\n is at least the minimum of their orders. -/\ntheorem le_order_add (\u03c6 \u03c8 : R\u27e6X\u27e7) : min (order \u03c6) (order \u03c8) \u2264 order (\u03c6 + \u03c8) := by\n refine' le_order _ _ _\n simp (config := { contextual := true }) [coeff_of_lt_order]\n#align power_series.le_order_add PowerSeries.le_order_add\n\nprivate theorem order_add_of_order_eq.aux (\u03c6 \u03c8 : R\u27e6X\u27e7) (_h : order \u03c6 \u2260 order \u03c8)\n (H : order \u03c6 < order \u03c8) : order (\u03c6 + \u03c8) \u2264 order \u03c6 \u2293 order \u03c8 := by\n suffices order (\u03c6 + \u03c8) = order \u03c6 by\n rw [le_inf_iff, this]\n exact \u27e8le_rfl, le_of_lt H\u27e9\n \u00b7 rw [order_eq]\n constructor\n \u00b7 intro i hi\n rw [\u2190 hi] at H\n rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero]\n exact (order_eq_nat.1 hi.symm).1\n \u00b7 intro i hi\n rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H),\n zero_add]\n-- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux\n\n/-- The order of the sum of two formal power series\n is the minimum of their orders if their orders differ. -/\ntheorem order_add_of_order_eq (\u03c6 \u03c8 : R\u27e6X\u27e7) (h : order \u03c6 \u2260 order \u03c8) :\n order (\u03c6 + \u03c8) = order \u03c6 \u2293 order \u03c8 := by\n refine' le_antisymm _ (le_order_add _ _)\n by_cases H\u2081 : order \u03c6 < order \u03c8\n \u00b7 apply order_add_of_order_eq.aux _ _ h H\u2081\n by_cases H\u2082 : order \u03c8 < order \u03c6\n \u00b7 simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H\u2082\n exfalso; exact h (le_antisymm (not_lt.1 H\u2082) (not_lt.1 H\u2081))\n#align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq\n\n/-- The order of the product of two formal power series\n is at least the sum of their orders. -/\ntheorem order_mul_ge (\u03c6 \u03c8 : R\u27e6X\u27e7) : order \u03c6 + order \u03c8 \u2264 order (\u03c6 * \u03c8) := by\n apply le_order\n intro n hn; rw [coeff_mul, Finset.sum_eq_zero]\n rintro \u27e8i, j\u27e9 hij\n by_cases hi : \u2191i < order \u03c6\n \u00b7 rw [coeff_of_lt_order i hi, zero_mul]\n by_cases hj : \u2191j < order \u03c8\n \u00b7 rw [coeff_of_lt_order j hj, mul_zero]\n rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij\n exfalso\n apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj)\n rw [\u2190 Nat.cast_add, hij]\n#align power_series.order_mul_ge PowerSeries.order_mul_ge\n\n/-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/\ntheorem order_monomial (n : \u2115) (a : R) [Decidable (a = 0)] :\n order (monomial R n a) = if a = 0 then (\u22a4 : PartENat) else n := by\n split_ifs with h\n \u00b7 rw [h, order_eq_top, LinearMap.map_zero]\n \u00b7 rw [order_eq]\n constructor <;> intro i hi\n \u00b7 rw [PartENat.natCast_inj] at hi\n rwa [hi, coeff_monomial_same]\n \u00b7 rw [PartENat.coe_lt_coe] at hi\n rw [coeff_monomial, if_neg]\n exact ne_of_lt hi\n#align power_series.order_monomial PowerSeries.order_monomial\n\n/-- The order of the monomial `a*X^n` is `n` if `a \u2260 0`. -/\ntheorem order_monomial_of_ne_zero (n : \u2115) (a : R) (h : a \u2260 0) : order (monomial R n a) = n := by\n classical\n rw [order_monomial, if_neg h]\n#align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero\n\n/-- If `n` is strictly smaller than the order of `\u03c8`, then the `n`th coefficient of its product\nwith any other power series is `0`. -/\ntheorem coeff_mul_of_lt_order {\u03c6 \u03c8 : R\u27e6X\u27e7} {n : \u2115} (h : \u2191n < \u03c8.order) :\n coeff R n (\u03c6 * \u03c8) = 0 := by\n suffices coeff R n (\u03c6 * \u03c8) = \u2211 p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero]\n rw [coeff_mul]\n apply Finset.sum_congr rfl\n intro x hx\n refine' mul_eq_zero_of_right (coeff R x.fst \u03c6) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h))\n rw [mem_antidiagonal] at hx\n norm_cast\n omega\n#align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order\n\ntheorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {\u03c6 \u03c8 : R\u27e6X\u27e7} (n : \u2115)\n (h : \u2191n < \u03c8.order) : coeff R n (\u03c6 * (1 - \u03c8)) = coeff R n \u03c6 := by\n simp [coeff_mul_of_lt_order h, mul_sub]\n#align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order\n\ntheorem coeff_mul_prod_one_sub_of_lt_order {R \u03b9 : Type*} [CommRing R] (k : \u2115) (s : Finset \u03b9)\n (\u03c6 : R\u27e6X\u27e7) (f : \u03b9 \u2192 R\u27e6X\u27e7) :\n (\u2200 i \u2208 s, \u2191k < (f i).order) \u2192 coeff R k (\u03c6 * \u220f i in s, (1 - f i)) = coeff R k \u03c6 := by\n classical\n induction' s using Finset.induction_on with a s ha ih t\n \u00b7 simp\n \u00b7 intro t\n simp only [Finset.mem_insert, forall_eq_or_imp] at t\n rw [Finset.prod_insert ha, \u2190 mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1]\n exact ih t.2\n#align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order\n\n-- TODO: link with `X_pow_dvd_iff`\ntheorem X_pow_order_dvd (h : (order \u03c6).Dom) : X ^ (order \u03c6).get h \u2223 \u03c6 := by\n refine' \u27e8PowerSeries.mk fun n => coeff R (n + (order \u03c6).get h) \u03c6, _\u27e9\n ext n\n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite,\n Finset.sum_const_zero, add_zero]\n rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order \u03c6) h)]\n split_ifs with hn\n \u00b7 simp [tsub_add_cancel_of_le hn]\n \u00b7 simp only [Finset.sum_empty]\n refine' coeff_of_lt_order _ _\n simpa [PartENat.coe_lt_iff] using fun _ => hn\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd\n\ntheorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R\u27e6X\u27e7 (\u00b7 \u2223 \u00b7)] (\u03c6 : R\u27e6X\u27e7) :\n order \u03c6 = multiplicity X \u03c6 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simp\n induction' ho : order \u03c6 using PartENat.casesOn with n\n \u00b7 simp [h\u03c6] at ho\n have hn : \u03c6.order.get (order_finite_iff_ne_zero.mpr h\u03c6) = n := by simp [ho]\n rw [\u2190 hn]\n refine'\n le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr h\u03c6))\n (PartENat.find_le _ _ _)\n rintro \u27e8\u03c8, H\u27e9\n have := congr_arg (coeff R n) H\n rw [\u2190 (\u03c8.commute_X.pow_right _).eq, coeff_mul_of_lt_order, \u2190 hn] at this\n \u00b7 exact coeff_order _ this\n \u00b7 rw [X_pow_eq, order_monomial]\n split_ifs\n \u00b7 exact PartENat.natCast_lt_top _\n \u00b7 rw [\u2190 hn, PartENat.coe_lt_coe]\n exact Nat.lt_succ_self _\nset_option linter.uppercaseLean3 false in\n#align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X\n\n/-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\n dividing out the largest power of X that divides `f`, that is its order-/\ndef divided_by_X_pow_order {f : PowerSeries R} (hf : f \u2260 0) : R\u27e6X\u27e7 :=\n (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose\n\ntheorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R\u27e6X\u27e7} (hf : f \u2260 0) :\n X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f :=\n haveI dvd := X_pow_order_dvd (order_finite_iff_ne_zero.mpr hf)\n (exists_eq_mul_right_of_dvd dvd).choose_spec.symm\n\nend OrderBasic\n\nsection OrderZeroNeOne\n\nvariable [Semiring R] [Nontrivial R]\n\n/-- The order of the formal power series `1` is `0`. -/\n@[simp]\ntheorem order_one : order (1 : R\u27e6X\u27e7) = 0 := by\n simpa using order_monomial_of_ne_zero 0 (1 : R) one_ne_zero\n#align power_series.order_one PowerSeries.order_one\n\n/-- The order of an invertible power series is `0`. -/\ntheorem order_zero_of_unit {f : PowerSeries R} : IsUnit f \u2192 f.order = 0 := by\n rintro \u27e8\u27e8u, v, hu, hv\u27e9, hf\u27e9\n apply And.left\n rw [\u2190 add_eq_zero_iff, \u2190 hf, \u2190 nonpos_iff_eq_zero, \u2190 @order_one R _ _, \u2190 hu]\n exact order_mul_ge _ _\n\n/-- The order of the formal power series `X` is `1`. -/\n@[simp]\ntheorem order_X : order (X : R\u27e6X\u27e7) = 1 := by\n simpa only [Nat.cast_one] using order_monomial_of_ne_zero 1 (1 : R) one_ne_zero\nset_option linter.uppercaseLean3 false in\n#align power_series.order_X PowerSeries.order_X\n\n/-- The order of the formal power series `X^n` is `n`. -/\n@[simp]\ntheorem order_X_pow (n : \u2115) : order ((X : R\u27e6X\u27e7) ^ n) = n := by\n rw [X_pow_eq, order_monomial_of_ne_zero]\n exact one_ne_zero\nset_option linter.uppercaseLean3 false in\n#align power_series.order_X_pow PowerSeries.order_X_pow\n\nend OrderZeroNeOne\n\nsection OrderIsDomain\n\n-- TODO: generalize to `[Semiring R] [NoZeroDivisors R]`\nvariable [CommRing R] [IsDomain R]\n\n/-- The order of the product of two formal power series over an integral domain\n is the sum of their orders. -/\ntheorem order_mul (\u03c6 \u03c8 : R\u27e6X\u27e7) : order (\u03c6 * \u03c8) = order \u03c6 + order \u03c8 := by\n classical\n simp_rw [order_eq_multiplicity_X]\n exact multiplicity.mul X_prime\n#align power_series.order_mul PowerSeries.order_mul\n\n-- Dividing `X` by the maximal power of `X` dividing it leaves `1`.\n", "theoremStatement": "@[simp]\ntheorem divided_by_X_pow_order_of_X_eq_one : divided_by_X_pow_order X_ne_zero = (1 : R\u27e6X\u27e7)", "theoremName": "divided_by_X_pow_order_of_X_eq_one", "fileCreated": {"commit": "73d45f44c3", "date": "2024-02-29"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean", "positionMetadata": {"lineInFile": 360, "tokenPositionInFile": 14426, "theoremPositionInFile": 30}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rw [\u2190 mul_eq_left\u2080 X_ne_zero]\n simpa only [order_X, X_ne_zero, PartENat.get_one, pow_one, Ne.def,\n not_false_iff] using self_eq_X_pow_order_mul_divided_by_X_pow_order (@X_ne_zero R _ _)", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 194}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\n\nimport Mathlib.RingTheory.PowerSeries.Basic\nimport Mathlib.Algebra.CharP.Basic\n\n#align_import ring_theory.power_series.basic from \"leanprover-community/mathlib\"@\"2d5739b61641ee4e7e53eca5688a08f66f2e6a60\"\n\n/-! # Formal power series (in one variable) - Order\n\nThe `PowerSeries.order` of a formal power series `\u03c6` is the multiplicity of the variable `X` in `\u03c6`.\n\nIf the coefficients form an integral domain, then `PowerSeries.order` is an\nadditive valuation (`PowerSeries.order_mul`, `PowerSeries.le_order_add`).\n\nWe prove that if the commutative ring `R` of coefficients is an integral domain,\nthen the ring `R\u27e6X\u27e7` of formal power series in one variable over `R`\nis an integral domain.\n\nGiven a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\ndividing out the largest power of X that divides `f`, that is its order. This is useful when\nproving that `R\u27e6X\u27e7` is a normalization monoid, which is done in `PowerSeries.Inverse`.\n\n-/\nnoncomputable section\n\nopen BigOperators Polynomial\n\nopen Finset (antidiagonal mem_antidiagonal)\n\nnamespace PowerSeries\n\nopen Finsupp (single)\n\nvariable {R : Type*}\n\nsection OrderBasic\n\nopen multiplicity\n\nvariable [Semiring R] {\u03c6 : R\u27e6X\u27e7}\n\ntheorem exists_coeff_ne_zero_iff_ne_zero : (\u2203 n : \u2115, coeff R n \u03c6 \u2260 0) \u2194 \u03c6 \u2260 0 := by\n refine' not_iff_not.mp _\n push_neg\n -- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?\n simp [PowerSeries.ext_iff, (coeff R _).map_zero]\n#align power_series.exists_coeff_ne_zero_iff_ne_zero PowerSeries.exists_coeff_ne_zero_iff_ne_zero\n\n/-- The order of a formal power series `\u03c6` is the greatest `n : PartENat`\nsuch that `X^n` divides `\u03c6`. The order is `\u22a4` if and only if `\u03c6 = 0`. -/\ndef order (\u03c6 : R\u27e6X\u27e7) : PartENat :=\n letI := Classical.decEq R\n letI := Classical.decEq R\u27e6X\u27e7\n if h : \u03c6 = 0 then \u22a4 else Nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h)\n#align power_series.order PowerSeries.order\n\n/-- The order of the `0` power series is infinite. -/\n@[simp]\ntheorem order_zero : order (0 : R\u27e6X\u27e7) = \u22a4 :=\n dif_pos rfl\n#align power_series.order_zero PowerSeries.order_zero\n\ntheorem order_finite_iff_ne_zero : (order \u03c6).Dom \u2194 \u03c6 \u2260 0 := by\n simp only [order]\n constructor\n \u00b7 split_ifs with h <;> intro H\n \u00b7 simp only [PartENat.top_eq_none, Part.not_none_dom] at H\n \u00b7 exact h\n \u00b7 intro h\n simp [h]\n#align power_series.order_finite_iff_ne_zero PowerSeries.order_finite_iff_ne_zero\n\n/-- If the order of a formal power series is finite,\nthen the coefficient indexed by the order is nonzero. -/\ntheorem coeff_order (h : (order \u03c6).Dom) : coeff R (\u03c6.order.get h) \u03c6 \u2260 0 := by\n classical\n simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, PartENat.get_natCast']\n generalize_proofs h\n exact Nat.find_spec h\n#align power_series.coeff_order PowerSeries.coeff_order\n\n/-- If the `n`th coefficient of a formal power series is nonzero,\nthen the order of the power series is less than or equal to `n`. -/\ntheorem order_le (n : \u2115) (h : coeff R n \u03c6 \u2260 0) : order \u03c6 \u2264 n := by\n classical\n rw [order, dif_neg]\n \u00b7 simp only [PartENat.coe_le_coe]\n exact Nat.find_le h\n \u00b7 exact exists_coeff_ne_zero_iff_ne_zero.mp \u27e8n, h\u27e9\n#align power_series.order_le PowerSeries.order_le\n\n/-- The `n`th coefficient of a formal power series is `0` if `n` is strictly\nsmaller than the order of the power series. -/\ntheorem coeff_of_lt_order (n : \u2115) (h : \u2191n < order \u03c6) : coeff R n \u03c6 = 0 := by\n contrapose! h\n exact order_le _ h\n#align power_series.coeff_of_lt_order PowerSeries.coeff_of_lt_order\n\n/-- The `0` power series is the unique power series with infinite order. -/\n@[simp]\ntheorem order_eq_top {\u03c6 : R\u27e6X\u27e7} : \u03c6.order = \u22a4 \u2194 \u03c6 = 0 :=\n PartENat.not_dom_iff_eq_top.symm.trans order_finite_iff_ne_zero.not_left\n#align power_series.order_eq_top PowerSeries.order_eq_top\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem nat_le_order (\u03c6 : R\u27e6X\u27e7) (n : \u2115) (h : \u2200 i < n, coeff R i \u03c6 = 0) : \u2191n \u2264 order \u03c6 := by\n by_contra H; rw [not_le] at H\n have : (order \u03c6).Dom := PartENat.dom_of_le_natCast H.le\n rw [\u2190 PartENat.natCast_get this, PartENat.coe_lt_coe] at H\n exact coeff_order this (h _ H)\n#align power_series.nat_le_order PowerSeries.nat_le_order\n\n/-- The order of a formal power series is at least `n` if\nthe `i`th coefficient is `0` for all `i < n`. -/\ntheorem le_order (\u03c6 : R\u27e6X\u27e7) (n : PartENat) (h : \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0) :\n n \u2264 order \u03c6 := by\n induction n using PartENat.casesOn\n \u00b7 show _ \u2264 _\n rw [top_le_iff, order_eq_top]\n ext i\n exact h _ (PartENat.natCast_lt_top i)\n \u00b7 apply nat_le_order\n simpa only [PartENat.coe_lt_coe] using h\n#align power_series.le_order PowerSeries.le_order\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq_nat {\u03c6 : R\u27e6X\u27e7} {n : \u2115} :\n order \u03c6 = n \u2194 coeff R n \u03c6 \u2260 0 \u2227 \u2200 i, i < n \u2192 coeff R i \u03c6 = 0 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simpa [(coeff R _).map_zero] using (PartENat.natCast_ne_top _).symm\n simp [order, dif_neg h\u03c6, Nat.find_eq_iff]\n#align power_series.order_eq_nat PowerSeries.order_eq_nat\n\n/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,\nand the `i`th coefficient is `0` for all `i < n`. -/\ntheorem order_eq {\u03c6 : R\u27e6X\u27e7} {n : PartENat} :\n order \u03c6 = n \u2194 (\u2200 i : \u2115, \u2191i = n \u2192 coeff R i \u03c6 \u2260 0) \u2227 \u2200 i : \u2115, \u2191i < n \u2192 coeff R i \u03c6 = 0 := by\n induction n using PartENat.casesOn\n \u00b7 rw [order_eq_top]\n constructor\n \u00b7 rintro rfl\n constructor <;> intros\n \u00b7 exfalso\n exact PartENat.natCast_ne_top \u2039_\u203a \u2039_\u203a\n \u00b7 exact (coeff _ _).map_zero\n \u00b7 rintro \u27e8_h\u2081, h\u2082\u27e9\n ext i\n exact h\u2082 i (PartENat.natCast_lt_top i)\n \u00b7 simpa [PartENat.natCast_inj] using order_eq_nat\n#align power_series.order_eq PowerSeries.order_eq\n\n/-- The order of the sum of two formal power series\n is at least the minimum of their orders. -/\ntheorem le_order_add (\u03c6 \u03c8 : R\u27e6X\u27e7) : min (order \u03c6) (order \u03c8) \u2264 order (\u03c6 + \u03c8) := by\n refine' le_order _ _ _\n simp (config := { contextual := true }) [coeff_of_lt_order]\n#align power_series.le_order_add PowerSeries.le_order_add\n\nprivate theorem order_add_of_order_eq.aux (\u03c6 \u03c8 : R\u27e6X\u27e7) (_h : order \u03c6 \u2260 order \u03c8)\n (H : order \u03c6 < order \u03c8) : order (\u03c6 + \u03c8) \u2264 order \u03c6 \u2293 order \u03c8 := by\n suffices order (\u03c6 + \u03c8) = order \u03c6 by\n rw [le_inf_iff, this]\n exact \u27e8le_rfl, le_of_lt H\u27e9\n \u00b7 rw [order_eq]\n constructor\n \u00b7 intro i hi\n rw [\u2190 hi] at H\n rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero]\n exact (order_eq_nat.1 hi.symm).1\n \u00b7 intro i hi\n rw [(coeff _ _).map_add, coeff_of_lt_order i hi, coeff_of_lt_order i (lt_trans hi H),\n zero_add]\n-- #align power_series.order_add_of_order_eq.aux power_series.order_add_of_order_eq.aux\n\n/-- The order of the sum of two formal power series\n is the minimum of their orders if their orders differ. -/\ntheorem order_add_of_order_eq (\u03c6 \u03c8 : R\u27e6X\u27e7) (h : order \u03c6 \u2260 order \u03c8) :\n order (\u03c6 + \u03c8) = order \u03c6 \u2293 order \u03c8 := by\n refine' le_antisymm _ (le_order_add _ _)\n by_cases H\u2081 : order \u03c6 < order \u03c8\n \u00b7 apply order_add_of_order_eq.aux _ _ h H\u2081\n by_cases H\u2082 : order \u03c8 < order \u03c6\n \u00b7 simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H\u2082\n exfalso; exact h (le_antisymm (not_lt.1 H\u2082) (not_lt.1 H\u2081))\n#align power_series.order_add_of_order_eq PowerSeries.order_add_of_order_eq\n\n/-- The order of the product of two formal power series\n is at least the sum of their orders. -/\ntheorem order_mul_ge (\u03c6 \u03c8 : R\u27e6X\u27e7) : order \u03c6 + order \u03c8 \u2264 order (\u03c6 * \u03c8) := by\n apply le_order\n intro n hn; rw [coeff_mul, Finset.sum_eq_zero]\n rintro \u27e8i, j\u27e9 hij\n by_cases hi : \u2191i < order \u03c6\n \u00b7 rw [coeff_of_lt_order i hi, zero_mul]\n by_cases hj : \u2191j < order \u03c8\n \u00b7 rw [coeff_of_lt_order j hj, mul_zero]\n rw [not_lt] at hi hj; rw [mem_antidiagonal] at hij\n exfalso\n apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj)\n rw [\u2190 Nat.cast_add, hij]\n#align power_series.order_mul_ge PowerSeries.order_mul_ge\n\n/-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/\ntheorem order_monomial (n : \u2115) (a : R) [Decidable (a = 0)] :\n order (monomial R n a) = if a = 0 then (\u22a4 : PartENat) else n := by\n split_ifs with h\n \u00b7 rw [h, order_eq_top, LinearMap.map_zero]\n \u00b7 rw [order_eq]\n constructor <;> intro i hi\n \u00b7 rw [PartENat.natCast_inj] at hi\n rwa [hi, coeff_monomial_same]\n \u00b7 rw [PartENat.coe_lt_coe] at hi\n rw [coeff_monomial, if_neg]\n exact ne_of_lt hi\n#align power_series.order_monomial PowerSeries.order_monomial\n\n/-- The order of the monomial `a*X^n` is `n` if `a \u2260 0`. -/\ntheorem order_monomial_of_ne_zero (n : \u2115) (a : R) (h : a \u2260 0) : order (monomial R n a) = n := by\n classical\n rw [order_monomial, if_neg h]\n#align power_series.order_monomial_of_ne_zero PowerSeries.order_monomial_of_ne_zero\n\n/-- If `n` is strictly smaller than the order of `\u03c8`, then the `n`th coefficient of its product\nwith any other power series is `0`. -/\ntheorem coeff_mul_of_lt_order {\u03c6 \u03c8 : R\u27e6X\u27e7} {n : \u2115} (h : \u2191n < \u03c8.order) :\n coeff R n (\u03c6 * \u03c8) = 0 := by\n suffices coeff R n (\u03c6 * \u03c8) = \u2211 p in antidiagonal n, 0 by rw [this, Finset.sum_const_zero]\n rw [coeff_mul]\n apply Finset.sum_congr rfl\n intro x hx\n refine' mul_eq_zero_of_right (coeff R x.fst \u03c6) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h))\n rw [mem_antidiagonal] at hx\n norm_cast\n omega\n#align power_series.coeff_mul_of_lt_order PowerSeries.coeff_mul_of_lt_order\n\ntheorem coeff_mul_one_sub_of_lt_order {R : Type*} [CommRing R] {\u03c6 \u03c8 : R\u27e6X\u27e7} (n : \u2115)\n (h : \u2191n < \u03c8.order) : coeff R n (\u03c6 * (1 - \u03c8)) = coeff R n \u03c6 := by\n simp [coeff_mul_of_lt_order h, mul_sub]\n#align power_series.coeff_mul_one_sub_of_lt_order PowerSeries.coeff_mul_one_sub_of_lt_order\n\ntheorem coeff_mul_prod_one_sub_of_lt_order {R \u03b9 : Type*} [CommRing R] (k : \u2115) (s : Finset \u03b9)\n (\u03c6 : R\u27e6X\u27e7) (f : \u03b9 \u2192 R\u27e6X\u27e7) :\n (\u2200 i \u2208 s, \u2191k < (f i).order) \u2192 coeff R k (\u03c6 * \u220f i in s, (1 - f i)) = coeff R k \u03c6 := by\n classical\n induction' s using Finset.induction_on with a s ha ih t\n \u00b7 simp\n \u00b7 intro t\n simp only [Finset.mem_insert, forall_eq_or_imp] at t\n rw [Finset.prod_insert ha, \u2190 mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1]\n exact ih t.2\n#align power_series.coeff_mul_prod_one_sub_of_lt_order PowerSeries.coeff_mul_prod_one_sub_of_lt_order\n\n-- TODO: link with `X_pow_dvd_iff`\ntheorem X_pow_order_dvd (h : (order \u03c6).Dom) : X ^ (order \u03c6).get h \u2223 \u03c6 := by\n refine' \u27e8PowerSeries.mk fun n => coeff R (n + (order \u03c6).get h) \u03c6, _\u27e9\n ext n\n simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, Finset.sum_ite,\n Finset.sum_const_zero, add_zero]\n rw [Finset.filter_fst_eq_antidiagonal n (Part.get (order \u03c6) h)]\n split_ifs with hn\n \u00b7 simp [tsub_add_cancel_of_le hn]\n \u00b7 simp only [Finset.sum_empty]\n refine' coeff_of_lt_order _ _\n simpa [PartENat.coe_lt_iff] using fun _ => hn\nset_option linter.uppercaseLean3 false in\n#align power_series.X_pow_order_dvd PowerSeries.X_pow_order_dvd\n\ntheorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R\u27e6X\u27e7 (\u00b7 \u2223 \u00b7)] (\u03c6 : R\u27e6X\u27e7) :\n order \u03c6 = multiplicity X \u03c6 := by\n classical\n rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)\n \u00b7 simp\n induction' ho : order \u03c6 using PartENat.casesOn with n\n \u00b7 simp [h\u03c6] at ho\n have hn : \u03c6.order.get (order_finite_iff_ne_zero.mpr h\u03c6) = n := by simp [ho]\n rw [\u2190 hn]\n refine'\n le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr h\u03c6))\n (PartENat.find_le _ _ _)\n rintro \u27e8\u03c8, H\u27e9\n have := congr_arg (coeff R n) H\n rw [\u2190 (\u03c8.commute_X.pow_right _).eq, coeff_mul_of_lt_order, \u2190 hn] at this\n \u00b7 exact coeff_order _ this\n \u00b7 rw [X_pow_eq, order_monomial]\n split_ifs\n \u00b7 exact PartENat.natCast_lt_top _\n \u00b7 rw [\u2190 hn, PartENat.coe_lt_coe]\n exact Nat.lt_succ_self _\nset_option linter.uppercaseLean3 false in\n#align power_series.order_eq_multiplicity_X PowerSeries.order_eq_multiplicity_X\n\n/-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by\n dividing out the largest power of X that divides `f`, that is its order-/\ndef divided_by_X_pow_order {f : PowerSeries R} (hf : f \u2260 0) : R\u27e6X\u27e7 :=\n (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose\n\ntheorem self_eq_X_pow_order_mul_divided_by_X_pow_order {f : R\u27e6X\u27e7} (hf : f \u2260 0) :\n X ^ f.order.get (order_finite_iff_ne_zero.mpr hf) * divided_by_X_pow_order hf = f :=\n haveI dvd := X_pow_order_dvd (order_finite_iff_ne_zero.mpr hf)\n (exists_eq_mul_right_of_dvd dvd).choose_spec.symm\n\nend OrderBasic\n\nsection OrderZeroNeOne\n\nvariable [Semiring R] [Nontrivial R]\n\n/-- The order of the formal power series `1` is `0`. -/\n@[simp]\ntheorem order_one : order (1 : R\u27e6X\u27e7) = 0 := by\n simpa using order_monomial_of_ne_zero 0 (1 : R) one_ne_zero\n#align power_series.order_one PowerSeries.order_one\n\n/-- The order of an invertible power series is `0`. -/\ntheorem order_zero_of_unit {f : PowerSeries R} : IsUnit f \u2192 f.order = 0 := by\n rintro \u27e8\u27e8u, v, hu, hv\u27e9, hf\u27e9\n apply And.left\n rw [\u2190 add_eq_zero_iff, \u2190 hf, \u2190 nonpos_iff_eq_zero, \u2190 @order_one R _ _, \u2190 hu]\n exact order_mul_ge _ _\n\n/-- The order of the formal power series `X` is `1`. -/\n@[simp]\ntheorem order_X : order (X : R\u27e6X\u27e7) = 1 := by\n simpa only [Nat.cast_one] using order_monomial_of_ne_zero 1 (1 : R) one_ne_zero\nset_option linter.uppercaseLean3 false in\n#align power_series.order_X PowerSeries.order_X\n\n/-- The order of the formal power series `X^n` is `n`. -/\n@[simp]\ntheorem order_X_pow (n : \u2115) : order ((X : R\u27e6X\u27e7) ^ n) = n := by\n rw [X_pow_eq, order_monomial_of_ne_zero]\n exact one_ne_zero\nset_option linter.uppercaseLean3 false in\n#align power_series.order_X_pow PowerSeries.order_X_pow\n\nend OrderZeroNeOne\n\nsection OrderIsDomain\n\n-- TODO: generalize to `[Semiring R] [NoZeroDivisors R]`\nvariable [CommRing R] [IsDomain R]\n\n/-- The order of the product of two formal power series over an integral domain\n is the sum of their orders. -/\ntheorem order_mul (\u03c6 \u03c8 : R\u27e6X\u27e7) : order (\u03c6 * \u03c8) = order \u03c6 + order \u03c8 := by\n classical\n simp_rw [order_eq_multiplicity_X]\n exact multiplicity.mul X_prime\n#align power_series.order_mul PowerSeries.order_mul\n\n-- Dividing `X` by the maximal power of `X` dividing it leaves `1`.\n@[simp]\ntheorem divided_by_X_pow_order_of_X_eq_one : divided_by_X_pow_order X_ne_zero = (1 : R\u27e6X\u27e7) := by\n rw [\u2190 mul_eq_left\u2080 X_ne_zero]\n simpa only [order_X, X_ne_zero, PartENat.get_one, pow_one, Ne.def,\n not_false_iff] using self_eq_X_pow_order_mul_divided_by_X_pow_order (@X_ne_zero R _ _)\n\n-- Dividing a power series by the maximal power of `X` dividing it, respects multiplication.\n", "theoremStatement": "theorem divided_by_X_pow_orderMul {f g : R\u27e6X\u27e7} (hf : f \u2260 0) (hg : g \u2260 0) :\n divided_by_X_pow_order hf * divided_by_X_pow_order hg =\n divided_by_X_pow_order (mul_ne_zero hf hg)", "theoremName": "divided_by_X_pow_orderMul", "fileCreated": {"commit": "73d45f44c3", "date": "2024-02-29"}, "theoremCreated": {"commit": "85a47191ab", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/PowerSeries/Order.lean", "positionMetadata": {"lineInFile": 367, "tokenPositionInFile": 14817, "theoremPositionInFile": 31}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n set df := f.order.get (order_finite_iff_ne_zero.mpr hf)\n set dg := g.order.get (order_finite_iff_ne_zero.mpr hg)\n set dfg := (f * g).order.get (order_finite_iff_ne_zero.mpr (mul_ne_zero hf hg)) with hdfg\n have H_add_d : df + dg = dfg := by simp_all only [PartENat.get_add, order_mul f g]\n have H := self_eq_X_pow_order_mul_divided_by_X_pow_order (mul_ne_zero hf hg)\n have : f * g = X ^ dfg * (divided_by_X_pow_order hf * divided_by_X_pow_order hg) := by\n calc\n f * g = X ^ df * divided_by_X_pow_order hf * (X ^ dg * divided_by_X_pow_order hg) := by\n rw [self_eq_X_pow_order_mul_divided_by_X_pow_order,\n self_eq_X_pow_order_mul_divided_by_X_pow_order]\n _ = X ^ df * X ^ dg * divided_by_X_pow_order hf * divided_by_X_pow_order hg := by ring\n _ = X ^ (df + dg) * divided_by_X_pow_order hf * divided_by_X_pow_order hg := by rw [pow_add]\n _ = X ^ dfg * divided_by_X_pow_order hf * divided_by_X_pow_order hg := by rw [H_add_d]\n _ = X ^ dfg * (divided_by_X_pow_order hf * divided_by_X_pow_order hg) := by rw [mul_assoc]\n simp [\u2190 hdfg, this] at H\n refine' (IsLeftCancelMulZero.mul_left_cancel_of_ne_zero (pow_ne_zero dfg X_ne_zero) _).symm\n convert H", "proofType": "tactic", "proofLengthLines": 18, "proofLengthTokens": 1199}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2021 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Subgraph\nimport Mathlib.Data.List.Rotate\n\n#align_import combinatorics.simple_graph.connectivity from \"leanprover-community/mathlib\"@\"b99e2d58a5e6861833fa8de11e51a81144258db4\"\n\n/-!\n\n# Graph connectivity\n\nIn a simple graph,\n\n* A *walk* is a finite sequence of adjacent vertices, and can be\n thought of equally well as a sequence of directed edges.\n\n* A *trail* is a walk whose edges each appear no more than once.\n\n* A *path* is a trail whose vertices appear no more than once.\n\n* A *cycle* is a nonempty trail whose first and last vertices are the\n same and whose vertices except for the first appear no more than once.\n\n**Warning:** graph theorists mean something different by \"path\" than\ndo homotopy theorists. A \"walk\" in graph theory is a \"path\" in\nhomotopy theory. Another warning: some graph theorists use \"path\" and\n\"simple path\" for \"walk\" and \"path.\"\n\nSome definitions and theorems have inspiration from multigraph\ncounterparts in [Chou1994].\n\n## Main definitions\n\n* `SimpleGraph.Walk` (with accompanying pattern definitions\n `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'`)\n\n* `SimpleGraph.Walk.IsTrail`, `SimpleGraph.Walk.IsPath`, and `SimpleGraph.Walk.IsCycle`.\n\n* `SimpleGraph.Path`\n\n* `SimpleGraph.Walk.map` and `SimpleGraph.Path.map` for the induced map on walks,\n given an (injective) graph homomorphism.\n\n* `SimpleGraph.Reachable` for the relation of whether there exists\n a walk between a given pair of vertices\n\n* `SimpleGraph.Preconnected` and `SimpleGraph.Connected` are predicates\n on simple graphs for whether every vertex can be reached from every other,\n and in the latter case, whether the vertex type is nonempty.\n\n* `SimpleGraph.ConnectedComponent` is the type of connected components of\n a given graph.\n\n* `SimpleGraph.IsBridge` for whether an edge is a bridge edge\n\n## Main statements\n\n* `SimpleGraph.isBridge_iff_mem_and_forall_cycle_not_mem` characterizes bridge edges in terms of\n there being no cycle containing them.\n\n## Tags\nwalks, trails, paths, circuits, cycles, bridge edges\n\n-/\n\nopen Function\n\nuniverse u v w\n\nnamespace SimpleGraph\n\nvariable {V : Type u} {V' : Type v} {V'' : Type w}\nvariable (G : SimpleGraph V) (G' : SimpleGraph V') (G'' : SimpleGraph V'')\n\n/-- A walk is a sequence of adjacent vertices. For vertices `u v : V`,\nthe type `walk u v` consists of all walks starting at `u` and ending at `v`.\n\nWe say that a walk *visits* the vertices it contains. The set of vertices a\nwalk visits is `SimpleGraph.Walk.support`.\n\nSee `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'` for patterns that\ncan be useful in definitions since they make the vertices explicit. -/\ninductive Walk : V \u2192 V \u2192 Type u\n | nil {u : V} : Walk u u\n | cons {u v w : V} (h : G.Adj u v) (p : Walk v w) : Walk u w\n deriving DecidableEq\n#align simple_graph.walk SimpleGraph.Walk\n\nattribute [refl] Walk.nil\n\n@[simps]\ninstance Walk.instInhabited (v : V) : Inhabited (G.Walk v v) := \u27e8Walk.nil\u27e9\n#align simple_graph.walk.inhabited SimpleGraph.Walk.instInhabited\n\n/-- The one-edge walk associated to a pair of adjacent vertices. -/\n@[match_pattern, reducible]\ndef Adj.toWalk {G : SimpleGraph V} {u v : V} (h : G.Adj u v) : G.Walk u v :=\n Walk.cons h Walk.nil\n#align simple_graph.adj.to_walk SimpleGraph.Adj.toWalk\n\nnamespace Walk\n\nvariable {G}\n\n/-- Pattern to get `Walk.nil` with the vertex as an explicit argument. -/\n@[match_pattern]\nabbrev nil' (u : V) : G.Walk u u := Walk.nil\n#align simple_graph.walk.nil' SimpleGraph.Walk.nil'\n\n/-- Pattern to get `Walk.cons` with the vertices as explicit arguments. -/\n@[match_pattern]\nabbrev cons' (u v w : V) (h : G.Adj u v) (p : G.Walk v w) : G.Walk u w := Walk.cons h p\n#align simple_graph.walk.cons' SimpleGraph.Walk.cons'\n\n/-- Change the endpoints of a walk using equalities. This is helpful for relaxing\ndefinitional equality constraints and to be able to state otherwise difficult-to-state\nlemmas. While this is a simple wrapper around `Eq.rec`, it gives a canonical way to write it.\n\nThe simp-normal form is for the `copy` to be pushed outward. That way calculations can\noccur within the \"copy context.\" -/\nprotected def copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : G.Walk u' v' :=\n hu \u25b8 hv \u25b8 p\n#align simple_graph.walk.copy SimpleGraph.Walk.copy\n\n@[simp]\ntheorem copy_rfl_rfl {u v} (p : G.Walk u v) : p.copy rfl rfl = p := rfl\n#align simple_graph.walk.copy_rfl_rfl SimpleGraph.Walk.copy_rfl_rfl\n\n@[simp]\ntheorem copy_copy {u v u' v' u'' v''} (p : G.Walk u v)\n (hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') :\n (p.copy hu hv).copy hu' hv' = p.copy (hu.trans hu') (hv.trans hv') := by\n subst_vars\n rfl\n#align simple_graph.walk.copy_copy SimpleGraph.Walk.copy_copy\n\n@[simp]\ntheorem copy_nil {u u'} (hu : u = u') : (Walk.nil : G.Walk u u).copy hu hu = Walk.nil := by\n subst_vars\n rfl\n#align simple_graph.walk.copy_nil SimpleGraph.Walk.copy_nil\n\ntheorem copy_cons {u v w u' w'} (h : G.Adj u v) (p : G.Walk v w) (hu : u = u') (hw : w = w') :\n (Walk.cons h p).copy hu hw = Walk.cons (hu \u25b8 h) (p.copy rfl hw) := by\n subst_vars\n rfl\n#align simple_graph.walk.copy_cons SimpleGraph.Walk.copy_cons\n\n@[simp]\ntheorem cons_copy {u v w v' w'} (h : G.Adj u v) (p : G.Walk v' w') (hv : v' = v) (hw : w' = w) :\n Walk.cons h (p.copy hv hw) = (Walk.cons (hv \u25b8 h) p).copy rfl hw := by\n subst_vars\n rfl\n#align simple_graph.walk.cons_copy SimpleGraph.Walk.cons_copy\n\ntheorem exists_eq_cons_of_ne {u v : V} (hne : u \u2260 v) :\n \u2200 (p : G.Walk u v), \u2203 (w : V) (h : G.Adj u w) (p' : G.Walk w v), p = cons h p'\n | nil => (hne rfl).elim\n | cons h p' => \u27e8_, h, p', rfl\u27e9\n#align simple_graph.walk.exists_eq_cons_of_ne SimpleGraph.Walk.exists_eq_cons_of_ne\n\n/-- The length of a walk is the number of edges/darts along it. -/\ndef length {u v : V} : G.Walk u v \u2192 \u2115\n | nil => 0\n | cons _ q => q.length.succ\n#align simple_graph.walk.length SimpleGraph.Walk.length\n\n/-- The concatenation of two compatible walks. -/\n@[trans]\ndef append {u v w : V} : G.Walk u v \u2192 G.Walk v w \u2192 G.Walk u w\n | nil, q => q\n | cons h p, q => cons h (p.append q)\n#align simple_graph.walk.append SimpleGraph.Walk.append\n\n/-- The reversed version of `SimpleGraph.Walk.cons`, concatenating an edge to\nthe end of a walk. -/\ndef concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : G.Walk u w := p.append (cons h nil)\n#align simple_graph.walk.concat SimpleGraph.Walk.concat\n\ntheorem concat_eq_append {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n p.concat h = p.append (cons h nil) := rfl\n#align simple_graph.walk.concat_eq_append SimpleGraph.Walk.concat_eq_append\n\n/-- The concatenation of the reverse of the first walk with the second walk. -/\nprotected def reverseAux {u v w : V} : G.Walk u v \u2192 G.Walk u w \u2192 G.Walk v w\n | nil, q => q\n | cons h p, q => Walk.reverseAux p (cons (G.symm h) q)\n#align simple_graph.walk.reverse_aux SimpleGraph.Walk.reverseAux\n\n/-- The walk in reverse. -/\n@[symm]\ndef reverse {u v : V} (w : G.Walk u v) : G.Walk v u := w.reverseAux nil\n#align simple_graph.walk.reverse SimpleGraph.Walk.reverse\n\n/-- Get the `n`th vertex from a walk, where `n` is generally expected to be\nbetween `0` and `p.length`, inclusive.\nIf `n` is greater than or equal to `p.length`, the result is the path's endpoint. -/\ndef getVert {u v : V} : G.Walk u v \u2192 \u2115 \u2192 V\n | nil, _ => u\n | cons _ _, 0 => u\n | cons _ q, n + 1 => q.getVert n\n#align simple_graph.walk.get_vert SimpleGraph.Walk.getVert\n\n@[simp]\ntheorem getVert_zero {u v} (w : G.Walk u v) : w.getVert 0 = u := by cases w <;> rfl\n#align simple_graph.walk.get_vert_zero SimpleGraph.Walk.getVert_zero\n\ntheorem getVert_of_length_le {u v} (w : G.Walk u v) {i : \u2115} (hi : w.length \u2264 i) :\n w.getVert i = v := by\n induction w generalizing i with\n | nil => rfl\n | cons _ _ ih =>\n cases i\n \u00b7 cases hi\n \u00b7 exact ih (Nat.succ_le_succ_iff.1 hi)\n#align simple_graph.walk.get_vert_of_length_le SimpleGraph.Walk.getVert_of_length_le\n\n@[simp]\ntheorem getVert_length {u v} (w : G.Walk u v) : w.getVert w.length = v :=\n w.getVert_of_length_le rfl.le\n#align simple_graph.walk.get_vert_length SimpleGraph.Walk.getVert_length\n\ntheorem adj_getVert_succ {u v} (w : G.Walk u v) {i : \u2115} (hi : i < w.length) :\n G.Adj (w.getVert i) (w.getVert (i + 1)) := by\n induction w generalizing i with\n | nil => cases hi\n | cons hxy _ ih =>\n cases i\n \u00b7 simp [getVert, hxy]\n \u00b7 exact ih (Nat.succ_lt_succ_iff.1 hi)\n#align simple_graph.walk.adj_get_vert_succ SimpleGraph.Walk.adj_getVert_succ\n\n@[simp]\ntheorem cons_append {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (q : G.Walk w x) :\n (cons h p).append q = cons h (p.append q) := rfl\n#align simple_graph.walk.cons_append SimpleGraph.Walk.cons_append\n\n@[simp]\ntheorem cons_nil_append {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h nil).append p = cons h p := rfl\n#align simple_graph.walk.cons_nil_append SimpleGraph.Walk.cons_nil_append\n\n@[simp]\ntheorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p := by\n induction p with\n | nil => rfl\n | cons _ _ ih => rw [cons_append, ih]\n#align simple_graph.walk.append_nil SimpleGraph.Walk.append_nil\n\n@[simp]\ntheorem nil_append {u v : V} (p : G.Walk u v) : nil.append p = p :=\n rfl\n#align simple_graph.walk.nil_append SimpleGraph.Walk.nil_append\n\ntheorem append_assoc {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk w x) :\n p.append (q.append r) = (p.append q).append r := by\n induction p with\n | nil => rfl\n | cons h p' ih =>\n dsimp only [append]\n rw [ih]\n#align simple_graph.walk.append_assoc SimpleGraph.Walk.append_assoc\n\n@[simp]\ntheorem append_copy_copy {u v w u' v' w'} (p : G.Walk u v) (q : G.Walk v w)\n (hu : u = u') (hv : v = v') (hw : w = w') :\n (p.copy hu hv).append (q.copy hv hw) = (p.append q).copy hu hw := by\n subst_vars\n rfl\n#align simple_graph.walk.append_copy_copy SimpleGraph.Walk.append_copy_copy\n\ntheorem concat_nil {u v : V} (h : G.Adj u v) : nil.concat h = cons h nil := rfl\n#align simple_graph.walk.concat_nil SimpleGraph.Walk.concat_nil\n\n@[simp]\ntheorem concat_cons {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (h' : G.Adj w x) :\n (cons h p).concat h' = cons h (p.concat h') := rfl\n#align simple_graph.walk.concat_cons SimpleGraph.Walk.concat_cons\n\ntheorem append_concat {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (h : G.Adj w x) :\n p.append (q.concat h) = (p.append q).concat h := append_assoc _ _ _\n#align simple_graph.walk.append_concat SimpleGraph.Walk.append_concat\n\ntheorem concat_append {u v w x : V} (p : G.Walk u v) (h : G.Adj v w) (q : G.Walk w x) :\n (p.concat h).append q = p.append (cons h q) := by\n rw [concat_eq_append, \u2190 append_assoc, cons_nil_append]\n#align simple_graph.walk.concat_append SimpleGraph.Walk.concat_append\n\n/-- A non-trivial `cons` walk is representable as a `concat` walk. -/\ntheorem exists_cons_eq_concat {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n \u2203 (x : V) (q : G.Walk u x) (h' : G.Adj x w), cons h p = q.concat h' := by\n induction p generalizing u with\n | nil => exact \u27e8_, nil, h, rfl\u27e9\n | cons h' p ih =>\n obtain \u27e8y, q, h'', hc\u27e9 := ih h'\n refine' \u27e8y, cons h q, h'', _\u27e9\n rw [concat_cons, hc]\n#align simple_graph.walk.exists_cons_eq_concat SimpleGraph.Walk.exists_cons_eq_concat\n\n/-- A non-trivial `concat` walk is representable as a `cons` walk. -/\ntheorem exists_concat_eq_cons {u v w : V} :\n \u2200 (p : G.Walk u v) (h : G.Adj v w),\n \u2203 (x : V) (h' : G.Adj u x) (q : G.Walk x w), p.concat h = cons h' q\n | nil, h => \u27e8_, h, nil, rfl\u27e9\n | cons h' p, h => \u27e8_, h', Walk.concat p h, concat_cons _ _ _\u27e9\n#align simple_graph.walk.exists_concat_eq_cons SimpleGraph.Walk.exists_concat_eq_cons\n\n@[simp]\ntheorem reverse_nil {u : V} : (nil : G.Walk u u).reverse = nil := rfl\n#align simple_graph.walk.reverse_nil SimpleGraph.Walk.reverse_nil\n\ntheorem reverse_singleton {u v : V} (h : G.Adj u v) : (cons h nil).reverse = cons (G.symm h) nil :=\n rfl\n#align simple_graph.walk.reverse_singleton SimpleGraph.Walk.reverse_singleton\n\n@[simp]\ntheorem cons_reverseAux {u v w x : V} (p : G.Walk u v) (q : G.Walk w x) (h : G.Adj w u) :\n (cons h p).reverseAux q = p.reverseAux (cons (G.symm h) q) := rfl\n#align simple_graph.walk.cons_reverse_aux SimpleGraph.Walk.cons_reverseAux\n\n@[simp]\nprotected theorem append_reverseAux {u v w x : V}\n (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk u x) :\n (p.append q).reverseAux r = q.reverseAux (p.reverseAux r) := by\n induction p with\n | nil => rfl\n | cons h _ ih => exact ih q (cons (G.symm h) r)\n#align simple_graph.walk.append_reverse_aux SimpleGraph.Walk.append_reverseAux\n\n@[simp]\nprotected theorem reverseAux_append {u v w x : V}\n (p : G.Walk u v) (q : G.Walk u w) (r : G.Walk w x) :\n (p.reverseAux q).append r = p.reverseAux (q.append r) := by\n induction p with\n | nil => rfl\n | cons h _ ih => simp [ih (cons (G.symm h) q)]\n#align simple_graph.walk.reverse_aux_append SimpleGraph.Walk.reverseAux_append\n\nprotected theorem reverseAux_eq_reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk u w) :\n p.reverseAux q = p.reverse.append q := by simp [reverse]\n#align simple_graph.walk.reverse_aux_eq_reverse_append SimpleGraph.Walk.reverseAux_eq_reverse_append\n\n@[simp]\ntheorem reverse_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h p).reverse = p.reverse.append (cons (G.symm h) nil) := by simp [reverse]\n#align simple_graph.walk.reverse_cons SimpleGraph.Walk.reverse_cons\n\n@[simp]\ntheorem reverse_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).reverse = p.reverse.copy hv hu := by\n subst_vars\n rfl\n#align simple_graph.walk.reverse_copy SimpleGraph.Walk.reverse_copy\n\n@[simp]\ntheorem reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :\n (p.append q).reverse = q.reverse.append p.reverse := by simp [reverse]\n#align simple_graph.walk.reverse_append SimpleGraph.Walk.reverse_append\n\n@[simp]\ntheorem reverse_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n (p.concat h).reverse = cons (G.symm h) p.reverse := by simp [concat_eq_append]\n#align simple_graph.walk.reverse_concat SimpleGraph.Walk.reverse_concat\n\n@[simp]\ntheorem reverse_reverse {u v : V} (p : G.Walk u v) : p.reverse.reverse = p := by\n induction p with\n | nil => rfl\n | cons _ _ ih => simp [ih]\n#align simple_graph.walk.reverse_reverse SimpleGraph.Walk.reverse_reverse\n\n@[simp]\ntheorem length_nil {u : V} : (nil : G.Walk u u).length = 0 := rfl\n#align simple_graph.walk.length_nil SimpleGraph.Walk.length_nil\n\n@[simp]\ntheorem length_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h p).length = p.length + 1 := rfl\n#align simple_graph.walk.length_cons SimpleGraph.Walk.length_cons\n\n@[simp]\ntheorem length_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).length = p.length := by\n subst_vars\n rfl\n#align simple_graph.walk.length_copy SimpleGraph.Walk.length_copy\n\n@[simp]\ntheorem length_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :\n (p.append q).length = p.length + q.length := by\n induction p with\n | nil => simp\n | cons _ _ ih => simp [ih, add_comm, add_left_comm, add_assoc]\n#align simple_graph.walk.length_append SimpleGraph.Walk.length_append\n\n@[simp]\ntheorem length_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n (p.concat h).length = p.length + 1 := length_append _ _\n#align simple_graph.walk.length_concat SimpleGraph.Walk.length_concat\n\n@[simp]\nprotected theorem length_reverseAux {u v w : V} (p : G.Walk u v) (q : G.Walk u w) :\n (p.reverseAux q).length = p.length + q.length := by\n induction p with\n | nil => simp!\n | cons _ _ ih => simp [ih, Nat.add_succ, Nat.succ_add]\n#align simple_graph.walk.length_reverse_aux SimpleGraph.Walk.length_reverseAux\n\n@[simp]\ntheorem length_reverse {u v : V} (p : G.Walk u v) : p.reverse.length = p.length := by simp [reverse]\n#align simple_graph.walk.length_reverse SimpleGraph.Walk.length_reverse\n\ntheorem eq_of_length_eq_zero {u v : V} : \u2200 {p : G.Walk u v}, p.length = 0 \u2192 u = v\n | nil, _ => rfl\n#align simple_graph.walk.eq_of_length_eq_zero SimpleGraph.Walk.eq_of_length_eq_zero\n\n@[simp]\ntheorem exists_length_eq_zero_iff {u v : V} : (\u2203 p : G.Walk u v, p.length = 0) \u2194 u = v := by\n constructor\n \u00b7 rintro \u27e8p, hp\u27e9\n exact eq_of_length_eq_zero hp\n \u00b7 rintro rfl\n exact \u27e8nil, rfl\u27e9\n#align simple_graph.walk.exists_length_eq_zero_iff SimpleGraph.Walk.exists_length_eq_zero_iff\n\n@[simp]\ntheorem length_eq_zero_iff {u : V} {p : G.Walk u u} : p.length = 0 \u2194 p = nil := by cases p <;> simp\n#align simple_graph.walk.length_eq_zero_iff SimpleGraph.Walk.length_eq_zero_iff\n\nsection ConcatRec\n\nvariable {motive : \u2200 u v : V, G.Walk u v \u2192 Sort*} (Hnil : \u2200 {u : V}, motive u u nil)\n (Hconcat : \u2200 {u v w : V} (p : G.Walk u v) (h : G.Adj v w), motive u v p \u2192 motive u w (p.concat h))\n\n/-- Auxiliary definition for `SimpleGraph.Walk.concatRec` -/\ndef concatRecAux {u v : V} : (p : G.Walk u v) \u2192 motive v u p.reverse\n | nil => Hnil\n | cons h p => reverse_cons h p \u25b8 Hconcat p.reverse h.symm (concatRecAux p)\n#align simple_graph.walk.concat_rec_aux SimpleGraph.Walk.concatRecAux\n\n/-- Recursor on walks by inducting on `SimpleGraph.Walk.concat`.\n\nThis is inducting from the opposite end of the walk compared\nto `SimpleGraph.Walk.rec`, which inducts on `SimpleGraph.Walk.cons`. -/\n@[elab_as_elim]\ndef concatRec {u v : V} (p : G.Walk u v) : motive u v p :=\n reverse_reverse p \u25b8 concatRecAux @Hnil @Hconcat p.reverse\n#align simple_graph.walk.concat_rec SimpleGraph.Walk.concatRec\n\n@[simp]\ntheorem concatRec_nil (u : V) :\n @concatRec _ _ motive @Hnil @Hconcat _ _ (nil : G.Walk u u) = Hnil := rfl\n#align simple_graph.walk.concat_rec_nil SimpleGraph.Walk.concatRec_nil\n\n@[simp]\ntheorem concatRec_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n @concatRec _ _ motive @Hnil @Hconcat _ _ (p.concat h) =\n Hconcat p h (concatRec @Hnil @Hconcat p) := by\n simp only [concatRec]\n apply eq_of_heq\n apply rec_heq_of_heq\n trans concatRecAux @Hnil @Hconcat (cons h.symm p.reverse)\n \u00b7 congr\n simp\n \u00b7 rw [concatRecAux, rec_heq_iff_heq]\n congr <;> simp [heq_rec_iff_heq]\n#align simple_graph.walk.concat_rec_concat SimpleGraph.Walk.concatRec_concat\n\nend ConcatRec\n\ntheorem concat_ne_nil {u v : V} (p : G.Walk u v) (h : G.Adj v u) : p.concat h \u2260 nil := by\n cases p <;> simp [concat]\n#align simple_graph.walk.concat_ne_nil SimpleGraph.Walk.concat_ne_nil\n\ntheorem concat_inj {u v v' w : V} {p : G.Walk u v} {h : G.Adj v w} {p' : G.Walk u v'}\n {h' : G.Adj v' w} (he : p.concat h = p'.concat h') : \u2203 hv : v = v', p.copy rfl hv = p' := by\n induction p with\n | nil =>\n cases p'\n \u00b7 exact \u27e8rfl, rfl\u27e9\n \u00b7 exfalso\n simp only [concat_nil, concat_cons, cons.injEq] at he\n obtain \u27e8rfl, he\u27e9 := he\n simp only [heq_iff_eq] at he\n exact concat_ne_nil _ _ he.symm\n | cons _ _ ih =>\n rw [concat_cons] at he\n cases p'\n \u00b7 exfalso\n simp only [concat_nil, cons.injEq] at he\n obtain \u27e8rfl, he\u27e9 := he\n rw [heq_iff_eq] at he\n exact concat_ne_nil _ _ he\n \u00b7 rw [concat_cons, cons.injEq] at he\n obtain \u27e8rfl, he\u27e9 := he\n rw [heq_iff_eq] at he\n obtain \u27e8rfl, rfl\u27e9 := ih he\n exact \u27e8rfl, rfl\u27e9\n#align simple_graph.walk.concat_inj SimpleGraph.Walk.concat_inj\n\n/-- The `support` of a walk is the list of vertices it visits in order. -/\ndef support {u v : V} : G.Walk u v \u2192 List V\n | nil => [u]\n | cons _ p => u :: p.support\n#align simple_graph.walk.support SimpleGraph.Walk.support\n\n/-- The `darts` of a walk is the list of darts it visits in order. -/\ndef darts {u v : V} : G.Walk u v \u2192 List G.Dart\n | nil => []\n | cons h p => \u27e8(u, _), h\u27e9 :: p.darts\n#align simple_graph.walk.darts SimpleGraph.Walk.darts\n\n/-- The `edges` of a walk is the list of edges it visits in order.\nThis is defined to be the list of edges underlying `SimpleGraph.Walk.darts`. -/\ndef edges {u v : V} (p : G.Walk u v) : List (Sym2 V) := p.darts.map Dart.edge\n#align simple_graph.walk.edges SimpleGraph.Walk.edges\n\n@[simp]\ntheorem support_nil {u : V} : (nil : G.Walk u u).support = [u] := rfl\n#align simple_graph.walk.support_nil SimpleGraph.Walk.support_nil\n\n@[simp]\ntheorem support_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h p).support = u :: p.support := rfl\n#align simple_graph.walk.support_cons SimpleGraph.Walk.support_cons\n\n@[simp]\ntheorem support_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n (p.concat h).support = p.support.concat w := by\n induction p <;> simp [*, concat_nil]\n#align simple_graph.walk.support_concat SimpleGraph.Walk.support_concat\n\n@[simp]\ntheorem support_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).support = p.support := by\n subst_vars\n rfl\n#align simple_graph.walk.support_copy SimpleGraph.Walk.support_copy\n\ntheorem support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n (p.append p').support = p.support ++ p'.support.tail := by\n induction p <;> cases p' <;> simp [*]\n#align simple_graph.walk.support_append SimpleGraph.Walk.support_append\n\n@[simp]\ntheorem support_reverse {u v : V} (p : G.Walk u v) : p.reverse.support = p.support.reverse := by\n induction p <;> simp [support_append, *]\n#align simple_graph.walk.support_reverse SimpleGraph.Walk.support_reverse\n\n@[simp]\ntheorem support_ne_nil {u v : V} (p : G.Walk u v) : p.support \u2260 [] := by cases p <;> simp\n#align simple_graph.walk.support_ne_nil SimpleGraph.Walk.support_ne_nil\n\ntheorem tail_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n (p.append p').support.tail = p.support.tail ++ p'.support.tail := by\n rw [support_append, List.tail_append_of_ne_nil _ _ (support_ne_nil _)]\n#align simple_graph.walk.tail_support_append SimpleGraph.Walk.tail_support_append\n\ntheorem support_eq_cons {u v : V} (p : G.Walk u v) : p.support = u :: p.support.tail := by\n cases p <;> simp\n#align simple_graph.walk.support_eq_cons SimpleGraph.Walk.support_eq_cons\n\n@[simp]\ntheorem start_mem_support {u v : V} (p : G.Walk u v) : u \u2208 p.support := by cases p <;> simp\n#align simple_graph.walk.start_mem_support SimpleGraph.Walk.start_mem_support\n\n@[simp]\ntheorem end_mem_support {u v : V} (p : G.Walk u v) : v \u2208 p.support := by induction p <;> simp [*]\n#align simple_graph.walk.end_mem_support SimpleGraph.Walk.end_mem_support\n\n@[simp]\ntheorem support_nonempty {u v : V} (p : G.Walk u v) : { w | w \u2208 p.support }.Nonempty :=\n \u27e8u, by simp\u27e9\n#align simple_graph.walk.support_nonempty SimpleGraph.Walk.support_nonempty\n\ntheorem mem_support_iff {u v w : V} (p : G.Walk u v) : w \u2208 p.support \u2194 w = u \u2228 w \u2208 p.support.tail :=\n by cases p <;> simp\n#align simple_graph.walk.mem_support_iff SimpleGraph.Walk.mem_support_iff\n\ntheorem mem_support_nil_iff {u v : V} : u \u2208 (nil : G.Walk v v).support \u2194 u = v := by simp\n#align simple_graph.walk.mem_support_nil_iff SimpleGraph.Walk.mem_support_nil_iff\n\n@[simp]\ntheorem mem_tail_support_append_iff {t u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n t \u2208 (p.append p').support.tail \u2194 t \u2208 p.support.tail \u2228 t \u2208 p'.support.tail := by\n rw [tail_support_append, List.mem_append]\n#align simple_graph.walk.mem_tail_support_append_iff SimpleGraph.Walk.mem_tail_support_append_iff\n\n@[simp]\ntheorem end_mem_tail_support_of_ne {u v : V} (h : u \u2260 v) (p : G.Walk u v) : v \u2208 p.support.tail := by\n obtain \u27e8_, _, _, rfl\u27e9 := exists_eq_cons_of_ne h p\n simp\n#align simple_graph.walk.end_mem_tail_support_of_ne SimpleGraph.Walk.end_mem_tail_support_of_ne\n\n@[simp, nolint unusedHavesSuffices]\ntheorem mem_support_append_iff {t u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n t \u2208 (p.append p').support \u2194 t \u2208 p.support \u2228 t \u2208 p'.support := by\n simp only [mem_support_iff, mem_tail_support_append_iff]\n obtain rfl | h := eq_or_ne t v <;> obtain rfl | h' := eq_or_ne t u <;>\n -- this `have` triggers the unusedHavesSuffices linter:\n (try have := h'.symm) <;> simp [*]\n#align simple_graph.walk.mem_support_append_iff SimpleGraph.Walk.mem_support_append_iff\n\n@[simp]\ntheorem subset_support_append_left {V : Type u} {G : SimpleGraph V} {u v w : V}\n (p : G.Walk u v) (q : G.Walk v w) : p.support \u2286 (p.append q).support := by\n simp only [Walk.support_append, List.subset_append_left]\n#align simple_graph.walk.subset_support_append_left SimpleGraph.Walk.subset_support_append_left\n\n@[simp]\ntheorem subset_support_append_right {V : Type u} {G : SimpleGraph V} {u v w : V}\n (p : G.Walk u v) (q : G.Walk v w) : q.support \u2286 (p.append q).support := by\n intro h\n simp (config := { contextual := true }) only [mem_support_append_iff, or_true_iff, imp_true_iff]\n#align simple_graph.walk.subset_support_append_right SimpleGraph.Walk.subset_support_append_right\n\ntheorem coe_support {u v : V} (p : G.Walk u v) : (p.support : Multiset V) = {u} + p.support.tail :=\n by cases p <;> rfl\n#align simple_graph.walk.coe_support SimpleGraph.Walk.coe_support\n\ntheorem coe_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n ((p.append p').support : Multiset V) = {u} + p.support.tail + p'.support.tail := by\n rw [support_append, \u2190 Multiset.coe_add, coe_support]\n#align simple_graph.walk.coe_support_append SimpleGraph.Walk.coe_support_append\n\ntheorem coe_support_append' [DecidableEq V] {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n ((p.append p').support : Multiset V) = p.support + p'.support - {v} := by\n rw [support_append, \u2190 Multiset.coe_add]\n simp only [coe_support]\n rw [add_comm ({v} : Multiset V)]\n simp only [\u2190 add_assoc, add_tsub_cancel_right]\n#align simple_graph.walk.coe_support_append' SimpleGraph.Walk.coe_support_append'\n\ntheorem chain_adj_support {u v w : V} (h : G.Adj u v) :\n \u2200 (p : G.Walk v w), List.Chain G.Adj u p.support\n | nil => List.Chain.cons h List.Chain.nil\n | cons h' p => List.Chain.cons h (chain_adj_support h' p)\n#align simple_graph.walk.chain_adj_support SimpleGraph.Walk.chain_adj_support\n\ntheorem chain'_adj_support {u v : V} : \u2200 (p : G.Walk u v), List.Chain' G.Adj p.support\n | nil => List.Chain.nil\n | cons h p => chain_adj_support h p\n#align simple_graph.walk.chain'_adj_support SimpleGraph.Walk.chain'_adj_support\n\ntheorem chain_dartAdj_darts {d : G.Dart} {v w : V} (h : d.snd = v) (p : G.Walk v w) :\n List.Chain G.DartAdj d p.darts := by\n induction p generalizing d with\n | nil => exact List.Chain.nil\n -- Porting note: needed to defer `h` and `rfl` to help elaboration\n | cons h' p ih => exact List.Chain.cons (by exact h) (ih (by rfl))\n#align simple_graph.walk.chain_dart_adj_darts SimpleGraph.Walk.chain_dartAdj_darts\n\ntheorem chain'_dartAdj_darts {u v : V} : \u2200 (p : G.Walk u v), List.Chain' G.DartAdj p.darts\n | nil => trivial\n -- Porting note: needed to defer `rfl` to help elaboration\n | cons h p => chain_dartAdj_darts (by rfl) p\n#align simple_graph.walk.chain'_dart_adj_darts SimpleGraph.Walk.chain'_dartAdj_darts\n\n/-- Every edge in a walk's edge list is an edge of the graph.\nIt is written in this form (rather than using `\u2286`) to avoid unsightly coercions. -/\ntheorem edges_subset_edgeSet {u v : V} :\n \u2200 (p : G.Walk u v) \u2983e : Sym2 V\u2984, e \u2208 p.edges \u2192 e \u2208 G.edgeSet\n | cons h' p', e, h => by\n cases h\n \u00b7 exact h'\n next h' => exact edges_subset_edgeSet p' h'\n#align simple_graph.walk.edges_subset_edge_set SimpleGraph.Walk.edges_subset_edgeSet\n\ntheorem adj_of_mem_edges {u v x y : V} (p : G.Walk u v) (h : s(x, y) \u2208 p.edges) : G.Adj x y :=\n edges_subset_edgeSet p h\n#align simple_graph.walk.adj_of_mem_edges SimpleGraph.Walk.adj_of_mem_edges\n\n@[simp]\ntheorem darts_nil {u : V} : (nil : G.Walk u u).darts = [] := rfl\n#align simple_graph.walk.darts_nil SimpleGraph.Walk.darts_nil\n\n@[simp]\ntheorem darts_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h p).darts = \u27e8(u, v), h\u27e9 :: p.darts := rfl\n#align simple_graph.walk.darts_cons SimpleGraph.Walk.darts_cons\n\n@[simp]\ntheorem darts_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n (p.concat h).darts = p.darts.concat \u27e8(v, w), h\u27e9 := by\n induction p <;> simp [*, concat_nil]\n#align simple_graph.walk.darts_concat SimpleGraph.Walk.darts_concat\n\n@[simp]\ntheorem darts_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).darts = p.darts := by\n subst_vars\n rfl\n#align simple_graph.walk.darts_copy SimpleGraph.Walk.darts_copy\n\n@[simp]\ntheorem darts_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n (p.append p').darts = p.darts ++ p'.darts := by\n induction p <;> simp [*]\n#align simple_graph.walk.darts_append SimpleGraph.Walk.darts_append\n\n@[simp]\ntheorem darts_reverse {u v : V} (p : G.Walk u v) :\n p.reverse.darts = (p.darts.map Dart.symm).reverse := by\n induction p <;> simp [*, Sym2.eq_swap]\n#align simple_graph.walk.darts_reverse SimpleGraph.Walk.darts_reverse\n\ntheorem mem_darts_reverse {u v : V} {d : G.Dart} {p : G.Walk u v} :\n d \u2208 p.reverse.darts \u2194 d.symm \u2208 p.darts := by simp\n#align simple_graph.walk.mem_darts_reverse SimpleGraph.Walk.mem_darts_reverse\n\ntheorem cons_map_snd_darts {u v : V} (p : G.Walk u v) : (u :: p.darts.map (\u00b7.snd)) = p.support := by\n induction p <;> simp! [*]\n#align simple_graph.walk.cons_map_snd_darts SimpleGraph.Walk.cons_map_snd_darts\n\ntheorem map_snd_darts {u v : V} (p : G.Walk u v) : p.darts.map (\u00b7.snd) = p.support.tail := by\n simpa using congr_arg List.tail (cons_map_snd_darts p)\n#align simple_graph.walk.map_snd_darts SimpleGraph.Walk.map_snd_darts\n\ntheorem map_fst_darts_append {u v : V} (p : G.Walk u v) :\n p.darts.map (\u00b7.fst) ++ [v] = p.support := by\n induction p <;> simp! [*]\n#align simple_graph.walk.map_fst_darts_append SimpleGraph.Walk.map_fst_darts_append\n\ntheorem map_fst_darts {u v : V} (p : G.Walk u v) : p.darts.map (\u00b7.fst) = p.support.dropLast := by\n simpa! using congr_arg List.dropLast (map_fst_darts_append p)\n#align simple_graph.walk.map_fst_darts SimpleGraph.Walk.map_fst_darts\n\n@[simp]\ntheorem edges_nil {u : V} : (nil : G.Walk u u).edges = [] := rfl\n#align simple_graph.walk.edges_nil SimpleGraph.Walk.edges_nil\n\n@[simp]\ntheorem edges_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h p).edges = s(u, v) :: p.edges := rfl\n#align simple_graph.walk.edges_cons SimpleGraph.Walk.edges_cons\n\n@[simp]\ntheorem edges_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n (p.concat h).edges = p.edges.concat s(v, w) := by simp [edges]\n#align simple_graph.walk.edges_concat SimpleGraph.Walk.edges_concat\n\n@[simp]\ntheorem edges_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).edges = p.edges := by\n subst_vars\n rfl\n#align simple_graph.walk.edges_copy SimpleGraph.Walk.edges_copy\n\n@[simp]\ntheorem edges_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n (p.append p').edges = p.edges ++ p'.edges := by simp [edges]\n#align simple_graph.walk.edges_append SimpleGraph.Walk.edges_append\n\n@[simp]\ntheorem edges_reverse {u v : V} (p : G.Walk u v) : p.reverse.edges = p.edges.reverse := by\n simp [edges, List.map_reverse]\n#align simple_graph.walk.edges_reverse SimpleGraph.Walk.edges_reverse\n\n@[simp]\ntheorem length_support {u v : V} (p : G.Walk u v) : p.support.length = p.length + 1 := by\n induction p <;> simp [*]\n#align simple_graph.walk.length_support SimpleGraph.Walk.length_support\n\n@[simp]\ntheorem length_darts {u v : V} (p : G.Walk u v) : p.darts.length = p.length := by\n induction p <;> simp [*]\n#align simple_graph.walk.length_darts SimpleGraph.Walk.length_darts\n\n@[simp]\ntheorem length_edges {u v : V} (p : G.Walk u v) : p.edges.length = p.length := by simp [edges]\n#align simple_graph.walk.length_edges SimpleGraph.Walk.length_edges\n\ntheorem dart_fst_mem_support_of_mem_darts {u v : V} :\n \u2200 (p : G.Walk u v) {d : G.Dart}, d \u2208 p.darts \u2192 d.fst \u2208 p.support\n | cons h p', d, hd => by\n simp only [support_cons, darts_cons, List.mem_cons] at hd \u22a2\n rcases hd with (rfl | hd)\n \u00b7 exact Or.inl rfl\n \u00b7 exact Or.inr (dart_fst_mem_support_of_mem_darts _ hd)\n#align simple_graph.walk.dart_fst_mem_support_of_mem_darts SimpleGraph.Walk.dart_fst_mem_support_of_mem_darts\n\ntheorem dart_snd_mem_support_of_mem_darts {u v : V} (p : G.Walk u v) {d : G.Dart}\n (h : d \u2208 p.darts) : d.snd \u2208 p.support := by\n simpa using p.reverse.dart_fst_mem_support_of_mem_darts (by simp [h] : d.symm \u2208 p.reverse.darts)\n#align simple_graph.walk.dart_snd_mem_support_of_mem_darts SimpleGraph.Walk.dart_snd_mem_support_of_mem_darts\n\ntheorem fst_mem_support_of_mem_edges {t u v w : V} (p : G.Walk v w) (he : s(t, u) \u2208 p.edges) :\n t \u2208 p.support := by\n obtain \u27e8d, hd, he\u27e9 := List.mem_map.mp he\n rw [dart_edge_eq_mk'_iff'] at he\n rcases he with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)\n \u00b7 exact dart_fst_mem_support_of_mem_darts _ hd\n \u00b7 exact dart_snd_mem_support_of_mem_darts _ hd\n#align simple_graph.walk.fst_mem_support_of_mem_edges SimpleGraph.Walk.fst_mem_support_of_mem_edges\n\ntheorem snd_mem_support_of_mem_edges {t u v w : V} (p : G.Walk v w) (he : s(t, u) \u2208 p.edges) :\n u \u2208 p.support := by\n rw [Sym2.eq_swap] at he\n exact p.fst_mem_support_of_mem_edges he\n#align simple_graph.walk.snd_mem_support_of_mem_edges SimpleGraph.Walk.snd_mem_support_of_mem_edges\n\ntheorem darts_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.Nodup) :\n p.darts.Nodup := by\n induction p with\n | nil => simp\n | cons _ p' ih =>\n simp only [darts_cons, support_cons, List.nodup_cons] at h \u22a2\n exact \u27e8fun h' => h.1 (dart_fst_mem_support_of_mem_darts p' h'), ih h.2\u27e9\n#align simple_graph.walk.darts_nodup_of_support_nodup SimpleGraph.Walk.darts_nodup_of_support_nodup\n\ntheorem edges_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.Nodup) :\n p.edges.Nodup := by\n induction p with\n | nil => simp\n | cons _ p' ih =>\n simp only [edges_cons, support_cons, List.nodup_cons] at h \u22a2\n exact \u27e8fun h' => h.1 (fst_mem_support_of_mem_edges p' h'), ih h.2\u27e9\n#align simple_graph.walk.edges_nodup_of_support_nodup SimpleGraph.Walk.edges_nodup_of_support_nodup\n\n/-- Predicate for the empty walk.\n\nSolves the dependent type problem where `p = G.Walk.nil` typechecks\nonly if `p` has defeq endpoints. -/\ninductive Nil : {v w : V} \u2192 G.Walk v w \u2192 Prop\n | nil {u : V} : Nil (nil : G.Walk u u)\n\nvariable {u v w : V}\n\n@[simp] lemma nil_nil : (nil : G.Walk u u).Nil := Nil.nil\n\n@[simp] lemma not_nil_cons {h : G.Adj u v} {p : G.Walk v w} : \u00ac (cons h p).Nil := nofun\n\ninstance (p : G.Walk v w) : Decidable p.Nil :=\n match p with\n | nil => isTrue .nil\n | cons _ _ => isFalse nofun\n\nprotected lemma Nil.eq {p : G.Walk v w} : p.Nil \u2192 v = w | .nil => rfl\n\nlemma not_nil_of_ne {p : G.Walk v w} : v \u2260 w \u2192 \u00ac p.Nil := mt Nil.eq\n\nlemma nil_iff_support_eq {p : G.Walk v w} : p.Nil \u2194 p.support = [v] := by\n cases p <;> simp\n\nlemma nil_iff_length_eq {p : G.Walk v w} : p.Nil \u2194 p.length = 0 := by\n cases p <;> simp\n\nlemma not_nil_iff {p : G.Walk v w} :\n \u00ac p.Nil \u2194 \u2203 (u : V) (h : G.Adj v u) (q : G.Walk u w), p = cons h q := by\n cases p <;> simp [*]\n\n@[elab_as_elim]\ndef notNilRec {motive : {u w : V} \u2192 (p : G.Walk u w) \u2192 (h : \u00ac p.Nil) \u2192 Sort*}\n (cons : {u v w : V} \u2192 (h : G.Adj u v) \u2192 (q : G.Walk v w) \u2192 motive (cons h q) not_nil_cons)\n (p : G.Walk u w) : (hp : \u00ac p.Nil) \u2192 motive p hp :=\n match p with\n | nil => fun hp => absurd .nil hp\n | .cons h q => fun _ => cons h q\n\n/-- The second vertex along a non-nil walk. -/\ndef sndOfNotNil (p : G.Walk v w) (hp : \u00ac p.Nil) : V :=\n p.notNilRec (@fun _ u _ _ _ => u) hp\n\n@[simp] lemma adj_sndOfNotNil {p : G.Walk v w} (hp : \u00ac p.Nil) :\n G.Adj v (p.sndOfNotNil hp) :=\n p.notNilRec (fun h _ => h) hp\n\n/-- The walk obtained by removing the first dart of a non-nil walk. -/\ndef tail (p : G.Walk u v) (hp : \u00ac p.Nil) : G.Walk (p.sndOfNotNil hp) v :=\n p.notNilRec (fun _ q => q) hp\n\n/-- The first dart of a walk. -/\n@[simps]\ndef firstDart (p : G.Walk v w) (hp : \u00ac p.Nil) : G.Dart where\n fst := v\n snd := p.sndOfNotNil hp\n is_adj := p.adj_sndOfNotNil hp\n\nlemma edge_firstDart (p : G.Walk v w) (hp : \u00ac p.Nil) :\n (p.firstDart hp).edge = s(v, p.sndOfNotNil hp) := rfl\n\nvariable {x y : V} -- TODO: rename to u, v, w instead?\n\n@[simp] lemma cons_tail_eq (p : G.Walk x y) (hp : \u00ac p.Nil) :\n cons (p.adj_sndOfNotNil hp) (p.tail hp) = p :=\n p.notNilRec (fun _ _ => rfl) hp\n\n@[simp] lemma cons_support_tail (p : G.Walk x y) (hp : \u00ac p.Nil) :\n x :: (p.tail hp).support = p.support := by\n rw [\u2190 support_cons, cons_tail_eq]\n\n@[simp] lemma length_tail_add_one {p : G.Walk x y} (hp : \u00ac p.Nil) :\n (p.tail hp).length + 1 = p.length := by\n rw [\u2190 length_cons, cons_tail_eq]\n\n@[simp] lemma nil_copy {x' y' : V} {p : G.Walk x y} (hx : x = x') (hy : y = y') :\n (p.copy hx hy).Nil = p.Nil := by\n subst_vars; rfl\n\n/-! ### Trails, paths, circuits, cycles -/\n\n/-- A *trail* is a walk with no repeating edges. -/\n@[mk_iff isTrail_def]\nstructure IsTrail {u v : V} (p : G.Walk u v) : Prop where\n edges_nodup : p.edges.Nodup\n#align simple_graph.walk.is_trail SimpleGraph.Walk.IsTrail\n#align simple_graph.walk.is_trail_def SimpleGraph.Walk.isTrail_def\n\n/-- A *path* is a walk with no repeating vertices.\nUse `SimpleGraph.Walk.IsPath.mk'` for a simpler constructor. -/\nstructure IsPath {u v : V} (p : G.Walk u v) extends IsTrail p : Prop where\n support_nodup : p.support.Nodup\n#align simple_graph.walk.is_path SimpleGraph.Walk.IsPath\n\n-- Porting note: used to use `extends to_trail : is_trail p` in structure\nprotected lemma IsPath.isTrail {p : Walk G u v}(h : IsPath p) : IsTrail p := h.toIsTrail\n#align simple_graph.walk.is_path.to_trail SimpleGraph.Walk.IsPath.isTrail\n\n/-- A *circuit* at `u : V` is a nonempty trail beginning and ending at `u`. -/\n@[mk_iff isCircuit_def]\nstructure IsCircuit {u : V} (p : G.Walk u u) extends IsTrail p : Prop where\n ne_nil : p \u2260 nil\n#align simple_graph.walk.is_circuit SimpleGraph.Walk.IsCircuit\n#align simple_graph.walk.is_circuit_def SimpleGraph.Walk.isCircuit_def\n\n-- Porting note: used to use `extends to_trail : is_trail p` in structure\nprotected lemma IsCircuit.isTrail {p : Walk G u u} (h : IsCircuit p) : IsTrail p := h.toIsTrail\n#align simple_graph.walk.is_circuit.to_trail SimpleGraph.Walk.IsCircuit.isTrail\n\n/-- A *cycle* at `u : V` is a circuit at `u` whose only repeating vertex\nis `u` (which appears exactly twice). -/\nstructure IsCycle {u : V} (p : G.Walk u u) extends IsCircuit p : Prop where\n support_nodup : p.support.tail.Nodup\n#align simple_graph.walk.is_cycle SimpleGraph.Walk.IsCycle\n\n-- Porting note: used to use `extends to_circuit : is_circuit p` in structure\nprotected lemma IsCycle.isCircuit {p : Walk G u u} (h : IsCycle p) : IsCircuit p := h.toIsCircuit\n#align simple_graph.walk.is_cycle.to_circuit SimpleGraph.Walk.IsCycle.isCircuit\n\n@[simp]\ntheorem isTrail_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).IsTrail \u2194 p.IsTrail := by\n subst_vars\n rfl\n#align simple_graph.walk.is_trail_copy SimpleGraph.Walk.isTrail_copy\n\ntheorem IsPath.mk' {u v : V} {p : G.Walk u v} (h : p.support.Nodup) : p.IsPath :=\n \u27e8\u27e8edges_nodup_of_support_nodup h\u27e9, h\u27e9\n#align simple_graph.walk.is_path.mk' SimpleGraph.Walk.IsPath.mk'\n\ntheorem isPath_def {u v : V} (p : G.Walk u v) : p.IsPath \u2194 p.support.Nodup :=\n \u27e8IsPath.support_nodup, IsPath.mk'\u27e9\n#align simple_graph.walk.is_path_def SimpleGraph.Walk.isPath_def\n\n@[simp]\ntheorem isPath_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).IsPath \u2194 p.IsPath := by\n subst_vars\n rfl\n#align simple_graph.walk.is_path_copy SimpleGraph.Walk.isPath_copy\n\n@[simp]\ntheorem isCircuit_copy {u u'} (p : G.Walk u u) (hu : u = u') :\n (p.copy hu hu).IsCircuit \u2194 p.IsCircuit := by\n subst_vars\n rfl\n#align simple_graph.walk.is_circuit_copy SimpleGraph.Walk.isCircuit_copy\n\ntheorem isCycle_def {u : V} (p : G.Walk u u) :\n p.IsCycle \u2194 p.IsTrail \u2227 p \u2260 nil \u2227 p.support.tail.Nodup :=\n Iff.intro (fun h => \u27e8h.1.1, h.1.2, h.2\u27e9) fun h => \u27e8\u27e8h.1, h.2.1\u27e9, h.2.2\u27e9\n#align simple_graph.walk.is_cycle_def SimpleGraph.Walk.isCycle_def\n\n@[simp]\ntheorem isCycle_copy {u u'} (p : G.Walk u u) (hu : u = u') :\n (p.copy hu hu).IsCycle \u2194 p.IsCycle := by\n subst_vars\n rfl\n#align simple_graph.walk.is_cycle_copy SimpleGraph.Walk.isCycle_copy\n\n@[simp]\ntheorem IsTrail.nil {u : V} : (nil : G.Walk u u).IsTrail :=\n \u27e8by simp [edges]\u27e9\n#align simple_graph.walk.is_trail.nil SimpleGraph.Walk.IsTrail.nil\n\ntheorem IsTrail.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} :\n (cons h p).IsTrail \u2192 p.IsTrail := by simp [isTrail_def]\n#align simple_graph.walk.is_trail.of_cons SimpleGraph.Walk.IsTrail.of_cons\n\n@[simp]\ntheorem cons_isTrail_iff {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h p).IsTrail \u2194 p.IsTrail \u2227 s(u, v) \u2209 p.edges := by simp [isTrail_def, and_comm]\n#align simple_graph.walk.cons_is_trail_iff SimpleGraph.Walk.cons_isTrail_iff\n\ntheorem IsTrail.reverse {u v : V} (p : G.Walk u v) (h : p.IsTrail) : p.reverse.IsTrail := by\n simpa [isTrail_def] using h\n#align simple_graph.walk.is_trail.reverse SimpleGraph.Walk.IsTrail.reverse\n\n@[simp]\ntheorem reverse_isTrail_iff {u v : V} (p : G.Walk u v) : p.reverse.IsTrail \u2194 p.IsTrail := by\n constructor <;>\n \u00b7 intro h\n convert h.reverse _\n try rw [reverse_reverse]\n#align simple_graph.walk.reverse_is_trail_iff SimpleGraph.Walk.reverse_isTrail_iff\n\ntheorem IsTrail.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w}\n (h : (p.append q).IsTrail) : p.IsTrail := by\n rw [isTrail_def, edges_append, List.nodup_append] at h\n exact \u27e8h.1\u27e9\n#align simple_graph.walk.is_trail.of_append_left SimpleGraph.Walk.IsTrail.of_append_left\n\ntheorem IsTrail.of_append_right {u v w : V} {p : G.Walk u v} {q : G.Walk v w}\n (h : (p.append q).IsTrail) : q.IsTrail := by\n rw [isTrail_def, edges_append, List.nodup_append] at h\n exact \u27e8h.2.1\u27e9\n#align simple_graph.walk.is_trail.of_append_right SimpleGraph.Walk.IsTrail.of_append_right\n\ntheorem IsTrail.count_edges_le_one [DecidableEq V] {u v : V} {p : G.Walk u v} (h : p.IsTrail)\n (e : Sym2 V) : p.edges.count e \u2264 1 :=\n List.nodup_iff_count_le_one.mp h.edges_nodup e\n#align simple_graph.walk.is_trail.count_edges_le_one SimpleGraph.Walk.IsTrail.count_edges_le_one\n\ntheorem IsTrail.count_edges_eq_one [DecidableEq V] {u v : V} {p : G.Walk u v} (h : p.IsTrail)\n {e : Sym2 V} (he : e \u2208 p.edges) : p.edges.count e = 1 :=\n List.count_eq_one_of_mem h.edges_nodup he\n#align simple_graph.walk.is_trail.count_edges_eq_one SimpleGraph.Walk.IsTrail.count_edges_eq_one\n\ntheorem IsPath.nil {u : V} : (nil : G.Walk u u).IsPath := by constructor <;> simp\n#align simple_graph.walk.is_path.nil SimpleGraph.Walk.IsPath.nil\n\ntheorem IsPath.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} :\n (cons h p).IsPath \u2192 p.IsPath := by simp [isPath_def]\n#align simple_graph.walk.is_path.of_cons SimpleGraph.Walk.IsPath.of_cons\n\n@[simp]\ntheorem cons_isPath_iff {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n (cons h p).IsPath \u2194 p.IsPath \u2227 u \u2209 p.support := by\n constructor <;> simp (config := { contextual := true }) [isPath_def]\n#align simple_graph.walk.cons_is_path_iff SimpleGraph.Walk.cons_isPath_iff\n\nprotected lemma IsPath.cons {p : Walk G v w} (hp : p.IsPath) (hu : u \u2209 p.support) {h : G.Adj u v} :\n (cons h p).IsPath :=\n (cons_isPath_iff _ _).2 \u27e8hp, hu\u27e9\n\n@[simp]\ntheorem isPath_iff_eq_nil {u : V} (p : G.Walk u u) : p.IsPath \u2194 p = nil := by\n cases p <;> simp [IsPath.nil]\n#align simple_graph.walk.is_path_iff_eq_nil SimpleGraph.Walk.isPath_iff_eq_nil\n\ntheorem IsPath.reverse {u v : V} {p : G.Walk u v} (h : p.IsPath) : p.reverse.IsPath := by\n simpa [isPath_def] using h\n#align simple_graph.walk.is_path.reverse SimpleGraph.Walk.IsPath.reverse\n\n@[simp]\ntheorem isPath_reverse_iff {u v : V} (p : G.Walk u v) : p.reverse.IsPath \u2194 p.IsPath := by\n constructor <;> intro h <;> convert h.reverse; simp\n#align simple_graph.walk.is_path_reverse_iff SimpleGraph.Walk.isPath_reverse_iff\n\ntheorem IsPath.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w} :\n (p.append q).IsPath \u2192 p.IsPath := by\n simp only [isPath_def, support_append]\n exact List.Nodup.of_append_left\n#align simple_graph.walk.is_path.of_append_left SimpleGraph.Walk.IsPath.of_append_left\n\ntheorem IsPath.of_append_right {u v w : V} {p : G.Walk u v} {q : G.Walk v w}\n (h : (p.append q).IsPath) : q.IsPath := by\n rw [\u2190 isPath_reverse_iff] at h \u22a2\n rw [reverse_append] at h\n apply h.of_append_left\n#align simple_graph.walk.is_path.of_append_right SimpleGraph.Walk.IsPath.of_append_right\n\n@[simp]\ntheorem IsCycle.not_of_nil {u : V} : \u00ac(nil : G.Walk u u).IsCycle := fun h => h.ne_nil rfl\n#align simple_graph.walk.is_cycle.not_of_nil SimpleGraph.Walk.IsCycle.not_of_nil\n\nlemma IsCycle.ne_bot : \u2200 {p : G.Walk u u}, p.IsCycle \u2192 G \u2260 \u22a5\n | nil, hp => by cases hp.ne_nil rfl\n | cons h _, hp => by rintro rfl; exact h\n\nlemma IsCycle.three_le_length {v : V} {p : G.Walk v v} (hp : p.IsCycle) : 3 \u2264 p.length := by\n have \u27e8\u27e8hp, hp'\u27e9, _\u27e9 := hp\n match p with\n | .nil => simp at hp'\n | .cons h .nil => simp at h\n | .cons _ (.cons _ .nil) => simp at hp\n | .cons _ (.cons _ (.cons _ _)) => simp_rw [SimpleGraph.Walk.length_cons]; omega\n\ntheorem cons_isCycle_iff {u v : V} (p : G.Walk v u) (h : G.Adj u v) :\n (Walk.cons h p).IsCycle \u2194 p.IsPath \u2227 \u00acs(u, v) \u2208 p.edges := by\n simp only [Walk.isCycle_def, Walk.isPath_def, Walk.isTrail_def, edges_cons, List.nodup_cons,\n support_cons, List.tail_cons]\n have : p.support.Nodup \u2192 p.edges.Nodup := edges_nodup_of_support_nodup\n tauto\n#align simple_graph.walk.cons_is_cycle_iff SimpleGraph.Walk.cons_isCycle_iff\n\nlemma IsPath.tail {p : G.Walk u v} (hp : p.IsPath) (hp' : \u00ac p.Nil) : (p.tail hp').IsPath := by\n rw [Walk.isPath_def] at hp \u22a2\n rw [\u2190 cons_support_tail _ hp', List.nodup_cons] at hp\n exact hp.2\n\n/-! ### About paths -/\n\ninstance [DecidableEq V] {u v : V} (p : G.Walk u v) : Decidable p.IsPath := by\n rw [isPath_def]\n infer_instance\n\ntheorem IsPath.length_lt [Fintype V] {u v : V} {p : G.Walk u v} (hp : p.IsPath) :\n p.length < Fintype.card V := by\n rw [Nat.lt_iff_add_one_le, \u2190 length_support]\n exact hp.support_nodup.length_le_card\n#align simple_graph.walk.is_path.length_lt SimpleGraph.Walk.IsPath.length_lt\n\n\n/-! ### Walk decompositions -/\n\nsection WalkDecomp\n\nvariable [DecidableEq V]\n\n/-- Given a vertex in the support of a path, give the path up until (and including) that vertex. -/\ndef takeUntil {v w : V} : \u2200 (p : G.Walk v w) (u : V), u \u2208 p.support \u2192 G.Walk v u\n | nil, u, h => by rw [mem_support_nil_iff.mp h]\n | cons r p, u, h =>\n if hx : v = u then\n by subst u; exact Walk.nil\n else\n cons r (takeUntil p u <| by cases h; exact (hx rfl).elim; assumption)\n#align simple_graph.walk.take_until SimpleGraph.Walk.takeUntil\n\n/-- Given a vertex in the support of a path, give the path from (and including) that vertex to\nthe end. In other words, drop vertices from the front of a path until (and not including)\nthat vertex. -/\ndef dropUntil {v w : V} : \u2200 (p : G.Walk v w) (u : V), u \u2208 p.support \u2192 G.Walk u w\n | nil, u, h => by rw [mem_support_nil_iff.mp h]\n | cons r p, u, h =>\n if hx : v = u then by\n subst u\n exact cons r p\n else dropUntil p u <| by cases h; exact (hx rfl).elim; assumption\n#align simple_graph.walk.drop_until SimpleGraph.Walk.dropUntil\n\n/-- The `takeUntil` and `dropUntil` functions split a walk into two pieces.\nThe lemma `SimpleGraph.Walk.count_support_takeUntil_eq_one` specifies where this split occurs. -/\n@[simp]\ntheorem take_spec {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.takeUntil u h).append (p.dropUntil u h) = p := by\n induction p\n \u00b7 rw [mem_support_nil_iff] at h\n subst u\n rfl\n \u00b7 cases h\n \u00b7 simp!\n \u00b7 simp! only\n split_ifs with h' <;> subst_vars <;> simp [*]\n#align simple_graph.walk.take_spec SimpleGraph.Walk.take_spec\n\ntheorem mem_support_iff_exists_append {V : Type u} {G : SimpleGraph V} {u v w : V}\n {p : G.Walk u v} : w \u2208 p.support \u2194 \u2203 (q : G.Walk u w) (r : G.Walk w v), p = q.append r := by\n classical\n constructor\n \u00b7 exact fun h => \u27e8_, _, (p.take_spec h).symm\u27e9\n \u00b7 rintro \u27e8q, r, rfl\u27e9\n simp only [mem_support_append_iff, end_mem_support, start_mem_support, or_self_iff]\n#align simple_graph.walk.mem_support_iff_exists_append SimpleGraph.Walk.mem_support_iff_exists_append\n\n@[simp]\ntheorem count_support_takeUntil_eq_one {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.takeUntil u h).support.count u = 1 := by\n induction p\n \u00b7 rw [mem_support_nil_iff] at h\n subst u\n simp!\n \u00b7 cases h\n \u00b7 simp!\n \u00b7 simp! only\n split_ifs with h' <;> rw [eq_comm] at h' <;> subst_vars <;> simp! [*, List.count_cons]\n#align simple_graph.walk.count_support_take_until_eq_one SimpleGraph.Walk.count_support_takeUntil_eq_one\n\ntheorem count_edges_takeUntil_le_one {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) (x : V) :\n (p.takeUntil u h).edges.count s(u, x) \u2264 1 := by\n induction' p with u' u' v' w' ha p' ih\n \u00b7 rw [mem_support_nil_iff] at h\n subst u\n simp!\n \u00b7 cases h\n \u00b7 simp!\n \u00b7 simp! only\n split_ifs with h'\n \u00b7 subst h'\n simp\n \u00b7 rw [edges_cons, List.count_cons]\n split_ifs with h''\n \u00b7 rw [Sym2.eq_iff] at h''\n obtain \u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9 := h''\n \u00b7 exact (h' rfl).elim\n \u00b7 cases p' <;> simp!\n \u00b7 apply ih\n#align simple_graph.walk.count_edges_take_until_le_one SimpleGraph.Walk.count_edges_takeUntil_le_one\n\n@[simp]\ntheorem takeUntil_copy {u v w v' w'} (p : G.Walk v w) (hv : v = v') (hw : w = w')\n (h : u \u2208 (p.copy hv hw).support) :\n (p.copy hv hw).takeUntil u h = (p.takeUntil u (by subst_vars; exact h)).copy hv rfl := by\n subst_vars\n rfl\n#align simple_graph.walk.take_until_copy SimpleGraph.Walk.takeUntil_copy\n\n@[simp]\ntheorem dropUntil_copy {u v w v' w'} (p : G.Walk v w) (hv : v = v') (hw : w = w')\n (h : u \u2208 (p.copy hv hw).support) :\n (p.copy hv hw).dropUntil u h = (p.dropUntil u (by subst_vars; exact h)).copy rfl hw := by\n subst_vars\n rfl\n#align simple_graph.walk.drop_until_copy SimpleGraph.Walk.dropUntil_copy\n\ntheorem support_takeUntil_subset {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.takeUntil u h).support \u2286 p.support := fun x hx => by\n rw [\u2190 take_spec p h, mem_support_append_iff]\n exact Or.inl hx\n#align simple_graph.walk.support_take_until_subset SimpleGraph.Walk.support_takeUntil_subset\n\ntheorem support_dropUntil_subset {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.dropUntil u h).support \u2286 p.support := fun x hx => by\n rw [\u2190 take_spec p h, mem_support_append_iff]\n exact Or.inr hx\n#align simple_graph.walk.support_drop_until_subset SimpleGraph.Walk.support_dropUntil_subset\n\ntheorem darts_takeUntil_subset {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.takeUntil u h).darts \u2286 p.darts := fun x hx => by\n rw [\u2190 take_spec p h, darts_append, List.mem_append]\n exact Or.inl hx\n#align simple_graph.walk.darts_take_until_subset SimpleGraph.Walk.darts_takeUntil_subset\n\ntheorem darts_dropUntil_subset {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.dropUntil u h).darts \u2286 p.darts := fun x hx => by\n rw [\u2190 take_spec p h, darts_append, List.mem_append]\n exact Or.inr hx\n#align simple_graph.walk.darts_drop_until_subset SimpleGraph.Walk.darts_dropUntil_subset\n\ntheorem edges_takeUntil_subset {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.takeUntil u h).edges \u2286 p.edges :=\n List.map_subset _ (p.darts_takeUntil_subset h)\n#align simple_graph.walk.edges_take_until_subset SimpleGraph.Walk.edges_takeUntil_subset\n\ntheorem edges_dropUntil_subset {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.dropUntil u h).edges \u2286 p.edges :=\n List.map_subset _ (p.darts_dropUntil_subset h)\n#align simple_graph.walk.edges_drop_until_subset SimpleGraph.Walk.edges_dropUntil_subset\n\ntheorem length_takeUntil_le {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.takeUntil u h).length \u2264 p.length := by\n have := congr_arg Walk.length (p.take_spec h)\n rw [length_append] at this\n exact Nat.le.intro this\n#align simple_graph.walk.length_take_until_le SimpleGraph.Walk.length_takeUntil_le\n\ntheorem length_dropUntil_le {u v w : V} (p : G.Walk v w) (h : u \u2208 p.support) :\n (p.dropUntil u h).length \u2264 p.length := by\n have := congr_arg Walk.length (p.take_spec h)\n rw [length_append, add_comm] at this\n exact Nat.le.intro this\n#align simple_graph.walk.length_drop_until_le SimpleGraph.Walk.length_dropUntil_le\n\nprotected theorem IsTrail.takeUntil {u v w : V} {p : G.Walk v w} (hc : p.IsTrail)\n (h : u \u2208 p.support) : (p.takeUntil u h).IsTrail :=\n IsTrail.of_append_left (by rwa [\u2190 take_spec _ h] at hc)\n#align simple_graph.walk.is_trail.take_until SimpleGraph.Walk.IsTrail.takeUntil\n\nprotected theorem IsTrail.dropUntil {u v w : V} {p : G.Walk v w} (hc : p.IsTrail)\n (h : u \u2208 p.support) : (p.dropUntil u h).IsTrail :=\n IsTrail.of_append_right (by rwa [\u2190 take_spec _ h] at hc)\n#align simple_graph.walk.is_trail.drop_until SimpleGraph.Walk.IsTrail.dropUntil\n\nprotected theorem IsPath.takeUntil {u v w : V} {p : G.Walk v w} (hc : p.IsPath)\n (h : u \u2208 p.support) : (p.takeUntil u h).IsPath :=\n IsPath.of_append_left (by rwa [\u2190 take_spec _ h] at hc)\n#align simple_graph.walk.is_path.take_until SimpleGraph.Walk.IsPath.takeUntil\n\n-- Porting note: p was previously accidentally an explicit argument\nprotected theorem IsPath.dropUntil {u v w : V} {p : G.Walk v w} (hc : p.IsPath)\n (h : u \u2208 p.support) : (p.dropUntil u h).IsPath :=\n IsPath.of_append_right (by rwa [\u2190 take_spec _ h] at hc)\n#align simple_graph.walk.is_path.drop_until SimpleGraph.Walk.IsPath.dropUntil\n\n/-- Rotate a loop walk such that it is centered at the given vertex. -/\ndef rotate {u v : V} (c : G.Walk v v) (h : u \u2208 c.support) : G.Walk u u :=\n (c.dropUntil u h).append (c.takeUntil u h)\n#align simple_graph.walk.rotate SimpleGraph.Walk.rotate\n\n@[simp]\ntheorem support_rotate {u v : V} (c : G.Walk v v) (h : u \u2208 c.support) :\n (c.rotate h).support.tail ~r c.support.tail := by\n simp only [rotate, tail_support_append]\n apply List.IsRotated.trans List.isRotated_append\n rw [\u2190 tail_support_append, take_spec]\n#align simple_graph.walk.support_rotate SimpleGraph.Walk.support_rotate\n\ntheorem rotate_darts {u v : V} (c : G.Walk v v) (h : u \u2208 c.support) :\n (c.rotate h).darts ~r c.darts := by\n simp only [rotate, darts_append]\n apply List.IsRotated.trans List.isRotated_append\n rw [\u2190 darts_append, take_spec]\n#align simple_graph.walk.rotate_darts SimpleGraph.Walk.rotate_darts\n\ntheorem rotate_edges {u v : V} (c : G.Walk v v) (h : u \u2208 c.support) :\n (c.rotate h).edges ~r c.edges :=\n (rotate_darts c h).map _\n#align simple_graph.walk.rotate_edges SimpleGraph.Walk.rotate_edges\n\nprotected theorem IsTrail.rotate {u v : V} {c : G.Walk v v} (hc : c.IsTrail) (h : u \u2208 c.support) :\n (c.rotate h).IsTrail := by\n rw [isTrail_def, (c.rotate_edges h).perm.nodup_iff]\n exact hc.edges_nodup\n#align simple_graph.walk.is_trail.rotate SimpleGraph.Walk.IsTrail.rotate\n\nprotected theorem IsCircuit.rotate {u v : V} {c : G.Walk v v} (hc : c.IsCircuit)\n (h : u \u2208 c.support) : (c.rotate h).IsCircuit := by\n refine \u27e8hc.isTrail.rotate _, ?_\u27e9\n cases c\n \u00b7 exact (hc.ne_nil rfl).elim\n \u00b7 intro hn\n have hn' := congr_arg length hn\n rw [rotate, length_append, add_comm, \u2190 length_append, take_spec] at hn'\n simp at hn'\n#align simple_graph.walk.is_circuit.rotate SimpleGraph.Walk.IsCircuit.rotate\n\nprotected theorem IsCycle.rotate {u v : V} {c : G.Walk v v} (hc : c.IsCycle) (h : u \u2208 c.support) :\n (c.rotate h).IsCycle := by\n refine \u27e8hc.isCircuit.rotate _, ?_\u27e9\n rw [List.IsRotated.nodup_iff (support_rotate _ _)]\n exact hc.support_nodup\n#align simple_graph.walk.is_cycle.rotate SimpleGraph.Walk.IsCycle.rotate\n\nend WalkDecomp\n\n/-- Given a set `S` and a walk `w` from `u` to `v` such that `u \u2208 S` but `v \u2209 S`,\nthere exists a dart in the walk whose start is in `S` but whose end is not. -/\ntheorem exists_boundary_dart {u v : V} (p : G.Walk u v) (S : Set V) (uS : u \u2208 S) (vS : v \u2209 S) :\n \u2203 d : G.Dart, d \u2208 p.darts \u2227 d.fst \u2208 S \u2227 d.snd \u2209 S := by\n induction' p with _ x y w a p' ih\n \u00b7 cases vS uS\n \u00b7 by_cases h : y \u2208 S\n \u00b7 obtain \u27e8d, hd, hcd\u27e9 := ih h vS\n exact \u27e8d, List.Mem.tail _ hd, hcd\u27e9\n \u00b7 exact \u27e8\u27e8(x, y), a\u27e9, List.Mem.head _, uS, h\u27e9\n#align simple_graph.walk.exists_boundary_dart SimpleGraph.Walk.exists_boundary_dart\n\nend Walk\n\n\n/-! ### Type of paths -/\n\n/-- The type for paths between two vertices. -/\nabbrev Path (u v : V) := { p : G.Walk u v // p.IsPath }\n#align simple_graph.path SimpleGraph.Path\n\nnamespace Path\n\nvariable {G G'}\n\n@[simp]\nprotected theorem isPath {u v : V} (p : G.Path u v) : (p : G.Walk u v).IsPath := p.property\n#align simple_graph.path.is_path SimpleGraph.Path.isPath\n\n@[simp]\nprotected theorem isTrail {u v : V} (p : G.Path u v) : (p : G.Walk u v).IsTrail :=\n p.property.isTrail\n#align simple_graph.path.is_trail SimpleGraph.Path.isTrail\n\n/-- The length-0 path at a vertex. -/\n@[refl, simps]\nprotected def nil {u : V} : G.Path u u :=\n \u27e8Walk.nil, Walk.IsPath.nil\u27e9\n#align simple_graph.path.nil SimpleGraph.Path.nil\n\n/-- The length-1 path between a pair of adjacent vertices. -/\n@[simps]\ndef singleton {u v : V} (h : G.Adj u v) : G.Path u v :=\n \u27e8Walk.cons h Walk.nil, by simp [h.ne]\u27e9\n#align simple_graph.path.singleton SimpleGraph.Path.singleton\n\ntheorem mk'_mem_edges_singleton {u v : V} (h : G.Adj u v) :\n s(u, v) \u2208 (singleton h : G.Walk u v).edges := by simp [singleton]\n#align simple_graph.path.mk_mem_edges_singleton SimpleGraph.Path.mk'_mem_edges_singleton\n\n/-- The reverse of a path is another path. See also `SimpleGraph.Walk.reverse`. -/\n@[symm, simps]\ndef reverse {u v : V} (p : G.Path u v) : G.Path v u :=\n \u27e8Walk.reverse p, p.property.reverse\u27e9\n#align simple_graph.path.reverse SimpleGraph.Path.reverse\n\ntheorem count_support_eq_one [DecidableEq V] {u v w : V} {p : G.Path u v}\n (hw : w \u2208 (p : G.Walk u v).support) : (p : G.Walk u v).support.count w = 1 :=\n List.count_eq_one_of_mem p.property.support_nodup hw\n#align simple_graph.path.count_support_eq_one SimpleGraph.Path.count_support_eq_one\n\ntheorem count_edges_eq_one [DecidableEq V] {u v : V} {p : G.Path u v} (e : Sym2 V)\n (hw : e \u2208 (p : G.Walk u v).edges) : (p : G.Walk u v).edges.count e = 1 :=\n List.count_eq_one_of_mem p.property.isTrail.edges_nodup hw\n#align simple_graph.path.count_edges_eq_one SimpleGraph.Path.count_edges_eq_one\n\n@[simp]\ntheorem nodup_support {u v : V} (p : G.Path u v) : (p : G.Walk u v).support.Nodup :=\n (Walk.isPath_def _).mp p.property\n#align simple_graph.path.nodup_support SimpleGraph.Path.nodup_support\n\ntheorem loop_eq {v : V} (p : G.Path v v) : p = Path.nil := by\n obtain \u27e8_ | _, h\u27e9 := p\n \u00b7 rfl\n \u00b7 simp at h\n#align simple_graph.path.loop_eq SimpleGraph.Path.loop_eq\n\ntheorem not_mem_edges_of_loop {v : V} {e : Sym2 V} {p : G.Path v v} : \u00ace \u2208 (p : G.Walk v v).edges :=\n by simp [p.loop_eq]\n#align simple_graph.path.not_mem_edges_of_loop SimpleGraph.Path.not_mem_edges_of_loop\n\ntheorem cons_isCycle {u v : V} (p : G.Path v u) (h : G.Adj u v)\n (he : \u00acs(u, v) \u2208 (p : G.Walk v u).edges) : (Walk.cons h \u2191p).IsCycle := by\n simp [Walk.isCycle_def, Walk.cons_isTrail_iff, he]\n#align simple_graph.path.cons_is_cycle SimpleGraph.Path.cons_isCycle\n\nend Path\n\n\n/-! ### Walks to paths -/\n\nnamespace Walk\n\nvariable {G} [DecidableEq V]\n\n/-- Given a walk, produces a walk from it by bypassing subwalks between repeated vertices.\nThe result is a path, as shown in `SimpleGraph.Walk.bypass_isPath`.\nThis is packaged up in `SimpleGraph.Walk.toPath`. -/\ndef bypass {u v : V} : G.Walk u v \u2192 G.Walk u v\n | nil => nil\n | cons ha p =>\n let p' := p.bypass\n if hs : u \u2208 p'.support then\n p'.dropUntil u hs\n else\n cons ha p'\n#align simple_graph.walk.bypass SimpleGraph.Walk.bypass\n\n@[simp]\ntheorem bypass_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n (p.copy hu hv).bypass = p.bypass.copy hu hv := by\n subst_vars\n rfl\n#align simple_graph.walk.bypass_copy SimpleGraph.Walk.bypass_copy\n\ntheorem bypass_isPath {u v : V} (p : G.Walk u v) : p.bypass.IsPath := by\n induction p with\n | nil => simp!\n | cons _ p' ih =>\n simp only [bypass]\n split_ifs with hs\n \u00b7 exact ih.dropUntil hs\n \u00b7 simp [*, cons_isPath_iff]\n#align simple_graph.walk.bypass_is_path SimpleGraph.Walk.bypass_isPath\n\ntheorem length_bypass_le {u v : V} (p : G.Walk u v) : p.bypass.length \u2264 p.length := by\n induction p with\n | nil => rfl\n | cons _ _ ih =>\n simp only [bypass]\n split_ifs\n \u00b7 trans\n apply length_dropUntil_le\n rw [length_cons]\n exact le_add_right ih\n \u00b7 rw [length_cons, length_cons]\n exact add_le_add_right ih 1\n#align simple_graph.walk.length_bypass_le SimpleGraph.Walk.length_bypass_le\n\n", "theoremStatement": "lemma bypass_eq_self_of_length_le {u v : V} (p : G.Walk u v) (h : p.length \u2264 p.bypass.length) :\n p.bypass = p", "theoremName": "bypass_eq_self_of_length_le", "fileCreated": {"commit": "67ba3174dc", "date": "2023-03-01"}, "theoremCreated": {"commit": "2e8aeee458", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "positionMetadata": {"lineInFile": 1479, "tokenPositionInFile": 60347, "theoremPositionInFile": 202}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n induction p with\n | nil => rfl\n | cons h p ih =>\n simp only [Walk.bypass]\n split_ifs with hb\n \u00b7 exfalso\n simp only [hb, Walk.bypass, Walk.length_cons, dif_pos] at h\n apply Nat.not_succ_le_self p.length\n calc p.length + 1\n _ \u2264 (p.bypass.dropUntil _ _).length := h\n _ \u2264 p.bypass.length := Walk.length_dropUntil_le p.bypass hb\n _ \u2264 p.length := Walk.length_bypass_le _\n \u00b7 simp only [hb, Walk.bypass, Walk.length_cons, not_false_iff, dif_neg, add_le_add_iff_right]\n at h\n rw [ih h]", "proofType": "tactic", "proofLengthLines": 16, "proofLengthTokens": 542}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller, Vincent Beffara\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Connectivity\nimport Mathlib.Data.Nat.Lattice\n\n#align_import combinatorics.simple_graph.metric from \"leanprover-community/mathlib\"@\"352ecfe114946c903338006dd3287cb5a9955ff2\"\n\n/-!\n# Graph metric\n\nThis module defines the `SimpleGraph.dist` function, which takes\npairs of vertices to the length of the shortest walk between them.\n\n## Main definitions\n\n- `SimpleGraph.dist` is the graph metric.\n\n## Todo\n\n- Provide an additional computable version of `SimpleGraph.dist`\n for when `G` is connected.\n\n- Evaluate `Nat` vs `ENat` for the codomain of `dist`, or potentially\n having an additional `edist` when the objects under consideration are\n disconnected graphs.\n\n- When directed graphs exist, a directed notion of distance,\n likely `ENat`-valued.\n\n## Tags\n\ngraph metric, distance\n\n-/\n\n\nnamespace SimpleGraph\n\nvariable {V : Type*} (G : SimpleGraph V)\n\n/-! ## Metric -/\n\n\n/-- The distance between two vertices is the length of the shortest walk between them.\nIf no such walk exists, this uses the junk value of `0`. -/\nnoncomputable def dist (u v : V) : \u2115 :=\n sInf (Set.range (Walk.length : G.Walk u v \u2192 \u2115))\n#align simple_graph.dist SimpleGraph.dist\n\nvariable {G}\n\nprotected theorem Reachable.exists_walk_of_dist {u v : V} (hr : G.Reachable u v) :\n \u2203 p : G.Walk u v, p.length = G.dist u v :=\n Nat.sInf_mem (Set.range_nonempty_iff_nonempty.mpr hr)\n#align simple_graph.reachable.exists_walk_of_dist SimpleGraph.Reachable.exists_walk_of_dist\n\nprotected theorem Connected.exists_walk_of_dist (hconn : G.Connected) (u v : V) :\n \u2203 p : G.Walk u v, p.length = G.dist u v :=\n (hconn u v).exists_walk_of_dist\n#align simple_graph.connected.exists_walk_of_dist SimpleGraph.Connected.exists_walk_of_dist\n\ntheorem dist_le {u v : V} (p : G.Walk u v) : G.dist u v \u2264 p.length :=\n Nat.sInf_le \u27e8p, rfl\u27e9\n#align simple_graph.dist_le SimpleGraph.dist_le\n\n@[simp]\ntheorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} :\n G.dist u v = 0 \u2194 u = v \u2228 \u00acG.Reachable u v := by simp [dist, Nat.sInf_eq_zero, Reachable]\n#align simple_graph.dist_eq_zero_iff_eq_or_not_reachable SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable\n\ntheorem dist_self {v : V} : dist G v v = 0 := by simp\n#align simple_graph.dist_self SimpleGraph.dist_self\n\nprotected theorem Reachable.dist_eq_zero_iff {u v : V} (hr : G.Reachable u v) :\n G.dist u v = 0 \u2194 u = v := by simp [hr]\n#align simple_graph.reachable.dist_eq_zero_iff SimpleGraph.Reachable.dist_eq_zero_iff\n\nprotected theorem Reachable.pos_dist_of_ne {u v : V} (h : G.Reachable u v) (hne : u \u2260 v) :\n 0 < G.dist u v :=\n Nat.pos_of_ne_zero (by simp [h, hne])\n#align simple_graph.reachable.pos_dist_of_ne SimpleGraph.Reachable.pos_dist_of_ne\n\nprotected theorem Connected.dist_eq_zero_iff (hconn : G.Connected) {u v : V} :\n G.dist u v = 0 \u2194 u = v := by simp [hconn u v]\n#align simple_graph.connected.dist_eq_zero_iff SimpleGraph.Connected.dist_eq_zero_iff\n\nprotected theorem Connected.pos_dist_of_ne {u v : V} (hconn : G.Connected) (hne : u \u2260 v) :\n 0 < G.dist u v :=\n Nat.pos_of_ne_zero (by intro h; exact False.elim (hne (hconn.dist_eq_zero_iff.mp h)))\n#align simple_graph.connected.pos_dist_of_ne SimpleGraph.Connected.pos_dist_of_ne\n\ntheorem dist_eq_zero_of_not_reachable {u v : V} (h : \u00acG.Reachable u v) : G.dist u v = 0 := by\n simp [h]\n#align simple_graph.dist_eq_zero_of_not_reachable SimpleGraph.dist_eq_zero_of_not_reachable\n\ntheorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) :\n (Set.univ : Set (G.Walk u v)).Nonempty := by\n simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using\n Nat.nonempty_of_pos_sInf h\n#align simple_graph.nonempty_of_pos_dist SimpleGraph.nonempty_of_pos_dist\n\nprotected theorem Connected.dist_triangle (hconn : G.Connected) {u v w : V} :\n G.dist u w \u2264 G.dist u v + G.dist v w := by\n obtain \u27e8p, hp\u27e9 := hconn.exists_walk_of_dist u v\n obtain \u27e8q, hq\u27e9 := hconn.exists_walk_of_dist v w\n rw [\u2190 hp, \u2190 hq, \u2190 Walk.length_append]\n apply dist_le\n#align simple_graph.connected.dist_triangle SimpleGraph.Connected.dist_triangle\n\nprivate theorem dist_comm_aux {u v : V} (h : G.Reachable u v) : G.dist u v \u2264 G.dist v u := by\n obtain \u27e8p, hp\u27e9 := h.symm.exists_walk_of_dist\n rw [\u2190 hp, \u2190 Walk.length_reverse]\n apply dist_le\n\ntheorem dist_comm {u v : V} : G.dist u v = G.dist v u := by\n by_cases h : G.Reachable u v\n \u00b7 apply le_antisymm (dist_comm_aux h) (dist_comm_aux h.symm)\n \u00b7 have h' : \u00acG.Reachable v u := fun h' => absurd h'.symm h\n simp [h, h', dist_eq_zero_of_not_reachable]\n#align simple_graph.dist_comm SimpleGraph.dist_comm\n\n", "theoremStatement": "theorem Walk.isPath_of_length_eq_dist {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v) :\n p.IsPath", "theoremName": "Walk.isPath_of_length_eq_dist", "fileCreated": {"commit": "c97ad7df79", "date": "2023-03-02"}, "theoremCreated": {"commit": "2e8aeee458", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Combinatorics/SimpleGraph/Metric.lean", "positionMetadata": {"lineInFile": 125, "tokenPositionInFile": 4730, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n classical\n have : p.bypass = p := by\n apply Walk.bypass_eq_self_of_length_le\n calc p.length\n _ = G.dist u v := hp\n _ \u2264 p.bypass.length := dist_le p.bypass\n rw [\u2190 this]\n apply Walk.bypass_isPath", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 217}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller, Vincent Beffara\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Connectivity\nimport Mathlib.Data.Nat.Lattice\n\n#align_import combinatorics.simple_graph.metric from \"leanprover-community/mathlib\"@\"352ecfe114946c903338006dd3287cb5a9955ff2\"\n\n/-!\n# Graph metric\n\nThis module defines the `SimpleGraph.dist` function, which takes\npairs of vertices to the length of the shortest walk between them.\n\n## Main definitions\n\n- `SimpleGraph.dist` is the graph metric.\n\n## Todo\n\n- Provide an additional computable version of `SimpleGraph.dist`\n for when `G` is connected.\n\n- Evaluate `Nat` vs `ENat` for the codomain of `dist`, or potentially\n having an additional `edist` when the objects under consideration are\n disconnected graphs.\n\n- When directed graphs exist, a directed notion of distance,\n likely `ENat`-valued.\n\n## Tags\n\ngraph metric, distance\n\n-/\n\n\nnamespace SimpleGraph\n\nvariable {V : Type*} (G : SimpleGraph V)\n\n/-! ## Metric -/\n\n\n/-- The distance between two vertices is the length of the shortest walk between them.\nIf no such walk exists, this uses the junk value of `0`. -/\nnoncomputable def dist (u v : V) : \u2115 :=\n sInf (Set.range (Walk.length : G.Walk u v \u2192 \u2115))\n#align simple_graph.dist SimpleGraph.dist\n\nvariable {G}\n\nprotected theorem Reachable.exists_walk_of_dist {u v : V} (hr : G.Reachable u v) :\n \u2203 p : G.Walk u v, p.length = G.dist u v :=\n Nat.sInf_mem (Set.range_nonempty_iff_nonempty.mpr hr)\n#align simple_graph.reachable.exists_walk_of_dist SimpleGraph.Reachable.exists_walk_of_dist\n\nprotected theorem Connected.exists_walk_of_dist (hconn : G.Connected) (u v : V) :\n \u2203 p : G.Walk u v, p.length = G.dist u v :=\n (hconn u v).exists_walk_of_dist\n#align simple_graph.connected.exists_walk_of_dist SimpleGraph.Connected.exists_walk_of_dist\n\ntheorem dist_le {u v : V} (p : G.Walk u v) : G.dist u v \u2264 p.length :=\n Nat.sInf_le \u27e8p, rfl\u27e9\n#align simple_graph.dist_le SimpleGraph.dist_le\n\n@[simp]\ntheorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} :\n G.dist u v = 0 \u2194 u = v \u2228 \u00acG.Reachable u v := by simp [dist, Nat.sInf_eq_zero, Reachable]\n#align simple_graph.dist_eq_zero_iff_eq_or_not_reachable SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable\n\ntheorem dist_self {v : V} : dist G v v = 0 := by simp\n#align simple_graph.dist_self SimpleGraph.dist_self\n\nprotected theorem Reachable.dist_eq_zero_iff {u v : V} (hr : G.Reachable u v) :\n G.dist u v = 0 \u2194 u = v := by simp [hr]\n#align simple_graph.reachable.dist_eq_zero_iff SimpleGraph.Reachable.dist_eq_zero_iff\n\nprotected theorem Reachable.pos_dist_of_ne {u v : V} (h : G.Reachable u v) (hne : u \u2260 v) :\n 0 < G.dist u v :=\n Nat.pos_of_ne_zero (by simp [h, hne])\n#align simple_graph.reachable.pos_dist_of_ne SimpleGraph.Reachable.pos_dist_of_ne\n\nprotected theorem Connected.dist_eq_zero_iff (hconn : G.Connected) {u v : V} :\n G.dist u v = 0 \u2194 u = v := by simp [hconn u v]\n#align simple_graph.connected.dist_eq_zero_iff SimpleGraph.Connected.dist_eq_zero_iff\n\nprotected theorem Connected.pos_dist_of_ne {u v : V} (hconn : G.Connected) (hne : u \u2260 v) :\n 0 < G.dist u v :=\n Nat.pos_of_ne_zero (by intro h; exact False.elim (hne (hconn.dist_eq_zero_iff.mp h)))\n#align simple_graph.connected.pos_dist_of_ne SimpleGraph.Connected.pos_dist_of_ne\n\ntheorem dist_eq_zero_of_not_reachable {u v : V} (h : \u00acG.Reachable u v) : G.dist u v = 0 := by\n simp [h]\n#align simple_graph.dist_eq_zero_of_not_reachable SimpleGraph.dist_eq_zero_of_not_reachable\n\ntheorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) :\n (Set.univ : Set (G.Walk u v)).Nonempty := by\n simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using\n Nat.nonempty_of_pos_sInf h\n#align simple_graph.nonempty_of_pos_dist SimpleGraph.nonempty_of_pos_dist\n\nprotected theorem Connected.dist_triangle (hconn : G.Connected) {u v w : V} :\n G.dist u w \u2264 G.dist u v + G.dist v w := by\n obtain \u27e8p, hp\u27e9 := hconn.exists_walk_of_dist u v\n obtain \u27e8q, hq\u27e9 := hconn.exists_walk_of_dist v w\n rw [\u2190 hp, \u2190 hq, \u2190 Walk.length_append]\n apply dist_le\n#align simple_graph.connected.dist_triangle SimpleGraph.Connected.dist_triangle\n\nprivate theorem dist_comm_aux {u v : V} (h : G.Reachable u v) : G.dist u v \u2264 G.dist v u := by\n obtain \u27e8p, hp\u27e9 := h.symm.exists_walk_of_dist\n rw [\u2190 hp, \u2190 Walk.length_reverse]\n apply dist_le\n\ntheorem dist_comm {u v : V} : G.dist u v = G.dist v u := by\n by_cases h : G.Reachable u v\n \u00b7 apply le_antisymm (dist_comm_aux h) (dist_comm_aux h.symm)\n \u00b7 have h' : \u00acG.Reachable v u := fun h' => absurd h'.symm h\n simp [h, h', dist_eq_zero_of_not_reachable]\n#align simple_graph.dist_comm SimpleGraph.dist_comm\n\ntheorem Walk.isPath_of_length_eq_dist {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v) :\n p.IsPath := by\n classical\n have : p.bypass = p := by\n apply Walk.bypass_eq_self_of_length_le\n calc p.length\n _ = G.dist u v := hp\n _ \u2264 p.bypass.length := dist_le p.bypass\n rw [\u2190 this]\n apply Walk.bypass_isPath\n\n", "theoremStatement": "lemma Reachable.exists_path_of_dist {u v : V} (hr : G.Reachable u v) :\n \u2203 (p : G.Walk u v), p.IsPath \u2227 p.length = G.dist u v", "theoremName": "Reachable.exists_path_of_dist", "fileCreated": {"commit": "c97ad7df79", "date": "2023-03-02"}, "theoremCreated": {"commit": "2e8aeee458", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Combinatorics/SimpleGraph/Metric.lean", "positionMetadata": {"lineInFile": 136, "tokenPositionInFile": 5061, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n obtain \u27e8p, h\u27e9 := hr.exists_walk_of_dist\n exact \u27e8p, p.isPath_of_length_eq_dist h, h\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 89}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller, Vincent Beffara\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Connectivity\nimport Mathlib.Data.Nat.Lattice\n\n#align_import combinatorics.simple_graph.metric from \"leanprover-community/mathlib\"@\"352ecfe114946c903338006dd3287cb5a9955ff2\"\n\n/-!\n# Graph metric\n\nThis module defines the `SimpleGraph.dist` function, which takes\npairs of vertices to the length of the shortest walk between them.\n\n## Main definitions\n\n- `SimpleGraph.dist` is the graph metric.\n\n## Todo\n\n- Provide an additional computable version of `SimpleGraph.dist`\n for when `G` is connected.\n\n- Evaluate `Nat` vs `ENat` for the codomain of `dist`, or potentially\n having an additional `edist` when the objects under consideration are\n disconnected graphs.\n\n- When directed graphs exist, a directed notion of distance,\n likely `ENat`-valued.\n\n## Tags\n\ngraph metric, distance\n\n-/\n\n\nnamespace SimpleGraph\n\nvariable {V : Type*} (G : SimpleGraph V)\n\n/-! ## Metric -/\n\n\n/-- The distance between two vertices is the length of the shortest walk between them.\nIf no such walk exists, this uses the junk value of `0`. -/\nnoncomputable def dist (u v : V) : \u2115 :=\n sInf (Set.range (Walk.length : G.Walk u v \u2192 \u2115))\n#align simple_graph.dist SimpleGraph.dist\n\nvariable {G}\n\nprotected theorem Reachable.exists_walk_of_dist {u v : V} (hr : G.Reachable u v) :\n \u2203 p : G.Walk u v, p.length = G.dist u v :=\n Nat.sInf_mem (Set.range_nonempty_iff_nonempty.mpr hr)\n#align simple_graph.reachable.exists_walk_of_dist SimpleGraph.Reachable.exists_walk_of_dist\n\nprotected theorem Connected.exists_walk_of_dist (hconn : G.Connected) (u v : V) :\n \u2203 p : G.Walk u v, p.length = G.dist u v :=\n (hconn u v).exists_walk_of_dist\n#align simple_graph.connected.exists_walk_of_dist SimpleGraph.Connected.exists_walk_of_dist\n\ntheorem dist_le {u v : V} (p : G.Walk u v) : G.dist u v \u2264 p.length :=\n Nat.sInf_le \u27e8p, rfl\u27e9\n#align simple_graph.dist_le SimpleGraph.dist_le\n\n@[simp]\ntheorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} :\n G.dist u v = 0 \u2194 u = v \u2228 \u00acG.Reachable u v := by simp [dist, Nat.sInf_eq_zero, Reachable]\n#align simple_graph.dist_eq_zero_iff_eq_or_not_reachable SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable\n\ntheorem dist_self {v : V} : dist G v v = 0 := by simp\n#align simple_graph.dist_self SimpleGraph.dist_self\n\nprotected theorem Reachable.dist_eq_zero_iff {u v : V} (hr : G.Reachable u v) :\n G.dist u v = 0 \u2194 u = v := by simp [hr]\n#align simple_graph.reachable.dist_eq_zero_iff SimpleGraph.Reachable.dist_eq_zero_iff\n\nprotected theorem Reachable.pos_dist_of_ne {u v : V} (h : G.Reachable u v) (hne : u \u2260 v) :\n 0 < G.dist u v :=\n Nat.pos_of_ne_zero (by simp [h, hne])\n#align simple_graph.reachable.pos_dist_of_ne SimpleGraph.Reachable.pos_dist_of_ne\n\nprotected theorem Connected.dist_eq_zero_iff (hconn : G.Connected) {u v : V} :\n G.dist u v = 0 \u2194 u = v := by simp [hconn u v]\n#align simple_graph.connected.dist_eq_zero_iff SimpleGraph.Connected.dist_eq_zero_iff\n\nprotected theorem Connected.pos_dist_of_ne {u v : V} (hconn : G.Connected) (hne : u \u2260 v) :\n 0 < G.dist u v :=\n Nat.pos_of_ne_zero (by intro h; exact False.elim (hne (hconn.dist_eq_zero_iff.mp h)))\n#align simple_graph.connected.pos_dist_of_ne SimpleGraph.Connected.pos_dist_of_ne\n\ntheorem dist_eq_zero_of_not_reachable {u v : V} (h : \u00acG.Reachable u v) : G.dist u v = 0 := by\n simp [h]\n#align simple_graph.dist_eq_zero_of_not_reachable SimpleGraph.dist_eq_zero_of_not_reachable\n\ntheorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) :\n (Set.univ : Set (G.Walk u v)).Nonempty := by\n simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using\n Nat.nonempty_of_pos_sInf h\n#align simple_graph.nonempty_of_pos_dist SimpleGraph.nonempty_of_pos_dist\n\nprotected theorem Connected.dist_triangle (hconn : G.Connected) {u v w : V} :\n G.dist u w \u2264 G.dist u v + G.dist v w := by\n obtain \u27e8p, hp\u27e9 := hconn.exists_walk_of_dist u v\n obtain \u27e8q, hq\u27e9 := hconn.exists_walk_of_dist v w\n rw [\u2190 hp, \u2190 hq, \u2190 Walk.length_append]\n apply dist_le\n#align simple_graph.connected.dist_triangle SimpleGraph.Connected.dist_triangle\n\nprivate theorem dist_comm_aux {u v : V} (h : G.Reachable u v) : G.dist u v \u2264 G.dist v u := by\n obtain \u27e8p, hp\u27e9 := h.symm.exists_walk_of_dist\n rw [\u2190 hp, \u2190 Walk.length_reverse]\n apply dist_le\n\ntheorem dist_comm {u v : V} : G.dist u v = G.dist v u := by\n by_cases h : G.Reachable u v\n \u00b7 apply le_antisymm (dist_comm_aux h) (dist_comm_aux h.symm)\n \u00b7 have h' : \u00acG.Reachable v u := fun h' => absurd h'.symm h\n simp [h, h', dist_eq_zero_of_not_reachable]\n#align simple_graph.dist_comm SimpleGraph.dist_comm\n\ntheorem Walk.isPath_of_length_eq_dist {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v) :\n p.IsPath := by\n classical\n have : p.bypass = p := by\n apply Walk.bypass_eq_self_of_length_le\n calc p.length\n _ = G.dist u v := hp\n _ \u2264 p.bypass.length := dist_le p.bypass\n rw [\u2190 this]\n apply Walk.bypass_isPath\n\nlemma Reachable.exists_path_of_dist {u v : V} (hr : G.Reachable u v) :\n \u2203 (p : G.Walk u v), p.IsPath \u2227 p.length = G.dist u v := by\n obtain \u27e8p, h\u27e9 := hr.exists_walk_of_dist\n exact \u27e8p, p.isPath_of_length_eq_dist h, h\u27e9\n\n", "theoremStatement": "lemma Connected.exists_path_of_dist (hconn : G.Connected) (u v : V) :\n \u2203 (p : G.Walk u v), p.IsPath \u2227 p.length = G.dist u v", "theoremName": "Connected.exists_path_of_dist", "fileCreated": {"commit": "c97ad7df79", "date": "2023-03-02"}, "theoremCreated": {"commit": "2e8aeee458", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Combinatorics/SimpleGraph/Metric.lean", "positionMetadata": {"lineInFile": 141, "tokenPositionInFile": 5283, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n obtain \u27e8p, h\u27e9 := hconn.exists_walk_of_dist u v\n exact \u27e8p, p.isPath_of_length_eq_dist h, h\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 96}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp, Yury Kudryashov\n-/\nimport Mathlib.Analysis.Convex.Combination\nimport Mathlib.Analysis.Convex.Strict\nimport Mathlib.Topology.Connected.PathConnected\nimport Mathlib.Topology.Algebra.Affine\nimport Mathlib.Topology.Algebra.Module.Basic\n\n#align_import analysis.convex.topology from \"leanprover-community/mathlib\"@\"0e3aacdc98d25e0afe035c452d876d28cbffaa7e\"\n\n/-!\n# Topological properties of convex sets\n\nWe prove the following facts:\n\n* `Convex.interior` : interior of a convex set is convex;\n* `Convex.closure` : closure of a convex set is convex;\n* `Set.Finite.isCompact_convexHull` : convex hull of a finite set is compact;\n* `Set.Finite.isClosed_convexHull` : convex hull of a finite set is closed.\n-/\n\nassert_not_exists Norm\n\nopen Metric Bornology Set Pointwise Convex\n\nvariable {\u03b9 \ud835\udd5c E : Type*}\n\ntheorem Real.convex_iff_isPreconnected {s : Set \u211d} : Convex \u211d s \u2194 IsPreconnected s :=\n convex_iff_ordConnected.trans isPreconnected_iff_ordConnected.symm\n#align real.convex_iff_is_preconnected Real.convex_iff_isPreconnected\n\nalias \u27e8_, IsPreconnected.convex\u27e9 := Real.convex_iff_isPreconnected\n#align is_preconnected.convex IsPreconnected.convex\n\n/-! ### Standard simplex -/\n\n\nsection stdSimplex\n\nvariable [Fintype \u03b9]\n\n/-- Every vector in `stdSimplex \ud835\udd5c \u03b9` has `max`-norm at most `1`. -/\ntheorem stdSimplex_subset_closedBall : stdSimplex \u211d \u03b9 \u2286 Metric.closedBall 0 1 := fun f hf \u21a6 by\n rw [Metric.mem_closedBall, dist_pi_le_iff zero_le_one]\n intro x\n rw [Pi.zero_apply, Real.dist_0_eq_abs, abs_of_nonneg <| hf.1 x]\n exact (mem_Icc_of_mem_stdSimplex hf x).2\n#align std_simplex_subset_closed_ball stdSimplex_subset_closedBall\n\nvariable (\u03b9)\n\n/-- `stdSimplex \u211d \u03b9` is bounded. -/\ntheorem bounded_stdSimplex : IsBounded (stdSimplex \u211d \u03b9) :=\n (Metric.isBounded_iff_subset_closedBall 0).2 \u27e81, stdSimplex_subset_closedBall\u27e9\n#align bounded_std_simplex bounded_stdSimplex\n\n/-- `stdSimplex \u211d \u03b9` is closed. -/\ntheorem isClosed_stdSimplex : IsClosed (stdSimplex \u211d \u03b9) :=\n (stdSimplex_eq_inter \u211d \u03b9).symm \u25b8\n IsClosed.inter (isClosed_iInter fun i => isClosed_le continuous_const (continuous_apply i))\n (isClosed_eq (continuous_finset_sum _ fun x _ => continuous_apply x) continuous_const)\n#align is_closed_std_simplex isClosed_stdSimplex\n\n/-- `stdSimplex \u211d \u03b9` is compact. -/\ntheorem isCompact_stdSimplex : IsCompact (stdSimplex \u211d \u03b9) :=\n Metric.isCompact_iff_isClosed_bounded.2 \u27e8isClosed_stdSimplex \u03b9, bounded_stdSimplex \u03b9\u27e9\n#align is_compact_std_simplex isCompact_stdSimplex\n\ninstance stdSimplex.instCompactSpace_coe : CompactSpace \u21a5(stdSimplex \u211d \u03b9) :=\n isCompact_iff_compactSpace.mp <| isCompact_stdSimplex _\n\n/-- The standard one-dimensional simplex in `\u211d\u00b2 = Fin 2 \u2192 \u211d`\nis homeomorphic to the unit interval. -/\n@[simps! (config := .asFn)]\ndef stdSimplexHomeomorphUnitInterval : stdSimplex \u211d (Fin 2) \u2243\u209c unitInterval where\n toEquiv := stdSimplexEquivIcc \u211d\n continuous_toFun := .subtype_mk ((continuous_apply 0).comp continuous_subtype_val) _\n continuous_invFun := by\n apply Continuous.subtype_mk\n exact (continuous_pi <| Fin.forall_fin_two.2\n \u27e8continuous_subtype_val, continuous_const.sub continuous_subtype_val\u27e9)\n\nend stdSimplex\n\n/-! ### Topological vector spaces -/\nsection TopologicalSpace\n\nvariable [LinearOrderedRing \ud835\udd5c] [DenselyOrdered \ud835\udd5c] [TopologicalSpace \ud835\udd5c] [OrderTopology \ud835\udd5c]\n [AddCommGroup E] [TopologicalSpace E] [ContinuousAdd E] [Module \ud835\udd5c E] [ContinuousSMul \ud835\udd5c E]\n {x y : E}\n\ntheorem segment_subset_closure_openSegment : [x -[\ud835\udd5c] y] \u2286 closure (openSegment \ud835\udd5c x y) := by\n rw [segment_eq_image, openSegment_eq_image, \u2190 closure_Ioo (zero_ne_one' \ud835\udd5c)]\n exact image_closure_subset_closure_image (by continuity)\n#align segment_subset_closure_open_segment segment_subset_closure_openSegment\n\nend TopologicalSpace\n\nsection PseudoMetricSpace\n\nvariable [LinearOrderedRing \ud835\udd5c] [DenselyOrdered \ud835\udd5c] [PseudoMetricSpace \ud835\udd5c] [OrderTopology \ud835\udd5c]\n [ProperSpace \ud835\udd5c] [CompactIccSpace \ud835\udd5c] [AddCommGroup E] [TopologicalSpace E] [T2Space E]\n [ContinuousAdd E] [Module \ud835\udd5c E] [ContinuousSMul \ud835\udd5c E]\n\n@[simp]\ntheorem closure_openSegment (x y : E) : closure (openSegment \ud835\udd5c x y) = [x -[\ud835\udd5c] y] := by\n rw [segment_eq_image, openSegment_eq_image, \u2190 closure_Ioo (zero_ne_one' \ud835\udd5c)]\n exact (image_closure_of_isCompact (isBounded_Ioo _ _).isCompact_closure <|\n Continuous.continuousOn <| by continuity).symm\n#align closure_open_segment closure_openSegment\n\nend PseudoMetricSpace\n\nsection ContinuousConstSMul\n\nvariable [LinearOrderedField \ud835\udd5c] [AddCommGroup E] [Module \ud835\udd5c E] [TopologicalSpace E]\n [TopologicalAddGroup E] [ContinuousConstSMul \ud835\udd5c E]\n\n/-- If `s` is a convex set, then `a \u2022 interior s + b \u2022 closure s \u2286 interior s` for all `0 < a`,\n`0 \u2264 b`, `a + b = 1`. See also `Convex.combo_interior_self_subset_interior` for a weaker version. -/\ntheorem Convex.combo_interior_closure_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {a b : \ud835\udd5c}\n (ha : 0 < a) (hb : 0 \u2264 b) (hab : a + b = 1) : a \u2022 interior s + b \u2022 closure s \u2286 interior s :=\n interior_smul\u2080 ha.ne' s \u25b8\n calc\n interior (a \u2022 s) + b \u2022 closure s \u2286 interior (a \u2022 s) + closure (b \u2022 s) :=\n add_subset_add Subset.rfl (smul_closure_subset b s)\n _ = interior (a \u2022 s) + b \u2022 s := by rw [isOpen_interior.add_closure (b \u2022 s)]\n _ \u2286 interior (a \u2022 s + b \u2022 s) := subset_interior_add_left\n _ \u2286 interior s := interior_mono <| hs.set_combo_subset ha.le hb hab\n\n#align convex.combo_interior_closure_subset_interior Convex.combo_interior_closure_subset_interior\n\n/-- If `s` is a convex set, then `a \u2022 interior s + b \u2022 s \u2286 interior s` for all `0 < a`, `0 \u2264 b`,\n`a + b = 1`. See also `Convex.combo_interior_closure_subset_interior` for a stronger version. -/\ntheorem Convex.combo_interior_self_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {a b : \ud835\udd5c}\n (ha : 0 < a) (hb : 0 \u2264 b) (hab : a + b = 1) : a \u2022 interior s + b \u2022 s \u2286 interior s :=\n calc\n a \u2022 interior s + b \u2022 s \u2286 a \u2022 interior s + b \u2022 closure s :=\n add_subset_add Subset.rfl <| image_subset _ subset_closure\n _ \u2286 interior s := hs.combo_interior_closure_subset_interior ha hb hab\n\n#align convex.combo_interior_self_subset_interior Convex.combo_interior_self_subset_interior\n\n/-- If `s` is a convex set, then `a \u2022 closure s + b \u2022 interior s \u2286 interior s` for all `0 \u2264 a`,\n`0 < b`, `a + b = 1`. See also `Convex.combo_self_interior_subset_interior` for a weaker version. -/\ntheorem Convex.combo_closure_interior_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {a b : \ud835\udd5c}\n (ha : 0 \u2264 a) (hb : 0 < b) (hab : a + b = 1) : a \u2022 closure s + b \u2022 interior s \u2286 interior s := by\n rw [add_comm]\n exact hs.combo_interior_closure_subset_interior hb ha (add_comm a b \u25b8 hab)\n#align convex.combo_closure_interior_subset_interior Convex.combo_closure_interior_subset_interior\n\n/-- If `s` is a convex set, then `a \u2022 s + b \u2022 interior s \u2286 interior s` for all `0 \u2264 a`, `0 < b`,\n`a + b = 1`. See also `Convex.combo_closure_interior_subset_interior` for a stronger version. -/\ntheorem Convex.combo_self_interior_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {a b : \ud835\udd5c}\n (ha : 0 \u2264 a) (hb : 0 < b) (hab : a + b = 1) : a \u2022 s + b \u2022 interior s \u2286 interior s := by\n rw [add_comm]\n exact hs.combo_interior_self_subset_interior hb ha (add_comm a b \u25b8 hab)\n#align convex.combo_self_interior_subset_interior Convex.combo_self_interior_subset_interior\n\ntheorem Convex.combo_interior_closure_mem_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 interior s) (hy : y \u2208 closure s) {a b : \ud835\udd5c} (ha : 0 < a) (hb : 0 \u2264 b)\n (hab : a + b = 1) : a \u2022 x + b \u2022 y \u2208 interior s :=\n hs.combo_interior_closure_subset_interior ha hb hab <|\n add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy)\n#align convex.combo_interior_closure_mem_interior Convex.combo_interior_closure_mem_interior\n\ntheorem Convex.combo_interior_self_mem_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 interior s) (hy : y \u2208 s) {a b : \ud835\udd5c} (ha : 0 < a) (hb : 0 \u2264 b) (hab : a + b = 1) :\n a \u2022 x + b \u2022 y \u2208 interior s :=\n hs.combo_interior_closure_mem_interior hx (subset_closure hy) ha hb hab\n#align convex.combo_interior_self_mem_interior Convex.combo_interior_self_mem_interior\n\ntheorem Convex.combo_closure_interior_mem_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 closure s) (hy : y \u2208 interior s) {a b : \ud835\udd5c} (ha : 0 \u2264 a) (hb : 0 < b)\n (hab : a + b = 1) : a \u2022 x + b \u2022 y \u2208 interior s :=\n hs.combo_closure_interior_subset_interior ha hb hab <|\n add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy)\n#align convex.combo_closure_interior_mem_interior Convex.combo_closure_interior_mem_interior\n\ntheorem Convex.combo_self_interior_mem_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E} (hx : x \u2208 s)\n (hy : y \u2208 interior s) {a b : \ud835\udd5c} (ha : 0 \u2264 a) (hb : 0 < b) (hab : a + b = 1) :\n a \u2022 x + b \u2022 y \u2208 interior s :=\n hs.combo_closure_interior_mem_interior (subset_closure hx) hy ha hb hab\n#align convex.combo_self_interior_mem_interior Convex.combo_self_interior_mem_interior\n\ntheorem Convex.openSegment_interior_closure_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 interior s) (hy : y \u2208 closure s) : openSegment \ud835\udd5c x y \u2286 interior s := by\n rintro _ \u27e8a, b, ha, hb, hab, rfl\u27e9\n exact hs.combo_interior_closure_mem_interior hx hy ha hb.le hab\n#align convex.open_segment_interior_closure_subset_interior Convex.openSegment_interior_closure_subset_interior\n\ntheorem Convex.openSegment_interior_self_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 interior s) (hy : y \u2208 s) : openSegment \ud835\udd5c x y \u2286 interior s :=\n hs.openSegment_interior_closure_subset_interior hx (subset_closure hy)\n#align convex.open_segment_interior_self_subset_interior Convex.openSegment_interior_self_subset_interior\n\ntheorem Convex.openSegment_closure_interior_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 closure s) (hy : y \u2208 interior s) : openSegment \ud835\udd5c x y \u2286 interior s := by\n rintro _ \u27e8a, b, ha, hb, hab, rfl\u27e9\n exact hs.combo_closure_interior_mem_interior hx hy ha.le hb hab\n#align convex.open_segment_closure_interior_subset_interior Convex.openSegment_closure_interior_subset_interior\n\ntheorem Convex.openSegment_self_interior_subset_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 s) (hy : y \u2208 interior s) : openSegment \ud835\udd5c x y \u2286 interior s :=\n hs.openSegment_closure_interior_subset_interior (subset_closure hx) hy\n#align convex.open_segment_self_interior_subset_interior Convex.openSegment_self_interior_subset_interior\n\n/-- If `x \u2208 closure s` and `y \u2208 interior s`, then the segment `(x, y]` is included in `interior s`.\n-/\ntheorem Convex.add_smul_sub_mem_interior' {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E}\n (hx : x \u2208 closure s) (hy : y \u2208 interior s) {t : \ud835\udd5c} (ht : t \u2208 Ioc (0 : \ud835\udd5c) 1) :\n x + t \u2022 (y - x) \u2208 interior s := by\n simpa only [sub_smul, smul_sub, one_smul, add_sub, add_comm] using\n hs.combo_interior_closure_mem_interior hy hx ht.1 (sub_nonneg.mpr ht.2)\n (add_sub_cancel _ _)\n#align convex.add_smul_sub_mem_interior' Convex.add_smul_sub_mem_interior'\n\n/-- If `x \u2208 s` and `y \u2208 interior s`, then the segment `(x, y]` is included in `interior s`. -/\ntheorem Convex.add_smul_sub_mem_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E} (hx : x \u2208 s)\n (hy : y \u2208 interior s) {t : \ud835\udd5c} (ht : t \u2208 Ioc (0 : \ud835\udd5c) 1) : x + t \u2022 (y - x) \u2208 interior s :=\n hs.add_smul_sub_mem_interior' (subset_closure hx) hy ht\n#align convex.add_smul_sub_mem_interior Convex.add_smul_sub_mem_interior\n\n/-- If `x \u2208 closure s` and `x + y \u2208 interior s`, then `x + t y \u2208 interior s` for `t \u2208 (0, 1]`. -/\ntheorem Convex.add_smul_mem_interior' {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E} (hx : x \u2208 closure s)\n (hy : x + y \u2208 interior s) {t : \ud835\udd5c} (ht : t \u2208 Ioc (0 : \ud835\udd5c) 1) : x + t \u2022 y \u2208 interior s := by\n simpa only [add_sub_cancel_left] using hs.add_smul_sub_mem_interior' hx hy ht\n#align convex.add_smul_mem_interior' Convex.add_smul_mem_interior'\n\n/-- If `x \u2208 s` and `x + y \u2208 interior s`, then `x + t y \u2208 interior s` for `t \u2208 (0, 1]`. -/\ntheorem Convex.add_smul_mem_interior {s : Set E} (hs : Convex \ud835\udd5c s) {x y : E} (hx : x \u2208 s)\n (hy : x + y \u2208 interior s) {t : \ud835\udd5c} (ht : t \u2208 Ioc (0 : \ud835\udd5c) 1) : x + t \u2022 y \u2208 interior s :=\n hs.add_smul_mem_interior' (subset_closure hx) hy ht\n#align convex.add_smul_mem_interior Convex.add_smul_mem_interior\n\n/-- In a topological vector space, the interior of a convex set is convex. -/\nprotected theorem Convex.interior {s : Set E} (hs : Convex \ud835\udd5c s) : Convex \ud835\udd5c (interior s) :=\n convex_iff_openSegment_subset.mpr fun _ hx _ hy =>\n hs.openSegment_closure_interior_subset_interior (interior_subset_closure hx) hy\n#align convex.interior Convex.interior\n\n/-- In a topological vector space, the closure of a convex set is convex. -/\nprotected theorem Convex.closure {s : Set E} (hs : Convex \ud835\udd5c s) : Convex \ud835\udd5c (closure s) :=\n fun x hx y hy a b ha hb hab =>\n let f : E \u2192 E \u2192 E := fun x' y' => a \u2022 x' + b \u2022 y'\n have hf : Continuous (Function.uncurry f) :=\n (continuous_fst.const_smul _).add (continuous_snd.const_smul _)\n show f x y \u2208 closure s from map_mem_closure\u2082 hf hx hy fun _ hx' _ hy' => hs hx' hy' ha hb hab\n#align convex.closure Convex.closure\n\nopen AffineMap\n\n/-- A convex set `s` is strictly convex provided that for any two distinct points of\n`s \\ interior s`, the line passing through these points has nonempty intersection with\n`interior s`. -/\nprotected theorem Convex.strictConvex' {s : Set E} (hs : Convex \ud835\udd5c s)\n (h : (s \\ interior s).Pairwise fun x y => \u2203 c : \ud835\udd5c, lineMap x y c \u2208 interior s) :\n StrictConvex \ud835\udd5c s := by\n refine' strictConvex_iff_openSegment_subset.2 _\n intro x hx y hy hne\n by_cases hx' : x \u2208 interior s\n \u00b7 exact hs.openSegment_interior_self_subset_interior hx' hy\n by_cases hy' : y \u2208 interior s\n \u00b7 exact hs.openSegment_self_interior_subset_interior hx hy'\n rcases h \u27e8hx, hx'\u27e9 \u27e8hy, hy'\u27e9 hne with \u27e8c, hc\u27e9\n refine' (openSegment_subset_union x y \u27e8c, rfl\u27e9).trans (insert_subset_iff.2 \u27e8hc, union_subset _ _\u27e9)\n exacts [hs.openSegment_self_interior_subset_interior hx hc,\n hs.openSegment_interior_self_subset_interior hc hy]\n#align convex.strict_convex' Convex.strictConvex'\n\n/-- A convex set `s` is strictly convex provided that for any two distinct points `x`, `y` of\n`s \\ interior s`, the segment with endpoints `x`, `y` has nonempty intersection with\n`interior s`. -/\nprotected theorem Convex.strictConvex {s : Set E} (hs : Convex \ud835\udd5c s)\n (h : (s \\ interior s).Pairwise fun x y => ([x -[\ud835\udd5c] y] \\ frontier s).Nonempty) :\n StrictConvex \ud835\udd5c s := by\n refine' hs.strictConvex' <| h.imp_on fun x hx y hy _ => _\n simp only [segment_eq_image_lineMap, \u2190 self_diff_frontier]\n rintro \u27e8_, \u27e8\u27e8c, hc, rfl\u27e9, hcs\u27e9\u27e9\n refine' \u27e8c, hs.segment_subset hx.1 hy.1 _, hcs\u27e9\n exact (segment_eq_image_lineMap \ud835\udd5c x y).symm \u25b8 mem_image_of_mem _ hc\n#align convex.strict_convex Convex.strictConvex\n\nend ContinuousConstSMul\n\nsection ContinuousSMul\n\nvariable [AddCommGroup E] [Module \u211d E] [TopologicalSpace E] [TopologicalAddGroup E]\n [ContinuousSMul \u211d E]\n\n/-- Convex hull of a finite set is compact. -/\ntheorem Set.Finite.isCompact_convexHull {s : Set E} (hs : s.Finite) :\n IsCompact (convexHull \u211d s) := by\n rw [hs.convexHull_eq_image]\n apply (@isCompact_stdSimplex _ hs.fintype).image\n haveI := hs.fintype\n apply LinearMap.continuous_on_pi\n#align set.finite.compact_convex_hull Set.Finite.isCompact_convexHull\n\n/-- Convex hull of a finite set is closed. -/\ntheorem Set.Finite.isClosed_convexHull [T2Space E] {s : Set E} (hs : s.Finite) :\n IsClosed (convexHull \u211d s) :=\n hs.isCompact_convexHull.isClosed\n#align set.finite.is_closed_convex_hull Set.Finite.isClosed_convexHull\n\nopen AffineMap\n\n/-- If we dilate the interior of a convex set about a point in its interior by a scale `t > 1`,\nthe result includes the closure of the original set.\n\nTODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. -/\ntheorem Convex.closure_subset_image_homothety_interior_of_one_lt {s : Set E} (hs : Convex \u211d s)\n {x : E} (hx : x \u2208 interior s) (t : \u211d) (ht : 1 < t) :\n closure s \u2286 homothety x t '' interior s := by\n intro y hy\n have hne : t \u2260 0 := (one_pos.trans ht).ne'\n refine'\n \u27e8homothety x t\u207b\u00b9 y, hs.openSegment_interior_closure_subset_interior hx hy _,\n (AffineEquiv.homothetyUnitsMulHom x (Units.mk0 t hne)).apply_symm_apply y\u27e9\n rw [openSegment_eq_image_lineMap, \u2190 inv_one, \u2190 inv_Ioi (zero_lt_one' \u211d), \u2190 image_inv, image_image,\n homothety_eq_lineMap]\n exact mem_image_of_mem _ ht\n#align convex.closure_subset_image_homothety_interior_of_one_lt Convex.closure_subset_image_homothety_interior_of_one_lt\n\n/-- If we dilate a convex set about a point in its interior by a scale `t > 1`, the interior of\nthe result includes the closure of the original set.\n\nTODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. -/\ntheorem Convex.closure_subset_interior_image_homothety_of_one_lt {s : Set E} (hs : Convex \u211d s)\n {x : E} (hx : x \u2208 interior s) (t : \u211d) (ht : 1 < t) :\n closure s \u2286 interior (homothety x t '' s) :=\n (hs.closure_subset_image_homothety_interior_of_one_lt hx t ht).trans <|\n (homothety_isOpenMap x t (one_pos.trans ht).ne').image_interior_subset _\n#align convex.closure_subset_interior_image_homothety_of_one_lt Convex.closure_subset_interior_image_homothety_of_one_lt\n\n/-- If we dilate a convex set about a point in its interior by a scale `t > 1`, the interior of\nthe result includes the closure of the original set.\n\nTODO Generalise this from convex sets to sets that are balanced / star-shaped about `x`. -/\ntheorem Convex.subset_interior_image_homothety_of_one_lt {s : Set E} (hs : Convex \u211d s) {x : E}\n (hx : x \u2208 interior s) (t : \u211d) (ht : 1 < t) : s \u2286 interior (homothety x t '' s) :=\n subset_closure.trans <| hs.closure_subset_interior_image_homothety_of_one_lt hx t ht\n#align convex.subset_interior_image_homothety_of_one_lt Convex.subset_interior_image_homothety_of_one_lt\n\ntheorem JoinedIn.of_segment_subset {E : Type*} [AddCommGroup E] [Module \u211d E]\n [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul \u211d E]\n {x y : E} {s : Set E} (h : [x -[\u211d] y] \u2286 s) : JoinedIn s x y := by\n have A : Continuous (fun t \u21a6 (1 - t) \u2022 x + t \u2022 y : \u211d \u2192 E) := by continuity\n apply JoinedIn.ofLine A.continuousOn (by simp) (by simp)\n convert h\n rw [segment_eq_image \u211d x y]\n\n/-- A nonempty convex set is path connected. -/\nprotected theorem Convex.isPathConnected {s : Set E} (hconv : Convex \u211d s) (hne : s.Nonempty) :\n IsPathConnected s := by\n refine' isPathConnected_iff.mpr \u27e8hne, _\u27e9\n intro x x_in y y_in\n exact JoinedIn.of_segment_subset ((segment_subset_iff \u211d).2 (hconv x_in y_in))\n#align convex.is_path_connected Convex.isPathConnected\n\n/-- A nonempty convex set is connected. -/\nprotected theorem Convex.isConnected {s : Set E} (h : Convex \u211d s) (hne : s.Nonempty) :\n IsConnected s :=\n (h.isPathConnected hne).isConnected\n#align convex.is_connected Convex.isConnected\n\n/-- A convex set is preconnected. -/\nprotected theorem Convex.isPreconnected {s : Set E} (h : Convex \u211d s) : IsPreconnected s :=\n s.eq_empty_or_nonempty.elim (fun h => h.symm \u25b8 isPreconnected_empty) fun hne =>\n (h.isConnected hne).isPreconnected\n#align convex.is_preconnected Convex.isPreconnected\n\n/-- Every topological vector space over \u211d is path connected.\n\nNot an instance, because it creates enormous TC subproblems (turn on `pp.all`).\n-/\nprotected theorem TopologicalAddGroup.pathConnectedSpace : PathConnectedSpace E :=\n pathConnectedSpace_iff_univ.mpr <| convex_univ.isPathConnected \u27e8(0 : E), trivial\u27e9\n#align topological_add_group.path_connected TopologicalAddGroup.pathConnectedSpace\n\nend ContinuousSMul\n\nsection ComplementsConnected\n\nvariable {E : Type*} [AddCommGroup E] [Module \u211d E] [TopologicalSpace E] [TopologicalAddGroup E]\n\nlocal notation \"\u03c0\" => Submodule.linearProjOfIsCompl _ _\n\nattribute [local instance 100] TopologicalAddGroup.pathConnectedSpace\n\n/-- Given two complementary subspaces `p` and `q` in `E`, if the complement of `{0}`\nis path connected in `p` then the complement of `q` is path connected in `E`. -/\n", "theoremStatement": "theorem isPathConnected_compl_of_isPathConnected_compl_zero [ContinuousSMul \u211d E]\n {p q : Submodule \u211d E} (hpq : IsCompl p q) (hpc : IsPathConnected ({0}\u1d9c : Set p)) :\n IsPathConnected (q\u1d9c : Set E)", "theoremName": "isPathConnected_compl_of_isPathConnected_compl_zero", "fileCreated": {"commit": "0993683f2d", "date": "2023-04-24"}, "theoremCreated": {"commit": "6efcbba115", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Analysis/Convex/Topology.lean", "positionMetadata": {"lineInFile": 396, "tokenPositionInFile": 20058, "theoremPositionInFile": 42}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rw [isPathConnected_iff] at hpc \u22a2\n constructor\n \u00b7 rcases hpc.1 with \u27e8a, ha\u27e9\n exact \u27e8a, mt (Submodule.eq_zero_of_coe_mem_of_disjoint hpq.disjoint) ha\u27e9\n \u00b7 intro x hx y hy\n have : \u03c0 hpq x \u2260 0 \u2227 \u03c0 hpq y \u2260 0 := by\n constructor <;> intro h <;> rw [Submodule.linearProjOfIsCompl_apply_eq_zero_iff hpq] at h <;>\n [exact hx h; exact hy h]\n rcases hpc.2 (\u03c0 hpq x) this.1 (\u03c0 hpq y) this.2 with \u27e8\u03b3\u2081, h\u03b3\u2081\u27e9\n let \u03b3\u2082 := PathConnectedSpace.somePath (\u03c0 hpq.symm x) (\u03c0 hpq.symm y)\n let \u03b3\u2081' : Path (_ : E) _ := \u03b3\u2081.map continuous_subtype_val\n let \u03b3\u2082' : Path (_ : E) _ := \u03b3\u2082.map continuous_subtype_val\n refine \u27e8(\u03b3\u2081'.add \u03b3\u2082').cast (Submodule.linear_proj_add_linearProjOfIsCompl_eq_self hpq x).symm\n (Submodule.linear_proj_add_linearProjOfIsCompl_eq_self hpq y).symm, fun t \u21a6 ?_\u27e9\n rw [Path.cast_coe, Path.add_apply]\n change \u03b3\u2081 t + (\u03b3\u2082 t : E) \u2209 q\n rw [\u2190 Submodule.linearProjOfIsCompl_apply_eq_zero_iff hpq, LinearMap.map_add,\n Submodule.linearProjOfIsCompl_apply_right, add_zero,\n Submodule.linearProjOfIsCompl_apply_eq_zero_iff]\n exact mt (Submodule.eq_zero_of_coe_mem_of_disjoint hpq.disjoint) (h\u03b3\u2081 t)", "proofType": "tactic", "proofLengthLines": 21, "proofLengthTokens": 1146}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2023 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Analysis.Convex.Topology\nimport Mathlib.LinearAlgebra.Dimension.DivisionRing\nimport Mathlib.Topology.Algebra.Module.Cardinality\n\n/-!\n# Connectedness of subsets of vector spaces\n\nWe show several results related to the (path)-connectedness of subsets of real vector spaces:\n* `Set.Countable.isPathConnected_compl_of_one_lt_rank` asserts that the complement of a countable\n set is path-connected in a space of dimension `> 1`.\n* `isPathConnected_compl_singleton_of_one_lt_rank` is the special case of the complement of a\n singleton.\n* `isPathConnected_sphere` shows that any sphere is path-connected in dimension `> 1`.\n* `isPathConnected_compl_of_one_lt_codim` shows that the complement of a subspace\n of codimension `> 1` is path-connected.\n\nStatements with connectedness instead of path-connectedness are also given.\n-/\n\nopen Convex Set Metric\n\nsection TopologicalVectorSpace\n\nvariable {E : Type*} [AddCommGroup E] [Module \u211d E]\n [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul \u211d E]\n\n/-- In a real vector space of dimension `> 1`, the complement of any countable set is path\nconnected. -/\ntheorem Set.Countable.isPathConnected_compl_of_one_lt_rank\n (h : 1 < Module.rank \u211d E) {s : Set E} (hs : s.Countable) :\n IsPathConnected s\u1d9c := by\n have : Nontrivial E := (rank_pos_iff_nontrivial (R := \u211d)).1 (zero_lt_one.trans h)\n -- the set `s\u1d9c` is dense, therefore nonempty. Pick `a \u2208 s\u1d9c`. We have to show that any\n -- `b \u2208 s\u1d9c` can be joined to `a`.\n obtain \u27e8a, ha\u27e9 : s\u1d9c.Nonempty := (hs.dense_compl \u211d).nonempty\n refine \u27e8a, ha, ?_\u27e9\n intro b hb\n rcases eq_or_ne a b with rfl|hab\n \u00b7 exact JoinedIn.refl ha\n /- Assume `b \u2260 a`. Write `a = c - x` and `b = c + x` for some nonzero `x`. Choose `y` which\n is linearly independent from `x`. Then the segments joining `a = c - x` to `c + ty` are pairwise\n disjoint for varying `t` (except for the endpoint `a`) so only countably many of them can\n intersect `s`. In the same way, there are countably many `t`s for which the segment\n from `b = c + x` to `c + ty` intersects `s`. Choosing `t` outside of these countable exceptions,\n one gets a path in the complement of `s` from `a` to `z = c + ty` and then to `b`.\n -/\n let c := (2 : \u211d)\u207b\u00b9 \u2022 (a + b)\n let x := (2 : \u211d)\u207b\u00b9 \u2022 (b - a)\n have Ia : c - x = a := by\n simp only [c, x, smul_add, smul_sub]\n abel_nf\n simp [zsmul_eq_smul_cast \u211d 2]\n have Ib : c + x = b := by\n simp only [c, x, smul_add, smul_sub]\n abel_nf\n simp [zsmul_eq_smul_cast \u211d 2]\n have x_ne_zero : x \u2260 0 := by simpa [x] using sub_ne_zero.2 hab.symm\n obtain \u27e8y, hy\u27e9 : \u2203 y, LinearIndependent \u211d ![x, y] :=\n exists_linearIndependent_pair_of_one_lt_rank h x_ne_zero\n have A : Set.Countable {t : \u211d | ([c + x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty} := by\n apply countable_setOf_nonempty_of_disjoint _ (fun t \u21a6 inter_subset_right _ _) hs\n intro t t' htt'\n apply disjoint_iff_inter_eq_empty.2\n have N : {c + x} \u2229 s = \u2205 := by\n simpa only [singleton_inter_eq_empty, mem_compl_iff, Ib] using hb\n rw [inter_assoc, inter_comm s, inter_assoc, inter_self, \u2190 inter_assoc, \u2190 subset_empty_iff, \u2190 N]\n apply inter_subset_inter_left\n apply Eq.subset\n apply segment_inter_eq_endpoint_of_linearIndependent_of_ne hy htt'.symm\n have B : Set.Countable {t : \u211d | ([c - x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty} := by\n apply countable_setOf_nonempty_of_disjoint _ (fun t \u21a6 inter_subset_right _ _) hs\n intro t t' htt'\n apply disjoint_iff_inter_eq_empty.2\n have N : {c - x} \u2229 s = \u2205 := by\n simpa only [singleton_inter_eq_empty, mem_compl_iff, Ia] using ha\n rw [inter_assoc, inter_comm s, inter_assoc, inter_self, \u2190 inter_assoc, \u2190 subset_empty_iff, \u2190 N]\n apply inter_subset_inter_left\n rw [sub_eq_add_neg _ x]\n apply Eq.subset\n apply segment_inter_eq_endpoint_of_linearIndependent_of_ne _ htt'.symm\n convert hy.units_smul ![-1, 1]\n simp [\u2190 List.ofFn_inj]\n obtain \u27e8t, ht\u27e9 : Set.Nonempty ({t : \u211d | ([c + x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty}\n \u222a {t : \u211d | ([c - x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty})\u1d9c := ((A.union B).dense_compl \u211d).nonempty\n let z := c + t \u2022 y\n simp only [compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_nonempty_iff_eq_empty]\n at ht\n have JA : JoinedIn s\u1d9c a z := by\n apply JoinedIn.of_segment_subset\n rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty]\n convert ht.2\n exact Ia.symm\n have JB : JoinedIn s\u1d9c b z := by\n apply JoinedIn.of_segment_subset\n rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty]\n convert ht.1\n exact Ib.symm\n exact JA.trans JB.symm\n\n/-- In a real vector space of dimension `> 1`, the complement of any countable set is\nconnected. -/\ntheorem Set.Countable.isConnected_compl_of_one_lt_rank (h : 1 < Module.rank \u211d E) {s : Set E}\n (hs : s.Countable) : IsConnected s\u1d9c :=\n (hs.isPathConnected_compl_of_one_lt_rank h).isConnected\n\n/-- In a real vector space of dimension `> 1`, the complement of any singleton is path-connected. -/\ntheorem isPathConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank \u211d E) (x : E) :\n IsPathConnected {x}\u1d9c :=\n Set.Countable.isPathConnected_compl_of_one_lt_rank h (countable_singleton x)\n\n/-- In a real vector space of dimension `> 1`, the complement of a singleton is connected. -/\ntheorem isConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank \u211d E) (x : E) :\n IsConnected {x}\u1d9c :=\n (isPathConnected_compl_singleton_of_one_lt_rank h x).isConnected\n\nend TopologicalVectorSpace\n\nsection NormedSpace\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n\n/-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is\npath connected. -/\ntheorem isPathConnected_sphere (h : 1 < Module.rank \u211d E) (x : E) {r : \u211d} (hr : 0 \u2264 r) :\n IsPathConnected (sphere x r) := by\n /- when `r > 0`, we write the sphere as the image of `{0}\u1d9c` under the map\n `y \u21a6 x + (r * \u2016y\u2016\u207b\u00b9) \u2022 y`. Since the image under a continuous map of a path connected set\n is path connected, this concludes the proof. -/\n rcases hr.eq_or_lt with rfl|rpos\n \u00b7 simpa using isPathConnected_singleton x\n let f : E \u2192 E := fun y \u21a6 x + (r * \u2016y\u2016\u207b\u00b9) \u2022 y\n have A : ContinuousOn f {0}\u1d9c := by\n intro y hy\n apply (continuousAt_const.add _).continuousWithinAt\n apply (continuousAt_const.mul (ContinuousAt.inv\u2080 continuousAt_id.norm ?_)).smul continuousAt_id\n simpa using hy\n have B : IsPathConnected ({0}\u1d9c : Set E) := isPathConnected_compl_singleton_of_one_lt_rank h 0\n have C : IsPathConnected (f '' {0}\u1d9c) := B.image' A\n have : f '' {0}\u1d9c = sphere x r := by\n apply Subset.antisymm\n \u00b7 rintro - \u27e8y, hy, rfl\u27e9\n have : \u2016y\u2016 \u2260 0 := by simpa using hy\n simp [f, norm_smul, abs_of_nonneg hr, mul_assoc, inv_mul_cancel this]\n \u00b7 intro y hy\n refine \u27e8y - x, ?_, ?_\u27e9\n \u00b7 intro H\n simp only [mem_singleton_iff, sub_eq_zero] at H\n simp only [H, mem_sphere_iff_norm, sub_self, norm_zero] at hy\n exact rpos.ne hy\n \u00b7 simp [f, mem_sphere_iff_norm.1 hy, mul_inv_cancel rpos.ne']\n rwa [this] at C\n\n/-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is connected. -/\ntheorem isConnected_sphere (h : 1 < Module.rank \u211d E) (x : E) {r : \u211d} (hr : 0 \u2264 r) :\n IsConnected (sphere x r) :=\n (isPathConnected_sphere h x hr).isConnected\n\n/-- In a real vector space of dimension `> 1`, any sphere is preconnected. -/\ntheorem isPreconnected_sphere (h : 1 < Module.rank \u211d E) (x : E) (r : \u211d) :\n IsPreconnected (sphere x r) := by\n rcases le_or_lt 0 r with hr|hr\n \u00b7 exact (isConnected_sphere h x hr).isPreconnected\n \u00b7 simpa [hr] using isPreconnected_empty\n\nend NormedSpace\n\nsection\n\nvariable {F : Type*} [AddCommGroup F] [Module \u211d F] [TopologicalSpace F]\n [TopologicalAddGroup F] [ContinuousSMul \u211d F]\n\n/-- Let `E` be a linear subspace in a real vector space.\nIf `E` has codimension at least two, its complement is path-connected. -/\n", "theoremStatement": "theorem isPathConnected_compl_of_one_lt_codim {E : Submodule \u211d F}\n (hcodim : 1 < Module.rank \u211d (F \u29f8 E)) : IsPathConnected (E\u1d9c : Set F)", "theoremName": "isPathConnected_compl_of_one_lt_codim", "fileCreated": {"commit": "23550c1e3b", "date": "2023-08-31"}, "theoremCreated": {"commit": "6efcbba115", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Analysis/NormedSpace/Connected.lean", "positionMetadata": {"lineInFile": 179, "tokenPositionInFile": 8020, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rcases E.exists_isCompl with \u27e8E', hE'\u27e9\n refine isPathConnected_compl_of_isPathConnected_compl_zero hE'.symm\n (isPathConnected_compl_singleton_of_one_lt_rank ?_ 0)\n rwa [\u2190 (E.quotientEquivOfIsCompl E' hE').rank_eq]", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 223}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2023 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Analysis.Convex.Topology\nimport Mathlib.LinearAlgebra.Dimension.DivisionRing\nimport Mathlib.Topology.Algebra.Module.Cardinality\n\n/-!\n# Connectedness of subsets of vector spaces\n\nWe show several results related to the (path)-connectedness of subsets of real vector spaces:\n* `Set.Countable.isPathConnected_compl_of_one_lt_rank` asserts that the complement of a countable\n set is path-connected in a space of dimension `> 1`.\n* `isPathConnected_compl_singleton_of_one_lt_rank` is the special case of the complement of a\n singleton.\n* `isPathConnected_sphere` shows that any sphere is path-connected in dimension `> 1`.\n* `isPathConnected_compl_of_one_lt_codim` shows that the complement of a subspace\n of codimension `> 1` is path-connected.\n\nStatements with connectedness instead of path-connectedness are also given.\n-/\n\nopen Convex Set Metric\n\nsection TopologicalVectorSpace\n\nvariable {E : Type*} [AddCommGroup E] [Module \u211d E]\n [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul \u211d E]\n\n/-- In a real vector space of dimension `> 1`, the complement of any countable set is path\nconnected. -/\ntheorem Set.Countable.isPathConnected_compl_of_one_lt_rank\n (h : 1 < Module.rank \u211d E) {s : Set E} (hs : s.Countable) :\n IsPathConnected s\u1d9c := by\n have : Nontrivial E := (rank_pos_iff_nontrivial (R := \u211d)).1 (zero_lt_one.trans h)\n -- the set `s\u1d9c` is dense, therefore nonempty. Pick `a \u2208 s\u1d9c`. We have to show that any\n -- `b \u2208 s\u1d9c` can be joined to `a`.\n obtain \u27e8a, ha\u27e9 : s\u1d9c.Nonempty := (hs.dense_compl \u211d).nonempty\n refine \u27e8a, ha, ?_\u27e9\n intro b hb\n rcases eq_or_ne a b with rfl|hab\n \u00b7 exact JoinedIn.refl ha\n /- Assume `b \u2260 a`. Write `a = c - x` and `b = c + x` for some nonzero `x`. Choose `y` which\n is linearly independent from `x`. Then the segments joining `a = c - x` to `c + ty` are pairwise\n disjoint for varying `t` (except for the endpoint `a`) so only countably many of them can\n intersect `s`. In the same way, there are countably many `t`s for which the segment\n from `b = c + x` to `c + ty` intersects `s`. Choosing `t` outside of these countable exceptions,\n one gets a path in the complement of `s` from `a` to `z = c + ty` and then to `b`.\n -/\n let c := (2 : \u211d)\u207b\u00b9 \u2022 (a + b)\n let x := (2 : \u211d)\u207b\u00b9 \u2022 (b - a)\n have Ia : c - x = a := by\n simp only [c, x, smul_add, smul_sub]\n abel_nf\n simp [zsmul_eq_smul_cast \u211d 2]\n have Ib : c + x = b := by\n simp only [c, x, smul_add, smul_sub]\n abel_nf\n simp [zsmul_eq_smul_cast \u211d 2]\n have x_ne_zero : x \u2260 0 := by simpa [x] using sub_ne_zero.2 hab.symm\n obtain \u27e8y, hy\u27e9 : \u2203 y, LinearIndependent \u211d ![x, y] :=\n exists_linearIndependent_pair_of_one_lt_rank h x_ne_zero\n have A : Set.Countable {t : \u211d | ([c + x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty} := by\n apply countable_setOf_nonempty_of_disjoint _ (fun t \u21a6 inter_subset_right _ _) hs\n intro t t' htt'\n apply disjoint_iff_inter_eq_empty.2\n have N : {c + x} \u2229 s = \u2205 := by\n simpa only [singleton_inter_eq_empty, mem_compl_iff, Ib] using hb\n rw [inter_assoc, inter_comm s, inter_assoc, inter_self, \u2190 inter_assoc, \u2190 subset_empty_iff, \u2190 N]\n apply inter_subset_inter_left\n apply Eq.subset\n apply segment_inter_eq_endpoint_of_linearIndependent_of_ne hy htt'.symm\n have B : Set.Countable {t : \u211d | ([c - x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty} := by\n apply countable_setOf_nonempty_of_disjoint _ (fun t \u21a6 inter_subset_right _ _) hs\n intro t t' htt'\n apply disjoint_iff_inter_eq_empty.2\n have N : {c - x} \u2229 s = \u2205 := by\n simpa only [singleton_inter_eq_empty, mem_compl_iff, Ia] using ha\n rw [inter_assoc, inter_comm s, inter_assoc, inter_self, \u2190 inter_assoc, \u2190 subset_empty_iff, \u2190 N]\n apply inter_subset_inter_left\n rw [sub_eq_add_neg _ x]\n apply Eq.subset\n apply segment_inter_eq_endpoint_of_linearIndependent_of_ne _ htt'.symm\n convert hy.units_smul ![-1, 1]\n simp [\u2190 List.ofFn_inj]\n obtain \u27e8t, ht\u27e9 : Set.Nonempty ({t : \u211d | ([c + x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty}\n \u222a {t : \u211d | ([c - x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty})\u1d9c := ((A.union B).dense_compl \u211d).nonempty\n let z := c + t \u2022 y\n simp only [compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_nonempty_iff_eq_empty]\n at ht\n have JA : JoinedIn s\u1d9c a z := by\n apply JoinedIn.of_segment_subset\n rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty]\n convert ht.2\n exact Ia.symm\n have JB : JoinedIn s\u1d9c b z := by\n apply JoinedIn.of_segment_subset\n rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty]\n convert ht.1\n exact Ib.symm\n exact JA.trans JB.symm\n\n/-- In a real vector space of dimension `> 1`, the complement of any countable set is\nconnected. -/\ntheorem Set.Countable.isConnected_compl_of_one_lt_rank (h : 1 < Module.rank \u211d E) {s : Set E}\n (hs : s.Countable) : IsConnected s\u1d9c :=\n (hs.isPathConnected_compl_of_one_lt_rank h).isConnected\n\n/-- In a real vector space of dimension `> 1`, the complement of any singleton is path-connected. -/\ntheorem isPathConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank \u211d E) (x : E) :\n IsPathConnected {x}\u1d9c :=\n Set.Countable.isPathConnected_compl_of_one_lt_rank h (countable_singleton x)\n\n/-- In a real vector space of dimension `> 1`, the complement of a singleton is connected. -/\ntheorem isConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank \u211d E) (x : E) :\n IsConnected {x}\u1d9c :=\n (isPathConnected_compl_singleton_of_one_lt_rank h x).isConnected\n\nend TopologicalVectorSpace\n\nsection NormedSpace\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n\n/-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is\npath connected. -/\ntheorem isPathConnected_sphere (h : 1 < Module.rank \u211d E) (x : E) {r : \u211d} (hr : 0 \u2264 r) :\n IsPathConnected (sphere x r) := by\n /- when `r > 0`, we write the sphere as the image of `{0}\u1d9c` under the map\n `y \u21a6 x + (r * \u2016y\u2016\u207b\u00b9) \u2022 y`. Since the image under a continuous map of a path connected set\n is path connected, this concludes the proof. -/\n rcases hr.eq_or_lt with rfl|rpos\n \u00b7 simpa using isPathConnected_singleton x\n let f : E \u2192 E := fun y \u21a6 x + (r * \u2016y\u2016\u207b\u00b9) \u2022 y\n have A : ContinuousOn f {0}\u1d9c := by\n intro y hy\n apply (continuousAt_const.add _).continuousWithinAt\n apply (continuousAt_const.mul (ContinuousAt.inv\u2080 continuousAt_id.norm ?_)).smul continuousAt_id\n simpa using hy\n have B : IsPathConnected ({0}\u1d9c : Set E) := isPathConnected_compl_singleton_of_one_lt_rank h 0\n have C : IsPathConnected (f '' {0}\u1d9c) := B.image' A\n have : f '' {0}\u1d9c = sphere x r := by\n apply Subset.antisymm\n \u00b7 rintro - \u27e8y, hy, rfl\u27e9\n have : \u2016y\u2016 \u2260 0 := by simpa using hy\n simp [f, norm_smul, abs_of_nonneg hr, mul_assoc, inv_mul_cancel this]\n \u00b7 intro y hy\n refine \u27e8y - x, ?_, ?_\u27e9\n \u00b7 intro H\n simp only [mem_singleton_iff, sub_eq_zero] at H\n simp only [H, mem_sphere_iff_norm, sub_self, norm_zero] at hy\n exact rpos.ne hy\n \u00b7 simp [f, mem_sphere_iff_norm.1 hy, mul_inv_cancel rpos.ne']\n rwa [this] at C\n\n/-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is connected. -/\ntheorem isConnected_sphere (h : 1 < Module.rank \u211d E) (x : E) {r : \u211d} (hr : 0 \u2264 r) :\n IsConnected (sphere x r) :=\n (isPathConnected_sphere h x hr).isConnected\n\n/-- In a real vector space of dimension `> 1`, any sphere is preconnected. -/\ntheorem isPreconnected_sphere (h : 1 < Module.rank \u211d E) (x : E) (r : \u211d) :\n IsPreconnected (sphere x r) := by\n rcases le_or_lt 0 r with hr|hr\n \u00b7 exact (isConnected_sphere h x hr).isPreconnected\n \u00b7 simpa [hr] using isPreconnected_empty\n\nend NormedSpace\n\nsection\n\nvariable {F : Type*} [AddCommGroup F] [Module \u211d F] [TopologicalSpace F]\n [TopologicalAddGroup F] [ContinuousSMul \u211d F]\n\n/-- Let `E` be a linear subspace in a real vector space.\nIf `E` has codimension at least two, its complement is path-connected. -/\ntheorem isPathConnected_compl_of_one_lt_codim {E : Submodule \u211d F}\n (hcodim : 1 < Module.rank \u211d (F \u29f8 E)) : IsPathConnected (E\u1d9c : Set F) := by\n rcases E.exists_isCompl with \u27e8E', hE'\u27e9\n refine isPathConnected_compl_of_isPathConnected_compl_zero hE'.symm\n (isPathConnected_compl_singleton_of_one_lt_rank ?_ 0)\n rwa [\u2190 (E.quotientEquivOfIsCompl E' hE').rank_eq]\n\n/-- Let `E` be a linear subspace in a real vector space.\nIf `E` has codimension at least two, its complement is connected. -/\n", "theoremStatement": "theorem isConnected_compl_of_one_lt_codim {E : Submodule \u211d F} (hcodim : 1 < Module.rank \u211d (F \u29f8 E)) :\n IsConnected (E\u1d9c : Set F)", "theoremName": "isConnected_compl_of_one_lt_codim", "fileCreated": {"commit": "23550c1e3b", "date": "2023-08-31"}, "theoremCreated": {"commit": "6efcbba115", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Analysis/NormedSpace/Connected.lean", "positionMetadata": {"lineInFile": 188, "tokenPositionInFile": 8512, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "(isPathConnected_compl_of_one_lt_codim hcodim).isConnected", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 58}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2023 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Analysis.Convex.Topology\nimport Mathlib.LinearAlgebra.Dimension.DivisionRing\nimport Mathlib.Topology.Algebra.Module.Cardinality\n\n/-!\n# Connectedness of subsets of vector spaces\n\nWe show several results related to the (path)-connectedness of subsets of real vector spaces:\n* `Set.Countable.isPathConnected_compl_of_one_lt_rank` asserts that the complement of a countable\n set is path-connected in a space of dimension `> 1`.\n* `isPathConnected_compl_singleton_of_one_lt_rank` is the special case of the complement of a\n singleton.\n* `isPathConnected_sphere` shows that any sphere is path-connected in dimension `> 1`.\n* `isPathConnected_compl_of_one_lt_codim` shows that the complement of a subspace\n of codimension `> 1` is path-connected.\n\nStatements with connectedness instead of path-connectedness are also given.\n-/\n\nopen Convex Set Metric\n\nsection TopologicalVectorSpace\n\nvariable {E : Type*} [AddCommGroup E] [Module \u211d E]\n [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul \u211d E]\n\n/-- In a real vector space of dimension `> 1`, the complement of any countable set is path\nconnected. -/\ntheorem Set.Countable.isPathConnected_compl_of_one_lt_rank\n (h : 1 < Module.rank \u211d E) {s : Set E} (hs : s.Countable) :\n IsPathConnected s\u1d9c := by\n have : Nontrivial E := (rank_pos_iff_nontrivial (R := \u211d)).1 (zero_lt_one.trans h)\n -- the set `s\u1d9c` is dense, therefore nonempty. Pick `a \u2208 s\u1d9c`. We have to show that any\n -- `b \u2208 s\u1d9c` can be joined to `a`.\n obtain \u27e8a, ha\u27e9 : s\u1d9c.Nonempty := (hs.dense_compl \u211d).nonempty\n refine \u27e8a, ha, ?_\u27e9\n intro b hb\n rcases eq_or_ne a b with rfl|hab\n \u00b7 exact JoinedIn.refl ha\n /- Assume `b \u2260 a`. Write `a = c - x` and `b = c + x` for some nonzero `x`. Choose `y` which\n is linearly independent from `x`. Then the segments joining `a = c - x` to `c + ty` are pairwise\n disjoint for varying `t` (except for the endpoint `a`) so only countably many of them can\n intersect `s`. In the same way, there are countably many `t`s for which the segment\n from `b = c + x` to `c + ty` intersects `s`. Choosing `t` outside of these countable exceptions,\n one gets a path in the complement of `s` from `a` to `z = c + ty` and then to `b`.\n -/\n let c := (2 : \u211d)\u207b\u00b9 \u2022 (a + b)\n let x := (2 : \u211d)\u207b\u00b9 \u2022 (b - a)\n have Ia : c - x = a := by\n simp only [c, x, smul_add, smul_sub]\n abel_nf\n simp [zsmul_eq_smul_cast \u211d 2]\n have Ib : c + x = b := by\n simp only [c, x, smul_add, smul_sub]\n abel_nf\n simp [zsmul_eq_smul_cast \u211d 2]\n have x_ne_zero : x \u2260 0 := by simpa [x] using sub_ne_zero.2 hab.symm\n obtain \u27e8y, hy\u27e9 : \u2203 y, LinearIndependent \u211d ![x, y] :=\n exists_linearIndependent_pair_of_one_lt_rank h x_ne_zero\n have A : Set.Countable {t : \u211d | ([c + x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty} := by\n apply countable_setOf_nonempty_of_disjoint _ (fun t \u21a6 inter_subset_right _ _) hs\n intro t t' htt'\n apply disjoint_iff_inter_eq_empty.2\n have N : {c + x} \u2229 s = \u2205 := by\n simpa only [singleton_inter_eq_empty, mem_compl_iff, Ib] using hb\n rw [inter_assoc, inter_comm s, inter_assoc, inter_self, \u2190 inter_assoc, \u2190 subset_empty_iff, \u2190 N]\n apply inter_subset_inter_left\n apply Eq.subset\n apply segment_inter_eq_endpoint_of_linearIndependent_of_ne hy htt'.symm\n have B : Set.Countable {t : \u211d | ([c - x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty} := by\n apply countable_setOf_nonempty_of_disjoint _ (fun t \u21a6 inter_subset_right _ _) hs\n intro t t' htt'\n apply disjoint_iff_inter_eq_empty.2\n have N : {c - x} \u2229 s = \u2205 := by\n simpa only [singleton_inter_eq_empty, mem_compl_iff, Ia] using ha\n rw [inter_assoc, inter_comm s, inter_assoc, inter_self, \u2190 inter_assoc, \u2190 subset_empty_iff, \u2190 N]\n apply inter_subset_inter_left\n rw [sub_eq_add_neg _ x]\n apply Eq.subset\n apply segment_inter_eq_endpoint_of_linearIndependent_of_ne _ htt'.symm\n convert hy.units_smul ![-1, 1]\n simp [\u2190 List.ofFn_inj]\n obtain \u27e8t, ht\u27e9 : Set.Nonempty ({t : \u211d | ([c + x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty}\n \u222a {t : \u211d | ([c - x -[\u211d] c + t \u2022 y] \u2229 s).Nonempty})\u1d9c := ((A.union B).dense_compl \u211d).nonempty\n let z := c + t \u2022 y\n simp only [compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_nonempty_iff_eq_empty]\n at ht\n have JA : JoinedIn s\u1d9c a z := by\n apply JoinedIn.of_segment_subset\n rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty]\n convert ht.2\n exact Ia.symm\n have JB : JoinedIn s\u1d9c b z := by\n apply JoinedIn.of_segment_subset\n rw [subset_compl_iff_disjoint_right, disjoint_iff_inter_eq_empty]\n convert ht.1\n exact Ib.symm\n exact JA.trans JB.symm\n\n/-- In a real vector space of dimension `> 1`, the complement of any countable set is\nconnected. -/\ntheorem Set.Countable.isConnected_compl_of_one_lt_rank (h : 1 < Module.rank \u211d E) {s : Set E}\n (hs : s.Countable) : IsConnected s\u1d9c :=\n (hs.isPathConnected_compl_of_one_lt_rank h).isConnected\n\n/-- In a real vector space of dimension `> 1`, the complement of any singleton is path-connected. -/\ntheorem isPathConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank \u211d E) (x : E) :\n IsPathConnected {x}\u1d9c :=\n Set.Countable.isPathConnected_compl_of_one_lt_rank h (countable_singleton x)\n\n/-- In a real vector space of dimension `> 1`, the complement of a singleton is connected. -/\ntheorem isConnected_compl_singleton_of_one_lt_rank (h : 1 < Module.rank \u211d E) (x : E) :\n IsConnected {x}\u1d9c :=\n (isPathConnected_compl_singleton_of_one_lt_rank h x).isConnected\n\nend TopologicalVectorSpace\n\nsection NormedSpace\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n\n/-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is\npath connected. -/\ntheorem isPathConnected_sphere (h : 1 < Module.rank \u211d E) (x : E) {r : \u211d} (hr : 0 \u2264 r) :\n IsPathConnected (sphere x r) := by\n /- when `r > 0`, we write the sphere as the image of `{0}\u1d9c` under the map\n `y \u21a6 x + (r * \u2016y\u2016\u207b\u00b9) \u2022 y`. Since the image under a continuous map of a path connected set\n is path connected, this concludes the proof. -/\n rcases hr.eq_or_lt with rfl|rpos\n \u00b7 simpa using isPathConnected_singleton x\n let f : E \u2192 E := fun y \u21a6 x + (r * \u2016y\u2016\u207b\u00b9) \u2022 y\n have A : ContinuousOn f {0}\u1d9c := by\n intro y hy\n apply (continuousAt_const.add _).continuousWithinAt\n apply (continuousAt_const.mul (ContinuousAt.inv\u2080 continuousAt_id.norm ?_)).smul continuousAt_id\n simpa using hy\n have B : IsPathConnected ({0}\u1d9c : Set E) := isPathConnected_compl_singleton_of_one_lt_rank h 0\n have C : IsPathConnected (f '' {0}\u1d9c) := B.image' A\n have : f '' {0}\u1d9c = sphere x r := by\n apply Subset.antisymm\n \u00b7 rintro - \u27e8y, hy, rfl\u27e9\n have : \u2016y\u2016 \u2260 0 := by simpa using hy\n simp [f, norm_smul, abs_of_nonneg hr, mul_assoc, inv_mul_cancel this]\n \u00b7 intro y hy\n refine \u27e8y - x, ?_, ?_\u27e9\n \u00b7 intro H\n simp only [mem_singleton_iff, sub_eq_zero] at H\n simp only [H, mem_sphere_iff_norm, sub_self, norm_zero] at hy\n exact rpos.ne hy\n \u00b7 simp [f, mem_sphere_iff_norm.1 hy, mul_inv_cancel rpos.ne']\n rwa [this] at C\n\n/-- In a real vector space of dimension `> 1`, any sphere of nonnegative radius is connected. -/\ntheorem isConnected_sphere (h : 1 < Module.rank \u211d E) (x : E) {r : \u211d} (hr : 0 \u2264 r) :\n IsConnected (sphere x r) :=\n (isPathConnected_sphere h x hr).isConnected\n\n/-- In a real vector space of dimension `> 1`, any sphere is preconnected. -/\ntheorem isPreconnected_sphere (h : 1 < Module.rank \u211d E) (x : E) (r : \u211d) :\n IsPreconnected (sphere x r) := by\n rcases le_or_lt 0 r with hr|hr\n \u00b7 exact (isConnected_sphere h x hr).isPreconnected\n \u00b7 simpa [hr] using isPreconnected_empty\n\nend NormedSpace\n\nsection\n\nvariable {F : Type*} [AddCommGroup F] [Module \u211d F] [TopologicalSpace F]\n [TopologicalAddGroup F] [ContinuousSMul \u211d F]\n\n/-- Let `E` be a linear subspace in a real vector space.\nIf `E` has codimension at least two, its complement is path-connected. -/\ntheorem isPathConnected_compl_of_one_lt_codim {E : Submodule \u211d F}\n (hcodim : 1 < Module.rank \u211d (F \u29f8 E)) : IsPathConnected (E\u1d9c : Set F) := by\n rcases E.exists_isCompl with \u27e8E', hE'\u27e9\n refine isPathConnected_compl_of_isPathConnected_compl_zero hE'.symm\n (isPathConnected_compl_singleton_of_one_lt_rank ?_ 0)\n rwa [\u2190 (E.quotientEquivOfIsCompl E' hE').rank_eq]\n\n/-- Let `E` be a linear subspace in a real vector space.\nIf `E` has codimension at least two, its complement is connected. -/\ntheorem isConnected_compl_of_one_lt_codim {E : Submodule \u211d F} (hcodim : 1 < Module.rank \u211d (F \u29f8 E)) :\n IsConnected (E\u1d9c : Set F) :=\n (isPathConnected_compl_of_one_lt_codim hcodim).isConnected\n\n", "theoremStatement": "theorem Submodule.connectedComponentIn_eq_self_of_one_lt_codim (E : Submodule \u211d F)\n (hcodim : 1 < Module.rank \u211d (F \u29f8 E)) {x : F} (hx : x \u2209 E) :\n connectedComponentIn ((E : Set F)\u1d9c) x = (E : Set F)\u1d9c", "theoremName": "Submodule.connectedComponentIn_eq_self_of_one_lt_codim", "fileCreated": {"commit": "23550c1e3b", "date": "2023-08-31"}, "theoremCreated": {"commit": "6efcbba115", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Analysis/NormedSpace/Connected.lean", "positionMetadata": {"lineInFile": 192, "tokenPositionInFile": 8707, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "(isConnected_compl_of_one_lt_codim hcodim).2.connectedComponentIn hx", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 68}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Yury Kudryashov\n-/\nimport Mathlib.Analysis.Normed.Group.Basic\nimport Mathlib.MeasureTheory.Function.AEMeasurableSequence\nimport Mathlib.MeasureTheory.Group.Arithmetic\nimport Mathlib.MeasureTheory.Order.Lattice\nimport Mathlib.Topology.Instances.EReal\nimport Mathlib.Topology.MetricSpace.Thickening\nimport Mathlib.Topology.GDelta\nimport Mathlib.Topology.Order.Lattice\nimport Mathlib.Topology.Semicontinuous\n\n#align_import measure_theory.constructions.borel_space.basic from \"leanprover-community/mathlib\"@\"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce\"\n\n/-!\n# Borel (measurable) space\n\n## Main definitions\n\n* `borel \u03b1` : the least `\u03c3`-algebra that contains all open sets;\n* `class BorelSpace` : a space with `TopologicalSpace` and `MeasurableSpace` structures\n such that `\u2039MeasurableSpace \u03b1\u203a = borel \u03b1`;\n* `class OpensMeasurableSpace` : a space with `TopologicalSpace` and `MeasurableSpace`\n structures such that all open sets are measurable; equivalently, `borel \u03b1 \u2264 \u2039MeasurableSpace \u03b1\u203a`.\n* `BorelSpace` instances on `Empty`, `Unit`, `Bool`, `Nat`, `Int`, `Rat`;\n* `MeasurableSpace` and `BorelSpace` instances on `\u211d`, `\u211d\u22650`, `\u211d\u22650\u221e`.\n\n## Main statements\n\n* `IsOpen.measurableSet`, `IsClosed.measurableSet`: open and closed sets are measurable;\n* `Continuous.measurable` : a continuous function is measurable;\n* `Continuous.measurable2` : if `f : \u03b1 \u2192 \u03b2` and `g : \u03b1 \u2192 \u03b3` are measurable and `op : \u03b2 \u00d7 \u03b3 \u2192 \u03b4`\n is continuous, then `fun x => op (f x, g y)` is measurable;\n* `Measurable.add` etc : dot notation for arithmetic operations on `Measurable` predicates,\n and similarly for `dist` and `edist`;\n* `AEMeasurable.add` : similar dot notation for almost everywhere measurable functions;\n* `Measurable.ennreal*` : special cases for arithmetic operations on `\u211d\u22650\u221e`.\n-/\n\n\nnoncomputable section\n\nopen Set Filter MeasureTheory\n\nopen scoped Classical BigOperators Topology NNReal ENNReal MeasureTheory\n\nuniverse u v w x y\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b3\u2082 \u03b4 : Type*} {\u03b9 : Sort y} {s t u : Set \u03b1}\n\nopen MeasurableSpace TopologicalSpace\n\n/-- `MeasurableSpace` structure generated by `TopologicalSpace`. -/\ndef borel (\u03b1 : Type u) [TopologicalSpace \u03b1] : MeasurableSpace \u03b1 :=\n generateFrom { s : Set \u03b1 | IsOpen s }\n#align borel borel\n\ntheorem borel_anti : Antitone (@borel \u03b1) := fun _ _ h =>\n MeasurableSpace.generateFrom_le fun _ hs => .basic _ (h _ hs)\n#align borel_anti borel_anti\n\ntheorem borel_eq_top_of_discrete [TopologicalSpace \u03b1] [DiscreteTopology \u03b1] : borel \u03b1 = \u22a4 :=\n top_le_iff.1 fun s _ => GenerateMeasurable.basic s (isOpen_discrete s)\n#align borel_eq_top_of_discrete borel_eq_top_of_discrete\n\ntheorem borel_eq_top_of_countable [TopologicalSpace \u03b1] [T1Space \u03b1] [Countable \u03b1] : borel \u03b1 = \u22a4 := by\n refine' top_le_iff.1 fun s _ => biUnion_of_singleton s \u25b8 _\n apply MeasurableSet.biUnion s.to_countable\n intro x _\n apply MeasurableSet.of_compl\n apply GenerateMeasurable.basic\n exact isClosed_singleton.isOpen_compl\n#align borel_eq_top_of_countable borel_eq_top_of_countable\n\ntheorem borel_eq_generateFrom_of_subbasis {s : Set (Set \u03b1)} [t : TopologicalSpace \u03b1]\n [SecondCountableTopology \u03b1] (hs : t = .generateFrom s) : borel \u03b1 = .generateFrom s :=\n le_antisymm\n (generateFrom_le fun u (hu : t.IsOpen u) => by\n rw [hs] at hu\n induction hu with\n | basic u hu => exact GenerateMeasurable.basic u hu\n | univ => exact @MeasurableSet.univ \u03b1 (generateFrom s)\n | inter s\u2081 s\u2082 _ _ hs\u2081 hs\u2082 => exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082\n | sUnion f hf ih =>\n rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9\n rw [\u2190 vu]\n exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv))\n (generateFrom_le fun u hu =>\n GenerateMeasurable.basic _ <| show t.IsOpen u by rw [hs]; exact GenerateOpen.basic _ hu)\n#align borel_eq_generate_from_of_subbasis borel_eq_generateFrom_of_subbasis\n\ntheorem TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom [TopologicalSpace \u03b1]\n [SecondCountableTopology \u03b1] {s : Set (Set \u03b1)} (hs : IsTopologicalBasis s) :\n borel \u03b1 = .generateFrom s :=\n borel_eq_generateFrom_of_subbasis hs.eq_generateFrom\n#align topological_space.is_topological_basis.borel_eq_generate_from TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom\n\ntheorem isPiSystem_isOpen [TopologicalSpace \u03b1] : IsPiSystem ({s : Set \u03b1 | IsOpen s}) :=\n fun _s hs _t ht _ => IsOpen.inter hs ht\n#align is_pi_system_is_open isPiSystem_isOpen\n\nlemma isPiSystem_isClosed [TopologicalSpace \u03b1] : IsPiSystem ({s : Set \u03b1 | IsClosed s}) :=\n fun _s hs _t ht _ \u21a6 IsClosed.inter hs ht\n\ntheorem borel_eq_generateFrom_isClosed [TopologicalSpace \u03b1] :\n borel \u03b1 = .generateFrom { s | IsClosed s } :=\n le_antisymm\n (generateFrom_le fun _t ht =>\n @MeasurableSet.of_compl \u03b1 _ (generateFrom { s | IsClosed s })\n (GenerateMeasurable.basic _ <| isClosed_compl_iff.2 ht))\n (generateFrom_le fun _t ht =>\n @MeasurableSet.of_compl \u03b1 _ (borel \u03b1) (GenerateMeasurable.basic _ <| isOpen_compl_iff.2 ht))\n#align borel_eq_generate_from_is_closed borel_eq_generateFrom_isClosed\n\nsection OrderTopology\n\nvariable (\u03b1)\nvariable [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1]\n\ntheorem borel_eq_generateFrom_Iio : borel \u03b1 = .generateFrom (range Iio) := by\n refine' le_antisymm _ (generateFrom_le _)\n \u00b7 rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals \u03b1 _ _ _)]\n letI : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom (range Iio)\n have H : \u2200 a : \u03b1, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ \u27e8_, rfl\u27e9\n refine' generateFrom_le _\n rintro _ \u27e8a, rfl | rfl\u27e9\n \u00b7 rcases em (\u2203 b, a \u22d6 b) with \u27e8b, hb\u27e9 | hcovBy\n \u00b7 rw [hb.Ioi_eq, \u2190 compl_Iio]\n exact (H _).compl\n \u00b7 rcases isOpen_biUnion_countable (Ioi a) Ioi fun _ _ \u21a6 isOpen_Ioi with \u27e8t, hat, htc, htU\u27e9\n have : Ioi a = \u22c3 b \u2208 t, Ici b := by\n refine Subset.antisymm ?_ <| iUnion\u2082_subset fun b hb \u21a6 Ici_subset_Ioi.2 (hat hb)\n refine Subset.trans ?_ <| iUnion\u2082_mono fun _ _ \u21a6 Ioi_subset_Ici_self\n simpa [CovBy, htU, subset_def] using hcovBy\n simp only [this, \u2190 compl_Iio]\n exact .biUnion htc <| fun _ _ \u21a6 (H _).compl\n \u00b7 apply H\n \u00b7 rw [forall_mem_range]\n intro a\n exact GenerateMeasurable.basic _ isOpen_Iio\n#align borel_eq_generate_from_Iio borel_eq_generateFrom_Iio\n\ntheorem borel_eq_generateFrom_Ioi : borel \u03b1 = .generateFrom (range Ioi) :=\n @borel_eq_generateFrom_Iio \u03b1\u1d52\u1d48 _ (by infer_instance : SecondCountableTopology \u03b1) _ _\n#align borel_eq_generate_from_Ioi borel_eq_generateFrom_Ioi\n\ntheorem borel_eq_generateFrom_Iic :\n borel \u03b1 = MeasurableSpace.generateFrom (range Iic) := by\n rw [borel_eq_generateFrom_Ioi]\n refine' le_antisymm _ _\n \u00b7 refine' MeasurableSpace.generateFrom_le fun t ht => _\n obtain \u27e8u, rfl\u27e9 := ht\n rw [\u2190 compl_Iic]\n exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr \u27e8u, rfl\u27e9)).compl\n \u00b7 refine' MeasurableSpace.generateFrom_le fun t ht => _\n obtain \u27e8u, rfl\u27e9 := ht\n rw [\u2190 compl_Ioi]\n exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr \u27e8u, rfl\u27e9)).compl\n#align borel_eq_generate_from_Iic borel_eq_generateFrom_Iic\n\ntheorem borel_eq_generateFrom_Ici : borel \u03b1 = MeasurableSpace.generateFrom (range Ici) :=\n @borel_eq_generateFrom_Iic \u03b1\u1d52\u1d48 _ _ _ _\n#align borel_eq_generate_from_Ici borel_eq_generateFrom_Ici\n\nend OrderTopology\n\ntheorem borel_comap {f : \u03b1 \u2192 \u03b2} {t : TopologicalSpace \u03b2} :\n @borel \u03b1 (t.induced f) = (@borel \u03b2 t).comap f :=\n comap_generateFrom.symm\n#align borel_comap borel_comap\n\ntheorem Continuous.borel_measurable [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n (hf : Continuous f) : @Measurable \u03b1 \u03b2 (borel \u03b1) (borel \u03b2) f :=\n Measurable.of_le_map <|\n generateFrom_le fun s hs => GenerateMeasurable.basic (f \u207b\u00b9' s) (hs.preimage hf)\n#align continuous.borel_measurable Continuous.borel_measurable\n\n/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that\nall open sets are measurable. -/\nclass OpensMeasurableSpace (\u03b1 : Type*) [TopologicalSpace \u03b1] [h : MeasurableSpace \u03b1] : Prop where\n /-- Borel-measurable sets are measurable. -/\n borel_le : borel \u03b1 \u2264 h\n#align opens_measurable_space OpensMeasurableSpace\n#align opens_measurable_space.borel_le OpensMeasurableSpace.borel_le\n\n/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that\nthe `\u03c3`-algebra of measurable sets is exactly the `\u03c3`-algebra generated by open sets. -/\nclass BorelSpace (\u03b1 : Type*) [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] : Prop where\n /-- The measurable sets are exactly the Borel-measurable sets. -/\n measurable_eq : \u2039MeasurableSpace \u03b1\u203a = borel \u03b1\n#align borel_space BorelSpace\n#align borel_space.measurable_eq BorelSpace.measurable_eq\n\nnamespace Mathlib.Tactic.Borelize\n\nopen Lean Elab Term Tactic Meta\n\n/-- The behaviour of `borelize \u03b1` depends on the existing assumptions on `\u03b1`.\n\n- if `\u03b1` is a topological space with instances `[MeasurableSpace \u03b1] [BorelSpace \u03b1]`, then\n `borelize \u03b1` replaces the former instance by `borel \u03b1`;\n- otherwise, `borelize \u03b1` adds instances `borel \u03b1 : MeasurableSpace \u03b1` and `\u27e8rfl\u27e9 : BorelSpace \u03b1`.\n\nFinally, `borelize \u03b1 \u03b2 \u03b3` runs `borelize \u03b1; borelize \u03b2; borelize \u03b3`.\n-/\nsyntax \"borelize\" (ppSpace colGt term:max)* : tactic\n\n/-- Add instances `borel e : MeasurableSpace e` and `\u27e8rfl\u27e9 : BorelSpace e`. -/\ndef addBorelInstance (e : Expr) : TacticM Unit := do\n let t \u2190 Lean.Elab.Term.exprToSyntax e\n evalTactic <| \u2190 `(tactic|\n refine_lift\n letI : MeasurableSpace $t := borel $t\n haveI : BorelSpace $t := \u27e8rfl\u27e9\n ?_)\n\n/-- Given a type `e`, an assumption `i : MeasurableSpace e`, and an instance `[BorelSpace e]`,\nreplace `i` with `borel e`. -/\ndef borelToRefl (e : Expr) (i : FVarId) : TacticM Unit := do\n let te \u2190 Lean.Elab.Term.exprToSyntax e\n evalTactic <| \u2190 `(tactic|\n have := @BorelSpace.measurable_eq $te _ _ _)\n try\n liftMetaTactic fun m => return [\u2190 subst m i]\n catch _ =>\n let et \u2190 synthInstance (\u2190 mkAppOptM ``TopologicalSpace #[e])\n throwError m!\"\\\n `\u2039TopologicalSpace {e}\u203a := {et}\\n\\\n depends on\\n\\\n {Expr.fvar i} : MeasurableSpace {e}`\\n\\\n so `borelize` isn't avaliable\"\n evalTactic <| \u2190 `(tactic|\n refine_lift\n letI : MeasurableSpace $te := borel $te\n ?_)\n\n/-- Given a type `$t`, if there is an assumption `[i : MeasurableSpace $t]`, then try to prove\n`[BorelSpace $t]` and replace `i` with `borel $t`. Otherwise, add instances\n`borel $t : MeasurableSpace $t` and `\u27e8rfl\u27e9 : BorelSpace $t`. -/\ndef borelize (t : Term) : TacticM Unit := withMainContext <| do\n let u \u2190 mkFreshLevelMVar\n let e \u2190 withoutRecover <| Tactic.elabTermEnsuringType t (mkSort (mkLevelSucc u))\n let i? \u2190 findLocalDeclWithType? (\u2190 mkAppOptM ``MeasurableSpace #[e])\n i?.elim (addBorelInstance e) (borelToRefl e)\n\nelab_rules : tactic\n | `(tactic| borelize $[$t:term]*) => t.forM borelize\n\nend Mathlib.Tactic.Borelize\n\ninstance (priority := 100) OrderDual.opensMeasurableSpace {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [h : OpensMeasurableSpace \u03b1] : OpensMeasurableSpace \u03b1\u1d52\u1d48 where\n borel_le := h.borel_le\n#align order_dual.opens_measurable_space OrderDual.opensMeasurableSpace\n\ninstance (priority := 100) OrderDual.borelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [h : BorelSpace \u03b1] : BorelSpace \u03b1\u1d52\u1d48 where\n measurable_eq := h.measurable_eq\n#align order_dual.borel_space OrderDual.borelSpace\n\n/-- In a `BorelSpace` all open sets are measurable. -/\ninstance (priority := 100) BorelSpace.opensMeasurable {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [BorelSpace \u03b1] : OpensMeasurableSpace \u03b1 :=\n \u27e8ge_of_eq <| BorelSpace.measurable_eq\u27e9\n#align borel_space.opens_measurable BorelSpace.opensMeasurable\n\ninstance Subtype.borelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [MeasurableSpace \u03b1]\n [h\u03b1 : BorelSpace \u03b1] (s : Set \u03b1) : BorelSpace s :=\n \u27e8by borelize \u03b1; symm; apply borel_comap\u27e9\n#align subtype.borel_space Subtype.borelSpace\n\ninstance Countable.instBorelSpace [Countable \u03b1] [MeasurableSpace \u03b1] [MeasurableSingletonClass \u03b1]\n [TopologicalSpace \u03b1] [DiscreteTopology \u03b1] : BorelSpace \u03b1 := by\n have : \u2200 s, @MeasurableSet \u03b1 inferInstance s := fun s \u21a6 s.to_countable.measurableSet\n have : \u2200 s, @MeasurableSet \u03b1 (borel \u03b1) s := fun s \u21a6 measurableSet_generateFrom (isOpen_discrete s)\n exact \u27e8by aesop\u27e9\n\ninstance Subtype.opensMeasurableSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [MeasurableSpace \u03b1]\n [h : OpensMeasurableSpace \u03b1] (s : Set \u03b1) : OpensMeasurableSpace s :=\n \u27e8by\n rw [borel_comap]\n exact comap_mono h.1\u27e9\n#align subtype.opens_measurable_space Subtype.opensMeasurableSpace\n\nlemma opensMeasurableSpace_iff_forall_measurableSet\n [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] :\n OpensMeasurableSpace \u03b1 \u2194 (\u2200 (s : Set \u03b1), IsOpen s \u2192 MeasurableSet s) := by\n refine \u27e8fun h s hs \u21a6 ?_, fun h \u21a6 \u27e8generateFrom_le h\u27e9\u27e9\n exact OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ hs\n\ninstance (priority := 100) BorelSpace.countablyGenerated {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [BorelSpace \u03b1] [SecondCountableTopology \u03b1] : CountablyGenerated \u03b1 := by\n obtain \u27e8b, bct, -, hb\u27e9 := exists_countable_basis \u03b1\n refine' \u27e8\u27e8b, bct, _\u27e9\u27e9\n borelize \u03b1\n exact hb.borel_eq_generateFrom\n#align borel_space.countably_generated BorelSpace.countablyGenerated\n\ntheorem MeasurableSet.induction_on_open [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1]\n {C : Set \u03b1 \u2192 Prop} (h_open : \u2200 U, IsOpen U \u2192 C U)\n (h_compl : \u2200 t, MeasurableSet t \u2192 C t \u2192 C t\u1d9c)\n (h_union :\n \u2200 f : \u2115 \u2192 Set \u03b1,\n Pairwise (Disjoint on f) \u2192 (\u2200 i, MeasurableSet (f i)) \u2192 (\u2200 i, C (f i)) \u2192 C (\u22c3 i, f i)) :\n \u2200 \u2983t\u2984, MeasurableSet t \u2192 C t :=\n MeasurableSpace.induction_on_inter BorelSpace.measurable_eq isPiSystem_isOpen\n (h_open _ isOpen_empty) h_open h_compl h_union\n#align measurable_set.induction_on_open MeasurableSet.induction_on_open\n\nsection\n\nvariable [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [OpensMeasurableSpace \u03b1] [TopologicalSpace \u03b2]\n [MeasurableSpace \u03b2] [OpensMeasurableSpace \u03b2] [TopologicalSpace \u03b3] [MeasurableSpace \u03b3]\n [BorelSpace \u03b3] [TopologicalSpace \u03b3\u2082] [MeasurableSpace \u03b3\u2082] [BorelSpace \u03b3\u2082] [MeasurableSpace \u03b4]\n\ntheorem IsOpen.measurableSet (h : IsOpen s) : MeasurableSet s :=\n OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ h\n#align is_open.measurable_set IsOpen.measurableSet\n\ninstance (priority := 500) {s : Set \u03b1} [HasCountableSeparatingOn \u03b1 IsOpen s] :\n HasCountableSeparatingOn \u03b1 MeasurableSet s :=\n .mono (fun _ \u21a6 IsOpen.measurableSet) Subset.rfl\n\n@[measurability]\ntheorem measurableSet_interior : MeasurableSet (interior s) :=\n isOpen_interior.measurableSet\n#align measurable_set_interior measurableSet_interior\n\ntheorem IsG\u03b4.measurableSet (h : IsG\u03b4 s) : MeasurableSet s := by\n rcases h with \u27e8S, hSo, hSc, rfl\u27e9\n exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet\nset_option linter.uppercaseLean3 false in\n#align is_G\u03b4.measurable_set IsG\u03b4.measurableSet\n\ntheorem measurableSet_of_continuousAt {\u03b2} [EMetricSpace \u03b2] (f : \u03b1 \u2192 \u03b2) :\n MeasurableSet { x | ContinuousAt f x } :=\n (IsG\u03b4.setOf_continuousAt f).measurableSet\n#align measurable_set_of_continuous_at measurableSet_of_continuousAt\n\ntheorem IsClosed.measurableSet (h : IsClosed s) : MeasurableSet s :=\n h.isOpen_compl.measurableSet.of_compl\n#align is_closed.measurable_set IsClosed.measurableSet\n\ntheorem IsCompact.measurableSet [T2Space \u03b1] (h : IsCompact s) : MeasurableSet s :=\n h.isClosed.measurableSet\n#align is_compact.measurable_set IsCompact.measurableSet\n\n/-- If two points are topologically inseparable,\nthen they can't be separated by a Borel measurable set. -/\ntheorem Inseparable.mem_measurableSet_iff {x y : \u03b3} (h : Inseparable x y) {s : Set \u03b3}\n (hs : MeasurableSet s) : x \u2208 s \u2194 y \u2208 s :=\n hs.induction_on_open (C := fun s \u21a6 (x \u2208 s \u2194 y \u2208 s)) (fun _ \u21a6 h.mem_open_iff) (fun s _ hs \u21a6 hs.not)\n fun _ _ _ h \u21a6 by simp [h]\n\n/-- If `K` is a compact set in an R\u2081 space and `s \u2287 K` is a Borel measurable superset,\nthen `s` includes the closure of `K` as well. -/\ntheorem IsCompact.closure_subset_measurableSet [R1Space \u03b3] {K s : Set \u03b3} (hK : IsCompact K)\n (hs : MeasurableSet s) (hKs : K \u2286 s) : closure K \u2286 s := by\n rw [hK.closure_eq_biUnion_inseparable, iUnion\u2082_subset_iff]\n exact fun x hx y hy \u21a6 (hy.mem_measurableSet_iff hs).1 (hKs hx)\n\n/-- In an R\u2081 topological space with Borel measure `\u03bc`,\nthe measure of the closure of a compact set `K` is equal to the measure of `K`.\n\nSee also `MeasureTheory.Measure.OuterRegular.measure_closure_eq_of_isCompact`\nfor a version that assumes `\u03bc` to be outer regular\nbut does not assume the `\u03c3`-algebra to be Borel. -/\ntheorem IsCompact.measure_closure [R1Space \u03b3] {K : Set \u03b3} (hK : IsCompact K) (\u03bc : Measure \u03b3) :\n \u03bc (closure K) = \u03bc K := by\n refine le_antisymm ?_ (measure_mono subset_closure)\n calc\n \u03bc (closure K) \u2264 \u03bc (toMeasurable \u03bc K) := measure_mono <|\n hK.closure_subset_measurableSet (measurableSet_toMeasurable ..) (subset_toMeasurable ..)\n _ = \u03bc K := measure_toMeasurable ..\n\n@[measurability]\ntheorem measurableSet_closure : MeasurableSet (closure s) :=\n isClosed_closure.measurableSet\n#align measurable_set_closure measurableSet_closure\n\ntheorem measurable_of_isOpen {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, IsOpen s \u2192 MeasurableSet (f \u207b\u00b9' s)) :\n Measurable f := by\n rw [\u2039BorelSpace \u03b3\u203a.measurable_eq]\n exact measurable_generateFrom hf\n#align measurable_of_is_open measurable_of_isOpen\n\ntheorem measurable_of_isClosed {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, IsClosed s \u2192 MeasurableSet (f \u207b\u00b9' s)) :\n Measurable f := by\n apply measurable_of_isOpen; intro s hs\n rw [\u2190 MeasurableSet.compl_iff, \u2190 preimage_compl]; apply hf; rw [isClosed_compl_iff]; exact hs\n#align measurable_of_is_closed measurable_of_isClosed\n\ntheorem measurable_of_isClosed' {f : \u03b4 \u2192 \u03b3}\n (hf : \u2200 s, IsClosed s \u2192 s.Nonempty \u2192 s \u2260 univ \u2192 MeasurableSet (f \u207b\u00b9' s)) : Measurable f := by\n apply measurable_of_isClosed; intro s hs\n rcases eq_empty_or_nonempty s with h1 | h1\n \u00b7 simp [h1]\n by_cases h2 : s = univ\n \u00b7 simp [h2]\n exact hf s hs h1 h2\n#align measurable_of_is_closed' measurable_of_isClosed'\n\ninstance nhds_isMeasurablyGenerated (a : \u03b1) : (\ud835\udcdd a).IsMeasurablyGenerated := by\n rw [nhds, iInf_subtype']\n refine' @Filter.iInf_isMeasurablyGenerated \u03b1 _ _ _ fun i => _\n exact i.2.2.measurableSet.principal_isMeasurablyGenerated\n#align nhds_is_measurably_generated nhds_isMeasurablyGenerated\n\n/-- If `s` is a measurable set, then `\ud835\udcdd[s] a` is a measurably generated filter for\neach `a`. This cannot be an `instance` because it depends on a non-instance `hs : MeasurableSet s`.\n-/\ntheorem MeasurableSet.nhdsWithin_isMeasurablyGenerated {s : Set \u03b1} (hs : MeasurableSet s) (a : \u03b1) :\n (\ud835\udcdd[s] a).IsMeasurablyGenerated :=\n haveI := hs.principal_isMeasurablyGenerated\n Filter.inf_isMeasurablyGenerated _ _\n#align measurable_set.nhds_within_is_measurably_generated MeasurableSet.nhdsWithin_isMeasurablyGenerated\n\n", "theoremStatement": "instance (priority := 100) OpensMeasurableSpace.separatesPoints [T0Space \u03b1] :\n SeparatesPoints \u03b1", "theoremName": null, "fileCreated": {"commit": "11332d53f1", "date": "2023-05-21"}, "theoremCreated": {"commit": "726f2a5ff9", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "positionMetadata": {"lineInFile": 421, "tokenPositionInFile": 19055, "theoremPositionInFile": 47}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rw [separatesPoints_iff]\n intro x y hxy\n apply Inseparable.eq\n rw [inseparable_iff_forall_open]\n exact fun s hs => hxy _ hs.measurableSet", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 146}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Yury Kudryashov\n-/\nimport Mathlib.Analysis.Normed.Group.Basic\nimport Mathlib.MeasureTheory.Function.AEMeasurableSequence\nimport Mathlib.MeasureTheory.Group.Arithmetic\nimport Mathlib.MeasureTheory.Order.Lattice\nimport Mathlib.Topology.Instances.EReal\nimport Mathlib.Topology.MetricSpace.Thickening\nimport Mathlib.Topology.GDelta\nimport Mathlib.Topology.Order.Lattice\nimport Mathlib.Topology.Semicontinuous\n\n#align_import measure_theory.constructions.borel_space.basic from \"leanprover-community/mathlib\"@\"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce\"\n\n/-!\n# Borel (measurable) space\n\n## Main definitions\n\n* `borel \u03b1` : the least `\u03c3`-algebra that contains all open sets;\n* `class BorelSpace` : a space with `TopologicalSpace` and `MeasurableSpace` structures\n such that `\u2039MeasurableSpace \u03b1\u203a = borel \u03b1`;\n* `class OpensMeasurableSpace` : a space with `TopologicalSpace` and `MeasurableSpace`\n structures such that all open sets are measurable; equivalently, `borel \u03b1 \u2264 \u2039MeasurableSpace \u03b1\u203a`.\n* `BorelSpace` instances on `Empty`, `Unit`, `Bool`, `Nat`, `Int`, `Rat`;\n* `MeasurableSpace` and `BorelSpace` instances on `\u211d`, `\u211d\u22650`, `\u211d\u22650\u221e`.\n\n## Main statements\n\n* `IsOpen.measurableSet`, `IsClosed.measurableSet`: open and closed sets are measurable;\n* `Continuous.measurable` : a continuous function is measurable;\n* `Continuous.measurable2` : if `f : \u03b1 \u2192 \u03b2` and `g : \u03b1 \u2192 \u03b3` are measurable and `op : \u03b2 \u00d7 \u03b3 \u2192 \u03b4`\n is continuous, then `fun x => op (f x, g y)` is measurable;\n* `Measurable.add` etc : dot notation for arithmetic operations on `Measurable` predicates,\n and similarly for `dist` and `edist`;\n* `AEMeasurable.add` : similar dot notation for almost everywhere measurable functions;\n* `Measurable.ennreal*` : special cases for arithmetic operations on `\u211d\u22650\u221e`.\n-/\n\n\nnoncomputable section\n\nopen Set Filter MeasureTheory\n\nopen scoped Classical BigOperators Topology NNReal ENNReal MeasureTheory\n\nuniverse u v w x y\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b3\u2082 \u03b4 : Type*} {\u03b9 : Sort y} {s t u : Set \u03b1}\n\nopen MeasurableSpace TopologicalSpace\n\n/-- `MeasurableSpace` structure generated by `TopologicalSpace`. -/\ndef borel (\u03b1 : Type u) [TopologicalSpace \u03b1] : MeasurableSpace \u03b1 :=\n generateFrom { s : Set \u03b1 | IsOpen s }\n#align borel borel\n\ntheorem borel_anti : Antitone (@borel \u03b1) := fun _ _ h =>\n MeasurableSpace.generateFrom_le fun _ hs => .basic _ (h _ hs)\n#align borel_anti borel_anti\n\ntheorem borel_eq_top_of_discrete [TopologicalSpace \u03b1] [DiscreteTopology \u03b1] : borel \u03b1 = \u22a4 :=\n top_le_iff.1 fun s _ => GenerateMeasurable.basic s (isOpen_discrete s)\n#align borel_eq_top_of_discrete borel_eq_top_of_discrete\n\ntheorem borel_eq_top_of_countable [TopologicalSpace \u03b1] [T1Space \u03b1] [Countable \u03b1] : borel \u03b1 = \u22a4 := by\n refine' top_le_iff.1 fun s _ => biUnion_of_singleton s \u25b8 _\n apply MeasurableSet.biUnion s.to_countable\n intro x _\n apply MeasurableSet.of_compl\n apply GenerateMeasurable.basic\n exact isClosed_singleton.isOpen_compl\n#align borel_eq_top_of_countable borel_eq_top_of_countable\n\ntheorem borel_eq_generateFrom_of_subbasis {s : Set (Set \u03b1)} [t : TopologicalSpace \u03b1]\n [SecondCountableTopology \u03b1] (hs : t = .generateFrom s) : borel \u03b1 = .generateFrom s :=\n le_antisymm\n (generateFrom_le fun u (hu : t.IsOpen u) => by\n rw [hs] at hu\n induction hu with\n | basic u hu => exact GenerateMeasurable.basic u hu\n | univ => exact @MeasurableSet.univ \u03b1 (generateFrom s)\n | inter s\u2081 s\u2082 _ _ hs\u2081 hs\u2082 => exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082\n | sUnion f hf ih =>\n rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9\n rw [\u2190 vu]\n exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv))\n (generateFrom_le fun u hu =>\n GenerateMeasurable.basic _ <| show t.IsOpen u by rw [hs]; exact GenerateOpen.basic _ hu)\n#align borel_eq_generate_from_of_subbasis borel_eq_generateFrom_of_subbasis\n\ntheorem TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom [TopologicalSpace \u03b1]\n [SecondCountableTopology \u03b1] {s : Set (Set \u03b1)} (hs : IsTopologicalBasis s) :\n borel \u03b1 = .generateFrom s :=\n borel_eq_generateFrom_of_subbasis hs.eq_generateFrom\n#align topological_space.is_topological_basis.borel_eq_generate_from TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom\n\ntheorem isPiSystem_isOpen [TopologicalSpace \u03b1] : IsPiSystem ({s : Set \u03b1 | IsOpen s}) :=\n fun _s hs _t ht _ => IsOpen.inter hs ht\n#align is_pi_system_is_open isPiSystem_isOpen\n\nlemma isPiSystem_isClosed [TopologicalSpace \u03b1] : IsPiSystem ({s : Set \u03b1 | IsClosed s}) :=\n fun _s hs _t ht _ \u21a6 IsClosed.inter hs ht\n\ntheorem borel_eq_generateFrom_isClosed [TopologicalSpace \u03b1] :\n borel \u03b1 = .generateFrom { s | IsClosed s } :=\n le_antisymm\n (generateFrom_le fun _t ht =>\n @MeasurableSet.of_compl \u03b1 _ (generateFrom { s | IsClosed s })\n (GenerateMeasurable.basic _ <| isClosed_compl_iff.2 ht))\n (generateFrom_le fun _t ht =>\n @MeasurableSet.of_compl \u03b1 _ (borel \u03b1) (GenerateMeasurable.basic _ <| isOpen_compl_iff.2 ht))\n#align borel_eq_generate_from_is_closed borel_eq_generateFrom_isClosed\n\nsection OrderTopology\n\nvariable (\u03b1)\nvariable [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1]\n\ntheorem borel_eq_generateFrom_Iio : borel \u03b1 = .generateFrom (range Iio) := by\n refine' le_antisymm _ (generateFrom_le _)\n \u00b7 rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals \u03b1 _ _ _)]\n letI : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom (range Iio)\n have H : \u2200 a : \u03b1, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ \u27e8_, rfl\u27e9\n refine' generateFrom_le _\n rintro _ \u27e8a, rfl | rfl\u27e9\n \u00b7 rcases em (\u2203 b, a \u22d6 b) with \u27e8b, hb\u27e9 | hcovBy\n \u00b7 rw [hb.Ioi_eq, \u2190 compl_Iio]\n exact (H _).compl\n \u00b7 rcases isOpen_biUnion_countable (Ioi a) Ioi fun _ _ \u21a6 isOpen_Ioi with \u27e8t, hat, htc, htU\u27e9\n have : Ioi a = \u22c3 b \u2208 t, Ici b := by\n refine Subset.antisymm ?_ <| iUnion\u2082_subset fun b hb \u21a6 Ici_subset_Ioi.2 (hat hb)\n refine Subset.trans ?_ <| iUnion\u2082_mono fun _ _ \u21a6 Ioi_subset_Ici_self\n simpa [CovBy, htU, subset_def] using hcovBy\n simp only [this, \u2190 compl_Iio]\n exact .biUnion htc <| fun _ _ \u21a6 (H _).compl\n \u00b7 apply H\n \u00b7 rw [forall_mem_range]\n intro a\n exact GenerateMeasurable.basic _ isOpen_Iio\n#align borel_eq_generate_from_Iio borel_eq_generateFrom_Iio\n\ntheorem borel_eq_generateFrom_Ioi : borel \u03b1 = .generateFrom (range Ioi) :=\n @borel_eq_generateFrom_Iio \u03b1\u1d52\u1d48 _ (by infer_instance : SecondCountableTopology \u03b1) _ _\n#align borel_eq_generate_from_Ioi borel_eq_generateFrom_Ioi\n\ntheorem borel_eq_generateFrom_Iic :\n borel \u03b1 = MeasurableSpace.generateFrom (range Iic) := by\n rw [borel_eq_generateFrom_Ioi]\n refine' le_antisymm _ _\n \u00b7 refine' MeasurableSpace.generateFrom_le fun t ht => _\n obtain \u27e8u, rfl\u27e9 := ht\n rw [\u2190 compl_Iic]\n exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr \u27e8u, rfl\u27e9)).compl\n \u00b7 refine' MeasurableSpace.generateFrom_le fun t ht => _\n obtain \u27e8u, rfl\u27e9 := ht\n rw [\u2190 compl_Ioi]\n exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr \u27e8u, rfl\u27e9)).compl\n#align borel_eq_generate_from_Iic borel_eq_generateFrom_Iic\n\ntheorem borel_eq_generateFrom_Ici : borel \u03b1 = MeasurableSpace.generateFrom (range Ici) :=\n @borel_eq_generateFrom_Iic \u03b1\u1d52\u1d48 _ _ _ _\n#align borel_eq_generate_from_Ici borel_eq_generateFrom_Ici\n\nend OrderTopology\n\ntheorem borel_comap {f : \u03b1 \u2192 \u03b2} {t : TopologicalSpace \u03b2} :\n @borel \u03b1 (t.induced f) = (@borel \u03b2 t).comap f :=\n comap_generateFrom.symm\n#align borel_comap borel_comap\n\ntheorem Continuous.borel_measurable [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n (hf : Continuous f) : @Measurable \u03b1 \u03b2 (borel \u03b1) (borel \u03b2) f :=\n Measurable.of_le_map <|\n generateFrom_le fun s hs => GenerateMeasurable.basic (f \u207b\u00b9' s) (hs.preimage hf)\n#align continuous.borel_measurable Continuous.borel_measurable\n\n/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that\nall open sets are measurable. -/\nclass OpensMeasurableSpace (\u03b1 : Type*) [TopologicalSpace \u03b1] [h : MeasurableSpace \u03b1] : Prop where\n /-- Borel-measurable sets are measurable. -/\n borel_le : borel \u03b1 \u2264 h\n#align opens_measurable_space OpensMeasurableSpace\n#align opens_measurable_space.borel_le OpensMeasurableSpace.borel_le\n\n/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that\nthe `\u03c3`-algebra of measurable sets is exactly the `\u03c3`-algebra generated by open sets. -/\nclass BorelSpace (\u03b1 : Type*) [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] : Prop where\n /-- The measurable sets are exactly the Borel-measurable sets. -/\n measurable_eq : \u2039MeasurableSpace \u03b1\u203a = borel \u03b1\n#align borel_space BorelSpace\n#align borel_space.measurable_eq BorelSpace.measurable_eq\n\nnamespace Mathlib.Tactic.Borelize\n\nopen Lean Elab Term Tactic Meta\n\n/-- The behaviour of `borelize \u03b1` depends on the existing assumptions on `\u03b1`.\n\n- if `\u03b1` is a topological space with instances `[MeasurableSpace \u03b1] [BorelSpace \u03b1]`, then\n `borelize \u03b1` replaces the former instance by `borel \u03b1`;\n- otherwise, `borelize \u03b1` adds instances `borel \u03b1 : MeasurableSpace \u03b1` and `\u27e8rfl\u27e9 : BorelSpace \u03b1`.\n\nFinally, `borelize \u03b1 \u03b2 \u03b3` runs `borelize \u03b1; borelize \u03b2; borelize \u03b3`.\n-/\nsyntax \"borelize\" (ppSpace colGt term:max)* : tactic\n\n/-- Add instances `borel e : MeasurableSpace e` and `\u27e8rfl\u27e9 : BorelSpace e`. -/\ndef addBorelInstance (e : Expr) : TacticM Unit := do\n let t \u2190 Lean.Elab.Term.exprToSyntax e\n evalTactic <| \u2190 `(tactic|\n refine_lift\n letI : MeasurableSpace $t := borel $t\n haveI : BorelSpace $t := \u27e8rfl\u27e9\n ?_)\n\n/-- Given a type `e`, an assumption `i : MeasurableSpace e`, and an instance `[BorelSpace e]`,\nreplace `i` with `borel e`. -/\ndef borelToRefl (e : Expr) (i : FVarId) : TacticM Unit := do\n let te \u2190 Lean.Elab.Term.exprToSyntax e\n evalTactic <| \u2190 `(tactic|\n have := @BorelSpace.measurable_eq $te _ _ _)\n try\n liftMetaTactic fun m => return [\u2190 subst m i]\n catch _ =>\n let et \u2190 synthInstance (\u2190 mkAppOptM ``TopologicalSpace #[e])\n throwError m!\"\\\n `\u2039TopologicalSpace {e}\u203a := {et}\\n\\\n depends on\\n\\\n {Expr.fvar i} : MeasurableSpace {e}`\\n\\\n so `borelize` isn't avaliable\"\n evalTactic <| \u2190 `(tactic|\n refine_lift\n letI : MeasurableSpace $te := borel $te\n ?_)\n\n/-- Given a type `$t`, if there is an assumption `[i : MeasurableSpace $t]`, then try to prove\n`[BorelSpace $t]` and replace `i` with `borel $t`. Otherwise, add instances\n`borel $t : MeasurableSpace $t` and `\u27e8rfl\u27e9 : BorelSpace $t`. -/\ndef borelize (t : Term) : TacticM Unit := withMainContext <| do\n let u \u2190 mkFreshLevelMVar\n let e \u2190 withoutRecover <| Tactic.elabTermEnsuringType t (mkSort (mkLevelSucc u))\n let i? \u2190 findLocalDeclWithType? (\u2190 mkAppOptM ``MeasurableSpace #[e])\n i?.elim (addBorelInstance e) (borelToRefl e)\n\nelab_rules : tactic\n | `(tactic| borelize $[$t:term]*) => t.forM borelize\n\nend Mathlib.Tactic.Borelize\n\ninstance (priority := 100) OrderDual.opensMeasurableSpace {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [h : OpensMeasurableSpace \u03b1] : OpensMeasurableSpace \u03b1\u1d52\u1d48 where\n borel_le := h.borel_le\n#align order_dual.opens_measurable_space OrderDual.opensMeasurableSpace\n\ninstance (priority := 100) OrderDual.borelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [h : BorelSpace \u03b1] : BorelSpace \u03b1\u1d52\u1d48 where\n measurable_eq := h.measurable_eq\n#align order_dual.borel_space OrderDual.borelSpace\n\n/-- In a `BorelSpace` all open sets are measurable. -/\ninstance (priority := 100) BorelSpace.opensMeasurable {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [BorelSpace \u03b1] : OpensMeasurableSpace \u03b1 :=\n \u27e8ge_of_eq <| BorelSpace.measurable_eq\u27e9\n#align borel_space.opens_measurable BorelSpace.opensMeasurable\n\ninstance Subtype.borelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [MeasurableSpace \u03b1]\n [h\u03b1 : BorelSpace \u03b1] (s : Set \u03b1) : BorelSpace s :=\n \u27e8by borelize \u03b1; symm; apply borel_comap\u27e9\n#align subtype.borel_space Subtype.borelSpace\n\ninstance Countable.instBorelSpace [Countable \u03b1] [MeasurableSpace \u03b1] [MeasurableSingletonClass \u03b1]\n [TopologicalSpace \u03b1] [DiscreteTopology \u03b1] : BorelSpace \u03b1 := by\n have : \u2200 s, @MeasurableSet \u03b1 inferInstance s := fun s \u21a6 s.to_countable.measurableSet\n have : \u2200 s, @MeasurableSet \u03b1 (borel \u03b1) s := fun s \u21a6 measurableSet_generateFrom (isOpen_discrete s)\n exact \u27e8by aesop\u27e9\n\ninstance Subtype.opensMeasurableSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [MeasurableSpace \u03b1]\n [h : OpensMeasurableSpace \u03b1] (s : Set \u03b1) : OpensMeasurableSpace s :=\n \u27e8by\n rw [borel_comap]\n exact comap_mono h.1\u27e9\n#align subtype.opens_measurable_space Subtype.opensMeasurableSpace\n\nlemma opensMeasurableSpace_iff_forall_measurableSet\n [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] :\n OpensMeasurableSpace \u03b1 \u2194 (\u2200 (s : Set \u03b1), IsOpen s \u2192 MeasurableSet s) := by\n refine \u27e8fun h s hs \u21a6 ?_, fun h \u21a6 \u27e8generateFrom_le h\u27e9\u27e9\n exact OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ hs\n\ninstance (priority := 100) BorelSpace.countablyGenerated {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [BorelSpace \u03b1] [SecondCountableTopology \u03b1] : CountablyGenerated \u03b1 := by\n obtain \u27e8b, bct, -, hb\u27e9 := exists_countable_basis \u03b1\n refine' \u27e8\u27e8b, bct, _\u27e9\u27e9\n borelize \u03b1\n exact hb.borel_eq_generateFrom\n#align borel_space.countably_generated BorelSpace.countablyGenerated\n\ntheorem MeasurableSet.induction_on_open [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1]\n {C : Set \u03b1 \u2192 Prop} (h_open : \u2200 U, IsOpen U \u2192 C U)\n (h_compl : \u2200 t, MeasurableSet t \u2192 C t \u2192 C t\u1d9c)\n (h_union :\n \u2200 f : \u2115 \u2192 Set \u03b1,\n Pairwise (Disjoint on f) \u2192 (\u2200 i, MeasurableSet (f i)) \u2192 (\u2200 i, C (f i)) \u2192 C (\u22c3 i, f i)) :\n \u2200 \u2983t\u2984, MeasurableSet t \u2192 C t :=\n MeasurableSpace.induction_on_inter BorelSpace.measurable_eq isPiSystem_isOpen\n (h_open _ isOpen_empty) h_open h_compl h_union\n#align measurable_set.induction_on_open MeasurableSet.induction_on_open\n\nsection\n\nvariable [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [OpensMeasurableSpace \u03b1] [TopologicalSpace \u03b2]\n [MeasurableSpace \u03b2] [OpensMeasurableSpace \u03b2] [TopologicalSpace \u03b3] [MeasurableSpace \u03b3]\n [BorelSpace \u03b3] [TopologicalSpace \u03b3\u2082] [MeasurableSpace \u03b3\u2082] [BorelSpace \u03b3\u2082] [MeasurableSpace \u03b4]\n\ntheorem IsOpen.measurableSet (h : IsOpen s) : MeasurableSet s :=\n OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ h\n#align is_open.measurable_set IsOpen.measurableSet\n\ninstance (priority := 500) {s : Set \u03b1} [HasCountableSeparatingOn \u03b1 IsOpen s] :\n HasCountableSeparatingOn \u03b1 MeasurableSet s :=\n .mono (fun _ \u21a6 IsOpen.measurableSet) Subset.rfl\n\n@[measurability]\ntheorem measurableSet_interior : MeasurableSet (interior s) :=\n isOpen_interior.measurableSet\n#align measurable_set_interior measurableSet_interior\n\ntheorem IsG\u03b4.measurableSet (h : IsG\u03b4 s) : MeasurableSet s := by\n rcases h with \u27e8S, hSo, hSc, rfl\u27e9\n exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet\nset_option linter.uppercaseLean3 false in\n#align is_G\u03b4.measurable_set IsG\u03b4.measurableSet\n\ntheorem measurableSet_of_continuousAt {\u03b2} [EMetricSpace \u03b2] (f : \u03b1 \u2192 \u03b2) :\n MeasurableSet { x | ContinuousAt f x } :=\n (IsG\u03b4.setOf_continuousAt f).measurableSet\n#align measurable_set_of_continuous_at measurableSet_of_continuousAt\n\ntheorem IsClosed.measurableSet (h : IsClosed s) : MeasurableSet s :=\n h.isOpen_compl.measurableSet.of_compl\n#align is_closed.measurable_set IsClosed.measurableSet\n\ntheorem IsCompact.measurableSet [T2Space \u03b1] (h : IsCompact s) : MeasurableSet s :=\n h.isClosed.measurableSet\n#align is_compact.measurable_set IsCompact.measurableSet\n\n/-- If two points are topologically inseparable,\nthen they can't be separated by a Borel measurable set. -/\ntheorem Inseparable.mem_measurableSet_iff {x y : \u03b3} (h : Inseparable x y) {s : Set \u03b3}\n (hs : MeasurableSet s) : x \u2208 s \u2194 y \u2208 s :=\n hs.induction_on_open (C := fun s \u21a6 (x \u2208 s \u2194 y \u2208 s)) (fun _ \u21a6 h.mem_open_iff) (fun s _ hs \u21a6 hs.not)\n fun _ _ _ h \u21a6 by simp [h]\n\n/-- If `K` is a compact set in an R\u2081 space and `s \u2287 K` is a Borel measurable superset,\nthen `s` includes the closure of `K` as well. -/\ntheorem IsCompact.closure_subset_measurableSet [R1Space \u03b3] {K s : Set \u03b3} (hK : IsCompact K)\n (hs : MeasurableSet s) (hKs : K \u2286 s) : closure K \u2286 s := by\n rw [hK.closure_eq_biUnion_inseparable, iUnion\u2082_subset_iff]\n exact fun x hx y hy \u21a6 (hy.mem_measurableSet_iff hs).1 (hKs hx)\n\n/-- In an R\u2081 topological space with Borel measure `\u03bc`,\nthe measure of the closure of a compact set `K` is equal to the measure of `K`.\n\nSee also `MeasureTheory.Measure.OuterRegular.measure_closure_eq_of_isCompact`\nfor a version that assumes `\u03bc` to be outer regular\nbut does not assume the `\u03c3`-algebra to be Borel. -/\ntheorem IsCompact.measure_closure [R1Space \u03b3] {K : Set \u03b3} (hK : IsCompact K) (\u03bc : Measure \u03b3) :\n \u03bc (closure K) = \u03bc K := by\n refine le_antisymm ?_ (measure_mono subset_closure)\n calc\n \u03bc (closure K) \u2264 \u03bc (toMeasurable \u03bc K) := measure_mono <|\n hK.closure_subset_measurableSet (measurableSet_toMeasurable ..) (subset_toMeasurable ..)\n _ = \u03bc K := measure_toMeasurable ..\n\n@[measurability]\ntheorem measurableSet_closure : MeasurableSet (closure s) :=\n isClosed_closure.measurableSet\n#align measurable_set_closure measurableSet_closure\n\ntheorem measurable_of_isOpen {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, IsOpen s \u2192 MeasurableSet (f \u207b\u00b9' s)) :\n Measurable f := by\n rw [\u2039BorelSpace \u03b3\u203a.measurable_eq]\n exact measurable_generateFrom hf\n#align measurable_of_is_open measurable_of_isOpen\n\ntheorem measurable_of_isClosed {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, IsClosed s \u2192 MeasurableSet (f \u207b\u00b9' s)) :\n Measurable f := by\n apply measurable_of_isOpen; intro s hs\n rw [\u2190 MeasurableSet.compl_iff, \u2190 preimage_compl]; apply hf; rw [isClosed_compl_iff]; exact hs\n#align measurable_of_is_closed measurable_of_isClosed\n\ntheorem measurable_of_isClosed' {f : \u03b4 \u2192 \u03b3}\n (hf : \u2200 s, IsClosed s \u2192 s.Nonempty \u2192 s \u2260 univ \u2192 MeasurableSet (f \u207b\u00b9' s)) : Measurable f := by\n apply measurable_of_isClosed; intro s hs\n rcases eq_empty_or_nonempty s with h1 | h1\n \u00b7 simp [h1]\n by_cases h2 : s = univ\n \u00b7 simp [h2]\n exact hf s hs h1 h2\n#align measurable_of_is_closed' measurable_of_isClosed'\n\ninstance nhds_isMeasurablyGenerated (a : \u03b1) : (\ud835\udcdd a).IsMeasurablyGenerated := by\n rw [nhds, iInf_subtype']\n refine' @Filter.iInf_isMeasurablyGenerated \u03b1 _ _ _ fun i => _\n exact i.2.2.measurableSet.principal_isMeasurablyGenerated\n#align nhds_is_measurably_generated nhds_isMeasurablyGenerated\n\n/-- If `s` is a measurable set, then `\ud835\udcdd[s] a` is a measurably generated filter for\neach `a`. This cannot be an `instance` because it depends on a non-instance `hs : MeasurableSet s`.\n-/\ntheorem MeasurableSet.nhdsWithin_isMeasurablyGenerated {s : Set \u03b1} (hs : MeasurableSet s) (a : \u03b1) :\n (\ud835\udcdd[s] a).IsMeasurablyGenerated :=\n haveI := hs.principal_isMeasurablyGenerated\n Filter.inf_isMeasurablyGenerated _ _\n#align measurable_set.nhds_within_is_measurably_generated MeasurableSet.nhdsWithin_isMeasurablyGenerated\n\ninstance (priority := 100) OpensMeasurableSpace.separatesPoints [T0Space \u03b1] :\n SeparatesPoints \u03b1 := by\n rw [separatesPoints_iff]\n intro x y hxy\n apply Inseparable.eq\n rw [inseparable_iff_forall_open]\n exact fun s hs => hxy _ hs.measurableSet\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OpensMeasurableSpace.toMeasurableSingletonClass [T1Space \u03b1] :\n MeasurableSingletonClass \u03b1 :=\n \u27e8fun _ => isClosed_singleton.measurableSet\u27e9\n#align opens_measurable_space.to_measurable_singleton_class OpensMeasurableSpace.toMeasurableSingletonClass\n48\ninstance Pi.opensMeasurableSpace {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [Countable \u03b9]\n [t' : \u2200 i, TopologicalSpace (\u03c0 i)] [\u2200 i, MeasurableSpace (\u03c0 i)]\n [\u2200 i, SecondCountableTopology (\u03c0 i)] [\u2200 i, OpensMeasurableSpace (\u03c0 i)] :\n OpensMeasurableSpace (\u2200 i, \u03c0 i) := by\n constructor\n have : Pi.topologicalSpace = .generateFrom { t | \u2203 (s : \u2200 a, Set (\u03c0 a)) (i : Finset \u03b9),\n (\u2200 a \u2208 i, s a \u2208 countableBasis (\u03c0 a)) \u2227 t = pi (\u2191i) s } := by\n rw [funext fun a => @eq_generateFrom_countableBasis (\u03c0 a) _ _, pi_generateFrom_eq]\n rw [borel_eq_generateFrom_of_subbasis this]\n apply generateFrom_le\n rintro _ \u27e8s, i, hi, rfl\u27e9\n refine' MeasurableSet.pi i.countable_toSet fun a ha => IsOpen.measurableSet _\n rw [eq_generateFrom_countableBasis (\u03c0 a)]\n exact .basic _ (hi a ha)\n#align pi.opens_measurable_space Pi.opensMeasurableSpace\n\n/-- The typeclass `SecondCountableTopologyEither \u03b1 \u03b2` registers the fact that at least one of\nthe two spaces has second countable topology. This is the right assumption to ensure that continuous\nmaps from `\u03b1` to `\u03b2` are strongly measurable. -/\nclass SecondCountableTopologyEither (\u03b1 \u03b2 : Type*) [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] :\n Prop where\n /-- The projection out of `SecondCountableTopologyEither` -/\n out : SecondCountableTopology \u03b1 \u2228 SecondCountableTopology \u03b2\n#align second_countable_topology_either SecondCountableTopologyEither\n\ninstance (priority := 100) secondCountableTopologyEither_of_left (\u03b1 \u03b2 : Type*) [TopologicalSpace \u03b1]\n [TopologicalSpace \u03b2] [SecondCountableTopology \u03b1] : SecondCountableTopologyEither \u03b1 \u03b2 where\n out := Or.inl (by infer_instance)\n#align second_countable_topology_either_of_left secondCountableTopologyEither_of_left\n\ninstance (priority := 100) secondCountableTopologyEither_of_right (\u03b1 \u03b2 : Type*)\n [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] [SecondCountableTopology \u03b2] :\n SecondCountableTopologyEither \u03b1 \u03b2 where\n out := Or.inr (by infer_instance)\n#align second_countable_topology_either_of_right secondCountableTopologyEither_of_right\n\n/-- If either `\u03b1` or `\u03b2` has second-countable topology, then the open sets in `\u03b1 \u00d7 \u03b2` belong to the\nproduct sigma-algebra. -/\ninstance Prod.opensMeasurableSpace [h : SecondCountableTopologyEither \u03b1 \u03b2] :\n OpensMeasurableSpace (\u03b1 \u00d7 \u03b2) := by\n apply opensMeasurableSpace_iff_forall_measurableSet.2 (fun s hs \u21a6 ?_)\n rcases h.out with h\u03b1|h\u03b2\n \u00b7 let F : Set \u03b1 \u2192 Set \u03b2 := fun a \u21a6 {y | \u2203 b, IsOpen b \u2227 y \u2208 b \u2227 a \u00d7\u02e2 b \u2286 s}\n have A : \u2200 a, IsOpen (F a) := by\n intro a\n apply isOpen_iff_forall_mem_open.2\n rintro y \u27e8b, b_open, yb, hb\u27e9\n exact \u27e8b, fun z zb \u21a6 \u27e8b, b_open, zb, hb\u27e9, b_open, yb\u27e9\n have : s = \u22c3 a \u2208 countableBasis \u03b1, a \u00d7\u02e2 F a := by\n apply Subset.antisymm\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n rcases isOpen_prod_iff.1 hs y1 y2 hy with \u27e8u, v, u_open, v_open, yu, yv, huv\u27e9\n obtain \u27e8a, ha, ya, au\u27e9 : \u2203 a \u2208 countableBasis \u03b1, y1 \u2208 a \u2227 a \u2286 u :=\n IsTopologicalBasis.exists_subset_of_mem_open (isBasis_countableBasis \u03b1) yu u_open\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop]\n exact \u27e8a, ya, ha, v, v_open, yv, (Set.prod_mono_left au).trans huv\u27e9\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop] at hy\n rcases hy with \u27e8a, ya, -, b, -, yb, hb\u27e9\n exact hb (mem_prod.2 \u27e8ya, yb\u27e9)\n rw [this]\n apply MeasurableSet.biUnion (countable_countableBasis \u03b1) (fun a ha \u21a6 ?_)\n exact (isOpen_of_mem_countableBasis ha).measurableSet.prod (A a).measurableSet\n \u00b7 let F : Set \u03b2 \u2192 Set \u03b1 := fun a \u21a6 {y | \u2203 b, IsOpen b \u2227 y \u2208 b \u2227 b \u00d7\u02e2 a \u2286 s}\n have A : \u2200 a, IsOpen (F a) := by\n intro a\n apply isOpen_iff_forall_mem_open.2\n rintro y \u27e8b, b_open, yb, hb\u27e9\n exact \u27e8b, fun z zb \u21a6 \u27e8b, b_open, zb, hb\u27e9, b_open, yb\u27e9\n have : s = \u22c3 a \u2208 countableBasis \u03b2, F a \u00d7\u02e2 a := by\n apply Subset.antisymm\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n rcases isOpen_prod_iff.1 hs y1 y2 hy with \u27e8u, v, u_open, v_open, yu, yv, huv\u27e9\n obtain \u27e8a, ha, ya, au\u27e9 : \u2203 a \u2208 countableBasis \u03b2, y2 \u2208 a \u2227 a \u2286 v :=\n IsTopologicalBasis.exists_subset_of_mem_open (isBasis_countableBasis \u03b2) yv v_open\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop]\n exact \u27e8a, \u27e8u, u_open, yu, (Set.prod_mono_right au).trans huv\u27e9, ha, ya\u27e9\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop] at hy\n rcases hy with \u27e8a, \u27e8b, -, yb, hb\u27e9, -, ya\u27e9\n exact hb (mem_prod.2 \u27e8yb, ya\u27e9)\n rw [this]\n apply MeasurableSet.biUnion (countable_countableBasis \u03b2) (fun a ha \u21a6 ?_)\n exact (A a).measurableSet.prod (isOpen_of_mem_countableBasis ha).measurableSet\n\nvariable {\u03b1' : Type*} [TopologicalSpace \u03b1'] [MeasurableSpace \u03b1']\n\ntheorem interior_ae_eq_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n interior s =\u1d50[\u03bc] s :=\n interior_subset.eventuallyLE.antisymm <| subset_closure.eventuallyLE.trans (ae_le_set.2 h)\n#align interior_ae_eq_of_null_frontier interior_ae_eq_of_null_frontier\n\ntheorem measure_interior_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n \u03bc (interior s) = \u03bc s :=\n measure_congr (interior_ae_eq_of_null_frontier h)\n#align measure_interior_of_null_frontier measure_interior_of_null_frontier\n\ntheorem nullMeasurableSet_of_null_frontier {s : Set \u03b1} {\u03bc : Measure \u03b1} (h : \u03bc (frontier s) = 0) :\n NullMeasurableSet s \u03bc :=\n \u27e8interior s, isOpen_interior.measurableSet, (interior_ae_eq_of_null_frontier h).symm\u27e9\n#align null_measurable_set_of_null_frontier nullMeasurableSet_of_null_frontier\n\ntheorem closure_ae_eq_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n closure s =\u1d50[\u03bc] s :=\n ((ae_le_set.2 h).trans interior_subset.eventuallyLE).antisymm <| subset_closure.eventuallyLE\n#align closure_ae_eq_of_null_frontier closure_ae_eq_of_null_frontier\n\ntheorem measure_closure_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n \u03bc (closure s) = \u03bc s :=\n measure_congr (closure_ae_eq_of_null_frontier h)\n#align measure_closure_of_null_frontier measure_closure_of_null_frontier\n\nsection Preorder\n\nvariable [Preorder \u03b1] [OrderClosedTopology \u03b1] {a b x : \u03b1}\n\n@[simp, measurability]\ntheorem measurableSet_Ici : MeasurableSet (Ici a) :=\n isClosed_Ici.measurableSet\n#align measurable_set_Ici measurableSet_Ici\n\n@[simp, measurability]\ntheorem measurableSet_Iic : MeasurableSet (Iic a) :=\n isClosed_Iic.measurableSet\n#align measurable_set_Iic measurableSet_Iic\n\n@[simp, measurability]\ntheorem measurableSet_Icc : MeasurableSet (Icc a b) :=\n isClosed_Icc.measurableSet\n#align measurable_set_Icc measurableSet_Icc\n\ninstance nhdsWithin_Ici_isMeasurablyGenerated : (\ud835\udcdd[Ici b] a).IsMeasurablyGenerated :=\n measurableSet_Ici.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Ici_is_measurably_generated nhdsWithin_Ici_isMeasurablyGenerated\n\ninstance nhdsWithin_Iic_isMeasurablyGenerated : (\ud835\udcdd[Iic b] a).IsMeasurablyGenerated :=\n measurableSet_Iic.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Iic_is_measurably_generated nhdsWithin_Iic_isMeasurablyGenerated\n\ninstance nhdsWithin_Icc_isMeasurablyGenerated : IsMeasurablyGenerated (\ud835\udcdd[Icc a b] x) := by\n rw [\u2190 Ici_inter_Iic, nhdsWithin_inter]\n infer_instance\n#align nhds_within_Icc_is_measurably_generated nhdsWithin_Icc_isMeasurablyGenerated\n\ninstance atTop_isMeasurablyGenerated : (Filter.atTop : Filter \u03b1).IsMeasurablyGenerated :=\n @Filter.iInf_isMeasurablyGenerated _ _ _ _ fun a =>\n (measurableSet_Ici : MeasurableSet (Ici a)).principal_isMeasurablyGenerated\n#align at_top_is_measurably_generated atTop_isMeasurablyGenerated\n\ninstance atBot_isMeasurablyGenerated : (Filter.atBot : Filter \u03b1).IsMeasurablyGenerated :=\n @Filter.iInf_isMeasurablyGenerated _ _ _ _ fun a =>\n (measurableSet_Iic : MeasurableSet (Iic a)).principal_isMeasurablyGenerated\n#align at_bot_is_measurably_generated atBot_isMeasurablyGenerated\n\ninstance [R1Space \u03b1] : IsMeasurablyGenerated (cocompact \u03b1) where\n exists_measurable_subset := by\n intro _ hs\n obtain \u27e8t, ht, hts\u27e9 := mem_cocompact.mp hs\n exact \u27e8(closure t)\u1d9c, ht.closure.compl_mem_cocompact, isClosed_closure.measurableSet.compl,\n (compl_subset_compl.2 subset_closure).trans hts\u27e9\n\nend Preorder\n\nsection PartialOrder\n\nvariable [PartialOrder \u03b1] [OrderClosedTopology \u03b1] [SecondCountableTopology \u03b1] {a b : \u03b1}\n\n@[measurability]\ntheorem measurableSet_le' : MeasurableSet { p : \u03b1 \u00d7 \u03b1 | p.1 \u2264 p.2 } :=\n OrderClosedTopology.isClosed_le'.measurableSet\n#align measurable_set_le' measurableSet_le'\n\n@[measurability]\ntheorem measurableSet_le {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n MeasurableSet { a | f a \u2264 g a } :=\n hf.prod_mk hg measurableSet_le'\n#align measurable_set_le measurableSet_le\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] [OrderClosedTopology \u03b1] {a b x : \u03b1}\n\n-- we open this locale only here to avoid issues with list being treated as intervals above\nopen Interval\n\n@[simp, measurability]\ntheorem measurableSet_Iio : MeasurableSet (Iio a) :=\n isOpen_Iio.measurableSet\n#align measurable_set_Iio measurableSet_Iio\n\n@[simp, measurability]\ntheorem measurableSet_Ioi : MeasurableSet (Ioi a) :=\n isOpen_Ioi.measurableSet\n#align measurable_set_Ioi measurableSet_Ioi\n\n@[simp, measurability]\ntheorem measurableSet_Ioo : MeasurableSet (Ioo a b) :=\n isOpen_Ioo.measurableSet\n#align measurable_set_Ioo measurableSet_Ioo\n\n@[simp, measurability]\ntheorem measurableSet_Ioc : MeasurableSet (Ioc a b) :=\n measurableSet_Ioi.inter measurableSet_Iic\n#align measurable_set_Ioc measurableSet_Ioc\n\n@[simp, measurability]\ntheorem measurableSet_Ico : MeasurableSet (Ico a b) :=\n measurableSet_Ici.inter measurableSet_Iio\n#align measurable_set_Ico measurableSet_Ico\n\ninstance nhdsWithin_Ioi_isMeasurablyGenerated : (\ud835\udcdd[Ioi b] a).IsMeasurablyGenerated :=\n measurableSet_Ioi.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Ioi_is_measurably_generated nhdsWithin_Ioi_isMeasurablyGenerated\n\ninstance nhdsWithin_Iio_isMeasurablyGenerated : (\ud835\udcdd[Iio b] a).IsMeasurablyGenerated :=\n measurableSet_Iio.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Iio_is_measurably_generated nhdsWithin_Iio_isMeasurablyGenerated\n\ninstance nhdsWithin_uIcc_isMeasurablyGenerated : IsMeasurablyGenerated (\ud835\udcdd[[[a, b]]] x) :=\n nhdsWithin_Icc_isMeasurablyGenerated\n#align nhds_within_uIcc_is_measurably_generated nhdsWithin_uIcc_isMeasurablyGenerated\n\n@[measurability]\ntheorem measurableSet_lt' [SecondCountableTopology \u03b1] : MeasurableSet { p : \u03b1 \u00d7 \u03b1 | p.1 < p.2 } :=\n (isOpen_lt continuous_fst continuous_snd).measurableSet\n#align measurable_set_lt' measurableSet_lt'\n\n@[measurability]\ntheorem measurableSet_lt [SecondCountableTopology \u03b1] {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f)\n (hg : Measurable g) : MeasurableSet { a | f a < g a } :=\n hf.prod_mk hg measurableSet_lt'\n#align measurable_set_lt measurableSet_lt\n\ntheorem nullMeasurableSet_lt [SecondCountableTopology \u03b1] {\u03bc : Measure \u03b4} {f g : \u03b4 \u2192 \u03b1}\n (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) : NullMeasurableSet { a | f a < g a } \u03bc :=\n (hf.prod_mk hg).nullMeasurable measurableSet_lt'\n#align null_measurable_set_lt nullMeasurableSet_lt\n\ntheorem nullMeasurableSet_lt' [SecondCountableTopology \u03b1] {\u03bc : Measure (\u03b1 \u00d7 \u03b1)} :\n NullMeasurableSet { p : \u03b1 \u00d7 \u03b1 | p.1 < p.2 } \u03bc :=\n measurableSet_lt'.nullMeasurableSet\n\ntheorem nullMeasurableSet_le [SecondCountableTopology \u03b1] {\u03bc : Measure \u03b4}\n {f g : \u03b4 \u2192 \u03b1} (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) :\n NullMeasurableSet { a | f a \u2264 g a } \u03bc :=\n (hf.prod_mk hg).nullMeasurable measurableSet_le'\n\ntheorem Set.OrdConnected.measurableSet (h : OrdConnected s) : MeasurableSet s := by\n let u := \u22c3 (x \u2208 s) (y \u2208 s), Ioo x y\n have huopen : IsOpen u := isOpen_biUnion fun _ _ => isOpen_biUnion fun _ _ => isOpen_Ioo\n have humeas : MeasurableSet u := huopen.measurableSet\n have hfinite : (s \\ u).Finite := s.finite_diff_iUnion_Ioo\n have : u \u2286 s := iUnion\u2082_subset fun x hx => iUnion\u2082_subset fun y hy =>\n Ioo_subset_Icc_self.trans (h.out hx hy)\n rw [\u2190 union_diff_cancel this]\n exact humeas.union hfinite.measurableSet\n#align set.ord_connected.measurable_set Set.OrdConnected.measurableSet\n\ntheorem IsPreconnected.measurableSet (h : IsPreconnected s) : MeasurableSet s :=\n h.ordConnected.measurableSet\n#align is_preconnected.measurable_set IsPreconnected.measurableSet\n\ntheorem generateFrom_Ico_mem_le_borel {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderClosedTopology \u03b1] (s t : Set \u03b1) :\n MeasurableSpace.generateFrom { S | \u2203 l \u2208 s, \u2203 u \u2208 t, l < u \u2227 Ico l u = S }\n \u2264 borel \u03b1 := by\n apply generateFrom_le\n borelize \u03b1\n rintro _ \u27e8a, -, b, -, -, rfl\u27e9\n exact measurableSet_Ico\n#align generate_from_Ico_mem_le_borel generateFrom_Ico_mem_le_borel\n\ntheorem Dense.borel_eq_generateFrom_Ico_mem_aux {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] {s : Set \u03b1} (hd : Dense s)\n (hbot : \u2200 x, IsBot x \u2192 x \u2208 s) (hIoo : \u2200 x y : \u03b1, x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S } := by\n set S : Set (Set \u03b1) := { S | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S }\n refine' le_antisymm _ (generateFrom_Ico_mem_le_borel _ _)\n letI : MeasurableSpace \u03b1 := generateFrom S\n rw [borel_eq_generateFrom_Iio]\n refine' generateFrom_le (forall_mem_range.2 fun a => _)\n rcases hd.exists_countable_dense_subset_bot_top with \u27e8t, hts, hc, htd, htb, -\u27e9\n by_cases ha : \u2200 b < a, (Ioo b a).Nonempty\n \u00b7 convert_to MeasurableSet (\u22c3 (l \u2208 t) (u \u2208 t) (_ : l < u) (_ : u \u2264 a), Ico l u)\n \u00b7 ext y\n simp only [mem_iUnion, mem_Iio, mem_Ico]\n constructor\n \u00b7 intro hy\n rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) y with \u27e8l, hlt, hly\u27e9\n rcases htd.exists_mem_open isOpen_Ioo (ha y hy) with \u27e8u, hut, hyu, hua\u27e9\n exact \u27e8l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu\u27e9\n \u00b7 rintro \u27e8l, -, u, -, -, hua, -, hyu\u27e9\n exact hyu.trans_le hua\n \u00b7 refine' MeasurableSet.biUnion hc fun a ha => MeasurableSet.biUnion hc fun b hb => _\n refine' MeasurableSet.iUnion fun hab => MeasurableSet.iUnion fun _ => _\n exact .basic _ \u27e8a, hts ha, b, hts hb, hab, mem_singleton _\u27e9\n \u00b7 simp only [not_forall, not_nonempty_iff_eq_empty] at ha\n replace ha : a \u2208 s := hIoo ha.choose a ha.choose_spec.fst ha.choose_spec.snd\n convert_to MeasurableSet (\u22c3 (l \u2208 t) (_ : l < a), Ico l a)\n \u00b7 symm\n simp only [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, inter_eq_right, subset_def, mem_iUnion,\n mem_Ici, mem_Iio]\n intro x hx\n rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) x with \u27e8z, hzt, hzx\u27e9\n exact \u27e8z, hzt, hzx.trans_lt hx, hzx\u27e9\n \u00b7 refine' .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => _\n exact .basic _ \u27e8x, hts hx, a, ha, hlt, mem_singleton _\u27e9\n#align dense.borel_eq_generate_from_Ico_mem_aux Dense.borel_eq_generateFrom_Ico_mem_aux\n\ntheorem Dense.borel_eq_generateFrom_Ico_mem {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] [DenselyOrdered \u03b1] [NoMinOrder \u03b1] {s : Set \u03b1}\n (hd : Dense s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S } :=\n hd.borel_eq_generateFrom_Ico_mem_aux (by simp) fun x y hxy H =>\n ((nonempty_Ioo.2 hxy).ne_empty H).elim\n#align dense.borel_eq_generate_from_Ico_mem Dense.borel_eq_generateFrom_Ico_mem\n\ntheorem borel_eq_generateFrom_Ico (\u03b1 : Type*) [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1]\n [LinearOrder \u03b1] [OrderTopology \u03b1] :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 (l u : \u03b1), l < u \u2227 Ico l u = S } := by\n simpa only [exists_prop, mem_univ, true_and_iff] using\n (@dense_univ \u03b1 _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n mem_univ _\n#align borel_eq_generate_from_Ico borel_eq_generateFrom_Ico\n\ntheorem Dense.borel_eq_generateFrom_Ioc_mem_aux {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] {s : Set \u03b1} (hd : Dense s)\n (hbot : \u2200 x, IsTop x \u2192 x \u2208 s) (hIoo : \u2200 x y : \u03b1, x < y \u2192 Ioo x y = \u2205 \u2192 x \u2208 s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ioc l u = S } := by\n convert hd.orderDual.borel_eq_generateFrom_Ico_mem_aux hbot fun x y hlt he => hIoo y x hlt _\n using 2\n \u00b7 ext s\n constructor <;> rintro \u27e8l, hl, u, hu, hlt, rfl\u27e9\n exacts [\u27e8u, hu, l, hl, hlt, dual_Ico\u27e9, \u27e8u, hu, l, hl, hlt, dual_Ioc\u27e9]\n \u00b7 erw [dual_Ioo]\n exact he\n#align dense.borel_eq_generate_from_Ioc_mem_aux Dense.borel_eq_generateFrom_Ioc_mem_aux\n\ntheorem Dense.borel_eq_generateFrom_Ioc_mem {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] [DenselyOrdered \u03b1] [NoMaxOrder \u03b1] {s : Set \u03b1}\n (hd : Dense s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ioc l u = S } :=\n hd.borel_eq_generateFrom_Ioc_mem_aux (by simp) fun x y hxy H =>\n ((nonempty_Ioo.2 hxy).ne_empty H).elim\n#align dense.borel_eq_generate_from_Ioc_mem Dense.borel_eq_generateFrom_Ioc_mem\n\ntheorem borel_eq_generateFrom_Ioc (\u03b1 : Type*) [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1]\n [LinearOrder \u03b1] [OrderTopology \u03b1] :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l u, l < u \u2227 Ioc l u = S } := by\n simpa only [exists_prop, mem_univ, true_and_iff] using\n (@dense_univ \u03b1 _).borel_eq_generateFrom_Ioc_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n mem_univ _\n#align borel_eq_generate_from_Ioc borel_eq_generateFrom_Ioc\n\nnamespace MeasureTheory.Measure\n\n/-- Two finite measures on a Borel space are equal if they agree on all closed-open intervals. If\n`\u03b1` is a conditionally complete linear order with no top element,\n`MeasureTheory.Measure.ext_of_Ico` is an extensionality lemma with weaker assumptions on `\u03bc` and\n`\u03bd`. -/\ntheorem ext_of_Ico_finite {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h\u03bc\u03bd : \u03bc univ = \u03bd univ) (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) :\n \u03bc = \u03bd := by\n refine'\n ext_of_generate_finite _ (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico \u03b1))\n (isPiSystem_Ico (id : \u03b1 \u2192 \u03b1) id) _ h\u03bc\u03bd\n \u00b7 rintro - \u27e8a, b, hlt, rfl\u27e9\n exact h hlt\n#align measure_theory.measure.ext_of_Ico_finite MeasureTheory.Measure.ext_of_Ico_finite\n\n/-- Two finite measures on a Borel space are equal if they agree on all open-closed intervals. If\n`\u03b1` is a conditionally complete linear order with no top element,\n`MeasureTheory.Measure.ext_of_Ioc` is an extensionality lemma with weaker assumptions on `\u03bc` and\n`\u03bd`. -/\ntheorem ext_of_Ioc_finite {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h\u03bc\u03bd : \u03bc univ = \u03bd univ) (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) :\n \u03bc = \u03bd := by\n refine' @ext_of_Ico_finite \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a \u03bc \u03bd _ h\u03bc\u03bd fun a b hab => _\n erw [dual_Ico (\u03b1 := \u03b1)]\n exact h hab\n#align measure_theory.measure.ext_of_Ioc_finite MeasureTheory.Measure.ext_of_Ioc_finite\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nclosed-open intervals. -/\ntheorem ext_of_Ico' {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] [NoMaxOrder \u03b1]\n (\u03bc \u03bd : Measure \u03b1) (h\u03bc : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) \u2260 \u221e)\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) : \u03bc = \u03bd := by\n rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, hsb, _\u27e9\n have : (\u22c3 (l \u2208 s) (u \u2208 s) (_ : l < u), {Ico l u} : Set (Set \u03b1)).Countable :=\n hsc.biUnion fun l _ => hsc.biUnion fun u _ => countable_iUnion fun _ => countable_singleton _\n simp only [\u2190 setOf_eq_eq_singleton, \u2190 setOf_exists] at this\n refine'\n Measure.ext_of_generateFrom_of_cover_subset\n (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico \u03b1)) (isPiSystem_Ico id id) _ this\n _ _ _\n \u00b7 rintro _ \u27e8l, -, u, -, h, rfl\u27e9\n exact \u27e8l, u, h, rfl\u27e9\n \u00b7 refine' sUnion_eq_univ_iff.2 fun x => _\n rcases hsd.exists_le' hsb x with \u27e8l, hls, hlx\u27e9\n rcases hsd.exists_gt x with \u27e8u, hus, hxu\u27e9\n exact \u27e8_, \u27e8l, hls, u, hus, hlx.trans_lt hxu, rfl\u27e9, hlx, hxu\u27e9\n \u00b7 rintro _ \u27e8l, -, u, -, hlt, rfl\u27e9\n exact h\u03bc hlt\n \u00b7 rintro _ \u27e8l, u, hlt, rfl\u27e9\n exact h hlt\n#align measure_theory.measure.ext_of_Ico' MeasureTheory.Measure.ext_of_Ico'\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nopen-closed intervals. -/\ntheorem ext_of_Ioc' {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] [NoMinOrder \u03b1]\n (\u03bc \u03bd : Measure \u03b1) (h\u03bc : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) \u2260 \u221e)\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) : \u03bc = \u03bd := by\n refine' @ext_of_Ico' \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a _ \u03bc \u03bd _ _ <;> intro a b hab <;> erw [dual_Ico (\u03b1 := \u03b1)]\n exacts [h\u03bc hab, h hab]\n#align measure_theory.measure.ext_of_Ioc' MeasureTheory.Measure.ext_of_Ioc'\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nclosed-open intervals. -/\ntheorem ext_of_Ico {\u03b1 : Type*} [TopologicalSpace \u03b1] {_m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [ConditionallyCompleteLinearOrder \u03b1] [OrderTopology \u03b1]\n [BorelSpace \u03b1] [NoMaxOrder \u03b1] (\u03bc \u03bd : Measure \u03b1) [IsLocallyFiniteMeasure \u03bc]\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) : \u03bc = \u03bd :=\n \u03bc.ext_of_Ico' \u03bd (fun _ _ _ => measure_Ico_lt_top.ne) h\n#align measure_theory.measure.ext_of_Ico MeasureTheory.Measure.ext_of_Ico\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nopen-closed intervals. -/\ntheorem ext_of_Ioc {\u03b1 : Type*} [TopologicalSpace \u03b1] {_m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [ConditionallyCompleteLinearOrder \u03b1] [OrderTopology \u03b1]\n [BorelSpace \u03b1] [NoMinOrder \u03b1] (\u03bc \u03bd : Measure \u03b1) [IsLocallyFiniteMeasure \u03bc]\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) : \u03bc = \u03bd :=\n \u03bc.ext_of_Ioc' \u03bd (fun _ _ _ => measure_Ioc_lt_top.ne) h\n#align measure_theory.measure.ext_of_Ioc MeasureTheory.Measure.ext_of_Ioc\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-infinite right-closed\nintervals. -/\ntheorem ext_of_Iic {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h : \u2200 a, \u03bc (Iic a) = \u03bd (Iic a)) : \u03bc = \u03bd := by\n refine' ext_of_Ioc_finite \u03bc \u03bd _ fun a b hlt => _\n \u00b7 rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, -, hst\u27e9\n have : DirectedOn (\u00b7 \u2264 \u00b7) s := directedOn_iff_directed.2 (Subtype.mono_coe _).directed_le\n simp only [\u2190 biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h]\n rw [\u2190 Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic,\n measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic, h a, h b]\n \u00b7 rw [\u2190 h a]\n exact (measure_lt_top \u03bc _).ne\n \u00b7 exact (measure_lt_top \u03bc _).ne\n#align measure_theory.measure.ext_of_Iic MeasureTheory.Measure.ext_of_Iic\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-closed right-infinite\nintervals. -/\ntheorem ext_of_Ici {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h : \u2200 a, \u03bc (Ici a) = \u03bd (Ici a)) : \u03bc = \u03bd :=\n @ext_of_Iic \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a _ _ _ h\n#align measure_theory.measure.ext_of_Ici MeasureTheory.Measure.ext_of_Ici\n\nend MeasureTheory.Measure\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] [OrderClosedTopology \u03b1] {a b : \u03b1}\n\n@[measurability]\ntheorem measurableSet_uIcc : MeasurableSet (uIcc a b) :=\n measurableSet_Icc\n#align measurable_set_uIcc measurableSet_uIcc\n\n@[measurability]\ntheorem measurableSet_uIoc : MeasurableSet (uIoc a b) :=\n measurableSet_Ioc\n#align measurable_set_uIoc measurableSet_uIoc\n\nvariable [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem Measurable.max {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun a => max (f a) (g a) := by\n simpa only [max_def'] using hf.piecewise (measurableSet_le hg hf) hg\n#align measurable.max Measurable.max\n\n@[measurability]\nnonrec theorem AEMeasurable.max {f g : \u03b4 \u2192 \u03b1} {\u03bc : Measure \u03b4} (hf : AEMeasurable f \u03bc)\n (hg : AEMeasurable g \u03bc) : AEMeasurable (fun a => max (f a) (g a)) \u03bc :=\n \u27e8fun a => max (hf.mk f a) (hg.mk g a), hf.measurable_mk.max hg.measurable_mk,\n EventuallyEq.comp\u2082 hf.ae_eq_mk _ hg.ae_eq_mk\u27e9\n#align ae_measurable.max AEMeasurable.max\n\n@[measurability]\ntheorem Measurable.min {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun a => min (f a) (g a) := by\n simpa only [min_def] using hf.piecewise (measurableSet_le hf hg) hg\n#align measurable.min Measurable.min\n\n@[measurability]\nnonrec theorem AEMeasurable.min {f g : \u03b4 \u2192 \u03b1} {\u03bc : Measure \u03b4} (hf : AEMeasurable f \u03bc)\n (hg : AEMeasurable g \u03bc) : AEMeasurable (fun a => min (f a) (g a)) \u03bc :=\n \u27e8fun a => min (hf.mk f a) (hg.mk g a), hf.measurable_mk.min hg.measurable_mk,\n EventuallyEq.comp\u2082 hf.ae_eq_mk _ hg.ae_eq_mk\u27e9\n#align ae_measurable.min AEMeasurable.min\n\nend LinearOrder\n\n/-- A continuous function from an `OpensMeasurableSpace` to a `BorelSpace`\nis measurable. -/\ntheorem Continuous.measurable {f : \u03b1 \u2192 \u03b3} (hf : Continuous f) : Measurable f :=\n hf.borel_measurable.mono OpensMeasurableSpace.borel_le (le_of_eq <| BorelSpace.measurable_eq)\n#align continuous.measurable Continuous.measurable\n\n/-- A continuous function from an `OpensMeasurableSpace` to a `BorelSpace`\nis ae-measurable. -/\ntheorem Continuous.aemeasurable {f : \u03b1 \u2192 \u03b3} (h : Continuous f) {\u03bc : Measure \u03b1} : AEMeasurable f \u03bc :=\n h.measurable.aemeasurable\n#align continuous.ae_measurable Continuous.aemeasurable\n\ntheorem ClosedEmbedding.measurable {f : \u03b1 \u2192 \u03b3} (hf : ClosedEmbedding f) : Measurable f :=\n hf.continuous.measurable\n#align closed_embedding.measurable ClosedEmbedding.measurable\n\n/-- If a function is defined piecewise in terms of functions which are continuous on their\nrespective pieces, then it is measurable. -/\ntheorem ContinuousOn.measurable_piecewise {f g : \u03b1 \u2192 \u03b3} {s : Set \u03b1} [\u2200 j : \u03b1, Decidable (j \u2208 s)]\n (hf : ContinuousOn f s) (hg : ContinuousOn g s\u1d9c) (hs : MeasurableSet s) :\n Measurable (s.piecewise f g) := by\n refine' measurable_of_isOpen fun t ht => _\n rw [piecewise_preimage, Set.ite]\n apply MeasurableSet.union\n \u00b7 rcases _root_.continuousOn_iff'.1 hf t ht with \u27e8u, u_open, hu\u27e9\n rw [hu]\n exact u_open.measurableSet.inter hs\n \u00b7 rcases _root_.continuousOn_iff'.1 hg t ht with \u27e8u, u_open, hu\u27e9\n rw [diff_eq_compl_inter, inter_comm, hu]\n exact u_open.measurableSet.inter hs.compl\n#align continuous_on.measurable_piecewise ContinuousOn.measurable_piecewise\n\n@[to_additive]\ninstance (priority := 100) ContinuousMul.measurableMul [Mul \u03b3] [ContinuousMul \u03b3] :\n MeasurableMul \u03b3 where\n measurable_const_mul _ := (continuous_const.mul continuous_id).measurable\n measurable_mul_const _ := (continuous_id.mul continuous_const).measurable\n#align has_continuous_mul.has_measurable_mul ContinuousMul.measurableMul\n#align has_continuous_add.has_measurable_add ContinuousAdd.measurableAdd\n\ninstance (priority := 100) ContinuousSub.measurableSub [Sub \u03b3] [ContinuousSub \u03b3] :\n MeasurableSub \u03b3 where\n measurable_const_sub _ := (continuous_const.sub continuous_id).measurable\n measurable_sub_const _ := (continuous_id.sub continuous_const).measurable\n#align has_continuous_sub.has_measurable_sub ContinuousSub.measurableSub\n\n@[to_additive]\ninstance (priority := 100) TopologicalGroup.measurableInv [Group \u03b3] [TopologicalGroup \u03b3] :\n MeasurableInv \u03b3 :=\n \u27e8continuous_inv.measurable\u27e9\n#align topological_group.has_measurable_inv TopologicalGroup.measurableInv\n#align topological_add_group.has_measurable_neg TopologicalAddGroup.measurableNeg\n\ninstance (priority := 100) ContinuousSMul.measurableSMul {M \u03b1} [TopologicalSpace M]\n [TopologicalSpace \u03b1] [MeasurableSpace M] [MeasurableSpace \u03b1] [OpensMeasurableSpace M]\n [BorelSpace \u03b1] [SMul M \u03b1] [ContinuousSMul M \u03b1] : MeasurableSMul M \u03b1 :=\n \u27e8fun _ => (continuous_const_smul _).measurable, fun _ =>\n (continuous_id.smul continuous_const).measurable\u27e9\n#align has_continuous_smul.has_measurable_smul ContinuousSMul.measurableSMul\n\nsection Lattice\n\ninstance (priority := 100) ContinuousSup.measurableSup [Sup \u03b3] [ContinuousSup \u03b3] :\n MeasurableSup \u03b3 where\n measurable_const_sup _ := (continuous_const.sup continuous_id).measurable\n measurable_sup_const _ := (continuous_id.sup continuous_const).measurable\n#align has_continuous_sup.has_measurable_sup ContinuousSup.measurableSup\n\ninstance (priority := 100) ContinuousSup.measurableSup\u2082 [SecondCountableTopology \u03b3] [Sup \u03b3]\n [ContinuousSup \u03b3] : MeasurableSup\u2082 \u03b3 :=\n \u27e8continuous_sup.measurable\u27e9\n#align has_continuous_sup.has_measurable_sup\u2082 ContinuousSup.measurableSup\u2082\n\ninstance (priority := 100) ContinuousInf.measurableInf [Inf \u03b3] [ContinuousInf \u03b3] :\n MeasurableInf \u03b3 where\n measurable_const_inf _ := (continuous_const.inf continuous_id).measurable\n measurable_inf_const _ := (continuous_id.inf continuous_const).measurable\n#align has_continuous_inf.has_measurable_inf ContinuousInf.measurableInf\n\ninstance (priority := 100) ContinuousInf.measurableInf\u2082 [SecondCountableTopology \u03b3] [Inf \u03b3]\n [ContinuousInf \u03b3] : MeasurableInf\u2082 \u03b3 :=\n \u27e8continuous_inf.measurable\u27e9\n#align has_continuous_inf.has_measurable_inf\u2082 ContinuousInf.measurableInf\u2082\n\nend Lattice\n\nsection Homeomorph\n\n@[measurability]\nprotected theorem Homeomorph.measurable (h : \u03b1 \u2243\u209c \u03b3) : Measurable h :=\n h.continuous.measurable\n#align homeomorph.measurable Homeomorph.measurable\n\n/-- A homeomorphism between two Borel spaces is a measurable equivalence. -/\ndef Homeomorph.toMeasurableEquiv (h : \u03b3 \u2243\u209c \u03b3\u2082) : \u03b3 \u2243\u1d50 \u03b3\u2082 where\n measurable_toFun := h.measurable\n measurable_invFun := h.symm.measurable\n toEquiv := h.toEquiv\n#align homeomorph.to_measurable_equiv Homeomorph.toMeasurableEquiv\n\nlemma Homeomorph.measurableEmbedding (h : \u03b3 \u2243\u209c \u03b3\u2082) : MeasurableEmbedding h :=\n h.toMeasurableEquiv.measurableEmbedding\n\n@[simp]\ntheorem Homeomorph.toMeasurableEquiv_coe (h : \u03b3 \u2243\u209c \u03b3\u2082) : (h.toMeasurableEquiv : \u03b3 \u2192 \u03b3\u2082) = h :=\n rfl\n#align homeomorph.to_measurable_equiv_coe Homeomorph.toMeasurableEquiv_coe\n\n@[simp]\ntheorem Homeomorph.toMeasurableEquiv_symm_coe (h : \u03b3 \u2243\u209c \u03b3\u2082) :\n (h.toMeasurableEquiv.symm : \u03b3\u2082 \u2192 \u03b3) = h.symm :=\n rfl\n#align homeomorph.to_measurable_equiv_symm_coe Homeomorph.toMeasurableEquiv_symm_coe\n\nend Homeomorph\n\n@[measurability]\ntheorem ContinuousMap.measurable (f : C(\u03b1, \u03b3)) : Measurable f :=\n f.continuous.measurable\n#align continuous_map.measurable ContinuousMap.measurable\n\ntheorem measurable_of_continuousOn_compl_singleton [T1Space \u03b1] {f : \u03b1 \u2192 \u03b3} (a : \u03b1)\n (hf : ContinuousOn f {a}\u1d9c) : Measurable f :=\n measurable_of_measurable_on_compl_singleton a\n (continuousOn_iff_continuous_restrict.1 hf).measurable\n#align measurable_of_continuous_on_compl_singleton measurable_of_continuousOn_compl_singleton\n\ntheorem Continuous.measurable2 [SecondCountableTopologyEither \u03b1 \u03b2] {f : \u03b4 \u2192 \u03b1}\n {g : \u03b4 \u2192 \u03b2} {c : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (h : Continuous fun p : \u03b1 \u00d7 \u03b2 => c p.1 p.2) (hf : Measurable f)\n (hg : Measurable g) : Measurable fun a => c (f a) (g a) :=\n h.measurable.comp (hf.prod_mk hg)\n#align continuous.measurable2 Continuous.measurable2\n\ntheorem Continuous.aemeasurable2 [SecondCountableTopologyEither \u03b1 \u03b2]\n {f : \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b2} {c : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {\u03bc : Measure \u03b4}\n (h : Continuous fun p : \u03b1 \u00d7 \u03b2 => c p.1 p.2) (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) :\n AEMeasurable (fun a => c (f a) (g a)) \u03bc :=\n h.measurable.comp_aemeasurable (hf.prod_mk hg)\n#align continuous.ae_measurable2 Continuous.aemeasurable2\n\ninstance (priority := 100) HasContinuousInv\u2080.measurableInv [GroupWithZero \u03b3] [T1Space \u03b3]\n [HasContinuousInv\u2080 \u03b3] : MeasurableInv \u03b3 :=\n \u27e8measurable_of_continuousOn_compl_singleton 0 continuousOn_inv\u2080\u27e9\n#align has_continuous_inv\u2080.has_measurable_inv HasContinuousInv\u2080.measurableInv\n\n@[to_additive]\ninstance (priority := 100) ContinuousMul.measurableMul\u2082 [SecondCountableTopology \u03b3] [Mul \u03b3]\n [ContinuousMul \u03b3] : MeasurableMul\u2082 \u03b3 :=\n \u27e8continuous_mul.measurable\u27e9\n#align has_continuous_mul.has_measurable_mul\u2082 ContinuousMul.measurableMul\u2082\n#align has_continuous_add.has_measurable_mul\u2082 ContinuousAdd.measurableMul\u2082\n\ninstance (priority := 100) ContinuousSub.measurableSub\u2082 [SecondCountableTopology \u03b3] [Sub \u03b3]\n [ContinuousSub \u03b3] : MeasurableSub\u2082 \u03b3 :=\n \u27e8continuous_sub.measurable\u27e9\n#align has_continuous_sub.has_measurable_sub\u2082 ContinuousSub.measurableSub\u2082\n\ninstance (priority := 100) ContinuousSMul.measurableSMul\u2082 {M \u03b1} [TopologicalSpace M]\n [MeasurableSpace M] [OpensMeasurableSpace M] [TopologicalSpace \u03b1]\n [SecondCountableTopologyEither M \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1] [SMul M \u03b1]\n [ContinuousSMul M \u03b1] : MeasurableSMul\u2082 M \u03b1 :=\n \u27e8continuous_smul.measurable\u27e9\n#align has_continuous_smul.has_measurable_smul\u2082 ContinuousSMul.measurableSMul\u2082\n\nend\n\nsection BorelSpace\n\nvariable [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1] [TopologicalSpace \u03b2]\n [MeasurableSpace \u03b2] [BorelSpace \u03b2] [TopologicalSpace \u03b3] [MeasurableSpace \u03b3] [BorelSpace \u03b3]\n [MeasurableSpace \u03b4]\n\ntheorem pi_le_borel_pi {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [\u2200 i, TopologicalSpace (\u03c0 i)]\n [\u2200 i, MeasurableSpace (\u03c0 i)] [\u2200 i, BorelSpace (\u03c0 i)] :\n MeasurableSpace.pi \u2264 borel (\u2200 i, \u03c0 i) := by\n have : \u2039\u2200 i, MeasurableSpace (\u03c0 i)\u203a = fun i => borel (\u03c0 i) :=\n funext fun i => BorelSpace.measurable_eq\n rw [this]\n exact iSup_le fun i => comap_le_iff_le_map.2 <| (continuous_apply i).borel_measurable\n#align pi_le_borel_pi pi_le_borel_pi\n\ntheorem prod_le_borel_prod : Prod.instMeasurableSpace \u2264 borel (\u03b1 \u00d7 \u03b2) := by\n rw [\u2039BorelSpace \u03b1\u203a.measurable_eq, \u2039BorelSpace \u03b2\u203a.measurable_eq]\n refine' sup_le _ _\n \u00b7 exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable\n \u00b7 exact comap_le_iff_le_map.mpr continuous_snd.borel_measurable\n#align prod_le_borel_prod prod_le_borel_prod\n\ninstance Pi.borelSpace {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [Countable \u03b9] [\u2200 i, TopologicalSpace (\u03c0 i)]\n [\u2200 i, MeasurableSpace (\u03c0 i)] [\u2200 i, SecondCountableTopology (\u03c0 i)] [\u2200 i, BorelSpace (\u03c0 i)] :\n BorelSpace (\u2200 i, \u03c0 i) :=\n \u27e8le_antisymm pi_le_borel_pi OpensMeasurableSpace.borel_le\u27e9\n#align pi.borel_space Pi.borelSpace\n\ninstance Prod.borelSpace [SecondCountableTopologyEither \u03b1 \u03b2] :\n BorelSpace (\u03b1 \u00d7 \u03b2) :=\n \u27e8le_antisymm prod_le_borel_prod OpensMeasurableSpace.borel_le\u27e9\n#align prod.borel_space Prod.borelSpace\n\n/-- Given a measurable embedding to a Borel space which is also a topological embedding, then the\nsource space is also a Borel space. -/\nlemma MeasurableEmbedding.borelSpace {\u03b1 \u03b2 : Type*} [MeasurableSpace \u03b1] [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b2] [TopologicalSpace \u03b2] [h\u03b2 : BorelSpace \u03b2] {e : \u03b1 \u2192 \u03b2}\n (h'e : MeasurableEmbedding e) (h''e : Inducing e) :\n BorelSpace \u03b1 := by\n constructor\n have : MeasurableSpace.comap e (borel \u03b2) = \u2039_\u203a := by simpa [h\u03b2.measurable_eq] using h'e.comap_eq\n rw [\u2190 this, \u2190 borel_comap, h''e.induced]\n\ninstance _root_.ULift.instBorelSpace : BorelSpace (ULift \u03b1) :=\n MeasurableEquiv.ulift.measurableEmbedding.borelSpace Homeomorph.ulift.inducing\n\ninstance DiscreteMeasurableSpace.toBorelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [DiscreteTopology \u03b1]\n [MeasurableSpace \u03b1] [DiscreteMeasurableSpace \u03b1] : BorelSpace \u03b1 := by\n constructor; ext; simp [MeasurableSpace.measurableSet_generateFrom, measurableSet_discrete]\n\nprotected theorem Embedding.measurableEmbedding {f : \u03b1 \u2192 \u03b2} (h\u2081 : Embedding f)\n (h\u2082 : MeasurableSet (range f)) : MeasurableEmbedding f :=\n show MeasurableEmbedding\n (((\u2191) : range f \u2192 \u03b2) \u2218 (Homeomorph.ofEmbedding f h\u2081).toMeasurableEquiv) from\n (MeasurableEmbedding.subtype_coe h\u2082).comp (MeasurableEquiv.measurableEmbedding _)\n#align embedding.measurable_embedding Embedding.measurableEmbedding\n\nprotected theorem ClosedEmbedding.measurableEmbedding {f : \u03b1 \u2192 \u03b2} (h : ClosedEmbedding f) :\n MeasurableEmbedding f :=\n h.toEmbedding.measurableEmbedding h.isClosed_range.measurableSet\n#align closed_embedding.measurable_embedding ClosedEmbedding.measurableEmbedding\n\nprotected theorem OpenEmbedding.measurableEmbedding {f : \u03b1 \u2192 \u03b2} (h : OpenEmbedding f) :\n MeasurableEmbedding f :=\n h.toEmbedding.measurableEmbedding h.isOpen_range.measurableSet\n#align open_embedding.measurable_embedding OpenEmbedding.measurableEmbedding\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] [OrderTopology \u03b1] [SecondCountableTopology \u03b1]\n\ntheorem measurable_of_Iio {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Iio x)) : Measurable f := by\n convert measurable_generateFrom (\u03b1 := \u03b4) _\n \u00b7 exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Iio _)\n \u00b7 rintro _ \u27e8x, rfl\u27e9; exact hf x\n#align measurable_of_Iio measurable_of_Iio\n\ntheorem UpperSemicontinuous.measurable [TopologicalSpace \u03b4] [OpensMeasurableSpace \u03b4] {f : \u03b4 \u2192 \u03b1}\n (hf : UpperSemicontinuous f) : Measurable f :=\n measurable_of_Iio fun y => (hf.isOpen_preimage y).measurableSet\n#align upper_semicontinuous.measurable UpperSemicontinuous.measurable\n\ntheorem measurable_of_Ioi {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Ioi x)) : Measurable f := by\n convert measurable_generateFrom (\u03b1 := \u03b4) _\n \u00b7 exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioi _)\n \u00b7 rintro _ \u27e8x, rfl\u27e9; exact hf x\n#align measurable_of_Ioi measurable_of_Ioi\n\ntheorem LowerSemicontinuous.measurable [TopologicalSpace \u03b4] [OpensMeasurableSpace \u03b4] {f : \u03b4 \u2192 \u03b1}\n (hf : LowerSemicontinuous f) : Measurable f :=\n measurable_of_Ioi fun y => (hf.isOpen_preimage y).measurableSet\n#align lower_semicontinuous.measurable LowerSemicontinuous.measurable\n\ntheorem measurable_of_Iic {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Iic x)) : Measurable f := by\n apply measurable_of_Ioi\n simp_rw [\u2190 compl_Iic, preimage_compl, MeasurableSet.compl_iff]\n assumption\n#align measurable_of_Iic measurable_of_Iic\n\ntheorem measurable_of_Ici {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Ici x)) : Measurable f := by\n apply measurable_of_Iio\n simp_rw [\u2190 compl_Ici, preimage_compl, MeasurableSet.compl_iff]\n assumption\n#align measurable_of_Ici measurable_of_Ici\n\n/-- If a function is the least upper bound of countably many measurable functions,\nthen it is measurable. -/\ntheorem Measurable.isLUB {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i))\n (hg : \u2200 b, IsLUB { a | \u2203 i, f i b = a } (g b)) : Measurable g := by\n change \u2200 b, IsLUB (range fun i => f i b) (g b) at hg\n rw [\u2039BorelSpace \u03b1\u203a.measurable_eq, borel_eq_generateFrom_Ioi \u03b1]\n apply measurable_generateFrom\n rintro _ \u27e8a, rfl\u27e9\n simp_rw [Set.preimage, mem_Ioi, lt_isLUB_iff (hg _), exists_range_iff, setOf_exists]\n exact MeasurableSet.iUnion fun i => hf i (isOpen_lt' _).measurableSet\n#align measurable.is_lub Measurable.isLUB\n\n/-- If a function is the least upper bound of countably many measurable functions on a measurable\nset `s`, and coincides with a measurable function outside of `s`, then it is measurable. -/\ntheorem Measurable.isLUB_of_mem {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g g' : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, Measurable (f i))\n {s : Set \u03b4} (hs : MeasurableSet s) (hg : \u2200 b \u2208 s, IsLUB { a | \u2203 i, f i b = a } (g b))\n (hg' : EqOn g g' s\u1d9c) (g'_meas : Measurable g') : Measurable g := by\n rcases isEmpty_or_nonempty \u03b9 with h\u03b9|\u27e8\u27e8i\u27e9\u27e9\n \u00b7 rcases eq_empty_or_nonempty s with rfl|\u27e8x, hx\u27e9\n \u00b7 convert g'_meas\n rwa [compl_empty, eqOn_univ] at hg'\n \u00b7 have A : \u2200 b \u2208 s, IsBot (g b) := by simpa using hg\n have B : \u2200 b \u2208 s, g b = g x := by\n intro b hb\n apply le_antisymm (A b hb (g x)) (A x hx (g b))\n have : g = s.piecewise (fun _y \u21a6 g x) g' := by\n ext b\n by_cases hb : b \u2208 s\n \u00b7 simp [hb, B]\n \u00b7 simp [hb, hg' hb]\n rw [this]\n exact Measurable.piecewise hs measurable_const g'_meas\n \u00b7 let f' : \u03b9 \u2192 \u03b4 \u2192 \u03b1 := fun i \u21a6 s.piecewise (f i) g'\n suffices \u2200 b, IsLUB { a | \u2203 i, f' i b = a } (g b) from\n Measurable.isLUB (fun i \u21a6 Measurable.piecewise hs (hf i) g'_meas) this\n intro b\n by_cases hb : b \u2208 s\n \u00b7 have A : \u2200 i, f' i b = f i b := fun i \u21a6 by simp [f', hb]\n simpa [A] using hg b hb\n \u00b7 have A : \u2200 i, f' i b = g' b := fun i \u21a6 by simp [f', hb]\n have : {a | \u2203 (_i : \u03b9), g' b = a} = {g' b} := by\n apply Subset.antisymm\n \u00b7 rintro - \u27e8_j, rfl\u27e9\n simp only [mem_singleton_iff]\n \u00b7 rintro - rfl\n exact \u27e8i, rfl\u27e9\n simp [A, this, hg' hb, isLUB_singleton]\n\ntheorem AEMeasurable.isLUB {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, IsLUB { a | \u2203 i, f i b = a } (g b)) :\n AEMeasurable g \u03bc := by\n nontriviality \u03b1\n haveI h\u03b1 : Nonempty \u03b1 := inferInstance\n cases' isEmpty_or_nonempty \u03b9 with h\u03b9 h\u03b9\n \u00b7 simp only [IsEmpty.exists_iff, setOf_false, isLUB_empty_iff] at hg\n exact aemeasurable_const' (hg.mono fun a ha => hg.mono fun b hb => (ha _).antisymm (hb _))\n let p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB { a | \u2203 i, f' i = a } (g x)\n let g_seq := (aeSeqSet hf p).piecewise g fun _ => h\u03b1.some\n have hg_seq : \u2200 b, IsLUB { a | \u2203 i, aeSeq hf p i b = a } (g_seq b) := by\n intro b\n simp only [g_seq, aeSeq, Set.piecewise]\n split_ifs with h\n \u00b7 have h_set_eq : { a : \u03b1 | \u2203 i : \u03b9, (hf i).mk (f i) b = a } =\n { a : \u03b1 | \u2203 i : \u03b9, f i b = a } := by\n ext x\n simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]\n rw [h_set_eq]\n exact aeSeq.fun_prop_of_mem_aeSeqSet hf h\n \u00b7 exact IsGreatest.isLUB \u27e8(@exists_const (h\u03b1.some = h\u03b1.some) \u03b9 _).2 rfl, fun x \u27e8i, hi\u27e9 => hi.ge\u27e9\n refine' \u27e8g_seq, Measurable.isLUB (aeSeq.measurable hf p) hg_seq, _\u27e9\n exact\n (ite_ae_eq_of_measure_compl_zero g (fun _ => h\u03b1.some) (aeSeqSet hf p)\n (aeSeq.measure_compl_aeSeqSet_eq_zero hf hg)).symm\n#align ae_measurable.is_lub AEMeasurable.isLUB\n\n/-- If a function is the greatest lower bound of countably many measurable functions,\nthen it is measurable. -/\ntheorem Measurable.isGLB {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i))\n (hg : \u2200 b, IsGLB { a | \u2203 i, f i b = a } (g b)) : Measurable g :=\n Measurable.isLUB (\u03b1 := \u03b1\u1d52\u1d48) hf hg\n#align measurable.is_glb Measurable.isGLB\n\n/-- If a function is the greatest lower bound of countably many measurable functions on a measurable\nset `s`, and coincides with a measurable function outside of `s`, then it is measurable. -/\ntheorem Measurable.isGLB_of_mem {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g g' : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, Measurable (f i))\n {s : Set \u03b4} (hs : MeasurableSet s) (hg : \u2200 b \u2208 s, IsGLB { a | \u2203 i, f i b = a } (g b))\n (hg' : EqOn g g' s\u1d9c) (g'_meas : Measurable g') : Measurable g :=\n Measurable.isLUB_of_mem (\u03b1 := \u03b1\u1d52\u1d48) hf hs hg hg' g'_meas\n\ntheorem AEMeasurable.isGLB {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, IsGLB { a | \u2203 i, f i b = a } (g b)) :\n AEMeasurable g \u03bc :=\n AEMeasurable.isLUB (\u03b1 := \u03b1\u1d52\u1d48) hf hg\n#align ae_measurable.is_glb AEMeasurable.isGLB\n\nprotected theorem Monotone.measurable [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {f : \u03b2 \u2192 \u03b1}\n (hf : Monotone f) : Measurable f :=\n suffices h : \u2200 x, OrdConnected (f \u207b\u00b9' Ioi x) from measurable_of_Ioi fun x => (h x).measurableSet\n fun _ => ordConnected_def.mpr fun _a ha _ _ _c hc => lt_of_lt_of_le ha (hf hc.1)\n#align monotone.measurable Monotone.measurable\n\ntheorem aemeasurable_restrict_of_monotoneOn [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {\u03bc : Measure \u03b2}\n {s : Set \u03b2} (hs : MeasurableSet s) {f : \u03b2 \u2192 \u03b1} (hf : MonotoneOn f s) :\n AEMeasurable f (\u03bc.restrict s) :=\n have : Monotone (f \u2218 (\u2191) : s \u2192 \u03b1) := fun \u27e8x, hx\u27e9 \u27e8y, hy\u27e9 => fun (hxy : x \u2264 y) => hf hx hy hxy\n aemeasurable_restrict_of_measurable_subtype hs this.measurable\n#align ae_measurable_restrict_of_monotone_on aemeasurable_restrict_of_monotoneOn\n\nprotected theorem Antitone.measurable [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {f : \u03b2 \u2192 \u03b1}\n (hf : Antitone f) : Measurable f :=\n @Monotone.measurable \u03b1\u1d52\u1d48 \u03b2 _ _ \u2039_\u203a _ _ _ _ _ \u2039_\u203a _ _ _ hf\n#align antitone.measurable Antitone.measurable\n\ntheorem aemeasurable_restrict_of_antitoneOn [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {\u03bc : Measure \u03b2}\n {s : Set \u03b2} (hs : MeasurableSet s) {f : \u03b2 \u2192 \u03b1} (hf : AntitoneOn f s) :\n AEMeasurable f (\u03bc.restrict s) :=\n @aemeasurable_restrict_of_monotoneOn \u03b1\u1d52\u1d48 \u03b2 _ _ \u2039_\u203a _ _ _ _ _ \u2039_\u203a _ _ _ _ hs _ hf\n#align ae_measurable_restrict_of_antitone_on aemeasurable_restrict_of_antitoneOn\n\ntheorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set \u03b1} (h : \u2200 x \u2208 s, s \u2208 \ud835\udcdd[>] x)\n (h' : \u2200 x \u2208 s, \u2203 y, x < y) : MeasurableSet s := by\n choose! M hM using h'\n suffices H : (s \\ interior s).Countable by\n have : s = interior s \u222a s \\ interior s := by rw [union_diff_cancel interior_subset]\n rw [this]\n exact isOpen_interior.measurableSet.union H.measurableSet\n have A : \u2200 x \u2208 s, \u2203 y \u2208 Ioi x, Ioo x y \u2286 s := fun x hx =>\n (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)\n choose! y hy h'y using A\n have B : Set.PairwiseDisjoint (s \\ interior s) fun x => Ioo x (y x) := by\n intro x hx x' hx' hxx'\n rcases lt_or_gt_of_ne hxx' with (h' | h')\n \u00b7 refine disjoint_left.2 fun z hz h'z => ?_\n have : x' \u2208 interior s :=\n mem_interior.2 \u27e8Ioo x (y x), h'y _ hx.1, isOpen_Ioo, \u27e8h', h'z.1.trans hz.2\u27e9\u27e9\n exact False.elim (hx'.2 this)\n \u00b7 refine disjoint_left.2 fun z hz h'z => ?_\n have : x \u2208 interior s :=\n mem_interior.2 \u27e8Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, \u27e8h', hz.1.trans h'z.2\u27e9\u27e9\n exact False.elim (hx.2 this)\n exact B.countable_of_Ioo fun x hx => hy x hx.1\n#align measurable_set_of_mem_nhds_within_Ioi_aux measurableSet_of_mem_nhdsWithin_Ioi_aux\n\n/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/\ntheorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set \u03b1} (h : \u2200 x \u2208 s, s \u2208 \ud835\udcdd[>] x) :\n MeasurableSet s := by\n by_cases H : \u2203 x \u2208 s, IsTop x\n \u00b7 rcases H with \u27e8x\u2080, x\u2080s, h\u2080\u27e9\n have : s = {x\u2080} \u222a s \\ {x\u2080} := by rw [union_diff_cancel (singleton_subset_iff.2 x\u2080s)]\n rw [this]\n refine' (measurableSet_singleton _).union _\n have A : \u2200 x \u2208 s \\ {x\u2080}, x < x\u2080 := fun x hx => lt_of_le_of_ne (h\u2080 _) (by simpa using hx.2)\n refine' measurableSet_of_mem_nhdsWithin_Ioi_aux (fun x hx => _) fun x hx => \u27e8x\u2080, A x hx\u27e9\n obtain \u27e8u, hu, us\u27e9 : \u2203 (u : \u03b1), u \u2208 Ioi x \u2227 Ioo x u \u2286 s :=\n (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).1 (h x hx.1)\n refine' (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).2 \u27e8u, hu, fun y hy => \u27e8us hy, _\u27e9\u27e9\n exact ne_of_lt (hy.2.trans_le (h\u2080 _))\n \u00b7 apply measurableSet_of_mem_nhdsWithin_Ioi_aux h\n simp only [IsTop] at H\n push_neg at H\n exact H\n#align measurable_set_of_mem_nhds_within_Ioi measurableSet_of_mem_nhdsWithin_Ioi\n\nlemma measurableSet_bddAbove_range {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n MeasurableSet {b | BddAbove (range (fun i \u21a6 f i b))} := by\n rcases isEmpty_or_nonempty \u03b1 with h\u03b1|h\u03b1\n \u00b7 have : \u2200 b, range (fun i \u21a6 f i b) = \u2205 := fun b \u21a6 eq_empty_of_isEmpty _\n simp [this]\n have A : \u2200 (i : \u03b9) (c : \u03b1), MeasurableSet {x | f i x \u2264 c} := by\n intro i c\n exact measurableSet_le (hf i) measurable_const\n have B : \u2200 (c : \u03b1), MeasurableSet {x | \u2200 i, f i x \u2264 c} := by\n intro c\n rw [setOf_forall]\n exact MeasurableSet.iInter (fun i \u21a6 A i c)\n obtain \u27e8u, hu\u27e9 : \u2203 (u : \u2115 \u2192 \u03b1), Tendsto u atTop atTop := exists_seq_tendsto (atTop : Filter \u03b1)\n have : {b | BddAbove (range (fun i \u21a6 f i b))} = {x | \u2203 n, \u2200 i, f i x \u2264 u n} := by\n apply Subset.antisymm\n \u00b7 rintro x \u27e8c, hc\u27e9\n obtain \u27e8n, hn\u27e9 : \u2203 n, c \u2264 u n := (tendsto_atTop.1 hu c).exists\n exact \u27e8n, fun i \u21a6 (hc ((mem_range_self i))).trans hn\u27e9\n \u00b7 rintro x \u27e8n, hn\u27e9\n refine \u27e8u n, ?_\u27e9\n rintro - \u27e8i, rfl\u27e9\n exact hn i\n rw [this, setOf_exists]\n exact MeasurableSet.iUnion (fun n \u21a6 B (u n))\n\nlemma measurableSet_bddBelow_range {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n MeasurableSet {b | BddBelow (range (fun i \u21a6 f i b))} :=\n measurableSet_bddAbove_range (\u03b1 := \u03b1\u1d52\u1d48) hf\n\nend LinearOrder\n\n@[measurability]\ntheorem Measurable.iSup_Prop {\u03b1} [MeasurableSpace \u03b1] [ConditionallyCompleteLattice \u03b1]\n (p : Prop) {f : \u03b4 \u2192 \u03b1} (hf : Measurable f) : Measurable fun b => \u2a06 _ : p, f b := by\n simp_rw [ciSup_eq_ite]\n split_ifs with h\n \u00b7 exact hf\n \u00b7 exact measurable_const\n#align measurable.supr_Prop Measurable.iSup_Prop\n\n@[measurability]\ntheorem Measurable.iInf_Prop {\u03b1} [MeasurableSpace \u03b1] [ConditionallyCompleteLattice \u03b1]\n (p : Prop) {f : \u03b4 \u2192 \u03b1} (hf : Measurable f) : Measurable fun b => \u2a05 _ : p, f b := by\n simp_rw [ciInf_eq_ite]\n split_ifs with h\n \u00b7 exact hf\n \u00b7 exact measurable_const\n#align measurable.infi_Prop Measurable.iInf_Prop\n\nsection ConditionallyCompleteLinearOrder\n\nvariable [ConditionallyCompleteLinearOrder \u03b1] [OrderTopology \u03b1] [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem measurable_iSup {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable (fun b \u21a6 \u2a06 i, f i b) := by\n rcases isEmpty_or_nonempty \u03b9 with h\u03b9|h\u03b9\n \u00b7 simp [iSup_of_empty']\n have A : MeasurableSet {b | BddAbove (range (fun i \u21a6 f i b))} :=\n measurableSet_bddAbove_range hf\n have : Measurable (fun (_b : \u03b4) \u21a6 sSup (\u2205 : Set \u03b1)) := measurable_const\n apply Measurable.isLUB_of_mem hf A _ _ this\n \u00b7 rintro b \u27e8c, hc\u27e9\n apply isLUB_ciSup\n refine \u27e8c, ?_\u27e9\n rintro d \u27e8i, rfl\u27e9\n exact hc (mem_range_self i)\n \u00b7 intro b hb\n apply csSup_of_not_bddAbove\n exact hb\n\n@[measurability]\ntheorem aemeasurable_iSup {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a06 i, f i b) \u03bc := by\n refine \u27e8fun b \u21a6 \u2a06 i, (hf i).mk (f i) b, measurable_iSup (fun i \u21a6 (hf i).measurable_mk), ?_\u27e9\n filter_upwards [ae_all_iff.2 (fun i \u21a6 (hf i).ae_eq_mk)] with b hb using by simp [hb]\n#align ae_measurable_supr aemeasurable_iSup\n\n@[measurability]\ntheorem measurable_iInf {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable fun b => \u2a05 i, f i b :=\n measurable_iSup (\u03b1 := \u03b1\u1d52\u1d48) hf\n#align measurable_infi measurable_iInf\n\n@[measurability]\ntheorem aemeasurable_iInf {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a05 i, f i b) \u03bc :=\n aemeasurable_iSup (\u03b1 := \u03b1\u1d52\u1d48) hf\n#align ae_measurable_infi aemeasurable_iInf\n\ntheorem measurable_sSup {\u03b9} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {s : Set \u03b9} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) :\n Measurable fun x => sSup ((fun i => f i x) '' s) := by\n have : Countable \u2191s := countable_coe_iff.2 hs\n convert measurable_iSup (f := (fun (i : s) \u21a6 f i)) (fun i \u21a6 hf i i.2) using 1\n ext b\n congr\n exact image_eq_range (fun i \u21a6 f i b) s\n#align measurable_cSup measurable_sSup\n\ntheorem measurable_sInf {\u03b9} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {s : Set \u03b9} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) :\n Measurable fun x => sInf ((fun i => f i x) '' s) :=\n measurable_sSup (\u03b1 := \u03b1\u1d52\u1d48) hs hf\n#align measurable_cInf measurable_sInf\n\ntheorem measurable_biSup {\u03b9} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) : Measurable fun b => \u2a06 i \u2208 s, f i b := by\n haveI : Encodable s := hs.toEncodable\n by_cases H : \u2200 i, i \u2208 s\n \u00b7 have : \u2200 b, \u2a06 i \u2208 s, f i b = \u2a06 (i : s), f i b :=\n fun b \u21a6 cbiSup_eq_of_forall (f := fun i \u21a6 f i b) H\n simp only [this]\n exact measurable_iSup (fun (i : s) \u21a6 hf i i.2)\n \u00b7 have : \u2200 b, \u2a06 i \u2208 s, f i b = (\u2a06 (i : s), f i b) \u2294 sSup \u2205 :=\n fun b \u21a6 cbiSup_eq_of_not_forall (f := fun i \u21a6 f i b) H\n simp only [this]\n apply Measurable.sup _ measurable_const\n exact measurable_iSup (fun (i : s) \u21a6 hf i i.2)\n#align measurable_bsupr measurable_biSup\n\ntheorem aemeasurable_biSup {\u03b9} {\u03bc : Measure \u03b4} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a06 i \u2208 s, f i b) \u03bc := by\n let g : \u03b9 \u2192 \u03b4 \u2192 \u03b1 := fun i \u21a6 if hi : i \u2208 s then (hf i hi).mk (f i) else fun _b \u21a6 sSup \u2205\n have : \u2200 i \u2208 s, Measurable (g i) := by\n intro i hi\n simpa [g, hi] using (hf i hi).measurable_mk\n refine \u27e8fun b \u21a6 \u2a06 (i) (_ : i \u2208 s), g i b, measurable_biSup s hs this, ?_\u27e9\n have : \u2200 i \u2208 s, \u2200\u1d50 b \u2202\u03bc, f i b = g i b :=\n fun i hi \u21a6 by simpa [g, hi] using (hf i hi).ae_eq_mk\n filter_upwards [(ae_ball_iff hs).2 this] with b hb\n exact iSup_congr fun i => iSup_congr (hb i)\n#align ae_measurable_bsupr aemeasurable_biSup\n\ntheorem measurable_biInf {\u03b9} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) : Measurable fun b => \u2a05 i \u2208 s, f i b :=\n measurable_biSup (\u03b1 := \u03b1\u1d52\u1d48) s hs hf\n#align measurable_binfi measurable_biInf\n\ntheorem aemeasurable_biInf {\u03b9} {\u03bc : Measure \u03b4} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a05 i \u2208 s, f i b) \u03bc :=\n aemeasurable_biSup (\u03b1 := \u03b1\u1d52\u1d48) s hs hf\n#align ae_measurable_binfi aemeasurable_biInf\n\n/-- `liminf` over a general filter is measurable. See `measurable_liminf` for the version over `\u2115`.\n-/\ntheorem measurable_liminf' {\u03b9 \u03b9'} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {v : Filter \u03b9} (hf : \u2200 i, Measurable (f i))\n {p : \u03b9' \u2192 Prop} {s : \u03b9' \u2192 Set \u03b9} (hv : v.HasCountableBasis p s) (hs : \u2200 j, (s j).Countable) :\n Measurable fun x => liminf (f \u00b7 x) v := by\n /- We would like to write the liminf as `\u2a06 (j : Subtype p), \u2a05 (i : s j), f i x`, as the\n measurability would follow from the measurability of infs and sups. Unfortunately, this is not\n true in general conditionally complete linear orders because of issues with empty sets or sets\n which are not bounded above or below. A slightly more complicated expression for the liminf,\n valid in general, is given in `Filter.HasBasis.liminf_eq_ite`. This expression, built from\n `if ... then ... else` and infs and sups, can be readily checked to be measurable. -/\n have : Countable (Subtype p) := hv.countable\n rcases isEmpty_or_nonempty (Subtype p) with hp|hp\n \u00b7 simp [hv.liminf_eq_sSup_iUnion_iInter]\n by_cases H : \u2203 (j : Subtype p), s j = \u2205\n \u00b7 simp_rw [hv.liminf_eq_ite, if_pos H, measurable_const]\n simp_rw [hv.liminf_eq_ite, if_neg H]\n have : \u2200 i, Countable (s i) := fun i \u21a6 countable_coe_iff.2 (hs i)\n let m : Subtype p \u2192 Set \u03b4 := fun j \u21a6 {x | BddBelow (range (fun (i : s j) \u21a6 f i x))}\n have m_meas : \u2200 j, MeasurableSet (m j) :=\n fun j \u21a6 measurableSet_bddBelow_range (fun (i : s j) \u21a6 hf i)\n have mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), x \u2209 m j} := by\n rw [setOf_forall]\n exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl)\n apply Measurable.piecewise mc_meas measurable_const\n apply measurable_iSup (fun j \u21a6 ?_)\n let reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x \u21a6 liminf_reparam (fun i \u21a6 f i x) s p\n let F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x \u21a6 \u2a05 (i : s j), f i x\n have F0_meas : \u2200 j, Measurable (F0 j) := fun j \u21a6 measurable_iInf (fun (i : s j) \u21a6 hf i)\n set F1 : \u03b4 \u2192 \u03b1 := fun x \u21a6 F0 (reparam x j) x with hF1\n change Measurable F1\n let g : \u2115 \u2192 Subtype p := Classical.choose (exists_surjective_nat (Subtype p))\n have Z : \u2200 x, \u2203 n, x \u2208 m (g n) \u2228 \u2200 k, x \u2209 m k := by\n intro x\n by_cases H : \u2203 k, x \u2208 m k\n \u00b7 rcases H with \u27e8k, hk\u27e9\n rcases Classical.choose_spec (exists_surjective_nat (Subtype p)) k with \u27e8n, rfl\u27e9\n exact \u27e8n, Or.inl hk\u27e9\n \u00b7 push_neg at H\n exact \u27e80, Or.inr H\u27e9\n have : F1 = fun x \u21a6 if x \u2208 m j then F0 j x else F0 (g (Nat.find (Z x))) x := by\n ext x\n have A : reparam x j = if x \u2208 m j then j else g (Nat.find (Z x)) := rfl\n split_ifs with hjx\n \u00b7 have : reparam x j = j := by rw [A, if_pos hjx]\n simp only [hF1, this]\n \u00b7 have : reparam x j = g (Nat.find (Z x)) := by rw [A, if_neg hjx]\n simp only [hF1, this]\n rw [this]\n apply Measurable.piecewise (m_meas j) (F0_meas j)\n apply Measurable.find (fun n \u21a6 F0_meas (g n)) (fun n \u21a6 ?_)\n exact (m_meas (g n)).union mc_meas\n#align measurable_liminf' measurable_liminf'\n\n/-- `limsup` over a general filter is measurable. See `measurable_limsup` for the version over `\u2115`.\n-/\ntheorem measurable_limsup' {\u03b9 \u03b9'} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {u : Filter \u03b9} (hf : \u2200 i, Measurable (f i))\n {p : \u03b9' \u2192 Prop} {s : \u03b9' \u2192 Set \u03b9} (hu : u.HasCountableBasis p s) (hs : \u2200 i, (s i).Countable) :\n Measurable fun x => limsup (fun i => f i x) u :=\n measurable_liminf' (\u03b1 := \u03b1\u1d52\u1d48) hf hu hs\n#align measurable_limsup' measurable_limsup'\n\n/-- `liminf` over `\u2115` is measurable. See `measurable_liminf'` for a version with a general filter.\n-/\n@[measurability]\ntheorem measurable_liminf {f : \u2115 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable fun x => liminf (fun i => f i x) atTop :=\n measurable_liminf' hf atTop_countable_basis fun _ => to_countable _\n#align measurable_liminf measurable_liminf\n\n/-- `limsup` over `\u2115` is measurable. See `measurable_limsup'` for a version with a general filter.\n-/\n@[measurability]\ntheorem measurable_limsup {f : \u2115 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable fun x => limsup (fun i => f i x) atTop :=\n measurable_limsup' hf atTop_countable_basis fun _ => to_countable _\n#align measurable_limsup measurable_limsup\n\nend ConditionallyCompleteLinearOrder\n\n/-- Convert a `Homeomorph` to a `MeasurableEquiv`. -/\ndef Homemorph.toMeasurableEquiv (h : \u03b1 \u2243\u209c \u03b2) : \u03b1 \u2243\u1d50 \u03b2 where\n toEquiv := h.toEquiv\n measurable_toFun := h.continuous_toFun.measurable\n measurable_invFun := h.continuous_invFun.measurable\n#align homemorph.to_measurable_equiv Homemorph.toMeasurableEquiv\n\nprotected theorem IsFiniteMeasureOnCompacts.map (\u03bc : Measure \u03b1) [IsFiniteMeasureOnCompacts \u03bc]\n (f : \u03b1 \u2243\u209c \u03b2) : IsFiniteMeasureOnCompacts (Measure.map f \u03bc) := by\n refine \u27e8fun K hK \u21a6 ?_\u27e9\n rw [\u2190 Homeomorph.toMeasurableEquiv_coe, MeasurableEquiv.map_apply]\n exact IsCompact.measure_lt_top (f.isCompact_preimage.2 hK)\n#align is_finite_measure_on_compacts.map IsFiniteMeasureOnCompacts.map\n\nend BorelSpace\n\ninstance Empty.borelSpace : BorelSpace Empty :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align empty.borel_space Empty.borelSpace\n\ninstance Unit.borelSpace : BorelSpace Unit :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align unit.borel_space Unit.borelSpace\n\ninstance Bool.borelSpace : BorelSpace Bool :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align bool.borel_space Bool.borelSpace\n\ninstance Nat.borelSpace : BorelSpace \u2115 :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align nat.borel_space Nat.borelSpace\n\ninstance Int.borelSpace : BorelSpace \u2124 :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align int.borel_space Int.borelSpace\n\ninstance Rat.borelSpace : BorelSpace \u211a :=\n \u27e8borel_eq_top_of_countable.symm\u27e9\n#align rat.borel_space Rat.borelSpace\n\n/- Instances on `Real` and `Complex` are special cases of `RCLike` but without these instances,\nLean fails to prove `BorelSpace (\u03b9 \u2192 \u211d)`, so we leave them here. -/\ninstance Real.measurableSpace : MeasurableSpace \u211d :=\n borel \u211d\n#align real.measurable_space Real.measurableSpace\n\ninstance Real.borelSpace : BorelSpace \u211d :=\n \u27e8rfl\u27e9\n#align real.borel_space Real.borelSpace\n\ninstance NNReal.measurableSpace : MeasurableSpace \u211d\u22650 :=\n Subtype.instMeasurableSpace\n#align nnreal.measurable_space NNReal.measurableSpace\n\ninstance NNReal.borelSpace : BorelSpace \u211d\u22650 :=\n Subtype.borelSpace _\n#align nnreal.borel_space NNReal.borelSpace\n\ninstance ENNReal.measurableSpace : MeasurableSpace \u211d\u22650\u221e :=\n borel \u211d\u22650\u221e\n#align ennreal.measurable_space ENNReal.measurableSpace\n\ninstance ENNReal.borelSpace : BorelSpace \u211d\u22650\u221e :=\n \u27e8rfl\u27e9\n#align ennreal.borel_space ENNReal.borelSpace\n\ninstance EReal.measurableSpace : MeasurableSpace EReal :=\n borel EReal\n#align ereal.measurable_space EReal.measurableSpace\n\ninstance EReal.borelSpace : BorelSpace EReal :=\n \u27e8rfl\u27e9\n#align ereal.borel_space EReal.borelSpace\n\n/-- One can cut out `\u211d\u22650\u221e` into the sets `{0}`, `Ico (t^n) (t^(n+1))` for `n : \u2124` and `{\u221e}`. This\ngives a way to compute the measure of a set in terms of sets on which a given function `f` does not\nfluctuate by more than `t`. -/\ntheorem measure_eq_measure_preimage_add_measure_tsum_Ico_zpow [MeasurableSpace \u03b1] (\u03bc : Measure \u03b1)\n {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : Measurable f) {s : Set \u03b1} (hs : MeasurableSet s) {t : \u211d\u22650} (ht : 1 < t) :\n \u03bc s =\n \u03bc (s \u2229 f \u207b\u00b9' {0}) + \u03bc (s \u2229 f \u207b\u00b9' {\u221e}) +\n \u2211' n : \u2124, \u03bc (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) := by\n have A : \u03bc s = \u03bc (s \u2229 f \u207b\u00b9' {0}) + \u03bc (s \u2229 f \u207b\u00b9' Ioi 0) := by\n rw [\u2190 measure_union]\n \u00b7 rw [\u2190 inter_union_distrib_left, \u2190 preimage_union, singleton_union, Ioi_insert,\n \u2190 _root_.bot_eq_zero, Ici_bot, preimage_univ, inter_univ]\n \u00b7 exact disjoint_singleton_left.mpr not_mem_Ioi_self\n |>.preimage f |>.inter_right' s |>.inter_left' s\n \u00b7 exact hs.inter (hf measurableSet_Ioi)\n have B : \u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u03bc (s \u2229 f \u207b\u00b9' {\u221e}) + \u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u221e) := by\n rw [\u2190 measure_union]\n \u00b7 rw [\u2190 inter_union_distrib_left]\n congr\n ext x\n simp only [mem_singleton_iff, mem_union, mem_Ioo, mem_Ioi, mem_preimage]\n obtain (H | H) : f x = \u221e \u2228 f x < \u221e := eq_or_lt_of_le le_top\n \u00b7 simp only [H, eq_self_iff_true, or_false_iff, ENNReal.zero_lt_top, not_top_lt, and_false]\n \u00b7 simp only [H, H.ne, and_true_iff, false_or_iff]\n \u00b7 refine disjoint_left.2 fun x hx h'x => ?_\n have : f x < \u221e := h'x.2.2\n exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm))\n \u00b7 exact hs.inter (hf measurableSet_Ioo)\n have C : \u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u221e) =\n \u2211' n : \u2124, \u03bc (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) := by\n rw [\u2190 measure_iUnion,\n ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow (ENNReal.one_lt_coe_iff.2 ht) ENNReal.coe_ne_top,\n preimage_iUnion, inter_iUnion]\n \u00b7 intro i j hij\n wlog h : i < j generalizing i j\n \u00b7 exact (this hij.symm (hij.lt_or_lt.resolve_left h)).symm\n refine disjoint_left.2 fun x hx h'x => lt_irrefl (f x) ?_\n calc\n f x < (t : \u211d\u22650\u221e) ^ (i + 1) := hx.2.2\n _ \u2264 (t : \u211d\u22650\u221e) ^ j := ENNReal.zpow_le_of_le (ENNReal.one_le_coe_iff.2 ht.le) h\n _ \u2264 f x := h'x.2.1\n \u00b7 intro n\n exact hs.inter (hf measurableSet_Ico)\n rw [A, B, C, add_assoc]\n#align measure_eq_measure_preimage_add_measure_tsum_Ico_zpow measure_eq_measure_preimage_add_measure_tsum_Ico_zpow\n\nsection PseudoMetricSpace\n\nvariable [PseudoMetricSpace \u03b1] [MeasurableSpace \u03b1] [OpensMeasurableSpace \u03b1]\nvariable [MeasurableSpace \u03b2] {x : \u03b1} {\u03b5 : \u211d}\n\nopen Metric\n\n@[measurability]\ntheorem measurableSet_ball : MeasurableSet (Metric.ball x \u03b5) :=\n Metric.isOpen_ball.measurableSet\n#align measurable_set_ball measurableSet_ball\n\n@[measurability]\ntheorem measurableSet_closedBall : MeasurableSet (Metric.closedBall x \u03b5) :=\n Metric.isClosed_ball.measurableSet\n#align measurable_set_closed_ball measurableSet_closedBall\n\n@[measurability]\ntheorem measurable_infDist {s : Set \u03b1} : Measurable fun x => infDist x s :=\n (continuous_infDist_pt s).measurable\n#align measurable_inf_dist measurable_infDist\n\n@[measurability]\ntheorem Measurable.infDist {f : \u03b2 \u2192 \u03b1} (hf : Measurable f) {s : Set \u03b1} :\n Measurable fun x => infDist (f x) s :=\n measurable_infDist.comp hf\n#align measurable.inf_dist Measurable.infDist\n\n@[measurability]\ntheorem measurable_infNndist {s : Set \u03b1} : Measurable fun x => infNndist x s :=\n (continuous_infNndist_pt s).measurable\n#align measurable_inf_nndist measurable_infNndist\n\n@[measurability]\ntheorem Measurable.infNndist {f : \u03b2 \u2192 \u03b1} (hf : Measurable f) {s : Set \u03b1} :\n Measurable fun x => infNndist (f x) s :=\n measurable_infNndist.comp hf\n#align measurable.inf_nndist Measurable.infNndist\n\nsection\n\nvariable [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem measurable_dist : Measurable fun p : \u03b1 \u00d7 \u03b1 => dist p.1 p.2 :=\n continuous_dist.measurable\n#align measurable_dist measurable_dist\n\n@[measurability]\ntheorem Measurable.dist {f g : \u03b2 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun b => dist (f b) (g b) :=\n (@continuous_dist \u03b1 _).measurable2 hf hg\n#align measurable.dist Measurable.dist\n\n@[measurability]\ntheorem measurable_nndist : Measurable fun p : \u03b1 \u00d7 \u03b1 => nndist p.1 p.2 :=\n continuous_nndist.measurable\n#align measurable_nndist measurable_nndist\n\n@[measurability]\ntheorem Measurable.nndist {f g : \u03b2 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun b => nndist (f b) (g b) :=\n (@continuous_nndist \u03b1 _).measurable2 hf hg\n#align measurable.nndist Measurable.nndist\n\nend\n\nend PseudoMetricSpace\n\nsection PseudoEMetricSpace\n\nvariable [PseudoEMetricSpace \u03b1] [MeasurableSpace \u03b1] [OpensMeasurableSpace \u03b1]\nvariable [MeasurableSpace \u03b2] {x : \u03b1} {\u03b5 : \u211d\u22650\u221e}\n\nopen EMetric\n\n@[measurability]\ntheorem measurableSet_eball : MeasurableSet (EMetric.ball x \u03b5) :=\n EMetric.isOpen_ball.measurableSet\n#align measurable_set_eball measurableSet_eball\n\n@[measurability]\ntheorem measurable_edist_right : Measurable (edist x) :=\n (continuous_const.edist continuous_id).measurable\n#align measurable_edist_right measurable_edist_right\n\n@[measurability]\ntheorem measurable_edist_left : Measurable fun y => edist y x :=\n (continuous_id.edist continuous_const).measurable\n#align measurable_edist_left measurable_edist_left\n\n@[measurability]\ntheorem measurable_infEdist {s : Set \u03b1} : Measurable fun x => infEdist x s :=\n continuous_infEdist.measurable\n#align measurable_inf_edist measurable_infEdist\n\n@[measurability]\ntheorem Measurable.infEdist {f : \u03b2 \u2192 \u03b1} (hf : Measurable f) {s : Set \u03b1} :\n Measurable fun x => infEdist (f x) s :=\n measurable_infEdist.comp hf\n#align measurable.inf_edist Measurable.infEdist\n\nopen Metric EMetric\n\n/-- If a set has a closed thickening with finite measure, then the measure of its `r`-closed\nthickenings converges to the measure of its closure as `r` tends to `0`. -/\ntheorem tendsto_measure_cthickening {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (cthickening R s) \u2260 \u221e) :\n Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc (closure s))) := by\n have A : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n rw [closure_eq_iInter_cthickening]\n exact\n tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet)\n (fun i j _ ij => cthickening_mono ij _) hs\n have B : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n apply Tendsto.congr' _ tendsto_const_nhds\n filter_upwards [self_mem_nhdsWithin (\u03b1 := \u211d)] with _ hr\n rw [cthickening_of_nonpos hr]\n convert B.sup A\n exact (nhds_left_sup_nhds_right' 0).symm\n#align tendsto_measure_cthickening tendsto_measure_cthickening\n\n/-- If a closed set has a closed thickening with finite measure, then the measure of its closed\n`r`-thickenings converge to its measure as `r` tends to `0`. -/\ntheorem tendsto_measure_cthickening_of_isClosed {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (cthickening R s) \u2260 \u221e) (h's : IsClosed s) :\n Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc s)) := by\n convert tendsto_measure_cthickening hs\n exact h's.closure_eq.symm\n#align tendsto_measure_cthickening_of_is_closed tendsto_measure_cthickening_of_isClosed\n\n/-- If a set has a thickening with finite measure, then the measures of its `r`-thickenings\nconverge to the measure of its closure as `r > 0` tends to `0`. -/\ntheorem tendsto_measure_thickening {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (thickening R s) \u2260 \u221e) :\n Tendsto (fun r => \u03bc (thickening r s)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n rw [closure_eq_iInter_thickening]\n exact tendsto_measure_biInter_gt (fun r _ => isOpen_thickening.measurableSet)\n (fun i j _ ij => thickening_mono ij _) hs\n\n/-- If a closed set has a thickening with finite measure, then the measure of its\n`r`-thickenings converge to its measure as `r > 0` tends to `0`. -/\ntheorem tendsto_measure_thickening_of_isClosed {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (thickening R s) \u2260 \u221e) (h's : IsClosed s) :\n Tendsto (fun r => \u03bc (thickening r s)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc s)) := by\n convert tendsto_measure_thickening hs\n exact h's.closure_eq.symm\n\nvariable [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem measurable_edist : Measurable fun p : \u03b1 \u00d7 \u03b1 => edist p.1 p.2 :=\n continuous_edist.measurable\n#align measurable_edist measurable_edist\n\n@[measurability]\ntheorem Measurable.edist {f g : \u03b2 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun b => edist (f b) (g b) :=\n (@continuous_edist \u03b1 _).measurable2 hf hg\n#align measurable.edist Measurable.edist\n\n@[measurability]\ntheorem AEMeasurable.edist {f g : \u03b2 \u2192 \u03b1} {\u03bc : Measure \u03b2} (hf : AEMeasurable f \u03bc)\n (hg : AEMeasurable g \u03bc) : AEMeasurable (fun a => edist (f a) (g a)) \u03bc :=\n (@continuous_edist \u03b1 _).aemeasurable2 hf hg\n#align ae_measurable.edist AEMeasurable.edist\n\nend PseudoEMetricSpace\n\n/-- Given a compact set in a proper space, the measure of its `r`-closed thickenings converges to\nits measure as `r` tends to `0`. -/\ntheorem tendsto_measure_cthickening_of_isCompact [MetricSpace \u03b1] [MeasurableSpace \u03b1]\n [OpensMeasurableSpace \u03b1] [ProperSpace \u03b1] {\u03bc : Measure \u03b1} [IsFiniteMeasureOnCompacts \u03bc]\n {s : Set \u03b1} (hs : IsCompact s) :\n Tendsto (fun r => \u03bc (Metric.cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc s)) :=\n tendsto_measure_cthickening_of_isClosed\n \u27e81, zero_lt_one, hs.isBounded.cthickening.measure_lt_top.ne\u27e9 hs.isClosed\n#align tendsto_measure_cthickening_of_is_compact tendsto_measure_cthickening_of_isCompact\n\n/-- If a measurable space is countably generated and separates points, it arises as\nthe borel sets of some second countable t4 topology (i.e. a separable metrizable one). -", "theoremStatement": "/\ntheorem exists_borelSpace_of_countablyGenerated_of_separatesPoints (\u03b1 : Type*)\n [m : MeasurableSpace \u03b1] [CountablyGenerated \u03b1] [SeparatesPoints \u03b1] :\n \u2203 \u03c4 : TopologicalSpace \u03b1, SecondCountableTopology \u03b1 \u2227 T4Space \u03b1 \u2227 BorelSpace", "theoremName": "exists_borelSpace_of_countablyGenerated_of_separatesPoints", "fileCreated": {"commit": "11332d53f1", "date": "2023-05-21"}, "theoremCreated": {"commit": "726f2a5ff9", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "positionMetadata": {"lineInFile": 1907, "tokenPositionInFile": 88883, "theoremPositionInFile": 217}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "= by\n rcases measurableEquiv_nat_bool_of_countablyGenerated \u03b1 with \u27e8s, \u27e8f\u27e9\u27e9\n letI := induced f inferInstance\n let F := f.toEquiv.toHomeomorphOfInducing $ inducing_induced _\n exact \u27e8inferInstance, F.secondCountableTopology, F.symm.t4Space,\n MeasurableEmbedding.borelSpace f.measurableEmbedding F.inducin", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 309}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Yury Kudryashov\n-/\nimport Mathlib.Analysis.Normed.Group.Basic\nimport Mathlib.MeasureTheory.Function.AEMeasurableSequence\nimport Mathlib.MeasureTheory.Group.Arithmetic\nimport Mathlib.MeasureTheory.Order.Lattice\nimport Mathlib.Topology.Instances.EReal\nimport Mathlib.Topology.MetricSpace.Thickening\nimport Mathlib.Topology.GDelta\nimport Mathlib.Topology.Order.Lattice\nimport Mathlib.Topology.Semicontinuous\n\n#align_import measure_theory.constructions.borel_space.basic from \"leanprover-community/mathlib\"@\"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce\"\n\n/-!\n# Borel (measurable) space\n\n## Main definitions\n\n* `borel \u03b1` : the least `\u03c3`-algebra that contains all open sets;\n* `class BorelSpace` : a space with `TopologicalSpace` and `MeasurableSpace` structures\n such that `\u2039MeasurableSpace \u03b1\u203a = borel \u03b1`;\n* `class OpensMeasurableSpace` : a space with `TopologicalSpace` and `MeasurableSpace`\n structures such that all open sets are measurable; equivalently, `borel \u03b1 \u2264 \u2039MeasurableSpace \u03b1\u203a`.\n* `BorelSpace` instances on `Empty`, `Unit`, `Bool`, `Nat`, `Int`, `Rat`;\n* `MeasurableSpace` and `BorelSpace` instances on `\u211d`, `\u211d\u22650`, `\u211d\u22650\u221e`.\n\n## Main statements\n\n* `IsOpen.measurableSet`, `IsClosed.measurableSet`: open and closed sets are measurable;\n* `Continuous.measurable` : a continuous function is measurable;\n* `Continuous.measurable2` : if `f : \u03b1 \u2192 \u03b2` and `g : \u03b1 \u2192 \u03b3` are measurable and `op : \u03b2 \u00d7 \u03b3 \u2192 \u03b4`\n is continuous, then `fun x => op (f x, g y)` is measurable;\n* `Measurable.add` etc : dot notation for arithmetic operations on `Measurable` predicates,\n and similarly for `dist` and `edist`;\n* `AEMeasurable.add` : similar dot notation for almost everywhere measurable functions;\n* `Measurable.ennreal*` : special cases for arithmetic operations on `\u211d\u22650\u221e`.\n-/\n\n\nnoncomputable section\n\nopen Set Filter MeasureTheory\n\nopen scoped Classical BigOperators Topology NNReal ENNReal MeasureTheory\n\nuniverse u v w x y\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b3\u2082 \u03b4 : Type*} {\u03b9 : Sort y} {s t u : Set \u03b1}\n\nopen MeasurableSpace TopologicalSpace\n\n/-- `MeasurableSpace` structure generated by `TopologicalSpace`. -/\ndef borel (\u03b1 : Type u) [TopologicalSpace \u03b1] : MeasurableSpace \u03b1 :=\n generateFrom { s : Set \u03b1 | IsOpen s }\n#align borel borel\n\ntheorem borel_anti : Antitone (@borel \u03b1) := fun _ _ h =>\n MeasurableSpace.generateFrom_le fun _ hs => .basic _ (h _ hs)\n#align borel_anti borel_anti\n\ntheorem borel_eq_top_of_discrete [TopologicalSpace \u03b1] [DiscreteTopology \u03b1] : borel \u03b1 = \u22a4 :=\n top_le_iff.1 fun s _ => GenerateMeasurable.basic s (isOpen_discrete s)\n#align borel_eq_top_of_discrete borel_eq_top_of_discrete\n\ntheorem borel_eq_top_of_countable [TopologicalSpace \u03b1] [T1Space \u03b1] [Countable \u03b1] : borel \u03b1 = \u22a4 := by\n refine' top_le_iff.1 fun s _ => biUnion_of_singleton s \u25b8 _\n apply MeasurableSet.biUnion s.to_countable\n intro x _\n apply MeasurableSet.of_compl\n apply GenerateMeasurable.basic\n exact isClosed_singleton.isOpen_compl\n#align borel_eq_top_of_countable borel_eq_top_of_countable\n\ntheorem borel_eq_generateFrom_of_subbasis {s : Set (Set \u03b1)} [t : TopologicalSpace \u03b1]\n [SecondCountableTopology \u03b1] (hs : t = .generateFrom s) : borel \u03b1 = .generateFrom s :=\n le_antisymm\n (generateFrom_le fun u (hu : t.IsOpen u) => by\n rw [hs] at hu\n induction hu with\n | basic u hu => exact GenerateMeasurable.basic u hu\n | univ => exact @MeasurableSet.univ \u03b1 (generateFrom s)\n | inter s\u2081 s\u2082 _ _ hs\u2081 hs\u2082 => exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082\n | sUnion f hf ih =>\n rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9\n rw [\u2190 vu]\n exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv))\n (generateFrom_le fun u hu =>\n GenerateMeasurable.basic _ <| show t.IsOpen u by rw [hs]; exact GenerateOpen.basic _ hu)\n#align borel_eq_generate_from_of_subbasis borel_eq_generateFrom_of_subbasis\n\ntheorem TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom [TopologicalSpace \u03b1]\n [SecondCountableTopology \u03b1] {s : Set (Set \u03b1)} (hs : IsTopologicalBasis s) :\n borel \u03b1 = .generateFrom s :=\n borel_eq_generateFrom_of_subbasis hs.eq_generateFrom\n#align topological_space.is_topological_basis.borel_eq_generate_from TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom\n\ntheorem isPiSystem_isOpen [TopologicalSpace \u03b1] : IsPiSystem ({s : Set \u03b1 | IsOpen s}) :=\n fun _s hs _t ht _ => IsOpen.inter hs ht\n#align is_pi_system_is_open isPiSystem_isOpen\n\nlemma isPiSystem_isClosed [TopologicalSpace \u03b1] : IsPiSystem ({s : Set \u03b1 | IsClosed s}) :=\n fun _s hs _t ht _ \u21a6 IsClosed.inter hs ht\n\ntheorem borel_eq_generateFrom_isClosed [TopologicalSpace \u03b1] :\n borel \u03b1 = .generateFrom { s | IsClosed s } :=\n le_antisymm\n (generateFrom_le fun _t ht =>\n @MeasurableSet.of_compl \u03b1 _ (generateFrom { s | IsClosed s })\n (GenerateMeasurable.basic _ <| isClosed_compl_iff.2 ht))\n (generateFrom_le fun _t ht =>\n @MeasurableSet.of_compl \u03b1 _ (borel \u03b1) (GenerateMeasurable.basic _ <| isOpen_compl_iff.2 ht))\n#align borel_eq_generate_from_is_closed borel_eq_generateFrom_isClosed\n\nsection OrderTopology\n\nvariable (\u03b1)\nvariable [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1]\n\ntheorem borel_eq_generateFrom_Iio : borel \u03b1 = .generateFrom (range Iio) := by\n refine' le_antisymm _ (generateFrom_le _)\n \u00b7 rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals \u03b1 _ _ _)]\n letI : MeasurableSpace \u03b1 := MeasurableSpace.generateFrom (range Iio)\n have H : \u2200 a : \u03b1, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ \u27e8_, rfl\u27e9\n refine' generateFrom_le _\n rintro _ \u27e8a, rfl | rfl\u27e9\n \u00b7 rcases em (\u2203 b, a \u22d6 b) with \u27e8b, hb\u27e9 | hcovBy\n \u00b7 rw [hb.Ioi_eq, \u2190 compl_Iio]\n exact (H _).compl\n \u00b7 rcases isOpen_biUnion_countable (Ioi a) Ioi fun _ _ \u21a6 isOpen_Ioi with \u27e8t, hat, htc, htU\u27e9\n have : Ioi a = \u22c3 b \u2208 t, Ici b := by\n refine Subset.antisymm ?_ <| iUnion\u2082_subset fun b hb \u21a6 Ici_subset_Ioi.2 (hat hb)\n refine Subset.trans ?_ <| iUnion\u2082_mono fun _ _ \u21a6 Ioi_subset_Ici_self\n simpa [CovBy, htU, subset_def] using hcovBy\n simp only [this, \u2190 compl_Iio]\n exact .biUnion htc <| fun _ _ \u21a6 (H _).compl\n \u00b7 apply H\n \u00b7 rw [forall_mem_range]\n intro a\n exact GenerateMeasurable.basic _ isOpen_Iio\n#align borel_eq_generate_from_Iio borel_eq_generateFrom_Iio\n\ntheorem borel_eq_generateFrom_Ioi : borel \u03b1 = .generateFrom (range Ioi) :=\n @borel_eq_generateFrom_Iio \u03b1\u1d52\u1d48 _ (by infer_instance : SecondCountableTopology \u03b1) _ _\n#align borel_eq_generate_from_Ioi borel_eq_generateFrom_Ioi\n\ntheorem borel_eq_generateFrom_Iic :\n borel \u03b1 = MeasurableSpace.generateFrom (range Iic) := by\n rw [borel_eq_generateFrom_Ioi]\n refine' le_antisymm _ _\n \u00b7 refine' MeasurableSpace.generateFrom_le fun t ht => _\n obtain \u27e8u, rfl\u27e9 := ht\n rw [\u2190 compl_Iic]\n exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr \u27e8u, rfl\u27e9)).compl\n \u00b7 refine' MeasurableSpace.generateFrom_le fun t ht => _\n obtain \u27e8u, rfl\u27e9 := ht\n rw [\u2190 compl_Ioi]\n exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr \u27e8u, rfl\u27e9)).compl\n#align borel_eq_generate_from_Iic borel_eq_generateFrom_Iic\n\ntheorem borel_eq_generateFrom_Ici : borel \u03b1 = MeasurableSpace.generateFrom (range Ici) :=\n @borel_eq_generateFrom_Iic \u03b1\u1d52\u1d48 _ _ _ _\n#align borel_eq_generate_from_Ici borel_eq_generateFrom_Ici\n\nend OrderTopology\n\ntheorem borel_comap {f : \u03b1 \u2192 \u03b2} {t : TopologicalSpace \u03b2} :\n @borel \u03b1 (t.induced f) = (@borel \u03b2 t).comap f :=\n comap_generateFrom.symm\n#align borel_comap borel_comap\n\ntheorem Continuous.borel_measurable [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n (hf : Continuous f) : @Measurable \u03b1 \u03b2 (borel \u03b1) (borel \u03b2) f :=\n Measurable.of_le_map <|\n generateFrom_le fun s hs => GenerateMeasurable.basic (f \u207b\u00b9' s) (hs.preimage hf)\n#align continuous.borel_measurable Continuous.borel_measurable\n\n/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that\nall open sets are measurable. -/\nclass OpensMeasurableSpace (\u03b1 : Type*) [TopologicalSpace \u03b1] [h : MeasurableSpace \u03b1] : Prop where\n /-- Borel-measurable sets are measurable. -/\n borel_le : borel \u03b1 \u2264 h\n#align opens_measurable_space OpensMeasurableSpace\n#align opens_measurable_space.borel_le OpensMeasurableSpace.borel_le\n\n/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that\nthe `\u03c3`-algebra of measurable sets is exactly the `\u03c3`-algebra generated by open sets. -/\nclass BorelSpace (\u03b1 : Type*) [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] : Prop where\n /-- The measurable sets are exactly the Borel-measurable sets. -/\n measurable_eq : \u2039MeasurableSpace \u03b1\u203a = borel \u03b1\n#align borel_space BorelSpace\n#align borel_space.measurable_eq BorelSpace.measurable_eq\n\nnamespace Mathlib.Tactic.Borelize\n\nopen Lean Elab Term Tactic Meta\n\n/-- The behaviour of `borelize \u03b1` depends on the existing assumptions on `\u03b1`.\n\n- if `\u03b1` is a topological space with instances `[MeasurableSpace \u03b1] [BorelSpace \u03b1]`, then\n `borelize \u03b1` replaces the former instance by `borel \u03b1`;\n- otherwise, `borelize \u03b1` adds instances `borel \u03b1 : MeasurableSpace \u03b1` and `\u27e8rfl\u27e9 : BorelSpace \u03b1`.\n\nFinally, `borelize \u03b1 \u03b2 \u03b3` runs `borelize \u03b1; borelize \u03b2; borelize \u03b3`.\n-/\nsyntax \"borelize\" (ppSpace colGt term:max)* : tactic\n\n/-- Add instances `borel e : MeasurableSpace e` and `\u27e8rfl\u27e9 : BorelSpace e`. -/\ndef addBorelInstance (e : Expr) : TacticM Unit := do\n let t \u2190 Lean.Elab.Term.exprToSyntax e\n evalTactic <| \u2190 `(tactic|\n refine_lift\n letI : MeasurableSpace $t := borel $t\n haveI : BorelSpace $t := \u27e8rfl\u27e9\n ?_)\n\n/-- Given a type `e`, an assumption `i : MeasurableSpace e`, and an instance `[BorelSpace e]`,\nreplace `i` with `borel e`. -/\ndef borelToRefl (e : Expr) (i : FVarId) : TacticM Unit := do\n let te \u2190 Lean.Elab.Term.exprToSyntax e\n evalTactic <| \u2190 `(tactic|\n have := @BorelSpace.measurable_eq $te _ _ _)\n try\n liftMetaTactic fun m => return [\u2190 subst m i]\n catch _ =>\n let et \u2190 synthInstance (\u2190 mkAppOptM ``TopologicalSpace #[e])\n throwError m!\"\\\n `\u2039TopologicalSpace {e}\u203a := {et}\\n\\\n depends on\\n\\\n {Expr.fvar i} : MeasurableSpace {e}`\\n\\\n so `borelize` isn't avaliable\"\n evalTactic <| \u2190 `(tactic|\n refine_lift\n letI : MeasurableSpace $te := borel $te\n ?_)\n\n/-- Given a type `$t`, if there is an assumption `[i : MeasurableSpace $t]`, then try to prove\n`[BorelSpace $t]` and replace `i` with `borel $t`. Otherwise, add instances\n`borel $t : MeasurableSpace $t` and `\u27e8rfl\u27e9 : BorelSpace $t`. -/\ndef borelize (t : Term) : TacticM Unit := withMainContext <| do\n let u \u2190 mkFreshLevelMVar\n let e \u2190 withoutRecover <| Tactic.elabTermEnsuringType t (mkSort (mkLevelSucc u))\n let i? \u2190 findLocalDeclWithType? (\u2190 mkAppOptM ``MeasurableSpace #[e])\n i?.elim (addBorelInstance e) (borelToRefl e)\n\nelab_rules : tactic\n | `(tactic| borelize $[$t:term]*) => t.forM borelize\n\nend Mathlib.Tactic.Borelize\n\ninstance (priority := 100) OrderDual.opensMeasurableSpace {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [h : OpensMeasurableSpace \u03b1] : OpensMeasurableSpace \u03b1\u1d52\u1d48 where\n borel_le := h.borel_le\n#align order_dual.opens_measurable_space OrderDual.opensMeasurableSpace\n\ninstance (priority := 100) OrderDual.borelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [h : BorelSpace \u03b1] : BorelSpace \u03b1\u1d52\u1d48 where\n measurable_eq := h.measurable_eq\n#align order_dual.borel_space OrderDual.borelSpace\n\n/-- In a `BorelSpace` all open sets are measurable. -/\ninstance (priority := 100) BorelSpace.opensMeasurable {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [BorelSpace \u03b1] : OpensMeasurableSpace \u03b1 :=\n \u27e8ge_of_eq <| BorelSpace.measurable_eq\u27e9\n#align borel_space.opens_measurable BorelSpace.opensMeasurable\n\ninstance Subtype.borelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [MeasurableSpace \u03b1]\n [h\u03b1 : BorelSpace \u03b1] (s : Set \u03b1) : BorelSpace s :=\n \u27e8by borelize \u03b1; symm; apply borel_comap\u27e9\n#align subtype.borel_space Subtype.borelSpace\n\ninstance Countable.instBorelSpace [Countable \u03b1] [MeasurableSpace \u03b1] [MeasurableSingletonClass \u03b1]\n [TopologicalSpace \u03b1] [DiscreteTopology \u03b1] : BorelSpace \u03b1 := by\n have : \u2200 s, @MeasurableSet \u03b1 inferInstance s := fun s \u21a6 s.to_countable.measurableSet\n have : \u2200 s, @MeasurableSet \u03b1 (borel \u03b1) s := fun s \u21a6 measurableSet_generateFrom (isOpen_discrete s)\n exact \u27e8by aesop\u27e9\n\ninstance Subtype.opensMeasurableSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [MeasurableSpace \u03b1]\n [h : OpensMeasurableSpace \u03b1] (s : Set \u03b1) : OpensMeasurableSpace s :=\n \u27e8by\n rw [borel_comap]\n exact comap_mono h.1\u27e9\n#align subtype.opens_measurable_space Subtype.opensMeasurableSpace\n\nlemma opensMeasurableSpace_iff_forall_measurableSet\n [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] :\n OpensMeasurableSpace \u03b1 \u2194 (\u2200 (s : Set \u03b1), IsOpen s \u2192 MeasurableSet s) := by\n refine \u27e8fun h s hs \u21a6 ?_, fun h \u21a6 \u27e8generateFrom_le h\u27e9\u27e9\n exact OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ hs\n\ninstance (priority := 100) BorelSpace.countablyGenerated {\u03b1 : Type*} [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b1] [BorelSpace \u03b1] [SecondCountableTopology \u03b1] : CountablyGenerated \u03b1 := by\n obtain \u27e8b, bct, -, hb\u27e9 := exists_countable_basis \u03b1\n refine' \u27e8\u27e8b, bct, _\u27e9\u27e9\n borelize \u03b1\n exact hb.borel_eq_generateFrom\n#align borel_space.countably_generated BorelSpace.countablyGenerated\n\ntheorem MeasurableSet.induction_on_open [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1]\n {C : Set \u03b1 \u2192 Prop} (h_open : \u2200 U, IsOpen U \u2192 C U)\n (h_compl : \u2200 t, MeasurableSet t \u2192 C t \u2192 C t\u1d9c)\n (h_union :\n \u2200 f : \u2115 \u2192 Set \u03b1,\n Pairwise (Disjoint on f) \u2192 (\u2200 i, MeasurableSet (f i)) \u2192 (\u2200 i, C (f i)) \u2192 C (\u22c3 i, f i)) :\n \u2200 \u2983t\u2984, MeasurableSet t \u2192 C t :=\n MeasurableSpace.induction_on_inter BorelSpace.measurable_eq isPiSystem_isOpen\n (h_open _ isOpen_empty) h_open h_compl h_union\n#align measurable_set.induction_on_open MeasurableSet.induction_on_open\n\nsection\n\nvariable [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [OpensMeasurableSpace \u03b1] [TopologicalSpace \u03b2]\n [MeasurableSpace \u03b2] [OpensMeasurableSpace \u03b2] [TopologicalSpace \u03b3] [MeasurableSpace \u03b3]\n [BorelSpace \u03b3] [TopologicalSpace \u03b3\u2082] [MeasurableSpace \u03b3\u2082] [BorelSpace \u03b3\u2082] [MeasurableSpace \u03b4]\n\ntheorem IsOpen.measurableSet (h : IsOpen s) : MeasurableSet s :=\n OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ h\n#align is_open.measurable_set IsOpen.measurableSet\n\ninstance (priority := 500) {s : Set \u03b1} [HasCountableSeparatingOn \u03b1 IsOpen s] :\n HasCountableSeparatingOn \u03b1 MeasurableSet s :=\n .mono (fun _ \u21a6 IsOpen.measurableSet) Subset.rfl\n\n@[measurability]\ntheorem measurableSet_interior : MeasurableSet (interior s) :=\n isOpen_interior.measurableSet\n#align measurable_set_interior measurableSet_interior\n\ntheorem IsG\u03b4.measurableSet (h : IsG\u03b4 s) : MeasurableSet s := by\n rcases h with \u27e8S, hSo, hSc, rfl\u27e9\n exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet\nset_option linter.uppercaseLean3 false in\n#align is_G\u03b4.measurable_set IsG\u03b4.measurableSet\n\ntheorem measurableSet_of_continuousAt {\u03b2} [EMetricSpace \u03b2] (f : \u03b1 \u2192 \u03b2) :\n MeasurableSet { x | ContinuousAt f x } :=\n (IsG\u03b4.setOf_continuousAt f).measurableSet\n#align measurable_set_of_continuous_at measurableSet_of_continuousAt\n\ntheorem IsClosed.measurableSet (h : IsClosed s) : MeasurableSet s :=\n h.isOpen_compl.measurableSet.of_compl\n#align is_closed.measurable_set IsClosed.measurableSet\n\ntheorem IsCompact.measurableSet [T2Space \u03b1] (h : IsCompact s) : MeasurableSet s :=\n h.isClosed.measurableSet\n#align is_compact.measurable_set IsCompact.measurableSet\n\n/-- If two points are topologically inseparable,\nthen they can't be separated by a Borel measurable set. -/\ntheorem Inseparable.mem_measurableSet_iff {x y : \u03b3} (h : Inseparable x y) {s : Set \u03b3}\n (hs : MeasurableSet s) : x \u2208 s \u2194 y \u2208 s :=\n hs.induction_on_open (C := fun s \u21a6 (x \u2208 s \u2194 y \u2208 s)) (fun _ \u21a6 h.mem_open_iff) (fun s _ hs \u21a6 hs.not)\n fun _ _ _ h \u21a6 by simp [h]\n\n/-- If `K` is a compact set in an R\u2081 space and `s \u2287 K` is a Borel measurable superset,\nthen `s` includes the closure of `K` as well. -/\ntheorem IsCompact.closure_subset_measurableSet [R1Space \u03b3] {K s : Set \u03b3} (hK : IsCompact K)\n (hs : MeasurableSet s) (hKs : K \u2286 s) : closure K \u2286 s := by\n rw [hK.closure_eq_biUnion_inseparable, iUnion\u2082_subset_iff]\n exact fun x hx y hy \u21a6 (hy.mem_measurableSet_iff hs).1 (hKs hx)\n\n/-- In an R\u2081 topological space with Borel measure `\u03bc`,\nthe measure of the closure of a compact set `K` is equal to the measure of `K`.\n\nSee also `MeasureTheory.Measure.OuterRegular.measure_closure_eq_of_isCompact`\nfor a version that assumes `\u03bc` to be outer regular\nbut does not assume the `\u03c3`-algebra to be Borel. -/\ntheorem IsCompact.measure_closure [R1Space \u03b3] {K : Set \u03b3} (hK : IsCompact K) (\u03bc : Measure \u03b3) :\n \u03bc (closure K) = \u03bc K := by\n refine le_antisymm ?_ (measure_mono subset_closure)\n calc\n \u03bc (closure K) \u2264 \u03bc (toMeasurable \u03bc K) := measure_mono <|\n hK.closure_subset_measurableSet (measurableSet_toMeasurable ..) (subset_toMeasurable ..)\n _ = \u03bc K := measure_toMeasurable ..\n\n@[measurability]\ntheorem measurableSet_closure : MeasurableSet (closure s) :=\n isClosed_closure.measurableSet\n#align measurable_set_closure measurableSet_closure\n\ntheorem measurable_of_isOpen {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, IsOpen s \u2192 MeasurableSet (f \u207b\u00b9' s)) :\n Measurable f := by\n rw [\u2039BorelSpace \u03b3\u203a.measurable_eq]\n exact measurable_generateFrom hf\n#align measurable_of_is_open measurable_of_isOpen\n\ntheorem measurable_of_isClosed {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, IsClosed s \u2192 MeasurableSet (f \u207b\u00b9' s)) :\n Measurable f := by\n apply measurable_of_isOpen; intro s hs\n rw [\u2190 MeasurableSet.compl_iff, \u2190 preimage_compl]; apply hf; rw [isClosed_compl_iff]; exact hs\n#align measurable_of_is_closed measurable_of_isClosed\n\ntheorem measurable_of_isClosed' {f : \u03b4 \u2192 \u03b3}\n (hf : \u2200 s, IsClosed s \u2192 s.Nonempty \u2192 s \u2260 univ \u2192 MeasurableSet (f \u207b\u00b9' s)) : Measurable f := by\n apply measurable_of_isClosed; intro s hs\n rcases eq_empty_or_nonempty s with h1 | h1\n \u00b7 simp [h1]\n by_cases h2 : s = univ\n \u00b7 simp [h2]\n exact hf s hs h1 h2\n#align measurable_of_is_closed' measurable_of_isClosed'\n\ninstance nhds_isMeasurablyGenerated (a : \u03b1) : (\ud835\udcdd a).IsMeasurablyGenerated := by\n rw [nhds, iInf_subtype']\n refine' @Filter.iInf_isMeasurablyGenerated \u03b1 _ _ _ fun i => _\n exact i.2.2.measurableSet.principal_isMeasurablyGenerated\n#align nhds_is_measurably_generated nhds_isMeasurablyGenerated\n\n/-- If `s` is a measurable set, then `\ud835\udcdd[s] a` is a measurably generated filter for\neach `a`. This cannot be an `instance` because it depends on a non-instance `hs : MeasurableSet s`.\n-/\ntheorem MeasurableSet.nhdsWithin_isMeasurablyGenerated {s : Set \u03b1} (hs : MeasurableSet s) (a : \u03b1) :\n (\ud835\udcdd[s] a).IsMeasurablyGenerated :=\n haveI := hs.principal_isMeasurablyGenerated\n Filter.inf_isMeasurablyGenerated _ _\n#align measurable_set.nhds_within_is_measurably_generated MeasurableSet.nhdsWithin_isMeasurablyGenerated\n\ninstance (priority := 100) OpensMeasurableSpace.separatesPoints [T0Space \u03b1] :\n SeparatesPoints \u03b1 := by\n rw [separatesPoints_iff]\n intro x y hxy\n apply Inseparable.eq\n rw [inseparable_iff_forall_open]\n exact fun s hs => hxy _ hs.measurableSet\n\n-- see Note [lower instance priority]\ninstance (priority := 100) OpensMeasurableSpace.toMeasurableSingletonClass [T1Space \u03b1] :\n MeasurableSingletonClass \u03b1 :=\n \u27e8fun _ => isClosed_singleton.measurableSet\u27e9\n#align opens_measurable_space.to_measurable_singleton_class OpensMeasurableSpace.toMeasurableSingletonClass\n\ninstance Pi.opensMeasurableSpace {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [Countable \u03b9]\n [t' : \u2200 i, TopologicalSpace (\u03c0 i)] [\u2200 i, MeasurableSpace (\u03c0 i)]\n [\u2200 i, SecondCountableTopology (\u03c0 i)] [\u2200 i, OpensMeasurableSpace (\u03c0 i)] :\n OpensMeasurableSpace (\u2200 i, \u03c0 i) := by\n constructor\n have : Pi.topologicalSpace = .generateFrom { t | \u2203 (s : \u2200 a, Set (\u03c0 a)) (i : Finset \u03b9),\n (\u2200 a \u2208 i, s a \u2208 countableBasis (\u03c0 a)) \u2227 t = pi (\u2191i) s } := by\n rw [funext fun a => @eq_generateFrom_countableBasis (\u03c0 a) _ _, pi_generateFrom_eq]\n rw [borel_eq_generateFrom_of_subbasis this]\n apply generateFrom_le\n rintro _ \u27e8s, i, hi, rfl\u27e9\n refine' MeasurableSet.pi i.countable_toSet fun a ha => IsOpen.measurableSet _\n rw [eq_generateFrom_countableBasis (\u03c0 a)]\n exact .basic _ (hi a ha)\n#align pi.opens_measurable_space Pi.opensMeasurableSpace\n\n/-- The typeclass `SecondCountableTopologyEither \u03b1 \u03b2` registers the fact that at least one of\nthe two spaces has second countable topology. This is the right assumption to ensure that continuous\nmaps from `\u03b1` to `\u03b2` are strongly measurable. -/\nclass SecondCountableTopologyEither (\u03b1 \u03b2 : Type*) [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] :\n Prop where\n /-- The projection out of `SecondCountableTopologyEither` -/\n out : SecondCountableTopology \u03b1 \u2228 SecondCountableTopology \u03b2\n#align second_countable_topology_either SecondCountableTopologyEither\n\ninstance (priority := 100) secondCountableTopologyEither_of_left (\u03b1 \u03b2 : Type*) [TopologicalSpace \u03b1]\n [TopologicalSpace \u03b2] [SecondCountableTopology \u03b1] : SecondCountableTopologyEither \u03b1 \u03b2 where\n out := Or.inl (by infer_instance)\n#align second_countable_topology_either_of_left secondCountableTopologyEither_of_left\n\ninstance (priority := 100) secondCountableTopologyEither_of_right (\u03b1 \u03b2 : Type*)\n [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] [SecondCountableTopology \u03b2] :\n SecondCountableTopologyEither \u03b1 \u03b2 where\n out := Or.inr (by infer_instance)\n#align second_countable_topology_either_of_right secondCountableTopologyEither_of_right\n\n/-- If either `\u03b1` or `\u03b2` has second-countable topology, then the open sets in `\u03b1 \u00d7 \u03b2` belong to the\nproduct sigma-algebra. -/\ninstance Prod.opensMeasurableSpace [h : SecondCountableTopologyEither \u03b1 \u03b2] :\n OpensMeasurableSpace (\u03b1 \u00d7 \u03b2) := by\n apply opensMeasurableSpace_iff_forall_measurableSet.2 (fun s hs \u21a6 ?_)\n rcases h.out with h\u03b1|h\u03b2\n \u00b7 let F : Set \u03b1 \u2192 Set \u03b2 := fun a \u21a6 {y | \u2203 b, IsOpen b \u2227 y \u2208 b \u2227 a \u00d7\u02e2 b \u2286 s}\n have A : \u2200 a, IsOpen (F a) := by\n intro a\n apply isOpen_iff_forall_mem_open.2\n rintro y \u27e8b, b_open, yb, hb\u27e9\n exact \u27e8b, fun z zb \u21a6 \u27e8b, b_open, zb, hb\u27e9, b_open, yb\u27e9\n have : s = \u22c3 a \u2208 countableBasis \u03b1, a \u00d7\u02e2 F a := by\n apply Subset.antisymm\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n rcases isOpen_prod_iff.1 hs y1 y2 hy with \u27e8u, v, u_open, v_open, yu, yv, huv\u27e9\n obtain \u27e8a, ha, ya, au\u27e9 : \u2203 a \u2208 countableBasis \u03b1, y1 \u2208 a \u2227 a \u2286 u :=\n IsTopologicalBasis.exists_subset_of_mem_open (isBasis_countableBasis \u03b1) yu u_open\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop]\n exact \u27e8a, ya, ha, v, v_open, yv, (Set.prod_mono_left au).trans huv\u27e9\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop] at hy\n rcases hy with \u27e8a, ya, -, b, -, yb, hb\u27e9\n exact hb (mem_prod.2 \u27e8ya, yb\u27e9)\n rw [this]\n apply MeasurableSet.biUnion (countable_countableBasis \u03b1) (fun a ha \u21a6 ?_)\n exact (isOpen_of_mem_countableBasis ha).measurableSet.prod (A a).measurableSet\n \u00b7 let F : Set \u03b2 \u2192 Set \u03b1 := fun a \u21a6 {y | \u2203 b, IsOpen b \u2227 y \u2208 b \u2227 b \u00d7\u02e2 a \u2286 s}\n have A : \u2200 a, IsOpen (F a) := by\n intro a\n apply isOpen_iff_forall_mem_open.2\n rintro y \u27e8b, b_open, yb, hb\u27e9\n exact \u27e8b, fun z zb \u21a6 \u27e8b, b_open, zb, hb\u27e9, b_open, yb\u27e9\n have : s = \u22c3 a \u2208 countableBasis \u03b2, F a \u00d7\u02e2 a := by\n apply Subset.antisymm\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n rcases isOpen_prod_iff.1 hs y1 y2 hy with \u27e8u, v, u_open, v_open, yu, yv, huv\u27e9\n obtain \u27e8a, ha, ya, au\u27e9 : \u2203 a \u2208 countableBasis \u03b2, y2 \u2208 a \u2227 a \u2286 v :=\n IsTopologicalBasis.exists_subset_of_mem_open (isBasis_countableBasis \u03b2) yv v_open\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop]\n exact \u27e8a, \u27e8u, u_open, yu, (Set.prod_mono_right au).trans huv\u27e9, ha, ya\u27e9\n \u00b7 rintro \u27e8y1, y2\u27e9 hy\n simp only [mem_iUnion, mem_prod, mem_setOf_eq, exists_and_left, exists_prop] at hy\n rcases hy with \u27e8a, \u27e8b, -, yb, hb\u27e9, -, ya\u27e9\n exact hb (mem_prod.2 \u27e8yb, ya\u27e9)\n rw [this]\n apply MeasurableSet.biUnion (countable_countableBasis \u03b2) (fun a ha \u21a6 ?_)\n exact (A a).measurableSet.prod (isOpen_of_mem_countableBasis ha).measurableSet\n\nvariable {\u03b1' : Type*} [TopologicalSpace \u03b1'] [MeasurableSpace \u03b1']\n\ntheorem interior_ae_eq_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n interior s =\u1d50[\u03bc] s :=\n interior_subset.eventuallyLE.antisymm <| subset_closure.eventuallyLE.trans (ae_le_set.2 h)\n#align interior_ae_eq_of_null_frontier interior_ae_eq_of_null_frontier\n\ntheorem measure_interior_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n \u03bc (interior s) = \u03bc s :=\n measure_congr (interior_ae_eq_of_null_frontier h)\n#align measure_interior_of_null_frontier measure_interior_of_null_frontier\n\ntheorem nullMeasurableSet_of_null_frontier {s : Set \u03b1} {\u03bc : Measure \u03b1} (h : \u03bc (frontier s) = 0) :\n NullMeasurableSet s \u03bc :=\n \u27e8interior s, isOpen_interior.measurableSet, (interior_ae_eq_of_null_frontier h).symm\u27e9\n#align null_measurable_set_of_null_frontier nullMeasurableSet_of_null_frontier\n\ntheorem closure_ae_eq_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n closure s =\u1d50[\u03bc] s :=\n ((ae_le_set.2 h).trans interior_subset.eventuallyLE).antisymm <| subset_closure.eventuallyLE\n#align closure_ae_eq_of_null_frontier closure_ae_eq_of_null_frontier\n\ntheorem measure_closure_of_null_frontier {\u03bc : Measure \u03b1'} {s : Set \u03b1'} (h : \u03bc (frontier s) = 0) :\n \u03bc (closure s) = \u03bc s :=\n measure_congr (closure_ae_eq_of_null_frontier h)\n#align measure_closure_of_null_frontier measure_closure_of_null_frontier\n\nsection Preorder\n\nvariable [Preorder \u03b1] [OrderClosedTopology \u03b1] {a b x : \u03b1}\n\n@[simp, measurability]\ntheorem measurableSet_Ici : MeasurableSet (Ici a) :=\n isClosed_Ici.measurableSet\n#align measurable_set_Ici measurableSet_Ici\n\n@[simp, measurability]\ntheorem measurableSet_Iic : MeasurableSet (Iic a) :=\n isClosed_Iic.measurableSet\n#align measurable_set_Iic measurableSet_Iic\n\n@[simp, measurability]\ntheorem measurableSet_Icc : MeasurableSet (Icc a b) :=\n isClosed_Icc.measurableSet\n#align measurable_set_Icc measurableSet_Icc\n\ninstance nhdsWithin_Ici_isMeasurablyGenerated : (\ud835\udcdd[Ici b] a).IsMeasurablyGenerated :=\n measurableSet_Ici.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Ici_is_measurably_generated nhdsWithin_Ici_isMeasurablyGenerated\n\ninstance nhdsWithin_Iic_isMeasurablyGenerated : (\ud835\udcdd[Iic b] a).IsMeasurablyGenerated :=\n measurableSet_Iic.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Iic_is_measurably_generated nhdsWithin_Iic_isMeasurablyGenerated\n\ninstance nhdsWithin_Icc_isMeasurablyGenerated : IsMeasurablyGenerated (\ud835\udcdd[Icc a b] x) := by\n rw [\u2190 Ici_inter_Iic, nhdsWithin_inter]\n infer_instance\n#align nhds_within_Icc_is_measurably_generated nhdsWithin_Icc_isMeasurablyGenerated\n\ninstance atTop_isMeasurablyGenerated : (Filter.atTop : Filter \u03b1).IsMeasurablyGenerated :=\n @Filter.iInf_isMeasurablyGenerated _ _ _ _ fun a =>\n (measurableSet_Ici : MeasurableSet (Ici a)).principal_isMeasurablyGenerated\n#align at_top_is_measurably_generated atTop_isMeasurablyGenerated\n\ninstance atBot_isMeasurablyGenerated : (Filter.atBot : Filter \u03b1).IsMeasurablyGenerated :=\n @Filter.iInf_isMeasurablyGenerated _ _ _ _ fun a =>\n (measurableSet_Iic : MeasurableSet (Iic a)).principal_isMeasurablyGenerated\n#align at_bot_is_measurably_generated atBot_isMeasurablyGenerated\n\ninstance [R1Space \u03b1] : IsMeasurablyGenerated (cocompact \u03b1) where\n exists_measurable_subset := by\n intro _ hs\n obtain \u27e8t, ht, hts\u27e9 := mem_cocompact.mp hs\n exact \u27e8(closure t)\u1d9c, ht.closure.compl_mem_cocompact, isClosed_closure.measurableSet.compl,\n (compl_subset_compl.2 subset_closure).trans hts\u27e9\n\nend Preorder\n\nsection PartialOrder\n\nvariable [PartialOrder \u03b1] [OrderClosedTopology \u03b1] [SecondCountableTopology \u03b1] {a b : \u03b1}\n\n@[measurability]\ntheorem measurableSet_le' : MeasurableSet { p : \u03b1 \u00d7 \u03b1 | p.1 \u2264 p.2 } :=\n OrderClosedTopology.isClosed_le'.measurableSet\n#align measurable_set_le' measurableSet_le'\n\n@[measurability]\ntheorem measurableSet_le {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n MeasurableSet { a | f a \u2264 g a } :=\n hf.prod_mk hg measurableSet_le'\n#align measurable_set_le measurableSet_le\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] [OrderClosedTopology \u03b1] {a b x : \u03b1}\n\n-- we open this locale only here to avoid issues with list being treated as intervals above\nopen Interval\n\n@[simp, measurability]\ntheorem measurableSet_Iio : MeasurableSet (Iio a) :=\n isOpen_Iio.measurableSet\n#align measurable_set_Iio measurableSet_Iio\n\n@[simp, measurability]\ntheorem measurableSet_Ioi : MeasurableSet (Ioi a) :=\n isOpen_Ioi.measurableSet\n#align measurable_set_Ioi measurableSet_Ioi\n\n@[simp, measurability]\ntheorem measurableSet_Ioo : MeasurableSet (Ioo a b) :=\n isOpen_Ioo.measurableSet\n#align measurable_set_Ioo measurableSet_Ioo\n\n@[simp, measurability]\ntheorem measurableSet_Ioc : MeasurableSet (Ioc a b) :=\n measurableSet_Ioi.inter measurableSet_Iic\n#align measurable_set_Ioc measurableSet_Ioc\n\n@[simp, measurability]\ntheorem measurableSet_Ico : MeasurableSet (Ico a b) :=\n measurableSet_Ici.inter measurableSet_Iio\n#align measurable_set_Ico measurableSet_Ico\n\ninstance nhdsWithin_Ioi_isMeasurablyGenerated : (\ud835\udcdd[Ioi b] a).IsMeasurablyGenerated :=\n measurableSet_Ioi.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Ioi_is_measurably_generated nhdsWithin_Ioi_isMeasurablyGenerated\n\ninstance nhdsWithin_Iio_isMeasurablyGenerated : (\ud835\udcdd[Iio b] a).IsMeasurablyGenerated :=\n measurableSet_Iio.nhdsWithin_isMeasurablyGenerated _\n#align nhds_within_Iio_is_measurably_generated nhdsWithin_Iio_isMeasurablyGenerated\n\ninstance nhdsWithin_uIcc_isMeasurablyGenerated : IsMeasurablyGenerated (\ud835\udcdd[[[a, b]]] x) :=\n nhdsWithin_Icc_isMeasurablyGenerated\n#align nhds_within_uIcc_is_measurably_generated nhdsWithin_uIcc_isMeasurablyGenerated\n\n@[measurability]\ntheorem measurableSet_lt' [SecondCountableTopology \u03b1] : MeasurableSet { p : \u03b1 \u00d7 \u03b1 | p.1 < p.2 } :=\n (isOpen_lt continuous_fst continuous_snd).measurableSet\n#align measurable_set_lt' measurableSet_lt'\n\n@[measurability]\ntheorem measurableSet_lt [SecondCountableTopology \u03b1] {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f)\n (hg : Measurable g) : MeasurableSet { a | f a < g a } :=\n hf.prod_mk hg measurableSet_lt'\n#align measurable_set_lt measurableSet_lt\n\ntheorem nullMeasurableSet_lt [SecondCountableTopology \u03b1] {\u03bc : Measure \u03b4} {f g : \u03b4 \u2192 \u03b1}\n (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) : NullMeasurableSet { a | f a < g a } \u03bc :=\n (hf.prod_mk hg).nullMeasurable measurableSet_lt'\n#align null_measurable_set_lt nullMeasurableSet_lt\n\ntheorem nullMeasurableSet_lt' [SecondCountableTopology \u03b1] {\u03bc : Measure (\u03b1 \u00d7 \u03b1)} :\n NullMeasurableSet { p : \u03b1 \u00d7 \u03b1 | p.1 < p.2 } \u03bc :=\n measurableSet_lt'.nullMeasurableSet\n\ntheorem nullMeasurableSet_le [SecondCountableTopology \u03b1] {\u03bc : Measure \u03b4}\n {f g : \u03b4 \u2192 \u03b1} (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) :\n NullMeasurableSet { a | f a \u2264 g a } \u03bc :=\n (hf.prod_mk hg).nullMeasurable measurableSet_le'\n\ntheorem Set.OrdConnected.measurableSet (h : OrdConnected s) : MeasurableSet s := by\n let u := \u22c3 (x \u2208 s) (y \u2208 s), Ioo x y\n have huopen : IsOpen u := isOpen_biUnion fun _ _ => isOpen_biUnion fun _ _ => isOpen_Ioo\n have humeas : MeasurableSet u := huopen.measurableSet\n have hfinite : (s \\ u).Finite := s.finite_diff_iUnion_Ioo\n have : u \u2286 s := iUnion\u2082_subset fun x hx => iUnion\u2082_subset fun y hy =>\n Ioo_subset_Icc_self.trans (h.out hx hy)\n rw [\u2190 union_diff_cancel this]\n exact humeas.union hfinite.measurableSet\n#align set.ord_connected.measurable_set Set.OrdConnected.measurableSet\n\ntheorem IsPreconnected.measurableSet (h : IsPreconnected s) : MeasurableSet s :=\n h.ordConnected.measurableSet\n#align is_preconnected.measurable_set IsPreconnected.measurableSet\n\ntheorem generateFrom_Ico_mem_le_borel {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderClosedTopology \u03b1] (s t : Set \u03b1) :\n MeasurableSpace.generateFrom { S | \u2203 l \u2208 s, \u2203 u \u2208 t, l < u \u2227 Ico l u = S }\n \u2264 borel \u03b1 := by\n apply generateFrom_le\n borelize \u03b1\n rintro _ \u27e8a, -, b, -, -, rfl\u27e9\n exact measurableSet_Ico\n#align generate_from_Ico_mem_le_borel generateFrom_Ico_mem_le_borel\n\ntheorem Dense.borel_eq_generateFrom_Ico_mem_aux {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] {s : Set \u03b1} (hd : Dense s)\n (hbot : \u2200 x, IsBot x \u2192 x \u2208 s) (hIoo : \u2200 x y : \u03b1, x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S } := by\n set S : Set (Set \u03b1) := { S | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S }\n refine' le_antisymm _ (generateFrom_Ico_mem_le_borel _ _)\n letI : MeasurableSpace \u03b1 := generateFrom S\n rw [borel_eq_generateFrom_Iio]\n refine' generateFrom_le (forall_mem_range.2 fun a => _)\n rcases hd.exists_countable_dense_subset_bot_top with \u27e8t, hts, hc, htd, htb, -\u27e9\n by_cases ha : \u2200 b < a, (Ioo b a).Nonempty\n \u00b7 convert_to MeasurableSet (\u22c3 (l \u2208 t) (u \u2208 t) (_ : l < u) (_ : u \u2264 a), Ico l u)\n \u00b7 ext y\n simp only [mem_iUnion, mem_Iio, mem_Ico]\n constructor\n \u00b7 intro hy\n rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) y with \u27e8l, hlt, hly\u27e9\n rcases htd.exists_mem_open isOpen_Ioo (ha y hy) with \u27e8u, hut, hyu, hua\u27e9\n exact \u27e8l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu\u27e9\n \u00b7 rintro \u27e8l, -, u, -, -, hua, -, hyu\u27e9\n exact hyu.trans_le hua\n \u00b7 refine' MeasurableSet.biUnion hc fun a ha => MeasurableSet.biUnion hc fun b hb => _\n refine' MeasurableSet.iUnion fun hab => MeasurableSet.iUnion fun _ => _\n exact .basic _ \u27e8a, hts ha, b, hts hb, hab, mem_singleton _\u27e9\n \u00b7 simp only [not_forall, not_nonempty_iff_eq_empty] at ha\n replace ha : a \u2208 s := hIoo ha.choose a ha.choose_spec.fst ha.choose_spec.snd\n convert_to MeasurableSet (\u22c3 (l \u2208 t) (_ : l < a), Ico l a)\n \u00b7 symm\n simp only [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, inter_eq_right, subset_def, mem_iUnion,\n mem_Ici, mem_Iio]\n intro x hx\n rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) x with \u27e8z, hzt, hzx\u27e9\n exact \u27e8z, hzt, hzx.trans_lt hx, hzx\u27e9\n \u00b7 refine' .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => _\n exact .basic _ \u27e8x, hts hx, a, ha, hlt, mem_singleton _\u27e9\n#align dense.borel_eq_generate_from_Ico_mem_aux Dense.borel_eq_generateFrom_Ico_mem_aux\n\ntheorem Dense.borel_eq_generateFrom_Ico_mem {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] [DenselyOrdered \u03b1] [NoMinOrder \u03b1] {s : Set \u03b1}\n (hd : Dense s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ico l u = S } :=\n hd.borel_eq_generateFrom_Ico_mem_aux (by simp) fun x y hxy H =>\n ((nonempty_Ioo.2 hxy).ne_empty H).elim\n#align dense.borel_eq_generate_from_Ico_mem Dense.borel_eq_generateFrom_Ico_mem\n\ntheorem borel_eq_generateFrom_Ico (\u03b1 : Type*) [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1]\n [LinearOrder \u03b1] [OrderTopology \u03b1] :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 (l u : \u03b1), l < u \u2227 Ico l u = S } := by\n simpa only [exists_prop, mem_univ, true_and_iff] using\n (@dense_univ \u03b1 _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n mem_univ _\n#align borel_eq_generate_from_Ico borel_eq_generateFrom_Ico\n\ntheorem Dense.borel_eq_generateFrom_Ioc_mem_aux {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] {s : Set \u03b1} (hd : Dense s)\n (hbot : \u2200 x, IsTop x \u2192 x \u2208 s) (hIoo : \u2200 x y : \u03b1, x < y \u2192 Ioo x y = \u2205 \u2192 x \u2208 s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ioc l u = S } := by\n convert hd.orderDual.borel_eq_generateFrom_Ico_mem_aux hbot fun x y hlt he => hIoo y x hlt _\n using 2\n \u00b7 ext s\n constructor <;> rintro \u27e8l, hl, u, hu, hlt, rfl\u27e9\n exacts [\u27e8u, hu, l, hl, hlt, dual_Ico\u27e9, \u27e8u, hu, l, hl, hlt, dual_Ioc\u27e9]\n \u00b7 erw [dual_Ioo]\n exact he\n#align dense.borel_eq_generate_from_Ioc_mem_aux Dense.borel_eq_generateFrom_Ioc_mem_aux\n\ntheorem Dense.borel_eq_generateFrom_Ioc_mem {\u03b1 : Type*} [TopologicalSpace \u03b1] [LinearOrder \u03b1]\n [OrderTopology \u03b1] [SecondCountableTopology \u03b1] [DenselyOrdered \u03b1] [NoMaxOrder \u03b1] {s : Set \u03b1}\n (hd : Dense s) :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l \u2208 s, \u2203 u \u2208 s, l < u \u2227 Ioc l u = S } :=\n hd.borel_eq_generateFrom_Ioc_mem_aux (by simp) fun x y hxy H =>\n ((nonempty_Ioo.2 hxy).ne_empty H).elim\n#align dense.borel_eq_generate_from_Ioc_mem Dense.borel_eq_generateFrom_Ioc_mem\n\ntheorem borel_eq_generateFrom_Ioc (\u03b1 : Type*) [TopologicalSpace \u03b1] [SecondCountableTopology \u03b1]\n [LinearOrder \u03b1] [OrderTopology \u03b1] :\n borel \u03b1 = .generateFrom { S : Set \u03b1 | \u2203 l u, l < u \u2227 Ioc l u = S } := by\n simpa only [exists_prop, mem_univ, true_and_iff] using\n (@dense_univ \u03b1 _).borel_eq_generateFrom_Ioc_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>\n mem_univ _\n#align borel_eq_generate_from_Ioc borel_eq_generateFrom_Ioc\n\nnamespace MeasureTheory.Measure\n\n/-- Two finite measures on a Borel space are equal if they agree on all closed-open intervals. If\n`\u03b1` is a conditionally complete linear order with no top element,\n`MeasureTheory.Measure.ext_of_Ico` is an extensionality lemma with weaker assumptions on `\u03bc` and\n`\u03bd`. -/\ntheorem ext_of_Ico_finite {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h\u03bc\u03bd : \u03bc univ = \u03bd univ) (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) :\n \u03bc = \u03bd := by\n refine'\n ext_of_generate_finite _ (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico \u03b1))\n (isPiSystem_Ico (id : \u03b1 \u2192 \u03b1) id) _ h\u03bc\u03bd\n \u00b7 rintro - \u27e8a, b, hlt, rfl\u27e9\n exact h hlt\n#align measure_theory.measure.ext_of_Ico_finite MeasureTheory.Measure.ext_of_Ico_finite\n\n/-- Two finite measures on a Borel space are equal if they agree on all open-closed intervals. If\n`\u03b1` is a conditionally complete linear order with no top element,\n`MeasureTheory.Measure.ext_of_Ioc` is an extensionality lemma with weaker assumptions on `\u03bc` and\n`\u03bd`. -/\ntheorem ext_of_Ioc_finite {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h\u03bc\u03bd : \u03bc univ = \u03bd univ) (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) :\n \u03bc = \u03bd := by\n refine' @ext_of_Ico_finite \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a \u03bc \u03bd _ h\u03bc\u03bd fun a b hab => _\n erw [dual_Ico (\u03b1 := \u03b1)]\n exact h hab\n#align measure_theory.measure.ext_of_Ioc_finite MeasureTheory.Measure.ext_of_Ioc_finite\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nclosed-open intervals. -/\ntheorem ext_of_Ico' {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] [NoMaxOrder \u03b1]\n (\u03bc \u03bd : Measure \u03b1) (h\u03bc : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) \u2260 \u221e)\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) : \u03bc = \u03bd := by\n rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, hsb, _\u27e9\n have : (\u22c3 (l \u2208 s) (u \u2208 s) (_ : l < u), {Ico l u} : Set (Set \u03b1)).Countable :=\n hsc.biUnion fun l _ => hsc.biUnion fun u _ => countable_iUnion fun _ => countable_singleton _\n simp only [\u2190 setOf_eq_eq_singleton, \u2190 setOf_exists] at this\n refine'\n Measure.ext_of_generateFrom_of_cover_subset\n (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico \u03b1)) (isPiSystem_Ico id id) _ this\n _ _ _\n \u00b7 rintro _ \u27e8l, -, u, -, h, rfl\u27e9\n exact \u27e8l, u, h, rfl\u27e9\n \u00b7 refine' sUnion_eq_univ_iff.2 fun x => _\n rcases hsd.exists_le' hsb x with \u27e8l, hls, hlx\u27e9\n rcases hsd.exists_gt x with \u27e8u, hus, hxu\u27e9\n exact \u27e8_, \u27e8l, hls, u, hus, hlx.trans_lt hxu, rfl\u27e9, hlx, hxu\u27e9\n \u00b7 rintro _ \u27e8l, -, u, -, hlt, rfl\u27e9\n exact h\u03bc hlt\n \u00b7 rintro _ \u27e8l, u, hlt, rfl\u27e9\n exact h hlt\n#align measure_theory.measure.ext_of_Ico' MeasureTheory.Measure.ext_of_Ico'\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nopen-closed intervals. -/\ntheorem ext_of_Ioc' {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] [NoMinOrder \u03b1]\n (\u03bc \u03bd : Measure \u03b1) (h\u03bc : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) \u2260 \u221e)\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) : \u03bc = \u03bd := by\n refine' @ext_of_Ico' \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a _ \u03bc \u03bd _ _ <;> intro a b hab <;> erw [dual_Ico (\u03b1 := \u03b1)]\n exacts [h\u03bc hab, h hab]\n#align measure_theory.measure.ext_of_Ioc' MeasureTheory.Measure.ext_of_Ioc'\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nclosed-open intervals. -/\ntheorem ext_of_Ico {\u03b1 : Type*} [TopologicalSpace \u03b1] {_m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [ConditionallyCompleteLinearOrder \u03b1] [OrderTopology \u03b1]\n [BorelSpace \u03b1] [NoMaxOrder \u03b1] (\u03bc \u03bd : Measure \u03b1) [IsLocallyFiniteMeasure \u03bc]\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) : \u03bc = \u03bd :=\n \u03bc.ext_of_Ico' \u03bd (fun _ _ _ => measure_Ico_lt_top.ne) h\n#align measure_theory.measure.ext_of_Ico MeasureTheory.Measure.ext_of_Ico\n\n/-- Two measures which are finite on closed-open intervals are equal if they agree on all\nopen-closed intervals. -/\ntheorem ext_of_Ioc {\u03b1 : Type*} [TopologicalSpace \u03b1] {_m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [ConditionallyCompleteLinearOrder \u03b1] [OrderTopology \u03b1]\n [BorelSpace \u03b1] [NoMinOrder \u03b1] (\u03bc \u03bd : Measure \u03b1) [IsLocallyFiniteMeasure \u03bc]\n (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) : \u03bc = \u03bd :=\n \u03bc.ext_of_Ioc' \u03bd (fun _ _ _ => measure_Ioc_lt_top.ne) h\n#align measure_theory.measure.ext_of_Ioc MeasureTheory.Measure.ext_of_Ioc\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-infinite right-closed\nintervals. -/\ntheorem ext_of_Iic {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h : \u2200 a, \u03bc (Iic a) = \u03bd (Iic a)) : \u03bc = \u03bd := by\n refine' ext_of_Ioc_finite \u03bc \u03bd _ fun a b hlt => _\n \u00b7 rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, -, hst\u27e9\n have : DirectedOn (\u00b7 \u2264 \u00b7) s := directedOn_iff_directed.2 (Subtype.mono_coe _).directed_le\n simp only [\u2190 biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h]\n rw [\u2190 Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic,\n measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic, h a, h b]\n \u00b7 rw [\u2190 h a]\n exact (measure_lt_top \u03bc _).ne\n \u00b7 exact (measure_lt_top \u03bc _).ne\n#align measure_theory.measure.ext_of_Iic MeasureTheory.Measure.ext_of_Iic\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-closed right-infinite\nintervals. -/\ntheorem ext_of_Ici {\u03b1 : Type*} [TopologicalSpace \u03b1] {m : MeasurableSpace \u03b1}\n [SecondCountableTopology \u03b1] [LinearOrder \u03b1] [OrderTopology \u03b1] [BorelSpace \u03b1] (\u03bc \u03bd : Measure \u03b1)\n [IsFiniteMeasure \u03bc] (h : \u2200 a, \u03bc (Ici a) = \u03bd (Ici a)) : \u03bc = \u03bd :=\n @ext_of_Iic \u03b1\u1d52\u1d48 _ _ _ _ _ \u2039_\u203a _ _ _ h\n#align measure_theory.measure.ext_of_Ici MeasureTheory.Measure.ext_of_Ici\n\nend MeasureTheory.Measure\n\nend LinearOrder\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] [OrderClosedTopology \u03b1] {a b : \u03b1}\n\n@[measurability]\ntheorem measurableSet_uIcc : MeasurableSet (uIcc a b) :=\n measurableSet_Icc\n#align measurable_set_uIcc measurableSet_uIcc\n\n@[measurability]\ntheorem measurableSet_uIoc : MeasurableSet (uIoc a b) :=\n measurableSet_Ioc\n#align measurable_set_uIoc measurableSet_uIoc\n\nvariable [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem Measurable.max {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun a => max (f a) (g a) := by\n simpa only [max_def'] using hf.piecewise (measurableSet_le hg hf) hg\n#align measurable.max Measurable.max\n\n@[measurability]\nnonrec theorem AEMeasurable.max {f g : \u03b4 \u2192 \u03b1} {\u03bc : Measure \u03b4} (hf : AEMeasurable f \u03bc)\n (hg : AEMeasurable g \u03bc) : AEMeasurable (fun a => max (f a) (g a)) \u03bc :=\n \u27e8fun a => max (hf.mk f a) (hg.mk g a), hf.measurable_mk.max hg.measurable_mk,\n EventuallyEq.comp\u2082 hf.ae_eq_mk _ hg.ae_eq_mk\u27e9\n#align ae_measurable.max AEMeasurable.max\n\n@[measurability]\ntheorem Measurable.min {f g : \u03b4 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun a => min (f a) (g a) := by\n simpa only [min_def] using hf.piecewise (measurableSet_le hf hg) hg\n#align measurable.min Measurable.min\n\n@[measurability]\nnonrec theorem AEMeasurable.min {f g : \u03b4 \u2192 \u03b1} {\u03bc : Measure \u03b4} (hf : AEMeasurable f \u03bc)\n (hg : AEMeasurable g \u03bc) : AEMeasurable (fun a => min (f a) (g a)) \u03bc :=\n \u27e8fun a => min (hf.mk f a) (hg.mk g a), hf.measurable_mk.min hg.measurable_mk,\n EventuallyEq.comp\u2082 hf.ae_eq_mk _ hg.ae_eq_mk\u27e9\n#align ae_measurable.min AEMeasurable.min\n\nend LinearOrder\n\n/-- A continuous function from an `OpensMeasurableSpace` to a `BorelSpace`\nis measurable. -/\ntheorem Continuous.measurable {f : \u03b1 \u2192 \u03b3} (hf : Continuous f) : Measurable f :=\n hf.borel_measurable.mono OpensMeasurableSpace.borel_le (le_of_eq <| BorelSpace.measurable_eq)\n#align continuous.measurable Continuous.measurable\n\n/-- A continuous function from an `OpensMeasurableSpace` to a `BorelSpace`\nis ae-measurable. -/\ntheorem Continuous.aemeasurable {f : \u03b1 \u2192 \u03b3} (h : Continuous f) {\u03bc : Measure \u03b1} : AEMeasurable f \u03bc :=\n h.measurable.aemeasurable\n#align continuous.ae_measurable Continuous.aemeasurable\n\ntheorem ClosedEmbedding.measurable {f : \u03b1 \u2192 \u03b3} (hf : ClosedEmbedding f) : Measurable f :=\n hf.continuous.measurable\n#align closed_embedding.measurable ClosedEmbedding.measurable\n\n/-- If a function is defined piecewise in terms of functions which are continuous on their\nrespective pieces, then it is measurable. -/\ntheorem ContinuousOn.measurable_piecewise {f g : \u03b1 \u2192 \u03b3} {s : Set \u03b1} [\u2200 j : \u03b1, Decidable (j \u2208 s)]\n (hf : ContinuousOn f s) (hg : ContinuousOn g s\u1d9c) (hs : MeasurableSet s) :\n Measurable (s.piecewise f g) := by\n refine' measurable_of_isOpen fun t ht => _\n rw [piecewise_preimage, Set.ite]\n apply MeasurableSet.union\n \u00b7 rcases _root_.continuousOn_iff'.1 hf t ht with \u27e8u, u_open, hu\u27e9\n rw [hu]\n exact u_open.measurableSet.inter hs\n \u00b7 rcases _root_.continuousOn_iff'.1 hg t ht with \u27e8u, u_open, hu\u27e9\n rw [diff_eq_compl_inter, inter_comm, hu]\n exact u_open.measurableSet.inter hs.compl\n#align continuous_on.measurable_piecewise ContinuousOn.measurable_piecewise\n\n@[to_additive]\ninstance (priority := 100) ContinuousMul.measurableMul [Mul \u03b3] [ContinuousMul \u03b3] :\n MeasurableMul \u03b3 where\n measurable_const_mul _ := (continuous_const.mul continuous_id).measurable\n measurable_mul_const _ := (continuous_id.mul continuous_const).measurable\n#align has_continuous_mul.has_measurable_mul ContinuousMul.measurableMul\n#align has_continuous_add.has_measurable_add ContinuousAdd.measurableAdd\n\ninstance (priority := 100) ContinuousSub.measurableSub [Sub \u03b3] [ContinuousSub \u03b3] :\n MeasurableSub \u03b3 where\n measurable_const_sub _ := (continuous_const.sub continuous_id).measurable\n measurable_sub_const _ := (continuous_id.sub continuous_const).measurable\n#align has_continuous_sub.has_measurable_sub ContinuousSub.measurableSub\n\n@[to_additive]\ninstance (priority := 100) TopologicalGroup.measurableInv [Group \u03b3] [TopologicalGroup \u03b3] :\n MeasurableInv \u03b3 :=\n \u27e8continuous_inv.measurable\u27e9\n#align topological_group.has_measurable_inv TopologicalGroup.measurableInv\n#align topological_add_group.has_measurable_neg TopologicalAddGroup.measurableNeg\n\ninstance (priority := 100) ContinuousSMul.measurableSMul {M \u03b1} [TopologicalSpace M]\n [TopologicalSpace \u03b1] [MeasurableSpace M] [MeasurableSpace \u03b1] [OpensMeasurableSpace M]\n [BorelSpace \u03b1] [SMul M \u03b1] [ContinuousSMul M \u03b1] : MeasurableSMul M \u03b1 :=\n \u27e8fun _ => (continuous_const_smul _).measurable, fun _ =>\n (continuous_id.smul continuous_const).measurable\u27e9\n#align has_continuous_smul.has_measurable_smul ContinuousSMul.measurableSMul\n\nsection Lattice\n\ninstance (priority := 100) ContinuousSup.measurableSup [Sup \u03b3] [ContinuousSup \u03b3] :\n MeasurableSup \u03b3 where\n measurable_const_sup _ := (continuous_const.sup continuous_id).measurable\n measurable_sup_const _ := (continuous_id.sup continuous_const).measurable\n#align has_continuous_sup.has_measurable_sup ContinuousSup.measurableSup\n\ninstance (priority := 100) ContinuousSup.measurableSup\u2082 [SecondCountableTopology \u03b3] [Sup \u03b3]\n [ContinuousSup \u03b3] : MeasurableSup\u2082 \u03b3 :=\n \u27e8continuous_sup.measurable\u27e9\n#align has_continuous_sup.has_measurable_sup\u2082 ContinuousSup.measurableSup\u2082\n\ninstance (priority := 100) ContinuousInf.measurableInf [Inf \u03b3] [ContinuousInf \u03b3] :\n MeasurableInf \u03b3 where\n measurable_const_inf _ := (continuous_const.inf continuous_id).measurable\n measurable_inf_const _ := (continuous_id.inf continuous_const).measurable\n#align has_continuous_inf.has_measurable_inf ContinuousInf.measurableInf\n\ninstance (priority := 100) ContinuousInf.measurableInf\u2082 [SecondCountableTopology \u03b3] [Inf \u03b3]\n [ContinuousInf \u03b3] : MeasurableInf\u2082 \u03b3 :=\n \u27e8continuous_inf.measurable\u27e9\n#align has_continuous_inf.has_measurable_inf\u2082 ContinuousInf.measurableInf\u2082\n\nend Lattice\n\nsection Homeomorph\n\n@[measurability]\nprotected theorem Homeomorph.measurable (h : \u03b1 \u2243\u209c \u03b3) : Measurable h :=\n h.continuous.measurable\n#align homeomorph.measurable Homeomorph.measurable\n\n/-- A homeomorphism between two Borel spaces is a measurable equivalence. -/\ndef Homeomorph.toMeasurableEquiv (h : \u03b3 \u2243\u209c \u03b3\u2082) : \u03b3 \u2243\u1d50 \u03b3\u2082 where\n measurable_toFun := h.measurable\n measurable_invFun := h.symm.measurable\n toEquiv := h.toEquiv\n#align homeomorph.to_measurable_equiv Homeomorph.toMeasurableEquiv\n\nlemma Homeomorph.measurableEmbedding (h : \u03b3 \u2243\u209c \u03b3\u2082) : MeasurableEmbedding h :=\n h.toMeasurableEquiv.measurableEmbedding\n\n@[simp]\ntheorem Homeomorph.toMeasurableEquiv_coe (h : \u03b3 \u2243\u209c \u03b3\u2082) : (h.toMeasurableEquiv : \u03b3 \u2192 \u03b3\u2082) = h :=\n rfl\n#align homeomorph.to_measurable_equiv_coe Homeomorph.toMeasurableEquiv_coe\n\n@[simp]\ntheorem Homeomorph.toMeasurableEquiv_symm_coe (h : \u03b3 \u2243\u209c \u03b3\u2082) :\n (h.toMeasurableEquiv.symm : \u03b3\u2082 \u2192 \u03b3) = h.symm :=\n rfl\n#align homeomorph.to_measurable_equiv_symm_coe Homeomorph.toMeasurableEquiv_symm_coe\n\nend Homeomorph\n\n@[measurability]\ntheorem ContinuousMap.measurable (f : C(\u03b1, \u03b3)) : Measurable f :=\n f.continuous.measurable\n#align continuous_map.measurable ContinuousMap.measurable\n\ntheorem measurable_of_continuousOn_compl_singleton [T1Space \u03b1] {f : \u03b1 \u2192 \u03b3} (a : \u03b1)\n (hf : ContinuousOn f {a}\u1d9c) : Measurable f :=\n measurable_of_measurable_on_compl_singleton a\n (continuousOn_iff_continuous_restrict.1 hf).measurable\n#align measurable_of_continuous_on_compl_singleton measurable_of_continuousOn_compl_singleton\n\ntheorem Continuous.measurable2 [SecondCountableTopologyEither \u03b1 \u03b2] {f : \u03b4 \u2192 \u03b1}\n {g : \u03b4 \u2192 \u03b2} {c : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (h : Continuous fun p : \u03b1 \u00d7 \u03b2 => c p.1 p.2) (hf : Measurable f)\n (hg : Measurable g) : Measurable fun a => c (f a) (g a) :=\n h.measurable.comp (hf.prod_mk hg)\n#align continuous.measurable2 Continuous.measurable2\n\ntheorem Continuous.aemeasurable2 [SecondCountableTopologyEither \u03b1 \u03b2]\n {f : \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b2} {c : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {\u03bc : Measure \u03b4}\n (h : Continuous fun p : \u03b1 \u00d7 \u03b2 => c p.1 p.2) (hf : AEMeasurable f \u03bc) (hg : AEMeasurable g \u03bc) :\n AEMeasurable (fun a => c (f a) (g a)) \u03bc :=\n h.measurable.comp_aemeasurable (hf.prod_mk hg)\n#align continuous.ae_measurable2 Continuous.aemeasurable2\n\ninstance (priority := 100) HasContinuousInv\u2080.measurableInv [GroupWithZero \u03b3] [T1Space \u03b3]\n [HasContinuousInv\u2080 \u03b3] : MeasurableInv \u03b3 :=\n \u27e8measurable_of_continuousOn_compl_singleton 0 continuousOn_inv\u2080\u27e9\n#align has_continuous_inv\u2080.has_measurable_inv HasContinuousInv\u2080.measurableInv\n\n@[to_additive]\ninstance (priority := 100) ContinuousMul.measurableMul\u2082 [SecondCountableTopology \u03b3] [Mul \u03b3]\n [ContinuousMul \u03b3] : MeasurableMul\u2082 \u03b3 :=\n \u27e8continuous_mul.measurable\u27e9\n#align has_continuous_mul.has_measurable_mul\u2082 ContinuousMul.measurableMul\u2082\n#align has_continuous_add.has_measurable_mul\u2082 ContinuousAdd.measurableMul\u2082\n\ninstance (priority := 100) ContinuousSub.measurableSub\u2082 [SecondCountableTopology \u03b3] [Sub \u03b3]\n [ContinuousSub \u03b3] : MeasurableSub\u2082 \u03b3 :=\n \u27e8continuous_sub.measurable\u27e9\n#align has_continuous_sub.has_measurable_sub\u2082 ContinuousSub.measurableSub\u2082\n\ninstance (priority := 100) ContinuousSMul.measurableSMul\u2082 {M \u03b1} [TopologicalSpace M]\n [MeasurableSpace M] [OpensMeasurableSpace M] [TopologicalSpace \u03b1]\n [SecondCountableTopologyEither M \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1] [SMul M \u03b1]\n [ContinuousSMul M \u03b1] : MeasurableSMul\u2082 M \u03b1 :=\n \u27e8continuous_smul.measurable\u27e9\n#align has_continuous_smul.has_measurable_smul\u2082 ContinuousSMul.measurableSMul\u2082\n\nend\n\nsection BorelSpace\n\nvariable [TopologicalSpace \u03b1] [MeasurableSpace \u03b1] [BorelSpace \u03b1] [TopologicalSpace \u03b2]\n [MeasurableSpace \u03b2] [BorelSpace \u03b2] [TopologicalSpace \u03b3] [MeasurableSpace \u03b3] [BorelSpace \u03b3]\n [MeasurableSpace \u03b4]\n\ntheorem pi_le_borel_pi {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [\u2200 i, TopologicalSpace (\u03c0 i)]\n [\u2200 i, MeasurableSpace (\u03c0 i)] [\u2200 i, BorelSpace (\u03c0 i)] :\n MeasurableSpace.pi \u2264 borel (\u2200 i, \u03c0 i) := by\n have : \u2039\u2200 i, MeasurableSpace (\u03c0 i)\u203a = fun i => borel (\u03c0 i) :=\n funext fun i => BorelSpace.measurable_eq\n rw [this]\n exact iSup_le fun i => comap_le_iff_le_map.2 <| (continuous_apply i).borel_measurable\n#align pi_le_borel_pi pi_le_borel_pi\n\ntheorem prod_le_borel_prod : Prod.instMeasurableSpace \u2264 borel (\u03b1 \u00d7 \u03b2) := by\n rw [\u2039BorelSpace \u03b1\u203a.measurable_eq, \u2039BorelSpace \u03b2\u203a.measurable_eq]\n refine' sup_le _ _\n \u00b7 exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable\n \u00b7 exact comap_le_iff_le_map.mpr continuous_snd.borel_measurable\n#align prod_le_borel_prod prod_le_borel_prod\n\ninstance Pi.borelSpace {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [Countable \u03b9] [\u2200 i, TopologicalSpace (\u03c0 i)]\n [\u2200 i, MeasurableSpace (\u03c0 i)] [\u2200 i, SecondCountableTopology (\u03c0 i)] [\u2200 i, BorelSpace (\u03c0 i)] :\n BorelSpace (\u2200 i, \u03c0 i) :=\n \u27e8le_antisymm pi_le_borel_pi OpensMeasurableSpace.borel_le\u27e9\n#align pi.borel_space Pi.borelSpace\n\ninstance Prod.borelSpace [SecondCountableTopologyEither \u03b1 \u03b2] :\n BorelSpace (\u03b1 \u00d7 \u03b2) :=\n \u27e8le_antisymm prod_le_borel_prod OpensMeasurableSpace.borel_le\u27e9\n#align prod.borel_space Prod.borelSpace\n\n/-- Given a measurable embedding to a Borel space which is also a topological embedding, then the\nsource space is also a Borel space. -/\nlemma MeasurableEmbedding.borelSpace {\u03b1 \u03b2 : Type*} [MeasurableSpace \u03b1] [TopologicalSpace \u03b1]\n [MeasurableSpace \u03b2] [TopologicalSpace \u03b2] [h\u03b2 : BorelSpace \u03b2] {e : \u03b1 \u2192 \u03b2}\n (h'e : MeasurableEmbedding e) (h''e : Inducing e) :\n BorelSpace \u03b1 := by\n constructor\n have : MeasurableSpace.comap e (borel \u03b2) = \u2039_\u203a := by simpa [h\u03b2.measurable_eq] using h'e.comap_eq\n rw [\u2190 this, \u2190 borel_comap, h''e.induced]\n\ninstance _root_.ULift.instBorelSpace : BorelSpace (ULift \u03b1) :=\n MeasurableEquiv.ulift.measurableEmbedding.borelSpace Homeomorph.ulift.inducing\n\ninstance DiscreteMeasurableSpace.toBorelSpace {\u03b1 : Type*} [TopologicalSpace \u03b1] [DiscreteTopology \u03b1]\n [MeasurableSpace \u03b1] [DiscreteMeasurableSpace \u03b1] : BorelSpace \u03b1 := by\n constructor; ext; simp [MeasurableSpace.measurableSet_generateFrom, measurableSet_discrete]\n\nprotected theorem Embedding.measurableEmbedding {f : \u03b1 \u2192 \u03b2} (h\u2081 : Embedding f)\n (h\u2082 : MeasurableSet (range f)) : MeasurableEmbedding f :=\n show MeasurableEmbedding\n (((\u2191) : range f \u2192 \u03b2) \u2218 (Homeomorph.ofEmbedding f h\u2081).toMeasurableEquiv) from\n (MeasurableEmbedding.subtype_coe h\u2082).comp (MeasurableEquiv.measurableEmbedding _)\n#align embedding.measurable_embedding Embedding.measurableEmbedding\n\nprotected theorem ClosedEmbedding.measurableEmbedding {f : \u03b1 \u2192 \u03b2} (h : ClosedEmbedding f) :\n MeasurableEmbedding f :=\n h.toEmbedding.measurableEmbedding h.isClosed_range.measurableSet\n#align closed_embedding.measurable_embedding ClosedEmbedding.measurableEmbedding\n\nprotected theorem OpenEmbedding.measurableEmbedding {f : \u03b1 \u2192 \u03b2} (h : OpenEmbedding f) :\n MeasurableEmbedding f :=\n h.toEmbedding.measurableEmbedding h.isOpen_range.measurableSet\n#align open_embedding.measurable_embedding OpenEmbedding.measurableEmbedding\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] [OrderTopology \u03b1] [SecondCountableTopology \u03b1]\n\ntheorem measurable_of_Iio {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Iio x)) : Measurable f := by\n convert measurable_generateFrom (\u03b1 := \u03b4) _\n \u00b7 exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Iio _)\n \u00b7 rintro _ \u27e8x, rfl\u27e9; exact hf x\n#align measurable_of_Iio measurable_of_Iio\n\ntheorem UpperSemicontinuous.measurable [TopologicalSpace \u03b4] [OpensMeasurableSpace \u03b4] {f : \u03b4 \u2192 \u03b1}\n (hf : UpperSemicontinuous f) : Measurable f :=\n measurable_of_Iio fun y => (hf.isOpen_preimage y).measurableSet\n#align upper_semicontinuous.measurable UpperSemicontinuous.measurable\n\ntheorem measurable_of_Ioi {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Ioi x)) : Measurable f := by\n convert measurable_generateFrom (\u03b1 := \u03b4) _\n \u00b7 exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioi _)\n \u00b7 rintro _ \u27e8x, rfl\u27e9; exact hf x\n#align measurable_of_Ioi measurable_of_Ioi\n\ntheorem LowerSemicontinuous.measurable [TopologicalSpace \u03b4] [OpensMeasurableSpace \u03b4] {f : \u03b4 \u2192 \u03b1}\n (hf : LowerSemicontinuous f) : Measurable f :=\n measurable_of_Ioi fun y => (hf.isOpen_preimage y).measurableSet\n#align lower_semicontinuous.measurable LowerSemicontinuous.measurable\n\ntheorem measurable_of_Iic {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Iic x)) : Measurable f := by\n apply measurable_of_Ioi\n simp_rw [\u2190 compl_Iic, preimage_compl, MeasurableSet.compl_iff]\n assumption\n#align measurable_of_Iic measurable_of_Iic\n\ntheorem measurable_of_Ici {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, MeasurableSet (f \u207b\u00b9' Ici x)) : Measurable f := by\n apply measurable_of_Iio\n simp_rw [\u2190 compl_Ici, preimage_compl, MeasurableSet.compl_iff]\n assumption\n#align measurable_of_Ici measurable_of_Ici\n\n/-- If a function is the least upper bound of countably many measurable functions,\nthen it is measurable. -/\ntheorem Measurable.isLUB {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i))\n (hg : \u2200 b, IsLUB { a | \u2203 i, f i b = a } (g b)) : Measurable g := by\n change \u2200 b, IsLUB (range fun i => f i b) (g b) at hg\n rw [\u2039BorelSpace \u03b1\u203a.measurable_eq, borel_eq_generateFrom_Ioi \u03b1]\n apply measurable_generateFrom\n rintro _ \u27e8a, rfl\u27e9\n simp_rw [Set.preimage, mem_Ioi, lt_isLUB_iff (hg _), exists_range_iff, setOf_exists]\n exact MeasurableSet.iUnion fun i => hf i (isOpen_lt' _).measurableSet\n#align measurable.is_lub Measurable.isLUB\n\n/-- If a function is the least upper bound of countably many measurable functions on a measurable\nset `s`, and coincides with a measurable function outside of `s`, then it is measurable. -/\ntheorem Measurable.isLUB_of_mem {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g g' : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, Measurable (f i))\n {s : Set \u03b4} (hs : MeasurableSet s) (hg : \u2200 b \u2208 s, IsLUB { a | \u2203 i, f i b = a } (g b))\n (hg' : EqOn g g' s\u1d9c) (g'_meas : Measurable g') : Measurable g := by\n rcases isEmpty_or_nonempty \u03b9 with h\u03b9|\u27e8\u27e8i\u27e9\u27e9\n \u00b7 rcases eq_empty_or_nonempty s with rfl|\u27e8x, hx\u27e9\n \u00b7 convert g'_meas\n rwa [compl_empty, eqOn_univ] at hg'\n \u00b7 have A : \u2200 b \u2208 s, IsBot (g b) := by simpa using hg\n have B : \u2200 b \u2208 s, g b = g x := by\n intro b hb\n apply le_antisymm (A b hb (g x)) (A x hx (g b))\n have : g = s.piecewise (fun _y \u21a6 g x) g' := by\n ext b\n by_cases hb : b \u2208 s\n \u00b7 simp [hb, B]\n \u00b7 simp [hb, hg' hb]\n rw [this]\n exact Measurable.piecewise hs measurable_const g'_meas\n \u00b7 let f' : \u03b9 \u2192 \u03b4 \u2192 \u03b1 := fun i \u21a6 s.piecewise (f i) g'\n suffices \u2200 b, IsLUB { a | \u2203 i, f' i b = a } (g b) from\n Measurable.isLUB (fun i \u21a6 Measurable.piecewise hs (hf i) g'_meas) this\n intro b\n by_cases hb : b \u2208 s\n \u00b7 have A : \u2200 i, f' i b = f i b := fun i \u21a6 by simp [f', hb]\n simpa [A] using hg b hb\n \u00b7 have A : \u2200 i, f' i b = g' b := fun i \u21a6 by simp [f', hb]\n have : {a | \u2203 (_i : \u03b9), g' b = a} = {g' b} := by\n apply Subset.antisymm\n \u00b7 rintro - \u27e8_j, rfl\u27e9\n simp only [mem_singleton_iff]\n \u00b7 rintro - rfl\n exact \u27e8i, rfl\u27e9\n simp [A, this, hg' hb, isLUB_singleton]\n\ntheorem AEMeasurable.isLUB {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, IsLUB { a | \u2203 i, f i b = a } (g b)) :\n AEMeasurable g \u03bc := by\n nontriviality \u03b1\n haveI h\u03b1 : Nonempty \u03b1 := inferInstance\n cases' isEmpty_or_nonempty \u03b9 with h\u03b9 h\u03b9\n \u00b7 simp only [IsEmpty.exists_iff, setOf_false, isLUB_empty_iff] at hg\n exact aemeasurable_const' (hg.mono fun a ha => hg.mono fun b hb => (ha _).antisymm (hb _))\n let p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := fun x f' => IsLUB { a | \u2203 i, f' i = a } (g x)\n let g_seq := (aeSeqSet hf p).piecewise g fun _ => h\u03b1.some\n have hg_seq : \u2200 b, IsLUB { a | \u2203 i, aeSeq hf p i b = a } (g_seq b) := by\n intro b\n simp only [g_seq, aeSeq, Set.piecewise]\n split_ifs with h\n \u00b7 have h_set_eq : { a : \u03b1 | \u2203 i : \u03b9, (hf i).mk (f i) b = a } =\n { a : \u03b1 | \u2203 i : \u03b9, f i b = a } := by\n ext x\n simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]\n rw [h_set_eq]\n exact aeSeq.fun_prop_of_mem_aeSeqSet hf h\n \u00b7 exact IsGreatest.isLUB \u27e8(@exists_const (h\u03b1.some = h\u03b1.some) \u03b9 _).2 rfl, fun x \u27e8i, hi\u27e9 => hi.ge\u27e9\n refine' \u27e8g_seq, Measurable.isLUB (aeSeq.measurable hf p) hg_seq, _\u27e9\n exact\n (ite_ae_eq_of_measure_compl_zero g (fun _ => h\u03b1.some) (aeSeqSet hf p)\n (aeSeq.measure_compl_aeSeqSet_eq_zero hf hg)).symm\n#align ae_measurable.is_lub AEMeasurable.isLUB\n\n/-- If a function is the greatest lower bound of countably many measurable functions,\nthen it is measurable. -/\ntheorem Measurable.isGLB {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i))\n (hg : \u2200 b, IsGLB { a | \u2203 i, f i b = a } (g b)) : Measurable g :=\n Measurable.isLUB (\u03b1 := \u03b1\u1d52\u1d48) hf hg\n#align measurable.is_glb Measurable.isGLB\n\n/-- If a function is the greatest lower bound of countably many measurable functions on a measurable\nset `s`, and coincides with a measurable function outside of `s`, then it is measurable. -/\ntheorem Measurable.isGLB_of_mem {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g g' : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, Measurable (f i))\n {s : Set \u03b4} (hs : MeasurableSet s) (hg : \u2200 b \u2208 s, IsGLB { a | \u2203 i, f i b = a } (g b))\n (hg' : EqOn g g' s\u1d9c) (g'_meas : Measurable g') : Measurable g :=\n Measurable.isLUB_of_mem (\u03b1 := \u03b1\u1d52\u1d48) hf hs hg hg' g'_meas\n\ntheorem AEMeasurable.isGLB {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, IsGLB { a | \u2203 i, f i b = a } (g b)) :\n AEMeasurable g \u03bc :=\n AEMeasurable.isLUB (\u03b1 := \u03b1\u1d52\u1d48) hf hg\n#align ae_measurable.is_glb AEMeasurable.isGLB\n\nprotected theorem Monotone.measurable [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {f : \u03b2 \u2192 \u03b1}\n (hf : Monotone f) : Measurable f :=\n suffices h : \u2200 x, OrdConnected (f \u207b\u00b9' Ioi x) from measurable_of_Ioi fun x => (h x).measurableSet\n fun _ => ordConnected_def.mpr fun _a ha _ _ _c hc => lt_of_lt_of_le ha (hf hc.1)\n#align monotone.measurable Monotone.measurable\n\ntheorem aemeasurable_restrict_of_monotoneOn [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {\u03bc : Measure \u03b2}\n {s : Set \u03b2} (hs : MeasurableSet s) {f : \u03b2 \u2192 \u03b1} (hf : MonotoneOn f s) :\n AEMeasurable f (\u03bc.restrict s) :=\n have : Monotone (f \u2218 (\u2191) : s \u2192 \u03b1) := fun \u27e8x, hx\u27e9 \u27e8y, hy\u27e9 => fun (hxy : x \u2264 y) => hf hx hy hxy\n aemeasurable_restrict_of_measurable_subtype hs this.measurable\n#align ae_measurable_restrict_of_monotone_on aemeasurable_restrict_of_monotoneOn\n\nprotected theorem Antitone.measurable [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {f : \u03b2 \u2192 \u03b1}\n (hf : Antitone f) : Measurable f :=\n @Monotone.measurable \u03b1\u1d52\u1d48 \u03b2 _ _ \u2039_\u203a _ _ _ _ _ \u2039_\u203a _ _ _ hf\n#align antitone.measurable Antitone.measurable\n\ntheorem aemeasurable_restrict_of_antitoneOn [LinearOrder \u03b2] [OrderClosedTopology \u03b2] {\u03bc : Measure \u03b2}\n {s : Set \u03b2} (hs : MeasurableSet s) {f : \u03b2 \u2192 \u03b1} (hf : AntitoneOn f s) :\n AEMeasurable f (\u03bc.restrict s) :=\n @aemeasurable_restrict_of_monotoneOn \u03b1\u1d52\u1d48 \u03b2 _ _ \u2039_\u203a _ _ _ _ _ \u2039_\u203a _ _ _ _ hs _ hf\n#align ae_measurable_restrict_of_antitone_on aemeasurable_restrict_of_antitoneOn\n\ntheorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set \u03b1} (h : \u2200 x \u2208 s, s \u2208 \ud835\udcdd[>] x)\n (h' : \u2200 x \u2208 s, \u2203 y, x < y) : MeasurableSet s := by\n choose! M hM using h'\n suffices H : (s \\ interior s).Countable by\n have : s = interior s \u222a s \\ interior s := by rw [union_diff_cancel interior_subset]\n rw [this]\n exact isOpen_interior.measurableSet.union H.measurableSet\n have A : \u2200 x \u2208 s, \u2203 y \u2208 Ioi x, Ioo x y \u2286 s := fun x hx =>\n (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)\n choose! y hy h'y using A\n have B : Set.PairwiseDisjoint (s \\ interior s) fun x => Ioo x (y x) := by\n intro x hx x' hx' hxx'\n rcases lt_or_gt_of_ne hxx' with (h' | h')\n \u00b7 refine disjoint_left.2 fun z hz h'z => ?_\n have : x' \u2208 interior s :=\n mem_interior.2 \u27e8Ioo x (y x), h'y _ hx.1, isOpen_Ioo, \u27e8h', h'z.1.trans hz.2\u27e9\u27e9\n exact False.elim (hx'.2 this)\n \u00b7 refine disjoint_left.2 fun z hz h'z => ?_\n have : x \u2208 interior s :=\n mem_interior.2 \u27e8Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, \u27e8h', hz.1.trans h'z.2\u27e9\u27e9\n exact False.elim (hx.2 this)\n exact B.countable_of_Ioo fun x hx => hy x hx.1\n#align measurable_set_of_mem_nhds_within_Ioi_aux measurableSet_of_mem_nhdsWithin_Ioi_aux\n\n/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/\ntheorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set \u03b1} (h : \u2200 x \u2208 s, s \u2208 \ud835\udcdd[>] x) :\n MeasurableSet s := by\n by_cases H : \u2203 x \u2208 s, IsTop x\n \u00b7 rcases H with \u27e8x\u2080, x\u2080s, h\u2080\u27e9\n have : s = {x\u2080} \u222a s \\ {x\u2080} := by rw [union_diff_cancel (singleton_subset_iff.2 x\u2080s)]\n rw [this]\n refine' (measurableSet_singleton _).union _\n have A : \u2200 x \u2208 s \\ {x\u2080}, x < x\u2080 := fun x hx => lt_of_le_of_ne (h\u2080 _) (by simpa using hx.2)\n refine' measurableSet_of_mem_nhdsWithin_Ioi_aux (fun x hx => _) fun x hx => \u27e8x\u2080, A x hx\u27e9\n obtain \u27e8u, hu, us\u27e9 : \u2203 (u : \u03b1), u \u2208 Ioi x \u2227 Ioo x u \u2286 s :=\n (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).1 (h x hx.1)\n refine' (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (A x hx)).2 \u27e8u, hu, fun y hy => \u27e8us hy, _\u27e9\u27e9\n exact ne_of_lt (hy.2.trans_le (h\u2080 _))\n \u00b7 apply measurableSet_of_mem_nhdsWithin_Ioi_aux h\n simp only [IsTop] at H\n push_neg at H\n exact H\n#align measurable_set_of_mem_nhds_within_Ioi measurableSet_of_mem_nhdsWithin_Ioi\n\nlemma measurableSet_bddAbove_range {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n MeasurableSet {b | BddAbove (range (fun i \u21a6 f i b))} := by\n rcases isEmpty_or_nonempty \u03b1 with h\u03b1|h\u03b1\n \u00b7 have : \u2200 b, range (fun i \u21a6 f i b) = \u2205 := fun b \u21a6 eq_empty_of_isEmpty _\n simp [this]\n have A : \u2200 (i : \u03b9) (c : \u03b1), MeasurableSet {x | f i x \u2264 c} := by\n intro i c\n exact measurableSet_le (hf i) measurable_const\n have B : \u2200 (c : \u03b1), MeasurableSet {x | \u2200 i, f i x \u2264 c} := by\n intro c\n rw [setOf_forall]\n exact MeasurableSet.iInter (fun i \u21a6 A i c)\n obtain \u27e8u, hu\u27e9 : \u2203 (u : \u2115 \u2192 \u03b1), Tendsto u atTop atTop := exists_seq_tendsto (atTop : Filter \u03b1)\n have : {b | BddAbove (range (fun i \u21a6 f i b))} = {x | \u2203 n, \u2200 i, f i x \u2264 u n} := by\n apply Subset.antisymm\n \u00b7 rintro x \u27e8c, hc\u27e9\n obtain \u27e8n, hn\u27e9 : \u2203 n, c \u2264 u n := (tendsto_atTop.1 hu c).exists\n exact \u27e8n, fun i \u21a6 (hc ((mem_range_self i))).trans hn\u27e9\n \u00b7 rintro x \u27e8n, hn\u27e9\n refine \u27e8u n, ?_\u27e9\n rintro - \u27e8i, rfl\u27e9\n exact hn i\n rw [this, setOf_exists]\n exact MeasurableSet.iUnion (fun n \u21a6 B (u n))\n\nlemma measurableSet_bddBelow_range {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n MeasurableSet {b | BddBelow (range (fun i \u21a6 f i b))} :=\n measurableSet_bddAbove_range (\u03b1 := \u03b1\u1d52\u1d48) hf\n\nend LinearOrder\n\n@[measurability]\ntheorem Measurable.iSup_Prop {\u03b1} [MeasurableSpace \u03b1] [ConditionallyCompleteLattice \u03b1]\n (p : Prop) {f : \u03b4 \u2192 \u03b1} (hf : Measurable f) : Measurable fun b => \u2a06 _ : p, f b := by\n simp_rw [ciSup_eq_ite]\n split_ifs with h\n \u00b7 exact hf\n \u00b7 exact measurable_const\n#align measurable.supr_Prop Measurable.iSup_Prop\n\n@[measurability]\ntheorem Measurable.iInf_Prop {\u03b1} [MeasurableSpace \u03b1] [ConditionallyCompleteLattice \u03b1]\n (p : Prop) {f : \u03b4 \u2192 \u03b1} (hf : Measurable f) : Measurable fun b => \u2a05 _ : p, f b := by\n simp_rw [ciInf_eq_ite]\n split_ifs with h\n \u00b7 exact hf\n \u00b7 exact measurable_const\n#align measurable.infi_Prop Measurable.iInf_Prop\n\nsection ConditionallyCompleteLinearOrder\n\nvariable [ConditionallyCompleteLinearOrder \u03b1] [OrderTopology \u03b1] [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem measurable_iSup {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable (fun b \u21a6 \u2a06 i, f i b) := by\n rcases isEmpty_or_nonempty \u03b9 with h\u03b9|h\u03b9\n \u00b7 simp [iSup_of_empty']\n have A : MeasurableSet {b | BddAbove (range (fun i \u21a6 f i b))} :=\n measurableSet_bddAbove_range hf\n have : Measurable (fun (_b : \u03b4) \u21a6 sSup (\u2205 : Set \u03b1)) := measurable_const\n apply Measurable.isLUB_of_mem hf A _ _ this\n \u00b7 rintro b \u27e8c, hc\u27e9\n apply isLUB_ciSup\n refine \u27e8c, ?_\u27e9\n rintro d \u27e8i, rfl\u27e9\n exact hc (mem_range_self i)\n \u00b7 intro b hb\n apply csSup_of_not_bddAbove\n exact hb\n\n@[measurability]\ntheorem aemeasurable_iSup {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a06 i, f i b) \u03bc := by\n refine \u27e8fun b \u21a6 \u2a06 i, (hf i).mk (f i) b, measurable_iSup (fun i \u21a6 (hf i).measurable_mk), ?_\u27e9\n filter_upwards [ae_all_iff.2 (fun i \u21a6 (hf i).ae_eq_mk)] with b hb using by simp [hb]\n#align ae_measurable_supr aemeasurable_iSup\n\n@[measurability]\ntheorem measurable_iInf {\u03b9} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable fun b => \u2a05 i, f i b :=\n measurable_iSup (\u03b1 := \u03b1\u1d52\u1d48) hf\n#align measurable_infi measurable_iInf\n\n@[measurability]\ntheorem aemeasurable_iInf {\u03b9} {\u03bc : Measure \u03b4} [Countable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1}\n (hf : \u2200 i, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a05 i, f i b) \u03bc :=\n aemeasurable_iSup (\u03b1 := \u03b1\u1d52\u1d48) hf\n#align ae_measurable_infi aemeasurable_iInf\n\ntheorem measurable_sSup {\u03b9} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {s : Set \u03b9} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) :\n Measurable fun x => sSup ((fun i => f i x) '' s) := by\n have : Countable \u2191s := countable_coe_iff.2 hs\n convert measurable_iSup (f := (fun (i : s) \u21a6 f i)) (fun i \u21a6 hf i i.2) using 1\n ext b\n congr\n exact image_eq_range (fun i \u21a6 f i b) s\n#align measurable_cSup measurable_sSup\n\ntheorem measurable_sInf {\u03b9} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {s : Set \u03b9} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) :\n Measurable fun x => sInf ((fun i => f i x) '' s) :=\n measurable_sSup (\u03b1 := \u03b1\u1d52\u1d48) hs hf\n#align measurable_cInf measurable_sInf\n\ntheorem measurable_biSup {\u03b9} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) : Measurable fun b => \u2a06 i \u2208 s, f i b := by\n haveI : Encodable s := hs.toEncodable\n by_cases H : \u2200 i, i \u2208 s\n \u00b7 have : \u2200 b, \u2a06 i \u2208 s, f i b = \u2a06 (i : s), f i b :=\n fun b \u21a6 cbiSup_eq_of_forall (f := fun i \u21a6 f i b) H\n simp only [this]\n exact measurable_iSup (fun (i : s) \u21a6 hf i i.2)\n \u00b7 have : \u2200 b, \u2a06 i \u2208 s, f i b = (\u2a06 (i : s), f i b) \u2294 sSup \u2205 :=\n fun b \u21a6 cbiSup_eq_of_not_forall (f := fun i \u21a6 f i b) H\n simp only [this]\n apply Measurable.sup _ measurable_const\n exact measurable_iSup (fun (i : s) \u21a6 hf i i.2)\n#align measurable_bsupr measurable_biSup\n\ntheorem aemeasurable_biSup {\u03b9} {\u03bc : Measure \u03b4} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a06 i \u2208 s, f i b) \u03bc := by\n let g : \u03b9 \u2192 \u03b4 \u2192 \u03b1 := fun i \u21a6 if hi : i \u2208 s then (hf i hi).mk (f i) else fun _b \u21a6 sSup \u2205\n have : \u2200 i \u2208 s, Measurable (g i) := by\n intro i hi\n simpa [g, hi] using (hf i hi).measurable_mk\n refine \u27e8fun b \u21a6 \u2a06 (i) (_ : i \u2208 s), g i b, measurable_biSup s hs this, ?_\u27e9\n have : \u2200 i \u2208 s, \u2200\u1d50 b \u2202\u03bc, f i b = g i b :=\n fun i hi \u21a6 by simpa [g, hi] using (hf i hi).ae_eq_mk\n filter_upwards [(ae_ball_iff hs).2 this] with b hb\n exact iSup_congr fun i => iSup_congr (hb i)\n#align ae_measurable_bsupr aemeasurable_biSup\n\ntheorem measurable_biInf {\u03b9} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, Measurable (f i)) : Measurable fun b => \u2a05 i \u2208 s, f i b :=\n measurable_biSup (\u03b1 := \u03b1\u1d52\u1d48) s hs hf\n#align measurable_binfi measurable_biInf\n\ntheorem aemeasurable_biInf {\u03b9} {\u03bc : Measure \u03b4} (s : Set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : s.Countable)\n (hf : \u2200 i \u2208 s, AEMeasurable (f i) \u03bc) : AEMeasurable (fun b => \u2a05 i \u2208 s, f i b) \u03bc :=\n aemeasurable_biSup (\u03b1 := \u03b1\u1d52\u1d48) s hs hf\n#align ae_measurable_binfi aemeasurable_biInf\n\n/-- `liminf` over a general filter is measurable. See `measurable_liminf` for the version over `\u2115`.\n-/\ntheorem measurable_liminf' {\u03b9 \u03b9'} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {v : Filter \u03b9} (hf : \u2200 i, Measurable (f i))\n {p : \u03b9' \u2192 Prop} {s : \u03b9' \u2192 Set \u03b9} (hv : v.HasCountableBasis p s) (hs : \u2200 j, (s j).Countable) :\n Measurable fun x => liminf (f \u00b7 x) v := by\n /- We would like to write the liminf as `\u2a06 (j : Subtype p), \u2a05 (i : s j), f i x`, as the\n measurability would follow from the measurability of infs and sups. Unfortunately, this is not\n true in general conditionally complete linear orders because of issues with empty sets or sets\n which are not bounded above or below. A slightly more complicated expression for the liminf,\n valid in general, is given in `Filter.HasBasis.liminf_eq_ite`. This expression, built from\n `if ... then ... else` and infs and sups, can be readily checked to be measurable. -/\n have : Countable (Subtype p) := hv.countable\n rcases isEmpty_or_nonempty (Subtype p) with hp|hp\n \u00b7 simp [hv.liminf_eq_sSup_iUnion_iInter]\n by_cases H : \u2203 (j : Subtype p), s j = \u2205\n \u00b7 simp_rw [hv.liminf_eq_ite, if_pos H, measurable_const]\n simp_rw [hv.liminf_eq_ite, if_neg H]\n have : \u2200 i, Countable (s i) := fun i \u21a6 countable_coe_iff.2 (hs i)\n let m : Subtype p \u2192 Set \u03b4 := fun j \u21a6 {x | BddBelow (range (fun (i : s j) \u21a6 f i x))}\n have m_meas : \u2200 j, MeasurableSet (m j) :=\n fun j \u21a6 measurableSet_bddBelow_range (fun (i : s j) \u21a6 hf i)\n have mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), x \u2209 m j} := by\n rw [setOf_forall]\n exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl)\n apply Measurable.piecewise mc_meas measurable_const\n apply measurable_iSup (fun j \u21a6 ?_)\n let reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x \u21a6 liminf_reparam (fun i \u21a6 f i x) s p\n let F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x \u21a6 \u2a05 (i : s j), f i x\n have F0_meas : \u2200 j, Measurable (F0 j) := fun j \u21a6 measurable_iInf (fun (i : s j) \u21a6 hf i)\n set F1 : \u03b4 \u2192 \u03b1 := fun x \u21a6 F0 (reparam x j) x with hF1\n change Measurable F1\n let g : \u2115 \u2192 Subtype p := Classical.choose (exists_surjective_nat (Subtype p))\n have Z : \u2200 x, \u2203 n, x \u2208 m (g n) \u2228 \u2200 k, x \u2209 m k := by\n intro x\n by_cases H : \u2203 k, x \u2208 m k\n \u00b7 rcases H with \u27e8k, hk\u27e9\n rcases Classical.choose_spec (exists_surjective_nat (Subtype p)) k with \u27e8n, rfl\u27e9\n exact \u27e8n, Or.inl hk\u27e9\n \u00b7 push_neg at H\n exact \u27e80, Or.inr H\u27e9\n have : F1 = fun x \u21a6 if x \u2208 m j then F0 j x else F0 (g (Nat.find (Z x))) x := by\n ext x\n have A : reparam x j = if x \u2208 m j then j else g (Nat.find (Z x)) := rfl\n split_ifs with hjx\n \u00b7 have : reparam x j = j := by rw [A, if_pos hjx]\n simp only [hF1, this]\n \u00b7 have : reparam x j = g (Nat.find (Z x)) := by rw [A, if_neg hjx]\n simp only [hF1, this]\n rw [this]\n apply Measurable.piecewise (m_meas j) (F0_meas j)\n apply Measurable.find (fun n \u21a6 F0_meas (g n)) (fun n \u21a6 ?_)\n exact (m_meas (g n)).union mc_meas\n#align measurable_liminf' measurable_liminf'\n\n/-- `limsup` over a general filter is measurable. See `measurable_limsup` for the version over `\u2115`.\n-/\ntheorem measurable_limsup' {\u03b9 \u03b9'} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {u : Filter \u03b9} (hf : \u2200 i, Measurable (f i))\n {p : \u03b9' \u2192 Prop} {s : \u03b9' \u2192 Set \u03b9} (hu : u.HasCountableBasis p s) (hs : \u2200 i, (s i).Countable) :\n Measurable fun x => limsup (fun i => f i x) u :=\n measurable_liminf' (\u03b1 := \u03b1\u1d52\u1d48) hf hu hs\n#align measurable_limsup' measurable_limsup'\n\n/-- `liminf` over `\u2115` is measurable. See `measurable_liminf'` for a version with a general filter.\n-/\n@[measurability]\ntheorem measurable_liminf {f : \u2115 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable fun x => liminf (fun i => f i x) atTop :=\n measurable_liminf' hf atTop_countable_basis fun _ => to_countable _\n#align measurable_liminf measurable_liminf\n\n/-- `limsup` over `\u2115` is measurable. See `measurable_limsup'` for a version with a general filter.\n-/\n@[measurability]\ntheorem measurable_limsup {f : \u2115 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, Measurable (f i)) :\n Measurable fun x => limsup (fun i => f i x) atTop :=\n measurable_limsup' hf atTop_countable_basis fun _ => to_countable _\n#align measurable_limsup measurable_limsup\n\nend ConditionallyCompleteLinearOrder\n\n/-- Convert a `Homeomorph` to a `MeasurableEquiv`. -/\ndef Homemorph.toMeasurableEquiv (h : \u03b1 \u2243\u209c \u03b2) : \u03b1 \u2243\u1d50 \u03b2 where\n toEquiv := h.toEquiv\n measurable_toFun := h.continuous_toFun.measurable\n measurable_invFun := h.continuous_invFun.measurable\n#align homemorph.to_measurable_equiv Homemorph.toMeasurableEquiv\n\nprotected theorem IsFiniteMeasureOnCompacts.map (\u03bc : Measure \u03b1) [IsFiniteMeasureOnCompacts \u03bc]\n (f : \u03b1 \u2243\u209c \u03b2) : IsFiniteMeasureOnCompacts (Measure.map f \u03bc) := by\n refine \u27e8fun K hK \u21a6 ?_\u27e9\n rw [\u2190 Homeomorph.toMeasurableEquiv_coe, MeasurableEquiv.map_apply]\n exact IsCompact.measure_lt_top (f.isCompact_preimage.2 hK)\n#align is_finite_measure_on_compacts.map IsFiniteMeasureOnCompacts.map\n\nend BorelSpace\n\ninstance Empty.borelSpace : BorelSpace Empty :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align empty.borel_space Empty.borelSpace\n\ninstance Unit.borelSpace : BorelSpace Unit :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align unit.borel_space Unit.borelSpace\n\ninstance Bool.borelSpace : BorelSpace Bool :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align bool.borel_space Bool.borelSpace\n\ninstance Nat.borelSpace : BorelSpace \u2115 :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align nat.borel_space Nat.borelSpace\n\ninstance Int.borelSpace : BorelSpace \u2124 :=\n \u27e8borel_eq_top_of_discrete.symm\u27e9\n#align int.borel_space Int.borelSpace\n\ninstance Rat.borelSpace : BorelSpace \u211a :=\n \u27e8borel_eq_top_of_countable.symm\u27e9\n#align rat.borel_space Rat.borelSpace\n\n/- Instances on `Real` and `Complex` are special cases of `RCLike` but without these instances,\nLean fails to prove `BorelSpace (\u03b9 \u2192 \u211d)`, so we leave them here. -/\ninstance Real.measurableSpace : MeasurableSpace \u211d :=\n borel \u211d\n#align real.measurable_space Real.measurableSpace\n\ninstance Real.borelSpace : BorelSpace \u211d :=\n \u27e8rfl\u27e9\n#align real.borel_space Real.borelSpace\n\ninstance NNReal.measurableSpace : MeasurableSpace \u211d\u22650 :=\n Subtype.instMeasurableSpace\n#align nnreal.measurable_space NNReal.measurableSpace\n\ninstance NNReal.borelSpace : BorelSpace \u211d\u22650 :=\n Subtype.borelSpace _\n#align nnreal.borel_space NNReal.borelSpace\n\ninstance ENNReal.measurableSpace : MeasurableSpace \u211d\u22650\u221e :=\n borel \u211d\u22650\u221e\n#align ennreal.measurable_space ENNReal.measurableSpace\n\ninstance ENNReal.borelSpace : BorelSpace \u211d\u22650\u221e :=\n \u27e8rfl\u27e9\n#align ennreal.borel_space ENNReal.borelSpace\n\ninstance EReal.measurableSpace : MeasurableSpace EReal :=\n borel EReal\n#align ereal.measurable_space EReal.measurableSpace\n\ninstance EReal.borelSpace : BorelSpace EReal :=\n \u27e8rfl\u27e9\n#align ereal.borel_space EReal.borelSpace\n\n/-- One can cut out `\u211d\u22650\u221e` into the sets `{0}`, `Ico (t^n) (t^(n+1))` for `n : \u2124` and `{\u221e}`. This\ngives a way to compute the measure of a set in terms of sets on which a given function `f` does not\nfluctuate by more than `t`. -/\ntheorem measure_eq_measure_preimage_add_measure_tsum_Ico_zpow [MeasurableSpace \u03b1] (\u03bc : Measure \u03b1)\n {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : Measurable f) {s : Set \u03b1} (hs : MeasurableSet s) {t : \u211d\u22650} (ht : 1 < t) :\n \u03bc s =\n \u03bc (s \u2229 f \u207b\u00b9' {0}) + \u03bc (s \u2229 f \u207b\u00b9' {\u221e}) +\n \u2211' n : \u2124, \u03bc (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) := by\n have A : \u03bc s = \u03bc (s \u2229 f \u207b\u00b9' {0}) + \u03bc (s \u2229 f \u207b\u00b9' Ioi 0) := by\n rw [\u2190 measure_union]\n \u00b7 rw [\u2190 inter_union_distrib_left, \u2190 preimage_union, singleton_union, Ioi_insert,\n \u2190 _root_.bot_eq_zero, Ici_bot, preimage_univ, inter_univ]\n \u00b7 exact disjoint_singleton_left.mpr not_mem_Ioi_self\n |>.preimage f |>.inter_right' s |>.inter_left' s\n \u00b7 exact hs.inter (hf measurableSet_Ioi)\n have B : \u03bc (s \u2229 f \u207b\u00b9' Ioi 0) = \u03bc (s \u2229 f \u207b\u00b9' {\u221e}) + \u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u221e) := by\n rw [\u2190 measure_union]\n \u00b7 rw [\u2190 inter_union_distrib_left]\n congr\n ext x\n simp only [mem_singleton_iff, mem_union, mem_Ioo, mem_Ioi, mem_preimage]\n obtain (H | H) : f x = \u221e \u2228 f x < \u221e := eq_or_lt_of_le le_top\n \u00b7 simp only [H, eq_self_iff_true, or_false_iff, ENNReal.zero_lt_top, not_top_lt, and_false]\n \u00b7 simp only [H, H.ne, and_true_iff, false_or_iff]\n \u00b7 refine disjoint_left.2 fun x hx h'x => ?_\n have : f x < \u221e := h'x.2.2\n exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm))\n \u00b7 exact hs.inter (hf measurableSet_Ioo)\n have C : \u03bc (s \u2229 f \u207b\u00b9' Ioo 0 \u221e) =\n \u2211' n : \u2124, \u03bc (s \u2229 f \u207b\u00b9' Ico ((t : \u211d\u22650\u221e) ^ n) ((t : \u211d\u22650\u221e) ^ (n + 1))) := by\n rw [\u2190 measure_iUnion,\n ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow (ENNReal.one_lt_coe_iff.2 ht) ENNReal.coe_ne_top,\n preimage_iUnion, inter_iUnion]\n \u00b7 intro i j hij\n wlog h : i < j generalizing i j\n \u00b7 exact (this hij.symm (hij.lt_or_lt.resolve_left h)).symm\n refine disjoint_left.2 fun x hx h'x => lt_irrefl (f x) ?_\n calc\n f x < (t : \u211d\u22650\u221e) ^ (i + 1) := hx.2.2\n _ \u2264 (t : \u211d\u22650\u221e) ^ j := ENNReal.zpow_le_of_le (ENNReal.one_le_coe_iff.2 ht.le) h\n _ \u2264 f x := h'x.2.1\n \u00b7 intro n\n exact hs.inter (hf measurableSet_Ico)\n rw [A, B, C, add_assoc]\n#align measure_eq_measure_preimage_add_measure_tsum_Ico_zpow measure_eq_measure_preimage_add_measure_tsum_Ico_zpow\n\nsection PseudoMetricSpace\n\nvariable [PseudoMetricSpace \u03b1] [MeasurableSpace \u03b1] [OpensMeasurableSpace \u03b1]\nvariable [MeasurableSpace \u03b2] {x : \u03b1} {\u03b5 : \u211d}\n\nopen Metric\n\n@[measurability]\ntheorem measurableSet_ball : MeasurableSet (Metric.ball x \u03b5) :=\n Metric.isOpen_ball.measurableSet\n#align measurable_set_ball measurableSet_ball\n\n@[measurability]\ntheorem measurableSet_closedBall : MeasurableSet (Metric.closedBall x \u03b5) :=\n Metric.isClosed_ball.measurableSet\n#align measurable_set_closed_ball measurableSet_closedBall\n\n@[measurability]\ntheorem measurable_infDist {s : Set \u03b1} : Measurable fun x => infDist x s :=\n (continuous_infDist_pt s).measurable\n#align measurable_inf_dist measurable_infDist\n\n@[measurability]\ntheorem Measurable.infDist {f : \u03b2 \u2192 \u03b1} (hf : Measurable f) {s : Set \u03b1} :\n Measurable fun x => infDist (f x) s :=\n measurable_infDist.comp hf\n#align measurable.inf_dist Measurable.infDist\n\n@[measurability]\ntheorem measurable_infNndist {s : Set \u03b1} : Measurable fun x => infNndist x s :=\n (continuous_infNndist_pt s).measurable\n#align measurable_inf_nndist measurable_infNndist\n\n@[measurability]\ntheorem Measurable.infNndist {f : \u03b2 \u2192 \u03b1} (hf : Measurable f) {s : Set \u03b1} :\n Measurable fun x => infNndist (f x) s :=\n measurable_infNndist.comp hf\n#align measurable.inf_nndist Measurable.infNndist\n\nsection\n\nvariable [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem measurable_dist : Measurable fun p : \u03b1 \u00d7 \u03b1 => dist p.1 p.2 :=\n continuous_dist.measurable\n#align measurable_dist measurable_dist\n\n@[measurability]\ntheorem Measurable.dist {f g : \u03b2 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun b => dist (f b) (g b) :=\n (@continuous_dist \u03b1 _).measurable2 hf hg\n#align measurable.dist Measurable.dist\n\n@[measurability]\ntheorem measurable_nndist : Measurable fun p : \u03b1 \u00d7 \u03b1 => nndist p.1 p.2 :=\n continuous_nndist.measurable\n#align measurable_nndist measurable_nndist\n\n@[measurability]\ntheorem Measurable.nndist {f g : \u03b2 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun b => nndist (f b) (g b) :=\n (@continuous_nndist \u03b1 _).measurable2 hf hg\n#align measurable.nndist Measurable.nndist\n\nend\n\nend PseudoMetricSpace\n\nsection PseudoEMetricSpace\n\nvariable [PseudoEMetricSpace \u03b1] [MeasurableSpace \u03b1] [OpensMeasurableSpace \u03b1]\nvariable [MeasurableSpace \u03b2] {x : \u03b1} {\u03b5 : \u211d\u22650\u221e}\n\nopen EMetric\n\n@[measurability]\ntheorem measurableSet_eball : MeasurableSet (EMetric.ball x \u03b5) :=\n EMetric.isOpen_ball.measurableSet\n#align measurable_set_eball measurableSet_eball\n\n@[measurability]\ntheorem measurable_edist_right : Measurable (edist x) :=\n (continuous_const.edist continuous_id).measurable\n#align measurable_edist_right measurable_edist_right\n\n@[measurability]\ntheorem measurable_edist_left : Measurable fun y => edist y x :=\n (continuous_id.edist continuous_const).measurable\n#align measurable_edist_left measurable_edist_left\n\n@[measurability]\ntheorem measurable_infEdist {s : Set \u03b1} : Measurable fun x => infEdist x s :=\n continuous_infEdist.measurable\n#align measurable_inf_edist measurable_infEdist\n\n@[measurability]\ntheorem Measurable.infEdist {f : \u03b2 \u2192 \u03b1} (hf : Measurable f) {s : Set \u03b1} :\n Measurable fun x => infEdist (f x) s :=\n measurable_infEdist.comp hf\n#align measurable.inf_edist Measurable.infEdist\n\nopen Metric EMetric\n\n/-- If a set has a closed thickening with finite measure, then the measure of its `r`-closed\nthickenings converges to the measure of its closure as `r` tends to `0`. -/\ntheorem tendsto_measure_cthickening {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (cthickening R s) \u2260 \u221e) :\n Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc (closure s))) := by\n have A : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n rw [closure_eq_iInter_cthickening]\n exact\n tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet)\n (fun i j _ ij => cthickening_mono ij _) hs\n have B : Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n apply Tendsto.congr' _ tendsto_const_nhds\n filter_upwards [self_mem_nhdsWithin (\u03b1 := \u211d)] with _ hr\n rw [cthickening_of_nonpos hr]\n convert B.sup A\n exact (nhds_left_sup_nhds_right' 0).symm\n#align tendsto_measure_cthickening tendsto_measure_cthickening\n\n/-- If a closed set has a closed thickening with finite measure, then the measure of its closed\n`r`-thickenings converge to its measure as `r` tends to `0`. -/\ntheorem tendsto_measure_cthickening_of_isClosed {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (cthickening R s) \u2260 \u221e) (h's : IsClosed s) :\n Tendsto (fun r => \u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc s)) := by\n convert tendsto_measure_cthickening hs\n exact h's.closure_eq.symm\n#align tendsto_measure_cthickening_of_is_closed tendsto_measure_cthickening_of_isClosed\n\n/-- If a set has a thickening with finite measure, then the measures of its `r`-thickenings\nconverge to the measure of its closure as `r > 0` tends to `0`. -/\ntheorem tendsto_measure_thickening {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (thickening R s) \u2260 \u221e) :\n Tendsto (fun r => \u03bc (thickening r s)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (closure s))) := by\n rw [closure_eq_iInter_thickening]\n exact tendsto_measure_biInter_gt (fun r _ => isOpen_thickening.measurableSet)\n (fun i j _ ij => thickening_mono ij _) hs\n\n/-- If a closed set has a thickening with finite measure, then the measure of its\n`r`-thickenings converge to its measure as `r > 0` tends to `0`. -/\ntheorem tendsto_measure_thickening_of_isClosed {\u03bc : Measure \u03b1} {s : Set \u03b1}\n (hs : \u2203 R > 0, \u03bc (thickening R s) \u2260 \u221e) (h's : IsClosed s) :\n Tendsto (fun r => \u03bc (thickening r s)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc s)) := by\n convert tendsto_measure_thickening hs\n exact h's.closure_eq.symm\n\nvariable [SecondCountableTopology \u03b1]\n\n@[measurability]\ntheorem measurable_edist : Measurable fun p : \u03b1 \u00d7 \u03b1 => edist p.1 p.2 :=\n continuous_edist.measurable\n#align measurable_edist measurable_edist\n\n@[measurability]\ntheorem Measurable.edist {f g : \u03b2 \u2192 \u03b1} (hf : Measurable f) (hg : Measurable g) :\n Measurable fun b => edist (f b) (g b) :=\n (@continuous_edist \u03b1 _).measurable2 hf hg\n#align measurable.edist Measurable.edist\n\n@[measurability]\ntheorem AEMeasurable.edist {f g : \u03b2 \u2192 \u03b1} {\u03bc : Measure \u03b2} (hf : AEMeasurable f \u03bc)\n (hg : AEMeasurable g \u03bc) : AEMeasurable (fun a => edist (f a) (g a)) \u03bc :=\n (@continuous_edist \u03b1 _).aemeasurable2 hf hg\n#align ae_measurable.edist AEMeasurable.edist\n\nend PseudoEMetricSpace\n\n/-- Given a compact set in a proper space, the measure of its `r`-closed thickenings converges to\nits measure as `r` tends to `0`. -/\ntheorem tendsto_measure_cthickening_of_isCompact [MetricSpace \u03b1] [MeasurableSpace \u03b1]\n [OpensMeasurableSpace \u03b1] [ProperSpace \u03b1] {\u03bc : Measure \u03b1} [IsFiniteMeasureOnCompacts \u03bc]\n {s : Set \u03b1} (hs : IsCompact s) :\n Tendsto (fun r => \u03bc (Metric.cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc s)) :=\n tendsto_measure_cthickening_of_isClosed\n \u27e81, zero_lt_one, hs.isBounded.cthickening.measure_lt_top.ne\u27e9 hs.isClosed\n#align tendsto_measure_cthickening_of_is_compact tendsto_measure_cthickening_of_isCompact\n\n/-- If a measurable space is countably generated and separates points, it arises as\nthe borel sets of some second countable t4 topology (i.e. a separable metrizable one). -/\ntheorem exists_borelSpace_of_countablyGenerated_of_separatesPoints (\u03b1 : Type*)\n [m : MeasurableSpace \u03b1] [CountablyGenerated \u03b1] [SeparatesPoints \u03b1] :\n \u2203 \u03c4 : TopologicalSpace \u03b1, SecondCountableTopology \u03b1 \u2227 T4Space \u03b1 \u2227 BorelSpace \u03b1 := by\n rcases measurableEquiv_nat_bool_of_countablyGenerated \u03b1 with \u27e8s, \u27e8f\u27e9\u27e9\n letI := induced f inferInstance\n let F := f.toEquiv.toHomeomorphOfInducing $ inducing_induced _\n exact \u27e8inferInstance, F.secondCountableTopology, F.symm.t4Space,\n MeasurableEmbedding.borelSpace f.measurableEmbedding F.inducing\u27e9\n\n/-- If a measurable space on `\u03b1` is countably generated and separates points, there is some\nsecond countable t4 topology on `\u03b1` (i.e. a separable metrizable one) for which every\nopen set is measurable. -/\n", "theoremStatement": "theorem exists_opensMeasurableSpace_of_hasCountableSeparatingOn (\u03b1 : Type*)\n [m : MeasurableSpace \u03b1] [HasCountableSeparatingOn \u03b1 MeasurableSet univ] :\n \u2203 \u03c4 : TopologicalSpace \u03b1, SecondCountableTopology \u03b1 \u2227 T4Space \u03b1 \u2227 OpensMeasurableSpace \u03b1", "theoremName": "exists_opensMeasurableSpace_of_hasCountableSeparatingOn", "fileCreated": {"commit": "11332d53f1", "date": "2023-05-21"}, "theoremCreated": {"commit": "726f2a5ff9", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "positionMetadata": {"lineInFile": 1920, "tokenPositionInFile": 89637, "theoremPositionInFile": 218}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rcases exists_countablyGenerated_le_of_hasCountableSeparatingOn \u03b1 with \u27e8m', _, _, m'le\u27e9\n rcases exists_borelSpace_of_countablyGenerated_of_separatesPoints (m := m') with \u27e8\u03c4, _, _, \u03c4m'\u27e9\n exact \u27e8\u03c4, \u2039_\u203a, \u2039_\u203a, @OpensMeasurableSpace.mk _ _ m (\u03c4m'.measurable_eq.symm.le.trans m'le)\u27e9", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 283}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.Algebra.Ring.Int\nimport Mathlib.Data.Fin.VecNotation\nimport Mathlib.Logic.Equiv.Defs\nimport Mathlib.Logic.Embedding.Set\n\n#align_import logic.equiv.fin from \"leanprover-community/mathlib\"@\"bd835ef554f37ef9b804f0903089211f89cb370b\"\n\n/-!\n# Equivalences for `Fin n`\n-/\n\nuniverse u\n\nvariable {m n : \u2115}\n\n/-- Equivalence between `Fin 0` and `Empty`. -/\ndef finZeroEquiv : Fin 0 \u2243 Empty :=\n Equiv.equivEmpty _\n#align fin_zero_equiv finZeroEquiv\n\n/-- Equivalence between `Fin 0` and `PEmpty`. -/\ndef finZeroEquiv' : Fin 0 \u2243 PEmpty.{u} :=\n Equiv.equivPEmpty _\n#align fin_zero_equiv' finZeroEquiv'\n\n/-- Equivalence between `Fin 1` and `Unit`. -/\ndef finOneEquiv : Fin 1 \u2243 Unit :=\n Equiv.equivPUnit _\n#align fin_one_equiv finOneEquiv\n\n/-- Equivalence between `Fin 2` and `Bool`. -/\ndef finTwoEquiv : Fin 2 \u2243 Bool where\n toFun := ![false, true]\n invFun b := b.casesOn 0 1\n left_inv := Fin.forall_fin_two.2 <| by simp\n right_inv := Bool.forall_bool.2 <| by simp\n#align fin_two_equiv finTwoEquiv\n\n/-- `\u03a0 i : Fin 2, \u03b1 i` is equivalent to `\u03b1 0 \u00d7 \u03b1 1`. See also `finTwoArrowEquiv` for a\nnon-dependent version and `prodEquivPiFinTwo` for a version with inputs `\u03b1 \u03b2 : Type u`. -/\n@[simps (config := .asFn)]\ndef piFinTwoEquiv (\u03b1 : Fin 2 \u2192 Type u) : (\u2200 i, \u03b1 i) \u2243 \u03b1 0 \u00d7 \u03b1 1\n where\n toFun f := (f 0, f 1)\n invFun p := Fin.cons p.1 <| Fin.cons p.2 finZeroElim\n left_inv _ := funext <| Fin.forall_fin_two.2 \u27e8rfl, rfl\u27e9\n right_inv := fun _ => rfl\n#align pi_fin_two_equiv piFinTwoEquiv\n#align pi_fin_two_equiv_symm_apply piFinTwoEquiv_symm_apply\n#align pi_fin_two_equiv_apply piFinTwoEquiv_apply\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Fin.preimage_apply_01_prod {\u03b1 : Fin 2 \u2192 Type u} (s : Set (\u03b1 0)) (t : Set (\u03b1 1)) :\n (fun f : \u2200 i, \u03b1 i => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t =\n Set.pi Set.univ (Fin.cons s <| Fin.cons t finZeroElim) := by\n ext f\n simp [Fin.forall_fin_two]\n#align fin.preimage_apply_01_prod Fin.preimage_apply_01_prod\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Fin.preimage_apply_01_prod' {\u03b1 : Type u} (s t : Set \u03b1) :\n (fun f : Fin 2 \u2192 \u03b1 => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t = Set.pi Set.univ ![s, t] :=\n @Fin.preimage_apply_01_prod (fun _ => \u03b1) s t\n#align fin.preimage_apply_01_prod' Fin.preimage_apply_01_prod'\n\n/-- A product space `\u03b1 \u00d7 \u03b2` is equivalent to the space `\u03a0 i : Fin 2, \u03b3 i`, where\n`\u03b3 = Fin.cons \u03b1 (Fin.cons \u03b2 finZeroElim)`. See also `piFinTwoEquiv` and\n`finTwoArrowEquiv`. -/\n@[simps! (config := .asFn)]\ndef prodEquivPiFinTwo (\u03b1 \u03b2 : Type u) : \u03b1 \u00d7 \u03b2 \u2243 \u2200 i : Fin 2, ![\u03b1, \u03b2] i :=\n (piFinTwoEquiv (Fin.cons \u03b1 (Fin.cons \u03b2 finZeroElim))).symm\n#align prod_equiv_pi_fin_two prodEquivPiFinTwo\n#align prod_equiv_pi_fin_two_apply prodEquivPiFinTwo_apply\n#align prod_equiv_pi_fin_two_symm_apply prodEquivPiFinTwo_symm_apply\n\n/-- The space of functions `Fin 2 \u2192 \u03b1` is equivalent to `\u03b1 \u00d7 \u03b1`. See also `piFinTwoEquiv` and\n`prodEquivPiFinTwo`. -/\n@[simps (config := .asFn)]\ndef finTwoArrowEquiv (\u03b1 : Type*) : (Fin 2 \u2192 \u03b1) \u2243 \u03b1 \u00d7 \u03b1 :=\n { piFinTwoEquiv fun _ => \u03b1 with invFun := fun x => ![x.1, x.2] }\n#align fin_two_arrow_equiv finTwoArrowEquiv\n#align fin_two_arrow_equiv_symm_apply finTwoArrowEquiv_symm_apply\n#align fin_two_arrow_equiv_apply finTwoArrowEquiv_apply\n\n/-- `\u03a0 i : Fin 2, \u03b1 i` is order equivalent to `\u03b1 0 \u00d7 \u03b1 1`. See also `OrderIso.finTwoArrowEquiv`\nfor a non-dependent version. -/\ndef OrderIso.piFinTwoIso (\u03b1 : Fin 2 \u2192 Type u) [\u2200 i, Preorder (\u03b1 i)] : (\u2200 i, \u03b1 i) \u2243o \u03b1 0 \u00d7 \u03b1 1\n where\n toEquiv := piFinTwoEquiv \u03b1\n map_rel_iff' := Iff.symm Fin.forall_fin_two\n#align order_iso.pi_fin_two_iso OrderIso.piFinTwoIso\n\n/-- The space of functions `Fin 2 \u2192 \u03b1` is order equivalent to `\u03b1 \u00d7 \u03b1`. See also\n`OrderIso.piFinTwoIso`. -/\ndef OrderIso.finTwoArrowIso (\u03b1 : Type*) [Preorder \u03b1] : (Fin 2 \u2192 \u03b1) \u2243o \u03b1 \u00d7 \u03b1 :=\n { OrderIso.piFinTwoIso fun _ => \u03b1 with toEquiv := finTwoArrowEquiv \u03b1 }\n#align order_iso.fin_two_arrow_iso OrderIso.finTwoArrowIso\n\n/-- The 'identity' equivalence between `Fin n` and `Fin m` when `n = m`. -/\ndef finCongr (h : m = n) : Fin m \u2243 Fin n :=\n (Fin.castIso h).toEquiv\n#align fin_congr finCongr\n\n@[simp] theorem finCongr_apply_mk (h : m = n) (k : \u2115) (w : k < m) :\n finCongr h \u27e8k, w\u27e9 = \u27e8k, h \u25b8 w\u27e9 :=\n rfl\n#align fin_congr_apply_mk finCongr_apply_mk\n\n@[simp] theorem finCongr_symm (h : m = n) : (finCongr h).symm = finCongr h.symm :=\n rfl\n#align fin_congr_symm finCongr_symm\n\n@[simp] theorem finCongr_apply_coe (h : m = n) (k : Fin m) : (finCongr h k : \u2115) = k :=\n rfl\n#align fin_congr_apply_coe finCongr_apply_coe\n\ntheorem finCongr_symm_apply_coe (h : m = n) (k : Fin n) : ((finCongr h).symm k : \u2115) = k :=\n rfl\n#align fin_congr_symm_apply_coe finCongr_symm_apply_coe\n\n/-- An equivalence that removes `i` and maps it to `none`.\nThis is a version of `Fin.predAbove` that produces `Option (Fin n)` instead of\nmapping both `i.cast_succ` and `i.succ` to `i`. -/\ndef finSuccEquiv' (i : Fin (n + 1)) : Fin (n + 1) \u2243 Option (Fin n)\n where\n toFun := i.insertNth none some\n invFun x := x.casesOn' i (Fin.succAbove i)\n left_inv x := Fin.succAboveCases i (by simp) (fun j => by simp) x\n right_inv x := by cases x <;> dsimp <;> simp\n#align fin_succ_equiv' finSuccEquiv'\n\n@[simp]\ntheorem finSuccEquiv'_at (i : Fin (n + 1)) : (finSuccEquiv' i) i = none := by\n simp [finSuccEquiv']\n#align fin_succ_equiv'_at finSuccEquiv'_at\n\n@[simp]\ntheorem finSuccEquiv'_succAbove (i : Fin (n + 1)) (j : Fin n) :\n finSuccEquiv' i (i.succAbove j) = some j :=\n @Fin.insertNth_apply_succAbove n (fun _ => Option (Fin n)) i _ _ _\n#align fin_succ_equiv'_succ_above finSuccEquiv'_succAbove\n\ntheorem finSuccEquiv'_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i) (Fin.castSucc m) = m := by\n rw [\u2190 Fin.succAbove_of_castSucc_lt _ _ h, finSuccEquiv'_succAbove]\n#align fin_succ_equiv'_below finSuccEquiv'_below\n\ntheorem finSuccEquiv'_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i) m.succ = some m := by\n rw [\u2190 Fin.succAbove_of_le_castSucc _ _ h, finSuccEquiv'_succAbove]\n#align fin_succ_equiv'_above finSuccEquiv'_above\n\n@[simp]\ntheorem finSuccEquiv'_symm_none (i : Fin (n + 1)) : (finSuccEquiv' i).symm none = i :=\n rfl\n#align fin_succ_equiv'_symm_none finSuccEquiv'_symm_none\n\n@[simp]\ntheorem finSuccEquiv'_symm_some (i : Fin (n + 1)) (j : Fin n) :\n (finSuccEquiv' i).symm (some j) = i.succAbove j :=\n rfl\n#align fin_succ_equiv'_symm_some finSuccEquiv'_symm_some\n\ntheorem finSuccEquiv'_symm_some_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i).symm (some m) = Fin.castSucc m :=\n Fin.succAbove_of_castSucc_lt i m h\n#align fin_succ_equiv'_symm_some_below finSuccEquiv'_symm_some_below\n\ntheorem finSuccEquiv'_symm_some_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i).symm (some m) = m.succ :=\n Fin.succAbove_of_le_castSucc i m h\n#align fin_succ_equiv'_symm_some_above finSuccEquiv'_symm_some_above\n\ntheorem finSuccEquiv'_symm_coe_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i).symm m = Fin.castSucc m :=\n finSuccEquiv'_symm_some_below h\n#align fin_succ_equiv'_symm_coe_below finSuccEquiv'_symm_coe_below\n\ntheorem finSuccEquiv'_symm_coe_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i).symm m = m.succ :=\n finSuccEquiv'_symm_some_above h\n#align fin_succ_equiv'_symm_coe_above finSuccEquiv'_symm_coe_above\n\n/-- Equivalence between `Fin (n + 1)` and `Option (Fin n)`.\nThis is a version of `Fin.pred` that produces `Option (Fin n)` instead of\nrequiring a proof that the input is not `0`. -/\ndef finSuccEquiv (n : \u2115) : Fin (n + 1) \u2243 Option (Fin n) :=\n finSuccEquiv' 0\n#align fin_succ_equiv finSuccEquiv\n\n@[simp]\ntheorem finSuccEquiv_zero : (finSuccEquiv n) 0 = none :=\n rfl\n#align fin_succ_equiv_zero finSuccEquiv_zero\n\n@[simp]\ntheorem finSuccEquiv_succ (m : Fin n) : (finSuccEquiv n) m.succ = some m :=\n finSuccEquiv'_above (Fin.zero_le _)\n#align fin_succ_equiv_succ finSuccEquiv_succ\n\n@[simp]\ntheorem finSuccEquiv_symm_none : (finSuccEquiv n).symm none = 0 :=\n finSuccEquiv'_symm_none _\n#align fin_succ_equiv_symm_none finSuccEquiv_symm_none\n\n@[simp]\ntheorem finSuccEquiv_symm_some (m : Fin n) : (finSuccEquiv n).symm (some m) = m.succ :=\n congr_fun Fin.succAbove_zero m\n#align fin_succ_equiv_symm_some finSuccEquiv_symm_some\n#align fin_succ_equiv_symm_coe finSuccEquiv_symm_some\n\n/-- The equiv version of `Fin.predAbove_zero`. -/\ntheorem finSuccEquiv'_zero : finSuccEquiv' (0 : Fin (n + 1)) = finSuccEquiv n :=\n rfl\n#align fin_succ_equiv'_zero finSuccEquiv'_zero\n\ntheorem finSuccEquiv'_last_apply_castSucc (i : Fin n) :\n finSuccEquiv' (Fin.last n) (Fin.castSucc i) = i := by\n rw [\u2190 Fin.succAbove_last, finSuccEquiv'_succAbove]\n\ntheorem finSuccEquiv'_last_apply {i : Fin (n + 1)} (h : i \u2260 Fin.last n) :\n finSuccEquiv' (Fin.last n) i = Fin.castLT i (Fin.val_lt_last h) := by\n rcases Fin.exists_castSucc_eq.2 h with \u27e8i, rfl\u27e9\n rw [finSuccEquiv'_last_apply_castSucc]\n rfl\n#align fin_succ_equiv'_last_apply finSuccEquiv'_last_apply\n\ntheorem finSuccEquiv'_ne_last_apply {i j : Fin (n + 1)} (hi : i \u2260 Fin.last n) (hj : j \u2260 i) :\n finSuccEquiv' i j = (i.castLT (Fin.val_lt_last hi)).predAbove j := by\n rcases Fin.exists_succAbove_eq hj with \u27e8j, rfl\u27e9\n rcases Fin.exists_castSucc_eq.2 hi with \u27e8i, rfl\u27e9\n simp\n#align fin_succ_equiv'_ne_last_apply finSuccEquiv'_ne_last_apply\n\n/-- `Fin.succAbove` as an order isomorphism between `Fin n` and `{x : Fin (n + 1) // x \u2260 p}`. -/\ndef finSuccAboveEquiv (p : Fin (n + 1)) : Fin n \u2243o { x : Fin (n + 1) // x \u2260 p } :=\n { Equiv.optionSubtype p \u27e8(finSuccEquiv' p).symm, rfl\u27e9 with\n map_rel_iff' := p.succAboveEmb.map_rel_iff' }\n#align fin_succ_above_equiv finSuccAboveEquiv\n\ntheorem finSuccAboveEquiv_apply (p : Fin (n + 1)) (i : Fin n) :\n finSuccAboveEquiv p i = \u27e8p.succAbove i, p.succAbove_ne i\u27e9 :=\n rfl\n#align fin_succ_above_equiv_apply finSuccAboveEquiv_apply\n\ntheorem finSuccAboveEquiv_symm_apply_last (x : { x : Fin (n + 1) // x \u2260 Fin.last n }) :\n (finSuccAboveEquiv (Fin.last n)).symm x = Fin.castLT x.1 (Fin.val_lt_last x.2) := by\n rw [\u2190 Option.some_inj]\n simpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_last_apply x.property\n#align fin_succ_above_equiv_symm_apply_last finSuccAboveEquiv_symm_apply_last\n\ntheorem finSuccAboveEquiv_symm_apply_ne_last {p : Fin (n + 1)} (h : p \u2260 Fin.last n)\n (x : { x : Fin (n + 1) // x \u2260 p }) :\n (finSuccAboveEquiv p).symm x = (p.castLT (Fin.val_lt_last h)).predAbove x := by\n rw [\u2190 Option.some_inj]\n simpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_ne_last_apply h x.property\n#align fin_succ_above_equiv_symm_apply_ne_last finSuccAboveEquiv_symm_apply_ne_last\n\n/-- `Equiv` between `Fin (n + 1)` and `Option (Fin n)` sending `Fin.last n` to `none` -/\ndef finSuccEquivLast : Fin (n + 1) \u2243 Option (Fin n) :=\n finSuccEquiv' (Fin.last n)\n#align fin_succ_equiv_last finSuccEquivLast\n\n@[simp]\ntheorem finSuccEquivLast_castSucc (i : Fin n) : finSuccEquivLast (Fin.castSucc i) = some i :=\n finSuccEquiv'_below i.2\n#align fin_succ_equiv_last_cast_succ finSuccEquivLast_castSucc\n\n@[simp]\ntheorem finSuccEquivLast_last : finSuccEquivLast (Fin.last n) = none := by\n simp [finSuccEquivLast]\n#align fin_succ_equiv_last_last finSuccEquivLast_last\n\n@[simp]\ntheorem finSuccEquivLast_symm_some (i : Fin n) :\n finSuccEquivLast.symm (some i) = Fin.castSucc i :=\n finSuccEquiv'_symm_some_below i.2\n#align fin_succ_equiv_last_symm_some finSuccEquivLast_symm_some\n#align fin_succ_equiv_last_symm_coe finSuccEquivLast_symm_some\n\n@[simp] theorem finSuccEquivLast_symm_none : finSuccEquivLast.symm none = Fin.last n :=\n finSuccEquiv'_symm_none _\n#align fin_succ_equiv_last_symm_none finSuccEquivLast_symm_none\n\n/-- Equivalence between `\u03a0 j : Fin (n + 1), \u03b1 j` and `\u03b1 i \u00d7 \u03a0 j : Fin n, \u03b1 (Fin.succAbove i j)`. -/\n@[simps (config := .asFn)]\ndef Equiv.piFinSuccAbove (\u03b1 : Fin (n + 1) \u2192 Type u) (i : Fin (n + 1)) :\n (\u2200 j, \u03b1 j) \u2243 \u03b1 i \u00d7 \u2200 j, \u03b1 (i.succAbove j) where\n toFun f := i.extractNth f\n invFun f := i.insertNth f.1 f.2\n left_inv f := by simp\n right_inv f := by simp\n#align equiv.pi_fin_succ_above_equiv Equiv.piFinSuccAbove\n#align equiv.pi_fin_succ_above_equiv_apply Equiv.piFinSuccAbove_apply\n#align equiv.pi_fin_succ_above_equiv_symm_apply Equiv.piFinSuccAbove_symm_apply\n\n/-- Order isomorphism between `\u03a0 j : Fin (n + 1), \u03b1 j` and\n`\u03b1 i \u00d7 \u03a0 j : Fin n, \u03b1 (Fin.succAbove i j)`. -/\ndef OrderIso.piFinSuccAboveIso (\u03b1 : Fin (n + 1) \u2192 Type u) [\u2200 i, LE (\u03b1 i)]\n (i : Fin (n + 1)) : (\u2200 j, \u03b1 j) \u2243o \u03b1 i \u00d7 \u2200 j, \u03b1 (i.succAbove j) where\n toEquiv := Equiv.piFinSuccAbove \u03b1 i\n map_rel_iff' := Iff.symm i.forall_iff_succAbove\n#align order_iso.pi_fin_succ_above_iso OrderIso.piFinSuccAboveIso\n\n/-- Equivalence between `Fin (n + 1) \u2192 \u03b2` and `\u03b2 \u00d7 (Fin n \u2192 \u03b2)`. -/\n@[simps! (config := .asFn)]\ndef Equiv.piFinSucc (n : \u2115) (\u03b2 : Type u) : (Fin (n + 1) \u2192 \u03b2) \u2243 \u03b2 \u00d7 (Fin n \u2192 \u03b2) :=\n Equiv.piFinSuccAbove (fun _ => \u03b2) 0\n#align equiv.pi_fin_succ Equiv.piFinSucc\n#align equiv.pi_fin_succ_apply Equiv.piFinSucc_apply\n#align equiv.pi_fin_succ_symm_apply Equiv.piFinSucc_symm_apply\n\n/-- An embedding `e : Fin (n+1) \u21aa \u03b9` corresponds to an embedding `f : Fin n \u21aa \u03b9` (corresponding\nthe last `n` coordinates of `e`) together with a value not taken by `f` (corresponding to `e 0`). -/\ndef Equiv.embeddingFinSucc (n : \u2115) (\u03b9 : Type*) :\n (Fin (n+1) \u21aa \u03b9) \u2243 (\u03a3 (e : Fin n \u21aa \u03b9), {i // i \u2209 Set.range e}) :=\n ((finSuccEquiv n).embeddingCongr (Equiv.refl \u03b9)).trans\n (Function.Embedding.optionEmbeddingEquiv (Fin n) \u03b9)\n\n", "theoremStatement": "@[simp] lemma Equiv.embeddingFinSucc_fst {n : \u2115} {\u03b9 : Type*} (e : Fin (n+1) \u21aa \u03b9) :\n ((Equiv.embeddingFinSucc n \u03b9 e).1 : Fin n \u2192 \u03b9) = e \u2218 Fin.succ", "theoremName": "Equiv.embeddingFinSucc_fst", "fileCreated": {"commit": "885e8510c4", "date": "2023-01-23"}, "theoremCreated": {"commit": "98843f5bd5", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Logic/Equiv/Fin.lean", "positionMetadata": {"lineInFile": 325, "tokenPositionInFile": 13691, "theoremPositionInFile": 49}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "rfl", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 3}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.Algebra.Ring.Int\nimport Mathlib.Data.Fin.VecNotation\nimport Mathlib.Logic.Equiv.Defs\nimport Mathlib.Logic.Embedding.Set\n\n#align_import logic.equiv.fin from \"leanprover-community/mathlib\"@\"bd835ef554f37ef9b804f0903089211f89cb370b\"\n\n/-!\n# Equivalences for `Fin n`\n-/\n\nuniverse u\n\nvariable {m n : \u2115}\n\n/-- Equivalence between `Fin 0` and `Empty`. -/\ndef finZeroEquiv : Fin 0 \u2243 Empty :=\n Equiv.equivEmpty _\n#align fin_zero_equiv finZeroEquiv\n\n/-- Equivalence between `Fin 0` and `PEmpty`. -/\ndef finZeroEquiv' : Fin 0 \u2243 PEmpty.{u} :=\n Equiv.equivPEmpty _\n#align fin_zero_equiv' finZeroEquiv'\n\n/-- Equivalence between `Fin 1` and `Unit`. -/\ndef finOneEquiv : Fin 1 \u2243 Unit :=\n Equiv.equivPUnit _\n#align fin_one_equiv finOneEquiv\n\n/-- Equivalence between `Fin 2` and `Bool`. -/\ndef finTwoEquiv : Fin 2 \u2243 Bool where\n toFun := ![false, true]\n invFun b := b.casesOn 0 1\n left_inv := Fin.forall_fin_two.2 <| by simp\n right_inv := Bool.forall_bool.2 <| by simp\n#align fin_two_equiv finTwoEquiv\n\n/-- `\u03a0 i : Fin 2, \u03b1 i` is equivalent to `\u03b1 0 \u00d7 \u03b1 1`. See also `finTwoArrowEquiv` for a\nnon-dependent version and `prodEquivPiFinTwo` for a version with inputs `\u03b1 \u03b2 : Type u`. -/\n@[simps (config := .asFn)]\ndef piFinTwoEquiv (\u03b1 : Fin 2 \u2192 Type u) : (\u2200 i, \u03b1 i) \u2243 \u03b1 0 \u00d7 \u03b1 1\n where\n toFun f := (f 0, f 1)\n invFun p := Fin.cons p.1 <| Fin.cons p.2 finZeroElim\n left_inv _ := funext <| Fin.forall_fin_two.2 \u27e8rfl, rfl\u27e9\n right_inv := fun _ => rfl\n#align pi_fin_two_equiv piFinTwoEquiv\n#align pi_fin_two_equiv_symm_apply piFinTwoEquiv_symm_apply\n#align pi_fin_two_equiv_apply piFinTwoEquiv_apply\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Fin.preimage_apply_01_prod {\u03b1 : Fin 2 \u2192 Type u} (s : Set (\u03b1 0)) (t : Set (\u03b1 1)) :\n (fun f : \u2200 i, \u03b1 i => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t =\n Set.pi Set.univ (Fin.cons s <| Fin.cons t finZeroElim) := by\n ext f\n simp [Fin.forall_fin_two]\n#align fin.preimage_apply_01_prod Fin.preimage_apply_01_prod\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Fin.preimage_apply_01_prod' {\u03b1 : Type u} (s t : Set \u03b1) :\n (fun f : Fin 2 \u2192 \u03b1 => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t = Set.pi Set.univ ![s, t] :=\n @Fin.preimage_apply_01_prod (fun _ => \u03b1) s t\n#align fin.preimage_apply_01_prod' Fin.preimage_apply_01_prod'\n\n/-- A product space `\u03b1 \u00d7 \u03b2` is equivalent to the space `\u03a0 i : Fin 2, \u03b3 i`, where\n`\u03b3 = Fin.cons \u03b1 (Fin.cons \u03b2 finZeroElim)`. See also `piFinTwoEquiv` and\n`finTwoArrowEquiv`. -/\n@[simps! (config := .asFn)]\ndef prodEquivPiFinTwo (\u03b1 \u03b2 : Type u) : \u03b1 \u00d7 \u03b2 \u2243 \u2200 i : Fin 2, ![\u03b1, \u03b2] i :=\n (piFinTwoEquiv (Fin.cons \u03b1 (Fin.cons \u03b2 finZeroElim))).symm\n#align prod_equiv_pi_fin_two prodEquivPiFinTwo\n#align prod_equiv_pi_fin_two_apply prodEquivPiFinTwo_apply\n#align prod_equiv_pi_fin_two_symm_apply prodEquivPiFinTwo_symm_apply\n\n/-- The space of functions `Fin 2 \u2192 \u03b1` is equivalent to `\u03b1 \u00d7 \u03b1`. See also `piFinTwoEquiv` and\n`prodEquivPiFinTwo`. -/\n@[simps (config := .asFn)]\ndef finTwoArrowEquiv (\u03b1 : Type*) : (Fin 2 \u2192 \u03b1) \u2243 \u03b1 \u00d7 \u03b1 :=\n { piFinTwoEquiv fun _ => \u03b1 with invFun := fun x => ![x.1, x.2] }\n#align fin_two_arrow_equiv finTwoArrowEquiv\n#align fin_two_arrow_equiv_symm_apply finTwoArrowEquiv_symm_apply\n#align fin_two_arrow_equiv_apply finTwoArrowEquiv_apply\n\n/-- `\u03a0 i : Fin 2, \u03b1 i` is order equivalent to `\u03b1 0 \u00d7 \u03b1 1`. See also `OrderIso.finTwoArrowEquiv`\nfor a non-dependent version. -/\ndef OrderIso.piFinTwoIso (\u03b1 : Fin 2 \u2192 Type u) [\u2200 i, Preorder (\u03b1 i)] : (\u2200 i, \u03b1 i) \u2243o \u03b1 0 \u00d7 \u03b1 1\n where\n toEquiv := piFinTwoEquiv \u03b1\n map_rel_iff' := Iff.symm Fin.forall_fin_two\n#align order_iso.pi_fin_two_iso OrderIso.piFinTwoIso\n\n/-- The space of functions `Fin 2 \u2192 \u03b1` is order equivalent to `\u03b1 \u00d7 \u03b1`. See also\n`OrderIso.piFinTwoIso`. -/\ndef OrderIso.finTwoArrowIso (\u03b1 : Type*) [Preorder \u03b1] : (Fin 2 \u2192 \u03b1) \u2243o \u03b1 \u00d7 \u03b1 :=\n { OrderIso.piFinTwoIso fun _ => \u03b1 with toEquiv := finTwoArrowEquiv \u03b1 }\n#align order_iso.fin_two_arrow_iso OrderIso.finTwoArrowIso\n\n/-- The 'identity' equivalence between `Fin n` and `Fin m` when `n = m`. -/\ndef finCongr (h : m = n) : Fin m \u2243 Fin n :=\n (Fin.castIso h).toEquiv\n#align fin_congr finCongr\n\n@[simp] theorem finCongr_apply_mk (h : m = n) (k : \u2115) (w : k < m) :\n finCongr h \u27e8k, w\u27e9 = \u27e8k, h \u25b8 w\u27e9 :=\n rfl\n#align fin_congr_apply_mk finCongr_apply_mk\n\n@[simp] theorem finCongr_symm (h : m = n) : (finCongr h).symm = finCongr h.symm :=\n rfl\n#align fin_congr_symm finCongr_symm\n\n@[simp] theorem finCongr_apply_coe (h : m = n) (k : Fin m) : (finCongr h k : \u2115) = k :=\n rfl\n#align fin_congr_apply_coe finCongr_apply_coe\n\ntheorem finCongr_symm_apply_coe (h : m = n) (k : Fin n) : ((finCongr h).symm k : \u2115) = k :=\n rfl\n#align fin_congr_symm_apply_coe finCongr_symm_apply_coe\n\n/-- An equivalence that removes `i` and maps it to `none`.\nThis is a version of `Fin.predAbove` that produces `Option (Fin n)` instead of\nmapping both `i.cast_succ` and `i.succ` to `i`. -/\ndef finSuccEquiv' (i : Fin (n + 1)) : Fin (n + 1) \u2243 Option (Fin n)\n where\n toFun := i.insertNth none some\n invFun x := x.casesOn' i (Fin.succAbove i)\n left_inv x := Fin.succAboveCases i (by simp) (fun j => by simp) x\n right_inv x := by cases x <;> dsimp <;> simp\n#align fin_succ_equiv' finSuccEquiv'\n\n@[simp]\ntheorem finSuccEquiv'_at (i : Fin (n + 1)) : (finSuccEquiv' i) i = none := by\n simp [finSuccEquiv']\n#align fin_succ_equiv'_at finSuccEquiv'_at\n\n@[simp]\ntheorem finSuccEquiv'_succAbove (i : Fin (n + 1)) (j : Fin n) :\n finSuccEquiv' i (i.succAbove j) = some j :=\n @Fin.insertNth_apply_succAbove n (fun _ => Option (Fin n)) i _ _ _\n#align fin_succ_equiv'_succ_above finSuccEquiv'_succAbove\n\ntheorem finSuccEquiv'_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i) (Fin.castSucc m) = m := by\n rw [\u2190 Fin.succAbove_of_castSucc_lt _ _ h, finSuccEquiv'_succAbove]\n#align fin_succ_equiv'_below finSuccEquiv'_below\n\ntheorem finSuccEquiv'_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i) m.succ = some m := by\n rw [\u2190 Fin.succAbove_of_le_castSucc _ _ h, finSuccEquiv'_succAbove]\n#align fin_succ_equiv'_above finSuccEquiv'_above\n\n@[simp]\ntheorem finSuccEquiv'_symm_none (i : Fin (n + 1)) : (finSuccEquiv' i).symm none = i :=\n rfl\n#align fin_succ_equiv'_symm_none finSuccEquiv'_symm_none\n\n@[simp]\ntheorem finSuccEquiv'_symm_some (i : Fin (n + 1)) (j : Fin n) :\n (finSuccEquiv' i).symm (some j) = i.succAbove j :=\n rfl\n#align fin_succ_equiv'_symm_some finSuccEquiv'_symm_some\n\ntheorem finSuccEquiv'_symm_some_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i).symm (some m) = Fin.castSucc m :=\n Fin.succAbove_of_castSucc_lt i m h\n#align fin_succ_equiv'_symm_some_below finSuccEquiv'_symm_some_below\n\ntheorem finSuccEquiv'_symm_some_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i).symm (some m) = m.succ :=\n Fin.succAbove_of_le_castSucc i m h\n#align fin_succ_equiv'_symm_some_above finSuccEquiv'_symm_some_above\n\ntheorem finSuccEquiv'_symm_coe_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i).symm m = Fin.castSucc m :=\n finSuccEquiv'_symm_some_below h\n#align fin_succ_equiv'_symm_coe_below finSuccEquiv'_symm_coe_below\n\ntheorem finSuccEquiv'_symm_coe_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i).symm m = m.succ :=\n finSuccEquiv'_symm_some_above h\n#align fin_succ_equiv'_symm_coe_above finSuccEquiv'_symm_coe_above\n\n/-- Equivalence between `Fin (n + 1)` and `Option (Fin n)`.\nThis is a version of `Fin.pred` that produces `Option (Fin n)` instead of\nrequiring a proof that the input is not `0`. -/\ndef finSuccEquiv (n : \u2115) : Fin (n + 1) \u2243 Option (Fin n) :=\n finSuccEquiv' 0\n#align fin_succ_equiv finSuccEquiv\n\n@[simp]\ntheorem finSuccEquiv_zero : (finSuccEquiv n) 0 = none :=\n rfl\n#align fin_succ_equiv_zero finSuccEquiv_zero\n\n@[simp]\ntheorem finSuccEquiv_succ (m : Fin n) : (finSuccEquiv n) m.succ = some m :=\n finSuccEquiv'_above (Fin.zero_le _)\n#align fin_succ_equiv_succ finSuccEquiv_succ\n\n@[simp]\ntheorem finSuccEquiv_symm_none : (finSuccEquiv n).symm none = 0 :=\n finSuccEquiv'_symm_none _\n#align fin_succ_equiv_symm_none finSuccEquiv_symm_none\n\n@[simp]\ntheorem finSuccEquiv_symm_some (m : Fin n) : (finSuccEquiv n).symm (some m) = m.succ :=\n congr_fun Fin.succAbove_zero m\n#align fin_succ_equiv_symm_some finSuccEquiv_symm_some\n#align fin_succ_equiv_symm_coe finSuccEquiv_symm_some\n\n/-- The equiv version of `Fin.predAbove_zero`. -/\ntheorem finSuccEquiv'_zero : finSuccEquiv' (0 : Fin (n + 1)) = finSuccEquiv n :=\n rfl\n#align fin_succ_equiv'_zero finSuccEquiv'_zero\n\ntheorem finSuccEquiv'_last_apply_castSucc (i : Fin n) :\n finSuccEquiv' (Fin.last n) (Fin.castSucc i) = i := by\n rw [\u2190 Fin.succAbove_last, finSuccEquiv'_succAbove]\n\ntheorem finSuccEquiv'_last_apply {i : Fin (n + 1)} (h : i \u2260 Fin.last n) :\n finSuccEquiv' (Fin.last n) i = Fin.castLT i (Fin.val_lt_last h) := by\n rcases Fin.exists_castSucc_eq.2 h with \u27e8i, rfl\u27e9\n rw [finSuccEquiv'_last_apply_castSucc]\n rfl\n#align fin_succ_equiv'_last_apply finSuccEquiv'_last_apply\n\ntheorem finSuccEquiv'_ne_last_apply {i j : Fin (n + 1)} (hi : i \u2260 Fin.last n) (hj : j \u2260 i) :\n finSuccEquiv' i j = (i.castLT (Fin.val_lt_last hi)).predAbove j := by\n rcases Fin.exists_succAbove_eq hj with \u27e8j, rfl\u27e9\n rcases Fin.exists_castSucc_eq.2 hi with \u27e8i, rfl\u27e9\n simp\n#align fin_succ_equiv'_ne_last_apply finSuccEquiv'_ne_last_apply\n\n/-- `Fin.succAbove` as an order isomorphism between `Fin n` and `{x : Fin (n + 1) // x \u2260 p}`. -/\ndef finSuccAboveEquiv (p : Fin (n + 1)) : Fin n \u2243o { x : Fin (n + 1) // x \u2260 p } :=\n { Equiv.optionSubtype p \u27e8(finSuccEquiv' p).symm, rfl\u27e9 with\n map_rel_iff' := p.succAboveEmb.map_rel_iff' }\n#align fin_succ_above_equiv finSuccAboveEquiv\n\ntheorem finSuccAboveEquiv_apply (p : Fin (n + 1)) (i : Fin n) :\n finSuccAboveEquiv p i = \u27e8p.succAbove i, p.succAbove_ne i\u27e9 :=\n rfl\n#align fin_succ_above_equiv_apply finSuccAboveEquiv_apply\n\ntheorem finSuccAboveEquiv_symm_apply_last (x : { x : Fin (n + 1) // x \u2260 Fin.last n }) :\n (finSuccAboveEquiv (Fin.last n)).symm x = Fin.castLT x.1 (Fin.val_lt_last x.2) := by\n rw [\u2190 Option.some_inj]\n simpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_last_apply x.property\n#align fin_succ_above_equiv_symm_apply_last finSuccAboveEquiv_symm_apply_last\n\ntheorem finSuccAboveEquiv_symm_apply_ne_last {p : Fin (n + 1)} (h : p \u2260 Fin.last n)\n (x : { x : Fin (n + 1) // x \u2260 p }) :\n (finSuccAboveEquiv p).symm x = (p.castLT (Fin.val_lt_last h)).predAbove x := by\n rw [\u2190 Option.some_inj]\n simpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_ne_last_apply h x.property\n#align fin_succ_above_equiv_symm_apply_ne_last finSuccAboveEquiv_symm_apply_ne_last\n\n/-- `Equiv` between `Fin (n + 1)` and `Option (Fin n)` sending `Fin.last n` to `none` -/\ndef finSuccEquivLast : Fin (n + 1) \u2243 Option (Fin n) :=\n finSuccEquiv' (Fin.last n)\n#align fin_succ_equiv_last finSuccEquivLast\n\n@[simp]\ntheorem finSuccEquivLast_castSucc (i : Fin n) : finSuccEquivLast (Fin.castSucc i) = some i :=\n finSuccEquiv'_below i.2\n#align fin_succ_equiv_last_cast_succ finSuccEquivLast_castSucc\n\n@[simp]\ntheorem finSuccEquivLast_last : finSuccEquivLast (Fin.last n) = none := by\n simp [finSuccEquivLast]\n#align fin_succ_equiv_last_last finSuccEquivLast_last\n\n@[simp]\ntheorem finSuccEquivLast_symm_some (i : Fin n) :\n finSuccEquivLast.symm (some i) = Fin.castSucc i :=\n finSuccEquiv'_symm_some_below i.2\n#align fin_succ_equiv_last_symm_some finSuccEquivLast_symm_some\n#align fin_succ_equiv_last_symm_coe finSuccEquivLast_symm_some\n\n@[simp] theorem finSuccEquivLast_symm_none : finSuccEquivLast.symm none = Fin.last n :=\n finSuccEquiv'_symm_none _\n#align fin_succ_equiv_last_symm_none finSuccEquivLast_symm_none\n\n/-- Equivalence between `\u03a0 j : Fin (n + 1), \u03b1 j` and `\u03b1 i \u00d7 \u03a0 j : Fin n, \u03b1 (Fin.succAbove i j)`. -/\n@[simps (config := .asFn)]\ndef Equiv.piFinSuccAbove (\u03b1 : Fin (n + 1) \u2192 Type u) (i : Fin (n + 1)) :\n (\u2200 j, \u03b1 j) \u2243 \u03b1 i \u00d7 \u2200 j, \u03b1 (i.succAbove j) where\n toFun f := i.extractNth f\n invFun f := i.insertNth f.1 f.2\n left_inv f := by simp\n right_inv f := by simp\n#align equiv.pi_fin_succ_above_equiv Equiv.piFinSuccAbove\n#align equiv.pi_fin_succ_above_equiv_apply Equiv.piFinSuccAbove_apply\n#align equiv.pi_fin_succ_above_equiv_symm_apply Equiv.piFinSuccAbove_symm_apply\n\n/-- Order isomorphism between `\u03a0 j : Fin (n + 1), \u03b1 j` and\n`\u03b1 i \u00d7 \u03a0 j : Fin n, \u03b1 (Fin.succAbove i j)`. -/\ndef OrderIso.piFinSuccAboveIso (\u03b1 : Fin (n + 1) \u2192 Type u) [\u2200 i, LE (\u03b1 i)]\n (i : Fin (n + 1)) : (\u2200 j, \u03b1 j) \u2243o \u03b1 i \u00d7 \u2200 j, \u03b1 (i.succAbove j) where\n toEquiv := Equiv.piFinSuccAbove \u03b1 i\n map_rel_iff' := Iff.symm i.forall_iff_succAbove\n#align order_iso.pi_fin_succ_above_iso OrderIso.piFinSuccAboveIso\n\n/-- Equivalence between `Fin (n + 1) \u2192 \u03b2` and `\u03b2 \u00d7 (Fin n \u2192 \u03b2)`. -/\n@[simps! (config := .asFn)]\ndef Equiv.piFinSucc (n : \u2115) (\u03b2 : Type u) : (Fin (n + 1) \u2192 \u03b2) \u2243 \u03b2 \u00d7 (Fin n \u2192 \u03b2) :=\n Equiv.piFinSuccAbove (fun _ => \u03b2) 0\n#align equiv.pi_fin_succ Equiv.piFinSucc\n#align equiv.pi_fin_succ_apply Equiv.piFinSucc_apply\n#align equiv.pi_fin_succ_symm_apply Equiv.piFinSucc_symm_apply\n\n/-- An embedding `e : Fin (n+1) \u21aa \u03b9` corresponds to an embedding `f : Fin n \u21aa \u03b9` (corresponding\nthe last `n` coordinates of `e`) together with a value not taken by `f` (corresponding to `e 0`). -/\ndef Equiv.embeddingFinSucc (n : \u2115) (\u03b9 : Type*) :\n (Fin (n+1) \u21aa \u03b9) \u2243 (\u03a3 (e : Fin n \u21aa \u03b9), {i // i \u2209 Set.range e}) :=\n ((finSuccEquiv n).embeddingCongr (Equiv.refl \u03b9)).trans\n (Function.Embedding.optionEmbeddingEquiv (Fin n) \u03b9)\n\n@[simp] lemma Equiv.embeddingFinSucc_fst {n : \u2115} {\u03b9 : Type*} (e : Fin (n+1) \u21aa \u03b9) :\n ((Equiv.embeddingFinSucc n \u03b9 e).1 : Fin n \u2192 \u03b9) = e \u2218 Fin.succ := rfl\n\n", "theoremStatement": "@[simp] lemma Equiv.embeddingFinSucc_snd {n : \u2115} {\u03b9 : Type*} (e : Fin (n+1) \u21aa \u03b9) :\n ((Equiv.embeddingFinSucc n \u03b9 e).2 : \u03b9) = e 0", "theoremName": "Equiv.embeddingFinSucc_snd", "fileCreated": {"commit": "885e8510c4", "date": "2023-01-23"}, "theoremCreated": {"commit": "98843f5bd5", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Logic/Equiv/Fin.lean", "positionMetadata": {"lineInFile": 328, "tokenPositionInFile": 13848, "theoremPositionInFile": 50}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "rfl", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 3}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.Algebra.Ring.Int\nimport Mathlib.Data.Fin.VecNotation\nimport Mathlib.Logic.Equiv.Defs\nimport Mathlib.Logic.Embedding.Set\n\n#align_import logic.equiv.fin from \"leanprover-community/mathlib\"@\"bd835ef554f37ef9b804f0903089211f89cb370b\"\n\n/-!\n# Equivalences for `Fin n`\n-/\n\nuniverse u\n\nvariable {m n : \u2115}\n\n/-- Equivalence between `Fin 0` and `Empty`. -/\ndef finZeroEquiv : Fin 0 \u2243 Empty :=\n Equiv.equivEmpty _\n#align fin_zero_equiv finZeroEquiv\n\n/-- Equivalence between `Fin 0` and `PEmpty`. -/\ndef finZeroEquiv' : Fin 0 \u2243 PEmpty.{u} :=\n Equiv.equivPEmpty _\n#align fin_zero_equiv' finZeroEquiv'\n\n/-- Equivalence between `Fin 1` and `Unit`. -/\ndef finOneEquiv : Fin 1 \u2243 Unit :=\n Equiv.equivPUnit _\n#align fin_one_equiv finOneEquiv\n\n/-- Equivalence between `Fin 2` and `Bool`. -/\ndef finTwoEquiv : Fin 2 \u2243 Bool where\n toFun := ![false, true]\n invFun b := b.casesOn 0 1\n left_inv := Fin.forall_fin_two.2 <| by simp\n right_inv := Bool.forall_bool.2 <| by simp\n#align fin_two_equiv finTwoEquiv\n\n/-- `\u03a0 i : Fin 2, \u03b1 i` is equivalent to `\u03b1 0 \u00d7 \u03b1 1`. See also `finTwoArrowEquiv` for a\nnon-dependent version and `prodEquivPiFinTwo` for a version with inputs `\u03b1 \u03b2 : Type u`. -/\n@[simps (config := .asFn)]\ndef piFinTwoEquiv (\u03b1 : Fin 2 \u2192 Type u) : (\u2200 i, \u03b1 i) \u2243 \u03b1 0 \u00d7 \u03b1 1\n where\n toFun f := (f 0, f 1)\n invFun p := Fin.cons p.1 <| Fin.cons p.2 finZeroElim\n left_inv _ := funext <| Fin.forall_fin_two.2 \u27e8rfl, rfl\u27e9\n right_inv := fun _ => rfl\n#align pi_fin_two_equiv piFinTwoEquiv\n#align pi_fin_two_equiv_symm_apply piFinTwoEquiv_symm_apply\n#align pi_fin_two_equiv_apply piFinTwoEquiv_apply\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Fin.preimage_apply_01_prod {\u03b1 : Fin 2 \u2192 Type u} (s : Set (\u03b1 0)) (t : Set (\u03b1 1)) :\n (fun f : \u2200 i, \u03b1 i => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t =\n Set.pi Set.univ (Fin.cons s <| Fin.cons t finZeroElim) := by\n ext f\n simp [Fin.forall_fin_two]\n#align fin.preimage_apply_01_prod Fin.preimage_apply_01_prod\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Fin.preimage_apply_01_prod' {\u03b1 : Type u} (s t : Set \u03b1) :\n (fun f : Fin 2 \u2192 \u03b1 => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t = Set.pi Set.univ ![s, t] :=\n @Fin.preimage_apply_01_prod (fun _ => \u03b1) s t\n#align fin.preimage_apply_01_prod' Fin.preimage_apply_01_prod'\n\n/-- A product space `\u03b1 \u00d7 \u03b2` is equivalent to the space `\u03a0 i : Fin 2, \u03b3 i`, where\n`\u03b3 = Fin.cons \u03b1 (Fin.cons \u03b2 finZeroElim)`. See also `piFinTwoEquiv` and\n`finTwoArrowEquiv`. -/\n@[simps! (config := .asFn)]\ndef prodEquivPiFinTwo (\u03b1 \u03b2 : Type u) : \u03b1 \u00d7 \u03b2 \u2243 \u2200 i : Fin 2, ![\u03b1, \u03b2] i :=\n (piFinTwoEquiv (Fin.cons \u03b1 (Fin.cons \u03b2 finZeroElim))).symm\n#align prod_equiv_pi_fin_two prodEquivPiFinTwo\n#align prod_equiv_pi_fin_two_apply prodEquivPiFinTwo_apply\n#align prod_equiv_pi_fin_two_symm_apply prodEquivPiFinTwo_symm_apply\n\n/-- The space of functions `Fin 2 \u2192 \u03b1` is equivalent to `\u03b1 \u00d7 \u03b1`. See also `piFinTwoEquiv` and\n`prodEquivPiFinTwo`. -/\n@[simps (config := .asFn)]\ndef finTwoArrowEquiv (\u03b1 : Type*) : (Fin 2 \u2192 \u03b1) \u2243 \u03b1 \u00d7 \u03b1 :=\n { piFinTwoEquiv fun _ => \u03b1 with invFun := fun x => ![x.1, x.2] }\n#align fin_two_arrow_equiv finTwoArrowEquiv\n#align fin_two_arrow_equiv_symm_apply finTwoArrowEquiv_symm_apply\n#align fin_two_arrow_equiv_apply finTwoArrowEquiv_apply\n\n/-- `\u03a0 i : Fin 2, \u03b1 i` is order equivalent to `\u03b1 0 \u00d7 \u03b1 1`. See also `OrderIso.finTwoArrowEquiv`\nfor a non-dependent version. -/\ndef OrderIso.piFinTwoIso (\u03b1 : Fin 2 \u2192 Type u) [\u2200 i, Preorder (\u03b1 i)] : (\u2200 i, \u03b1 i) \u2243o \u03b1 0 \u00d7 \u03b1 1\n where\n toEquiv := piFinTwoEquiv \u03b1\n map_rel_iff' := Iff.symm Fin.forall_fin_two\n#align order_iso.pi_fin_two_iso OrderIso.piFinTwoIso\n\n/-- The space of functions `Fin 2 \u2192 \u03b1` is order equivalent to `\u03b1 \u00d7 \u03b1`. See also\n`OrderIso.piFinTwoIso`. -/\ndef OrderIso.finTwoArrowIso (\u03b1 : Type*) [Preorder \u03b1] : (Fin 2 \u2192 \u03b1) \u2243o \u03b1 \u00d7 \u03b1 :=\n { OrderIso.piFinTwoIso fun _ => \u03b1 with toEquiv := finTwoArrowEquiv \u03b1 }\n#align order_iso.fin_two_arrow_iso OrderIso.finTwoArrowIso\n\n/-- The 'identity' equivalence between `Fin n` and `Fin m` when `n = m`. -/\ndef finCongr (h : m = n) : Fin m \u2243 Fin n :=\n (Fin.castIso h).toEquiv\n#align fin_congr finCongr\n\n@[simp] theorem finCongr_apply_mk (h : m = n) (k : \u2115) (w : k < m) :\n finCongr h \u27e8k, w\u27e9 = \u27e8k, h \u25b8 w\u27e9 :=\n rfl\n#align fin_congr_apply_mk finCongr_apply_mk\n\n@[simp] theorem finCongr_symm (h : m = n) : (finCongr h).symm = finCongr h.symm :=\n rfl\n#align fin_congr_symm finCongr_symm\n\n@[simp] theorem finCongr_apply_coe (h : m = n) (k : Fin m) : (finCongr h k : \u2115) = k :=\n rfl\n#align fin_congr_apply_coe finCongr_apply_coe\n\ntheorem finCongr_symm_apply_coe (h : m = n) (k : Fin n) : ((finCongr h).symm k : \u2115) = k :=\n rfl\n#align fin_congr_symm_apply_coe finCongr_symm_apply_coe\n\n/-- An equivalence that removes `i` and maps it to `none`.\nThis is a version of `Fin.predAbove` that produces `Option (Fin n)` instead of\nmapping both `i.cast_succ` and `i.succ` to `i`. -/\ndef finSuccEquiv' (i : Fin (n + 1)) : Fin (n + 1) \u2243 Option (Fin n)\n where\n toFun := i.insertNth none some\n invFun x := x.casesOn' i (Fin.succAbove i)\n left_inv x := Fin.succAboveCases i (by simp) (fun j => by simp) x\n right_inv x := by cases x <;> dsimp <;> simp\n#align fin_succ_equiv' finSuccEquiv'\n\n@[simp]\ntheorem finSuccEquiv'_at (i : Fin (n + 1)) : (finSuccEquiv' i) i = none := by\n simp [finSuccEquiv']\n#align fin_succ_equiv'_at finSuccEquiv'_at\n\n@[simp]\ntheorem finSuccEquiv'_succAbove (i : Fin (n + 1)) (j : Fin n) :\n finSuccEquiv' i (i.succAbove j) = some j :=\n @Fin.insertNth_apply_succAbove n (fun _ => Option (Fin n)) i _ _ _\n#align fin_succ_equiv'_succ_above finSuccEquiv'_succAbove\n\ntheorem finSuccEquiv'_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i) (Fin.castSucc m) = m := by\n rw [\u2190 Fin.succAbove_of_castSucc_lt _ _ h, finSuccEquiv'_succAbove]\n#align fin_succ_equiv'_below finSuccEquiv'_below\n\ntheorem finSuccEquiv'_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i) m.succ = some m := by\n rw [\u2190 Fin.succAbove_of_le_castSucc _ _ h, finSuccEquiv'_succAbove]\n#align fin_succ_equiv'_above finSuccEquiv'_above\n\n@[simp]\ntheorem finSuccEquiv'_symm_none (i : Fin (n + 1)) : (finSuccEquiv' i).symm none = i :=\n rfl\n#align fin_succ_equiv'_symm_none finSuccEquiv'_symm_none\n\n@[simp]\ntheorem finSuccEquiv'_symm_some (i : Fin (n + 1)) (j : Fin n) :\n (finSuccEquiv' i).symm (some j) = i.succAbove j :=\n rfl\n#align fin_succ_equiv'_symm_some finSuccEquiv'_symm_some\n\ntheorem finSuccEquiv'_symm_some_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i).symm (some m) = Fin.castSucc m :=\n Fin.succAbove_of_castSucc_lt i m h\n#align fin_succ_equiv'_symm_some_below finSuccEquiv'_symm_some_below\n\ntheorem finSuccEquiv'_symm_some_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i).symm (some m) = m.succ :=\n Fin.succAbove_of_le_castSucc i m h\n#align fin_succ_equiv'_symm_some_above finSuccEquiv'_symm_some_above\n\ntheorem finSuccEquiv'_symm_coe_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :\n (finSuccEquiv' i).symm m = Fin.castSucc m :=\n finSuccEquiv'_symm_some_below h\n#align fin_succ_equiv'_symm_coe_below finSuccEquiv'_symm_coe_below\n\ntheorem finSuccEquiv'_symm_coe_above {i : Fin (n + 1)} {m : Fin n} (h : i \u2264 Fin.castSucc m) :\n (finSuccEquiv' i).symm m = m.succ :=\n finSuccEquiv'_symm_some_above h\n#align fin_succ_equiv'_symm_coe_above finSuccEquiv'_symm_coe_above\n\n/-- Equivalence between `Fin (n + 1)` and `Option (Fin n)`.\nThis is a version of `Fin.pred` that produces `Option (Fin n)` instead of\nrequiring a proof that the input is not `0`. -/\ndef finSuccEquiv (n : \u2115) : Fin (n + 1) \u2243 Option (Fin n) :=\n finSuccEquiv' 0\n#align fin_succ_equiv finSuccEquiv\n\n@[simp]\ntheorem finSuccEquiv_zero : (finSuccEquiv n) 0 = none :=\n rfl\n#align fin_succ_equiv_zero finSuccEquiv_zero\n\n@[simp]\ntheorem finSuccEquiv_succ (m : Fin n) : (finSuccEquiv n) m.succ = some m :=\n finSuccEquiv'_above (Fin.zero_le _)\n#align fin_succ_equiv_succ finSuccEquiv_succ\n\n@[simp]\ntheorem finSuccEquiv_symm_none : (finSuccEquiv n).symm none = 0 :=\n finSuccEquiv'_symm_none _\n#align fin_succ_equiv_symm_none finSuccEquiv_symm_none\n\n@[simp]\ntheorem finSuccEquiv_symm_some (m : Fin n) : (finSuccEquiv n).symm (some m) = m.succ :=\n congr_fun Fin.succAbove_zero m\n#align fin_succ_equiv_symm_some finSuccEquiv_symm_some\n#align fin_succ_equiv_symm_coe finSuccEquiv_symm_some\n\n/-- The equiv version of `Fin.predAbove_zero`. -/\ntheorem finSuccEquiv'_zero : finSuccEquiv' (0 : Fin (n + 1)) = finSuccEquiv n :=\n rfl\n#align fin_succ_equiv'_zero finSuccEquiv'_zero\n\ntheorem finSuccEquiv'_last_apply_castSucc (i : Fin n) :\n finSuccEquiv' (Fin.last n) (Fin.castSucc i) = i := by\n rw [\u2190 Fin.succAbove_last, finSuccEquiv'_succAbove]\n\ntheorem finSuccEquiv'_last_apply {i : Fin (n + 1)} (h : i \u2260 Fin.last n) :\n finSuccEquiv' (Fin.last n) i = Fin.castLT i (Fin.val_lt_last h) := by\n rcases Fin.exists_castSucc_eq.2 h with \u27e8i, rfl\u27e9\n rw [finSuccEquiv'_last_apply_castSucc]\n rfl\n#align fin_succ_equiv'_last_apply finSuccEquiv'_last_apply\n\ntheorem finSuccEquiv'_ne_last_apply {i j : Fin (n + 1)} (hi : i \u2260 Fin.last n) (hj : j \u2260 i) :\n finSuccEquiv' i j = (i.castLT (Fin.val_lt_last hi)).predAbove j := by\n rcases Fin.exists_succAbove_eq hj with \u27e8j, rfl\u27e9\n rcases Fin.exists_castSucc_eq.2 hi with \u27e8i, rfl\u27e9\n simp\n#align fin_succ_equiv'_ne_last_apply finSuccEquiv'_ne_last_apply\n\n/-- `Fin.succAbove` as an order isomorphism between `Fin n` and `{x : Fin (n + 1) // x \u2260 p}`. -/\ndef finSuccAboveEquiv (p : Fin (n + 1)) : Fin n \u2243o { x : Fin (n + 1) // x \u2260 p } :=\n { Equiv.optionSubtype p \u27e8(finSuccEquiv' p).symm, rfl\u27e9 with\n map_rel_iff' := p.succAboveEmb.map_rel_iff' }\n#align fin_succ_above_equiv finSuccAboveEquiv\n\ntheorem finSuccAboveEquiv_apply (p : Fin (n + 1)) (i : Fin n) :\n finSuccAboveEquiv p i = \u27e8p.succAbove i, p.succAbove_ne i\u27e9 :=\n rfl\n#align fin_succ_above_equiv_apply finSuccAboveEquiv_apply\n\ntheorem finSuccAboveEquiv_symm_apply_last (x : { x : Fin (n + 1) // x \u2260 Fin.last n }) :\n (finSuccAboveEquiv (Fin.last n)).symm x = Fin.castLT x.1 (Fin.val_lt_last x.2) := by\n rw [\u2190 Option.some_inj]\n simpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_last_apply x.property\n#align fin_succ_above_equiv_symm_apply_last finSuccAboveEquiv_symm_apply_last\n\ntheorem finSuccAboveEquiv_symm_apply_ne_last {p : Fin (n + 1)} (h : p \u2260 Fin.last n)\n (x : { x : Fin (n + 1) // x \u2260 p }) :\n (finSuccAboveEquiv p).symm x = (p.castLT (Fin.val_lt_last h)).predAbove x := by\n rw [\u2190 Option.some_inj]\n simpa [finSuccAboveEquiv, OrderIso.symm] using finSuccEquiv'_ne_last_apply h x.property\n#align fin_succ_above_equiv_symm_apply_ne_last finSuccAboveEquiv_symm_apply_ne_last\n\n/-- `Equiv` between `Fin (n + 1)` and `Option (Fin n)` sending `Fin.last n` to `none` -/\ndef finSuccEquivLast : Fin (n + 1) \u2243 Option (Fin n) :=\n finSuccEquiv' (Fin.last n)\n#align fin_succ_equiv_last finSuccEquivLast\n\n@[simp]\ntheorem finSuccEquivLast_castSucc (i : Fin n) : finSuccEquivLast (Fin.castSucc i) = some i :=\n finSuccEquiv'_below i.2\n#align fin_succ_equiv_last_cast_succ finSuccEquivLast_castSucc\n\n@[simp]\ntheorem finSuccEquivLast_last : finSuccEquivLast (Fin.last n) = none := by\n simp [finSuccEquivLast]\n#align fin_succ_equiv_last_last finSuccEquivLast_last\n\n@[simp]\ntheorem finSuccEquivLast_symm_some (i : Fin n) :\n finSuccEquivLast.symm (some i) = Fin.castSucc i :=\n finSuccEquiv'_symm_some_below i.2\n#align fin_succ_equiv_last_symm_some finSuccEquivLast_symm_some\n#align fin_succ_equiv_last_symm_coe finSuccEquivLast_symm_some\n\n@[simp] theorem finSuccEquivLast_symm_none : finSuccEquivLast.symm none = Fin.last n :=\n finSuccEquiv'_symm_none _\n#align fin_succ_equiv_last_symm_none finSuccEquivLast_symm_none\n\n/-- Equivalence between `\u03a0 j : Fin (n + 1), \u03b1 j` and `\u03b1 i \u00d7 \u03a0 j : Fin n, \u03b1 (Fin.succAbove i j)`. -/\n@[simps (config := .asFn)]\ndef Equiv.piFinSuccAbove (\u03b1 : Fin (n + 1) \u2192 Type u) (i : Fin (n + 1)) :\n (\u2200 j, \u03b1 j) \u2243 \u03b1 i \u00d7 \u2200 j, \u03b1 (i.succAbove j) where\n toFun f := i.extractNth f\n invFun f := i.insertNth f.1 f.2\n left_inv f := by simp\n right_inv f := by simp\n#align equiv.pi_fin_succ_above_equiv Equiv.piFinSuccAbove\n#align equiv.pi_fin_succ_above_equiv_apply Equiv.piFinSuccAbove_apply\n#align equiv.pi_fin_succ_above_equiv_symm_apply Equiv.piFinSuccAbove_symm_apply\n\n/-- Order isomorphism between `\u03a0 j : Fin (n + 1), \u03b1 j` and\n`\u03b1 i \u00d7 \u03a0 j : Fin n, \u03b1 (Fin.succAbove i j)`. -/\ndef OrderIso.piFinSuccAboveIso (\u03b1 : Fin (n + 1) \u2192 Type u) [\u2200 i, LE (\u03b1 i)]\n (i : Fin (n + 1)) : (\u2200 j, \u03b1 j) \u2243o \u03b1 i \u00d7 \u2200 j, \u03b1 (i.succAbove j) where\n toEquiv := Equiv.piFinSuccAbove \u03b1 i\n map_rel_iff' := Iff.symm i.forall_iff_succAbove\n#align order_iso.pi_fin_succ_above_iso OrderIso.piFinSuccAboveIso\n\n/-- Equivalence between `Fin (n + 1) \u2192 \u03b2` and `\u03b2 \u00d7 (Fin n \u2192 \u03b2)`. -/\n@[simps! (config := .asFn)]\ndef Equiv.piFinSucc (n : \u2115) (\u03b2 : Type u) : (Fin (n + 1) \u2192 \u03b2) \u2243 \u03b2 \u00d7 (Fin n \u2192 \u03b2) :=\n Equiv.piFinSuccAbove (fun _ => \u03b2) 0\n#align equiv.pi_fin_succ Equiv.piFinSucc\n#align equiv.pi_fin_succ_apply Equiv.piFinSucc_apply\n#align equiv.pi_fin_succ_symm_apply Equiv.piFinSucc_symm_apply\n\n/-- An embedding `e : Fin (n+1) \u21aa \u03b9` corresponds to an embedding `f : Fin n \u21aa \u03b9` (corresponding\nthe last `n` coordinates of `e`) together with a value not taken by `f` (corresponding to `e 0`). -/\ndef Equiv.embeddingFinSucc (n : \u2115) (\u03b9 : Type*) :\n (Fin (n+1) \u21aa \u03b9) \u2243 (\u03a3 (e : Fin n \u21aa \u03b9), {i // i \u2209 Set.range e}) :=\n ((finSuccEquiv n).embeddingCongr (Equiv.refl \u03b9)).trans\n (Function.Embedding.optionEmbeddingEquiv (Fin n) \u03b9)\n\n@[simp] lemma Equiv.embeddingFinSucc_fst {n : \u2115} {\u03b9 : Type*} (e : Fin (n+1) \u21aa \u03b9) :\n ((Equiv.embeddingFinSucc n \u03b9 e).1 : Fin n \u2192 \u03b9) = e \u2218 Fin.succ := rfl\n\n@[simp] lemma Equiv.embeddingFinSucc_snd {n : \u2115} {\u03b9 : Type*} (e : Fin (n+1) \u21aa \u03b9) :\n ((Equiv.embeddingFinSucc n \u03b9 e).2 : \u03b9) = e 0 := rfl\n\n", "theoremStatement": "@[simp] lemma Equiv.coe_embeddingFinSucc_symm {n : \u2115} {\u03b9 : Type*}\n (f : \u03a3 (e : Fin n \u21aa \u03b9), {i // i \u2209 Set.range e}) :\n ((Equiv.embeddingFinSucc n \u03b9).symm f : Fin (n + 1) \u2192 \u03b9) = Fin.cons f.2.1 f.1", "theoremName": "Equiv.coe_embeddingFinSucc_symm", "fileCreated": {"commit": "885e8510c4", "date": "2023-01-23"}, "theoremCreated": {"commit": "98843f5bd5", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Logic/Equiv/Fin.lean", "positionMetadata": {"lineInFile": 331, "tokenPositionInFile": 13988, "theoremPositionInFile": 51}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n ext i\n exact Fin.cases rfl (fun j \u21a6 rfl) i", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 48}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Order.Cover\n\n#align_import algebra.support from \"leanprover-community/mathlib\"@\"29cb56a7b35f72758b05a30490e1f10bd62c35c1\"\n\n/-!\n# Support of a function\n\nIn this file we define `Function.support f = {x | f x \u2260 0}` and prove its basic properties.\nWe also define `Function.mulSupport f = {x | f x \u2260 1}`.\n-/\n\n\nopen Set\n\nnamespace Function\n\nvariable {\u03b1 \u03b2 A B M N P R S G M\u2080 G\u2080 : Type*} {\u03b9 : Sort*}\n\nsection One\n\nvariable [One M] [One N] [One P]\n\n/-- `support` of a function is the set of points `x` such that `f x \u2260 0`. -/\ndef support [Zero A] (f : \u03b1 \u2192 A) : Set \u03b1 :=\n { x | f x \u2260 0 }\n#align function.support Function.support\n\n/-- `mulSupport` of a function is the set of points `x` such that `f x \u2260 1`. -/\n@[to_additive existing]\ndef mulSupport (f : \u03b1 \u2192 M) : Set \u03b1 :=\n { x | f x \u2260 1 }\n#align function.mul_support Function.mulSupport\n\n@[to_additive]\ntheorem mulSupport_eq_preimage (f : \u03b1 \u2192 M) : mulSupport f = f \u207b\u00b9' {1}\u1d9c :=\n rfl\n#align function.mul_support_eq_preimage Function.mulSupport_eq_preimage\n#align function.support_eq_preimage Function.support_eq_preimage\n\n@[to_additive]\ntheorem nmem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2209 mulSupport f \u2194 f x = 1 :=\n not_not\n#align function.nmem_mul_support Function.nmem_mulSupport\n#align function.nmem_support Function.nmem_support\n\n@[to_additive]\ntheorem compl_mulSupport {f : \u03b1 \u2192 M} : (mulSupport f)\u1d9c = { x | f x = 1 } :=\n ext fun _ => nmem_mulSupport\n#align function.compl_mul_support Function.compl_mulSupport\n#align function.compl_support Function.compl_support\n\n@[to_additive (attr := simp)]\ntheorem mem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2208 mulSupport f \u2194 f x \u2260 1 :=\n Iff.rfl\n#align function.mem_mul_support Function.mem_mulSupport\n#align function.mem_support Function.mem_support\n\n@[to_additive (attr := simp)]\ntheorem mulSupport_subset_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} : mulSupport f \u2286 s \u2194 \u2200 x, f x \u2260 1 \u2192 x \u2208 s :=\n Iff.rfl\n#align function.mul_support_subset_iff Function.mulSupport_subset_iff\n#align function.support_subset_iff Function.support_subset_iff\n\n@[to_additive]\ntheorem mulSupport_subset_iff' {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f \u2286 s \u2194 \u2200 x \u2209 s, f x = 1 :=\n forall_congr' fun _ => not_imp_comm\n#align function.mul_support_subset_iff' Function.mulSupport_subset_iff'\n#align function.support_subset_iff' Function.support_subset_iff'\n\n@[to_additive]\ntheorem mulSupport_eq_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f = s \u2194 (\u2200 x, x \u2208 s \u2192 f x \u2260 1) \u2227 \u2200 x, x \u2209 s \u2192 f x = 1 := by\n simp (config := { contextual := true }) only [ext_iff, mem_mulSupport, ne_eq, iff_def,\n not_imp_comm, and_comm, forall_and]\n#align function.mul_support_eq_iff Function.mulSupport_eq_iff\n#align function.support_eq_iff Function.support_eq_iff\n\n", "theoremStatement": "@[to_additive]\ntheorem ext_iff_mulSupport {f g : \u03b1 \u2192 M} :\n f = g \u2194 f.mulSupport = g.mulSupport \u2227 \u2200 x \u2208 f.mulSupport, f x = g x", "theoremName": "ext_iff_mulSupport", "fileCreated": {"commit": "82ddb54f6c", "date": "2023-12-13"}, "theoremCreated": {"commit": "1342b9e13a", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Function/Support.lean", "positionMetadata": {"lineInFile": 85, "tokenPositionInFile": 2889, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "\u27e8fun h \u21a6 h \u25b8 \u27e8rfl, fun _ _ \u21a6 rfl\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 funext fun x \u21a6 by\n if hx : x \u2208 f.mulSupport then exact h\u2082 x hx\n else rw [nmem_mulSupport.1 hx, nmem_mulSupport.1 (mt (Set.ext_iff.1 h\u2081 x).2 hx)]\u27e9", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 201}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Order.Cover\n\n#align_import algebra.support from \"leanprover-community/mathlib\"@\"29cb56a7b35f72758b05a30490e1f10bd62c35c1\"\n\n/-!\n# Support of a function\n\nIn this file we define `Function.support f = {x | f x \u2260 0}` and prove its basic properties.\nWe also define `Function.mulSupport f = {x | f x \u2260 1}`.\n-/\n\n\nopen Set\n\nnamespace Function\n\nvariable {\u03b1 \u03b2 A B M N P R S G M\u2080 G\u2080 : Type*} {\u03b9 : Sort*}\n\nsection One\n\nvariable [One M] [One N] [One P]\n\n/-- `support` of a function is the set of points `x` such that `f x \u2260 0`. -/\ndef support [Zero A] (f : \u03b1 \u2192 A) : Set \u03b1 :=\n { x | f x \u2260 0 }\n#align function.support Function.support\n\n/-- `mulSupport` of a function is the set of points `x` such that `f x \u2260 1`. -/\n@[to_additive existing]\ndef mulSupport (f : \u03b1 \u2192 M) : Set \u03b1 :=\n { x | f x \u2260 1 }\n#align function.mul_support Function.mulSupport\n\n@[to_additive]\ntheorem mulSupport_eq_preimage (f : \u03b1 \u2192 M) : mulSupport f = f \u207b\u00b9' {1}\u1d9c :=\n rfl\n#align function.mul_support_eq_preimage Function.mulSupport_eq_preimage\n#align function.support_eq_preimage Function.support_eq_preimage\n\n@[to_additive]\ntheorem nmem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2209 mulSupport f \u2194 f x = 1 :=\n not_not\n#align function.nmem_mul_support Function.nmem_mulSupport\n#align function.nmem_support Function.nmem_support\n\n@[to_additive]\ntheorem compl_mulSupport {f : \u03b1 \u2192 M} : (mulSupport f)\u1d9c = { x | f x = 1 } :=\n ext fun _ => nmem_mulSupport\n#align function.compl_mul_support Function.compl_mulSupport\n#align function.compl_support Function.compl_support\n\n@[to_additive (attr := simp)]\ntheorem mem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2208 mulSupport f \u2194 f x \u2260 1 :=\n Iff.rfl\n#align function.mem_mul_support Function.mem_mulSupport\n#align function.mem_support Function.mem_support\n\n@[to_additive (attr := simp)]\ntheorem mulSupport_subset_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} : mulSupport f \u2286 s \u2194 \u2200 x, f x \u2260 1 \u2192 x \u2208 s :=\n Iff.rfl\n#align function.mul_support_subset_iff Function.mulSupport_subset_iff\n#align function.support_subset_iff Function.support_subset_iff\n\n@[to_additive]\ntheorem mulSupport_subset_iff' {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f \u2286 s \u2194 \u2200 x \u2209 s, f x = 1 :=\n forall_congr' fun _ => not_imp_comm\n#align function.mul_support_subset_iff' Function.mulSupport_subset_iff'\n#align function.support_subset_iff' Function.support_subset_iff'\n\n@[to_additive]\ntheorem mulSupport_eq_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f = s \u2194 (\u2200 x, x \u2208 s \u2192 f x \u2260 1) \u2227 \u2200 x, x \u2209 s \u2192 f x = 1 := by\n simp (config := { contextual := true }) only [ext_iff, mem_mulSupport, ne_eq, iff_def,\n not_imp_comm, and_comm, forall_and]\n#align function.mul_support_eq_iff Function.mulSupport_eq_iff\n#align function.support_eq_iff Function.support_eq_iff\n\n@[to_additive]\ntheorem ext_iff_mulSupport {f g : \u03b1 \u2192 M} :\n f = g \u2194 f.mulSupport = g.mulSupport \u2227 \u2200 x \u2208 f.mulSupport, f x = g x :=\n \u27e8fun h \u21a6 h \u25b8 \u27e8rfl, fun _ _ \u21a6 rfl\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 funext fun x \u21a6 by\n if hx : x \u2208 f.mulSupport then exact h\u2082 x hx\n else rw [nmem_mulSupport.1 hx, nmem_mulSupport.1 (mt (Set.ext_iff.1 h\u2081 x).2 hx)]\u27e9\n\n", "theoremStatement": "@[to_additive]\ntheorem mulSupport_update_of_ne_one [DecidableEq \u03b1] (f : \u03b1 \u2192 M) (x : \u03b1) {y : M} (hy : y \u2260 1) :\n mulSupport (update f x y) = insert x (mulSupport f)", "theoremName": "mulSupport_update_of_ne_one", "fileCreated": {"commit": "82ddb54f6c", "date": "2023-12-13"}, "theoremCreated": {"commit": "1342b9e13a", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Function/Support.lean", "positionMetadata": {"lineInFile": 92, "tokenPositionInFile": 3227, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 59}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Order.Cover\n\n#align_import algebra.support from \"leanprover-community/mathlib\"@\"29cb56a7b35f72758b05a30490e1f10bd62c35c1\"\n\n/-!\n# Support of a function\n\nIn this file we define `Function.support f = {x | f x \u2260 0}` and prove its basic properties.\nWe also define `Function.mulSupport f = {x | f x \u2260 1}`.\n-/\n\n\nopen Set\n\nnamespace Function\n\nvariable {\u03b1 \u03b2 A B M N P R S G M\u2080 G\u2080 : Type*} {\u03b9 : Sort*}\n\nsection One\n\nvariable [One M] [One N] [One P]\n\n/-- `support` of a function is the set of points `x` such that `f x \u2260 0`. -/\ndef support [Zero A] (f : \u03b1 \u2192 A) : Set \u03b1 :=\n { x | f x \u2260 0 }\n#align function.support Function.support\n\n/-- `mulSupport` of a function is the set of points `x` such that `f x \u2260 1`. -/\n@[to_additive existing]\ndef mulSupport (f : \u03b1 \u2192 M) : Set \u03b1 :=\n { x | f x \u2260 1 }\n#align function.mul_support Function.mulSupport\n\n@[to_additive]\ntheorem mulSupport_eq_preimage (f : \u03b1 \u2192 M) : mulSupport f = f \u207b\u00b9' {1}\u1d9c :=\n rfl\n#align function.mul_support_eq_preimage Function.mulSupport_eq_preimage\n#align function.support_eq_preimage Function.support_eq_preimage\n\n@[to_additive]\ntheorem nmem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2209 mulSupport f \u2194 f x = 1 :=\n not_not\n#align function.nmem_mul_support Function.nmem_mulSupport\n#align function.nmem_support Function.nmem_support\n\n@[to_additive]\ntheorem compl_mulSupport {f : \u03b1 \u2192 M} : (mulSupport f)\u1d9c = { x | f x = 1 } :=\n ext fun _ => nmem_mulSupport\n#align function.compl_mul_support Function.compl_mulSupport\n#align function.compl_support Function.compl_support\n\n@[to_additive (attr := simp)]\ntheorem mem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2208 mulSupport f \u2194 f x \u2260 1 :=\n Iff.rfl\n#align function.mem_mul_support Function.mem_mulSupport\n#align function.mem_support Function.mem_support\n\n@[to_additive (attr := simp)]\ntheorem mulSupport_subset_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} : mulSupport f \u2286 s \u2194 \u2200 x, f x \u2260 1 \u2192 x \u2208 s :=\n Iff.rfl\n#align function.mul_support_subset_iff Function.mulSupport_subset_iff\n#align function.support_subset_iff Function.support_subset_iff\n\n@[to_additive]\ntheorem mulSupport_subset_iff' {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f \u2286 s \u2194 \u2200 x \u2209 s, f x = 1 :=\n forall_congr' fun _ => not_imp_comm\n#align function.mul_support_subset_iff' Function.mulSupport_subset_iff'\n#align function.support_subset_iff' Function.support_subset_iff'\n\n@[to_additive]\ntheorem mulSupport_eq_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f = s \u2194 (\u2200 x, x \u2208 s \u2192 f x \u2260 1) \u2227 \u2200 x, x \u2209 s \u2192 f x = 1 := by\n simp (config := { contextual := true }) only [ext_iff, mem_mulSupport, ne_eq, iff_def,\n not_imp_comm, and_comm, forall_and]\n#align function.mul_support_eq_iff Function.mulSupport_eq_iff\n#align function.support_eq_iff Function.support_eq_iff\n\n@[to_additive]\ntheorem ext_iff_mulSupport {f g : \u03b1 \u2192 M} :\n f = g \u2194 f.mulSupport = g.mulSupport \u2227 \u2200 x \u2208 f.mulSupport, f x = g x :=\n \u27e8fun h \u21a6 h \u25b8 \u27e8rfl, fun _ _ \u21a6 rfl\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 funext fun x \u21a6 by\n if hx : x \u2208 f.mulSupport then exact h\u2082 x hx\n else rw [nmem_mulSupport.1 hx, nmem_mulSupport.1 (mt (Set.ext_iff.1 h\u2081 x).2 hx)]\u27e9\n\n@[to_additive]\ntheorem mulSupport_update_of_ne_one [DecidableEq \u03b1] (f : \u03b1 \u2192 M) (x : \u03b1) {y : M} (hy : y \u2260 1) :\n mulSupport (update f x y) = insert x (mulSupport f) := by\n ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]\n\n", "theoremStatement": "@[to_additive]\ntheorem mulSupport_update_one [DecidableEq \u03b1] (f : \u03b1 \u2192 M) (x : \u03b1) :\n mulSupport (update f x 1) = mulSupport f \\ {x}", "theoremName": "mulSupport_update_one", "fileCreated": {"commit": "82ddb54f6c", "date": "2023-12-13"}, "theoremCreated": {"commit": "1342b9e13a", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Function/Support.lean", "positionMetadata": {"lineInFile": 97, "tokenPositionInFile": 3457, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 59}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Algebra.Group.Prod\nimport Mathlib.Order.Cover\n\n#align_import algebra.support from \"leanprover-community/mathlib\"@\"29cb56a7b35f72758b05a30490e1f10bd62c35c1\"\n\n/-!\n# Support of a function\n\nIn this file we define `Function.support f = {x | f x \u2260 0}` and prove its basic properties.\nWe also define `Function.mulSupport f = {x | f x \u2260 1}`.\n-/\n\n\nopen Set\n\nnamespace Function\n\nvariable {\u03b1 \u03b2 A B M N P R S G M\u2080 G\u2080 : Type*} {\u03b9 : Sort*}\n\nsection One\n\nvariable [One M] [One N] [One P]\n\n/-- `support` of a function is the set of points `x` such that `f x \u2260 0`. -/\ndef support [Zero A] (f : \u03b1 \u2192 A) : Set \u03b1 :=\n { x | f x \u2260 0 }\n#align function.support Function.support\n\n/-- `mulSupport` of a function is the set of points `x` such that `f x \u2260 1`. -/\n@[to_additive existing]\ndef mulSupport (f : \u03b1 \u2192 M) : Set \u03b1 :=\n { x | f x \u2260 1 }\n#align function.mul_support Function.mulSupport\n\n@[to_additive]\ntheorem mulSupport_eq_preimage (f : \u03b1 \u2192 M) : mulSupport f = f \u207b\u00b9' {1}\u1d9c :=\n rfl\n#align function.mul_support_eq_preimage Function.mulSupport_eq_preimage\n#align function.support_eq_preimage Function.support_eq_preimage\n\n@[to_additive]\ntheorem nmem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2209 mulSupport f \u2194 f x = 1 :=\n not_not\n#align function.nmem_mul_support Function.nmem_mulSupport\n#align function.nmem_support Function.nmem_support\n\n@[to_additive]\ntheorem compl_mulSupport {f : \u03b1 \u2192 M} : (mulSupport f)\u1d9c = { x | f x = 1 } :=\n ext fun _ => nmem_mulSupport\n#align function.compl_mul_support Function.compl_mulSupport\n#align function.compl_support Function.compl_support\n\n@[to_additive (attr := simp)]\ntheorem mem_mulSupport {f : \u03b1 \u2192 M} {x : \u03b1} : x \u2208 mulSupport f \u2194 f x \u2260 1 :=\n Iff.rfl\n#align function.mem_mul_support Function.mem_mulSupport\n#align function.mem_support Function.mem_support\n\n@[to_additive (attr := simp)]\ntheorem mulSupport_subset_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} : mulSupport f \u2286 s \u2194 \u2200 x, f x \u2260 1 \u2192 x \u2208 s :=\n Iff.rfl\n#align function.mul_support_subset_iff Function.mulSupport_subset_iff\n#align function.support_subset_iff Function.support_subset_iff\n\n@[to_additive]\ntheorem mulSupport_subset_iff' {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f \u2286 s \u2194 \u2200 x \u2209 s, f x = 1 :=\n forall_congr' fun _ => not_imp_comm\n#align function.mul_support_subset_iff' Function.mulSupport_subset_iff'\n#align function.support_subset_iff' Function.support_subset_iff'\n\n@[to_additive]\ntheorem mulSupport_eq_iff {f : \u03b1 \u2192 M} {s : Set \u03b1} :\n mulSupport f = s \u2194 (\u2200 x, x \u2208 s \u2192 f x \u2260 1) \u2227 \u2200 x, x \u2209 s \u2192 f x = 1 := by\n simp (config := { contextual := true }) only [ext_iff, mem_mulSupport, ne_eq, iff_def,\n not_imp_comm, and_comm, forall_and]\n#align function.mul_support_eq_iff Function.mulSupport_eq_iff\n#align function.support_eq_iff Function.support_eq_iff\n\n@[to_additive]\ntheorem ext_iff_mulSupport {f g : \u03b1 \u2192 M} :\n f = g \u2194 f.mulSupport = g.mulSupport \u2227 \u2200 x \u2208 f.mulSupport, f x = g x :=\n \u27e8fun h \u21a6 h \u25b8 \u27e8rfl, fun _ _ \u21a6 rfl\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 funext fun x \u21a6 by\n if hx : x \u2208 f.mulSupport then exact h\u2082 x hx\n else rw [nmem_mulSupport.1 hx, nmem_mulSupport.1 (mt (Set.ext_iff.1 h\u2081 x).2 hx)]\u27e9\n\n@[to_additive]\ntheorem mulSupport_update_of_ne_one [DecidableEq \u03b1] (f : \u03b1 \u2192 M) (x : \u03b1) {y : M} (hy : y \u2260 1) :\n mulSupport (update f x y) = insert x (mulSupport f) := by\n ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]\n\n@[to_additive]\ntheorem mulSupport_update_one [DecidableEq \u03b1] (f : \u03b1 \u2192 M) (x : \u03b1) :\n mulSupport (update f x 1) = mulSupport f \\ {x} := by\n ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]\n\n", "theoremStatement": "@[to_additive]\ntheorem mulSupport_update_eq_ite [DecidableEq \u03b1] [DecidableEq M] (f : \u03b1 \u2192 M) (x : \u03b1) (y : M) :\n mulSupport (update f x y) = if y = 1 then mulSupport f \\ {x} else insert x (mulSupport f)", "theoremName": "mulSupport_update_eq_ite", "fileCreated": {"commit": "82ddb54f6c", "date": "2023-12-13"}, "theoremCreated": {"commit": "1342b9e13a", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Function/Support.lean", "positionMetadata": {"lineInFile": 102, "tokenPositionInFile": 3655, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rcases eq_or_ne y 1 with rfl | hy <;> simp [mulSupport_update_one, mulSupport_update_of_ne_one, *]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 103}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport Mathlib.RingTheory.QuotientNilpotent\nimport Mathlib.RingTheory.Smooth.Basic\nimport Mathlib.RingTheory.Unramified.Basic\n\n#align_import ring_theory.etale from \"leanprover-community/mathlib\"@\"73f96237417835f148a1f7bc1ff55f67119b7166\"\n\n/-!\n\n# Etale morphisms\n\nAn `R`-algebra `A` is formally \u00e9tale if for every `R`-algebra,\nevery square-zero ideal `I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists\nexactly one lift `A \u2192\u2090[R] B`.\nIt is \u00e9tale if it is formally \u00e9tale and of finite presentation.\n\nWe show that the property extends onto nilpotent ideals, and that these properties are stable\nunder `R`-algebra homomorphisms and compositions.\n\nWe show that \u00e9tale is stable under algebra isomorphisms, composition and\nlocalization at an element.\n\n## TODO:\n\n- Show that \u00e9tale is stable under base change.\n\n-/\n\n\n-- Porting note: added to make the syntax work below.\nopen scoped TensorProduct\n\nuniverse u\n\nnamespace Algebra\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R` algebra `A` is formally \u00e9tale if for every `R`-algebra, every square-zero ideal\n`I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists exactly one lift `A \u2192\u2090[R] B`. -/\n@[mk_iff]\nclass FormallyEtale : Prop where\n comp_bijective :\n \u2200 \u2983B : Type u\u2984 [CommRing B],\n \u2200 [Algebra R B] (I : Ideal B) (_ : I ^ 2 = \u22a5),\n Function.Bijective ((Ideal.Quotient.mk\u2090 R I).comp : (A \u2192\u2090[R] B) \u2192 A \u2192\u2090[R] B \u29f8 I)\n#align algebra.formally_etale Algebra.FormallyEtale\n\nend\n\nnamespace FormallyEtale\n\nsection\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)\n\ntheorem iff_unramified_and_smooth :\n FormallyEtale R A \u2194 FormallyUnramified R A \u2227 FormallySmooth R A := by\n rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]\n simp_rw [\u2190 forall_and, Function.Bijective]\n#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth\n\ninstance (priority := 100) to_unramified [h : FormallyEtale R A] :\n FormallyUnramified R A :=\n (FormallyEtale.iff_unramified_and_smooth.mp h).1\n#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified\n\ninstance (priority := 100) to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=\n (FormallyEtale.iff_unramified_and_smooth.mp h).2\n#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth\n\ntheorem of_unramified_and_smooth [h\u2081 : FormallyUnramified R A]\n [h\u2082 : FormallySmooth R A] : FormallyEtale R A :=\n FormallyEtale.iff_unramified_and_smooth.mpr \u27e8h\u2081, h\u2082\u27e9\n#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth\n\nend\n\nsection OfEquiv\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n\ntheorem of_equiv [FormallyEtale R A] (e : A \u2243\u2090[R] B) : FormallyEtale R B :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e\u27e9\n#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv\n\nend OfEquiv\n\nsection Comp\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [CommSemiring A] [Algebra R A]\nvariable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]\n\ntheorem comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.comp R A B, FormallySmooth.comp R A B\u27e9\n#align algebra.formally_etale.comp Algebra.FormallyEtale.comp\n\nend Comp\n\nsection BaseChange\n\nopen scoped TensorProduct\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable (B : Type u) [CommSemiring B] [Algebra R B]\n\ninstance base_change [FormallyEtale R A] : FormallyEtale B (B \u2297[R] A) :=\n FormallyEtale.iff_unramified_and_smooth.mpr \u27e8inferInstance, inferInstance\u27e9\n#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change\n\nend BaseChange\n\nsection Localization\n\nvariable {R S R\u2098 S\u2098 : Type u} [CommRing R] [CommRing S] [CommRing R\u2098] [CommRing S\u2098]\nvariable (M : Submonoid R)\nvariable [Algebra R S] [Algebra R S\u2098] [Algebra S S\u2098] [Algebra R R\u2098] [Algebra R\u2098 S\u2098]\nvariable [IsScalarTower R R\u2098 S\u2098] [IsScalarTower R S S\u2098]\nvariable [IsLocalization M R\u2098] [IsLocalization (M.map (algebraMap R S)) S\u2098]\n\n-- Porting note: no longer supported\n-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on\n\ntheorem of_isLocalization : FormallyEtale R R\u2098 :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.of_isLocalization M, FormallySmooth.of_isLocalization M\u27e9\n#align algebra.formally_etale.of_is_localization Algebra.FormallyEtale.of_isLocalization\n\ntheorem localization_base [FormallyEtale R S\u2098] : FormallyEtale R\u2098 S\u2098 :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.localization_base M, FormallySmooth.localization_base M\u27e9\n#align algebra.formally_etale.localization_base Algebra.FormallyEtale.localization_base\n\ntheorem localization_map [FormallyEtale R S] : FormallyEtale R\u2098 S\u2098 := by\n haveI : FormallyEtale S S\u2098 := FormallyEtale.of_isLocalization (M.map (algebraMap R S))\n haveI : FormallyEtale R S\u2098 := FormallyEtale.comp R S S\u2098\n exact FormallyEtale.localization_base M\n#align algebra.formally_etale.localization_map Algebra.FormallyEtale.localization_map\n\nend Localization\n\nend FormallyEtale\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is \u00e9tale if it is formally \u00e9tale and of finite presentation. -/\nclass Etale : Prop where\n formallyEtale : FormallyEtale R A := by infer_instance\n finitePresentation : FinitePresentation R A := by infer_instance\n\nend\n\nnamespace Etale\n\nattribute [instance] formallyEtale finitePresentation\n\nvariable {R : Type u} [CommRing R]\nvariable {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]\n\n/-- Being \u00e9tale is transported via algebra isomorphisms. -/\n", "theoremStatement": "theorem of_equiv [Etale R A] (e : A \u2243\u2090[R] B) : Etale R B", "theoremName": "of_equiv", "fileCreated": {"commit": "d991fe8f80", "date": "2024-04-08"}, "theoremCreated": {"commit": "c9ab842e13", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/Etale/Basic.lean", "positionMetadata": {"lineInFile": 178, "tokenPositionInFile": 6134, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "where\n formallyEtale := FormallyEtale.of_equiv e\n finitePresentation := FinitePresentation.equiv e", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 100}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport Mathlib.RingTheory.QuotientNilpotent\nimport Mathlib.RingTheory.Smooth.Basic\nimport Mathlib.RingTheory.Unramified.Basic\n\n#align_import ring_theory.etale from \"leanprover-community/mathlib\"@\"73f96237417835f148a1f7bc1ff55f67119b7166\"\n\n/-!\n\n# Etale morphisms\n\nAn `R`-algebra `A` is formally \u00e9tale if for every `R`-algebra,\nevery square-zero ideal `I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists\nexactly one lift `A \u2192\u2090[R] B`.\nIt is \u00e9tale if it is formally \u00e9tale and of finite presentation.\n\nWe show that the property extends onto nilpotent ideals, and that these properties are stable\nunder `R`-algebra homomorphisms and compositions.\n\nWe show that \u00e9tale is stable under algebra isomorphisms, composition and\nlocalization at an element.\n\n## TODO:\n\n- Show that \u00e9tale is stable under base change.\n\n-/\n\n\n-- Porting note: added to make the syntax work below.\nopen scoped TensorProduct\n\nuniverse u\n\nnamespace Algebra\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R` algebra `A` is formally \u00e9tale if for every `R`-algebra, every square-zero ideal\n`I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists exactly one lift `A \u2192\u2090[R] B`. -/\n@[mk_iff]\nclass FormallyEtale : Prop where\n comp_bijective :\n \u2200 \u2983B : Type u\u2984 [CommRing B],\n \u2200 [Algebra R B] (I : Ideal B) (_ : I ^ 2 = \u22a5),\n Function.Bijective ((Ideal.Quotient.mk\u2090 R I).comp : (A \u2192\u2090[R] B) \u2192 A \u2192\u2090[R] B \u29f8 I)\n#align algebra.formally_etale Algebra.FormallyEtale\n\nend\n\nnamespace FormallyEtale\n\nsection\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)\n\ntheorem iff_unramified_and_smooth :\n FormallyEtale R A \u2194 FormallyUnramified R A \u2227 FormallySmooth R A := by\n rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]\n simp_rw [\u2190 forall_and, Function.Bijective]\n#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth\n\ninstance (priority := 100) to_unramified [h : FormallyEtale R A] :\n FormallyUnramified R A :=\n (FormallyEtale.iff_unramified_and_smooth.mp h).1\n#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified\n\ninstance (priority := 100) to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=\n (FormallyEtale.iff_unramified_and_smooth.mp h).2\n#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth\n\ntheorem of_unramified_and_smooth [h\u2081 : FormallyUnramified R A]\n [h\u2082 : FormallySmooth R A] : FormallyEtale R A :=\n FormallyEtale.iff_unramified_and_smooth.mpr \u27e8h\u2081, h\u2082\u27e9\n#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth\n\nend\n\nsection OfEquiv\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n\ntheorem of_equiv [FormallyEtale R A] (e : A \u2243\u2090[R] B) : FormallyEtale R B :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e\u27e9\n#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv\n\nend OfEquiv\n\nsection Comp\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [CommSemiring A] [Algebra R A]\nvariable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]\n\ntheorem comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.comp R A B, FormallySmooth.comp R A B\u27e9\n#align algebra.formally_etale.comp Algebra.FormallyEtale.comp\n\nend Comp\n\nsection BaseChange\n\nopen scoped TensorProduct\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable (B : Type u) [CommSemiring B] [Algebra R B]\n\ninstance base_change [FormallyEtale R A] : FormallyEtale B (B \u2297[R] A) :=\n FormallyEtale.iff_unramified_and_smooth.mpr \u27e8inferInstance, inferInstance\u27e9\n#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change\n\nend BaseChange\n\nsection Localization\n\nvariable {R S R\u2098 S\u2098 : Type u} [CommRing R] [CommRing S] [CommRing R\u2098] [CommRing S\u2098]\nvariable (M : Submonoid R)\nvariable [Algebra R S] [Algebra R S\u2098] [Algebra S S\u2098] [Algebra R R\u2098] [Algebra R\u2098 S\u2098]\nvariable [IsScalarTower R R\u2098 S\u2098] [IsScalarTower R S S\u2098]\nvariable [IsLocalization M R\u2098] [IsLocalization (M.map (algebraMap R S)) S\u2098]\n\n-- Porting note: no longer supported\n-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on\n\ntheorem of_isLocalization : FormallyEtale R R\u2098 :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.of_isLocalization M, FormallySmooth.of_isLocalization M\u27e9\n#align algebra.formally_etale.of_is_localization Algebra.FormallyEtale.of_isLocalization\n\ntheorem localization_base [FormallyEtale R S\u2098] : FormallyEtale R\u2098 S\u2098 :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.localization_base M, FormallySmooth.localization_base M\u27e9\n#align algebra.formally_etale.localization_base Algebra.FormallyEtale.localization_base\n\ntheorem localization_map [FormallyEtale R S] : FormallyEtale R\u2098 S\u2098 := by\n haveI : FormallyEtale S S\u2098 := FormallyEtale.of_isLocalization (M.map (algebraMap R S))\n haveI : FormallyEtale R S\u2098 := FormallyEtale.comp R S S\u2098\n exact FormallyEtale.localization_base M\n#align algebra.formally_etale.localization_map Algebra.FormallyEtale.localization_map\n\nend Localization\n\nend FormallyEtale\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is \u00e9tale if it is formally \u00e9tale and of finite presentation. -/\nclass Etale : Prop where\n formallyEtale : FormallyEtale R A := by infer_instance\n finitePresentation : FinitePresentation R A := by infer_instance\n\nend\n\nnamespace Etale\n\nattribute [instance] formallyEtale finitePresentation\n\nvariable {R : Type u} [CommRing R]\nvariable {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]\n\n/-- Being \u00e9tale is transported via algebra isomorphisms. -/\ntheorem of_equiv [Etale R A] (e : A \u2243\u2090[R] B) : Etale R B where\n formallyEtale := FormallyEtale.of_equiv e\n finitePresentation := FinitePresentation.equiv e\n\nsection Comp\n\nvariable (R A B)\nvariable [Algebra A B] [IsScalarTower R A B]\n\n/-- \u00c9tale is stable under composition. -/\n", "theoremStatement": "theorem comp [Etale R A] [Etale A B] : Etale R B", "theoremName": "comp", "fileCreated": {"commit": "d991fe8f80", "date": "2024-04-08"}, "theoremCreated": {"commit": "c9ab842e13", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/Etale/Basic.lean", "positionMetadata": {"lineInFile": 188, "tokenPositionInFile": 6412, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "where\n formallyEtale := FormallyEtale.comp R A B\n finitePresentation := FinitePresentation.trans R A B", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 104}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport Mathlib.RingTheory.QuotientNilpotent\nimport Mathlib.RingTheory.Smooth.Basic\nimport Mathlib.RingTheory.Unramified.Basic\n\n#align_import ring_theory.etale from \"leanprover-community/mathlib\"@\"73f96237417835f148a1f7bc1ff55f67119b7166\"\n\n/-!\n\n# Etale morphisms\n\nAn `R`-algebra `A` is formally \u00e9tale if for every `R`-algebra,\nevery square-zero ideal `I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists\nexactly one lift `A \u2192\u2090[R] B`.\nIt is \u00e9tale if it is formally \u00e9tale and of finite presentation.\n\nWe show that the property extends onto nilpotent ideals, and that these properties are stable\nunder `R`-algebra homomorphisms and compositions.\n\nWe show that \u00e9tale is stable under algebra isomorphisms, composition and\nlocalization at an element.\n\n## TODO:\n\n- Show that \u00e9tale is stable under base change.\n\n-/\n\n\n-- Porting note: added to make the syntax work below.\nopen scoped TensorProduct\n\nuniverse u\n\nnamespace Algebra\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R` algebra `A` is formally \u00e9tale if for every `R`-algebra, every square-zero ideal\n`I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists exactly one lift `A \u2192\u2090[R] B`. -/\n@[mk_iff]\nclass FormallyEtale : Prop where\n comp_bijective :\n \u2200 \u2983B : Type u\u2984 [CommRing B],\n \u2200 [Algebra R B] (I : Ideal B) (_ : I ^ 2 = \u22a5),\n Function.Bijective ((Ideal.Quotient.mk\u2090 R I).comp : (A \u2192\u2090[R] B) \u2192 A \u2192\u2090[R] B \u29f8 I)\n#align algebra.formally_etale Algebra.FormallyEtale\n\nend\n\nnamespace FormallyEtale\n\nsection\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)\n\ntheorem iff_unramified_and_smooth :\n FormallyEtale R A \u2194 FormallyUnramified R A \u2227 FormallySmooth R A := by\n rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]\n simp_rw [\u2190 forall_and, Function.Bijective]\n#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth\n\ninstance (priority := 100) to_unramified [h : FormallyEtale R A] :\n FormallyUnramified R A :=\n (FormallyEtale.iff_unramified_and_smooth.mp h).1\n#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified\n\ninstance (priority := 100) to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=\n (FormallyEtale.iff_unramified_and_smooth.mp h).2\n#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth\n\ntheorem of_unramified_and_smooth [h\u2081 : FormallyUnramified R A]\n [h\u2082 : FormallySmooth R A] : FormallyEtale R A :=\n FormallyEtale.iff_unramified_and_smooth.mpr \u27e8h\u2081, h\u2082\u27e9\n#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth\n\nend\n\nsection OfEquiv\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n\ntheorem of_equiv [FormallyEtale R A] (e : A \u2243\u2090[R] B) : FormallyEtale R B :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e\u27e9\n#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv\n\nend OfEquiv\n\nsection Comp\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [CommSemiring A] [Algebra R A]\nvariable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]\n\ntheorem comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.comp R A B, FormallySmooth.comp R A B\u27e9\n#align algebra.formally_etale.comp Algebra.FormallyEtale.comp\n\nend Comp\n\nsection BaseChange\n\nopen scoped TensorProduct\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable (B : Type u) [CommSemiring B] [Algebra R B]\n\ninstance base_change [FormallyEtale R A] : FormallyEtale B (B \u2297[R] A) :=\n FormallyEtale.iff_unramified_and_smooth.mpr \u27e8inferInstance, inferInstance\u27e9\n#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change\n\nend BaseChange\n\nsection Localization\n\nvariable {R S R\u2098 S\u2098 : Type u} [CommRing R] [CommRing S] [CommRing R\u2098] [CommRing S\u2098]\nvariable (M : Submonoid R)\nvariable [Algebra R S] [Algebra R S\u2098] [Algebra S S\u2098] [Algebra R R\u2098] [Algebra R\u2098 S\u2098]\nvariable [IsScalarTower R R\u2098 S\u2098] [IsScalarTower R S S\u2098]\nvariable [IsLocalization M R\u2098] [IsLocalization (M.map (algebraMap R S)) S\u2098]\n\n-- Porting note: no longer supported\n-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on\n\ntheorem of_isLocalization : FormallyEtale R R\u2098 :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.of_isLocalization M, FormallySmooth.of_isLocalization M\u27e9\n#align algebra.formally_etale.of_is_localization Algebra.FormallyEtale.of_isLocalization\n\ntheorem localization_base [FormallyEtale R S\u2098] : FormallyEtale R\u2098 S\u2098 :=\n FormallyEtale.iff_unramified_and_smooth.mpr\n \u27e8FormallyUnramified.localization_base M, FormallySmooth.localization_base M\u27e9\n#align algebra.formally_etale.localization_base Algebra.FormallyEtale.localization_base\n\ntheorem localization_map [FormallyEtale R S] : FormallyEtale R\u2098 S\u2098 := by\n haveI : FormallyEtale S S\u2098 := FormallyEtale.of_isLocalization (M.map (algebraMap R S))\n haveI : FormallyEtale R S\u2098 := FormallyEtale.comp R S S\u2098\n exact FormallyEtale.localization_base M\n#align algebra.formally_etale.localization_map Algebra.FormallyEtale.localization_map\n\nend Localization\n\nend FormallyEtale\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is \u00e9tale if it is formally \u00e9tale and of finite presentation. -/\nclass Etale : Prop where\n formallyEtale : FormallyEtale R A := by infer_instance\n finitePresentation : FinitePresentation R A := by infer_instance\n\nend\n\nnamespace Etale\n\nattribute [instance] formallyEtale finitePresentation\n\nvariable {R : Type u} [CommRing R]\nvariable {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]\n\n/-- Being \u00e9tale is transported via algebra isomorphisms. -/\ntheorem of_equiv [Etale R A] (e : A \u2243\u2090[R] B) : Etale R B where\n formallyEtale := FormallyEtale.of_equiv e\n finitePresentation := FinitePresentation.equiv e\n\nsection Comp\n\nvariable (R A B)\nvariable [Algebra A B] [IsScalarTower R A B]\n\n/-- \u00c9tale is stable under composition. -/\ntheorem comp [Etale R A] [Etale A B] : Etale R B where\n formallyEtale := FormallyEtale.comp R A B\n finitePresentation := FinitePresentation.trans R A B\n\nend Comp\n\n/-- Localization at an element is \u00e9tale. -/\n", "theoremStatement": "theorem of_isLocalization_Away (r : R) [IsLocalization.Away r A] : Etale R A", "theoremName": "of_isLocalization_Away", "fileCreated": {"commit": "d991fe8f80", "date": "2024-04-08"}, "theoremCreated": {"commit": "c9ab842e13", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/Etale/Basic.lean", "positionMetadata": {"lineInFile": 195, "tokenPositionInFile": 6621, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "where\n formallyEtale := Algebra.FormallyEtale.of_isLocalization (Submonoid.powers r)\n finitePresentation := IsLocalization.Away.finitePresentation r", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 150}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport Mathlib.RingTheory.FinitePresentation\nimport Mathlib.RingTheory.Localization.Away.Basic\nimport Mathlib.RingTheory.Localization.Away.AdjoinRoot\nimport Mathlib.RingTheory.QuotientNilpotent\nimport Mathlib.RingTheory.TensorProduct.Basic\n\n/-!\n\n# Unramified morphisms\n\nAn `R`-algebra `A` is formally unramified if for every `R`-algebra,\nevery square-zero ideal `I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists\nat most one lift `A \u2192\u2090[R] B`.\nIt is unramified if it is formally unramified and of finite type.\n\nNote that there are multiple definitions in the literature. The definition we give is equivalent to\nthe one in the Stacks Project https://stacks.math.columbia.edu/tag/00US. Note that in EGA unramified\nis defined as formally unramified and of finite presentation.\n\nWe show that the property extends onto nilpotent ideals, and that it is stable\nunder `R`-algebra homomorphisms and compositions.\n\nWe show that unramified is stable under algebra isomorphisms, composition and\nlocalization at an element.\n\n# TODO\n\n- Show that unramified is stable under base change.\n\n-/\n\n-- Porting note: added to make the syntax work below.\nopen scoped TensorProduct\n\nuniverse u\n\nnamespace Algebra\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal\n`I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists at most one lift `A \u2192\u2090[R] B`. -/\n@[mk_iff]\nclass FormallyUnramified : Prop where\n comp_injective :\n \u2200 \u2983B : Type u\u2984 [CommRing B],\n \u2200 [Algebra R B] (I : Ideal B) (_ : I ^ 2 = \u22a5),\n Function.Injective ((Ideal.Quotient.mk\u2090 R I).comp : (A \u2192\u2090[R] B) \u2192 A \u2192\u2090[R] B \u29f8 I)\n#align algebra.formally_unramified Algebra.FormallyUnramified\n\nend\n\nnamespace FormallyUnramified\n\nsection\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)\n\ntheorem lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]\n [FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g\u2081 g\u2082 : A \u2192\u2090[R] B)\n (h : (Ideal.Quotient.mk\u2090 R I).comp g\u2081 = (Ideal.Quotient.mk\u2090 R I).comp g\u2082) : g\u2081 = g\u2082 := by\n revert g\u2081 g\u2082\n change Function.Injective (Ideal.Quotient.mk\u2090 R I).comp\n revert _RB\n apply Ideal.IsNilpotent.induction_on (R := B) I hI\n \u00b7 intro B _ I hI _; exact FormallyUnramified.comp_injective I hI\n \u00b7 intro B _ I J hIJ h\u2081 h\u2082 _ g\u2081 g\u2082 e\n apply h\u2081\n apply h\u2082\n ext x\n replace e := AlgHom.congr_fun e x\n dsimp only [AlgHom.comp_apply, Ideal.Quotient.mk\u2090_eq_mk] at e \u22a2\n rwa [Ideal.Quotient.eq, \u2190 map_sub, Ideal.mem_quotient_iff_mem hIJ, \u2190 Ideal.Quotient.eq]\n#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique\n\ntheorem ext [FormallyUnramified R A] (hI : IsNilpotent I) {g\u2081 g\u2082 : A \u2192\u2090[R] B}\n (H : \u2200 x, Ideal.Quotient.mk I (g\u2081 x) = Ideal.Quotient.mk I (g\u2082 x)) : g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique I hI g\u2081 g\u2082 (AlgHom.ext H)\n#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext\n\ntheorem lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]\n (f : B \u2192+* C) (hf : IsNilpotent <| RingHom.ker f) (g\u2081 g\u2082 : A \u2192\u2090[R] B)\n (h : f.comp \u2191g\u2081 = f.comp (g\u2082 : A \u2192+* B)) : g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique _ hf _ _\n (by\n ext x\n have := RingHom.congr_fun h x\n simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mk\u2090_eq_mk,\n RingHom.mem_ker, map_sub, sub_eq_zero])\n#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom\n\ntheorem ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B \u2192+* C)\n (hf : IsNilpotent <| RingHom.ker f) (g\u2081 g\u2082 : A \u2192\u2090[R] B) (h : \u2200 x, f (g\u2081 x) = f (g\u2082 x)) :\n g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique_of_ringHom f hf g\u2081 g\u2082 (RingHom.ext h)\n#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'\n\ntheorem lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]\n [Algebra R C] (f : B \u2192\u2090[R] C) (hf : IsNilpotent <| RingHom.ker (f : B \u2192+* C))\n (g\u2081 g\u2082 : A \u2192\u2090[R] B) (h : f.comp g\u2081 = f.comp g\u2082) : g\u2081 = g\u2082 :=\n FormallyUnramified.ext' _ hf g\u2081 g\u2082 (AlgHom.congr_fun h)\n#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'\n\nend\n\nsection OfEquiv\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n\ntheorem of_equiv [FormallyUnramified R A] (e : A \u2243\u2090[R] B) :\n FormallyUnramified R B := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e'\n rw [\u2190 f\u2081.comp_id, \u2190 f\u2082.comp_id, \u2190 e.comp_symm, \u2190 AlgHom.comp_assoc, \u2190 AlgHom.comp_assoc]\n congr 1\n refine' FormallyUnramified.comp_injective I hI _\n rw [\u2190 AlgHom.comp_assoc, e', AlgHom.comp_assoc]\n#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv\n\nend OfEquiv\n\nsection Comp\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [CommSemiring A] [Algebra R A]\nvariable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]\n\ntheorem comp [FormallyUnramified R A] [FormallyUnramified A B] :\n FormallyUnramified R B := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e\n have e' :=\n FormallyUnramified.lift_unique I \u27e82, hI\u27e9 (f\u2081.comp <| IsScalarTower.toAlgHom R A B)\n (f\u2082.comp <| IsScalarTower.toAlgHom R A B) (by rw [\u2190 AlgHom.comp_assoc, e, AlgHom.comp_assoc])\n letI := (f\u2081.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra\n let F\u2081 : B \u2192\u2090[A] C := { f\u2081 with commutes' := fun r => rfl }\n let F\u2082 : B \u2192\u2090[A] C := { f\u2082 with commutes' := AlgHom.congr_fun e'.symm }\n ext1 x\n change F\u2081 x = F\u2082 x\n congr\n exact FormallyUnramified.ext I \u27e82, hI\u27e9 (AlgHom.congr_fun e)\n#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp\n\ntheorem of_comp [FormallyUnramified R B] : FormallyUnramified A B := by\n constructor\n intro Q _ _ I e f\u2081 f\u2082 e'\n letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra\n letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl\n refine' AlgHom.restrictScalars_injective R _\n refine' FormallyUnramified.ext I \u27e82, e\u27e9 _\n intro x\n exact AlgHom.congr_fun e' x\n#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp\n\nend Comp\n\nsection BaseChange\n\nopen scoped TensorProduct\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable (B : Type u) [CommSemiring B] [Algebra R B]\n\ninstance base_change [FormallyUnramified R A] :\n FormallyUnramified B (B \u2297[R] A) := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e\n letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra\n haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl\n ext : 1\n \u00b7 exact Subsingleton.elim _ _\n \u00b7 exact FormallyUnramified.ext I \u27e82, hI\u27e9 fun x => AlgHom.congr_fun e (1 \u2297\u209c x)\n#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change\n\nend BaseChange\n\nsection Localization\n\nvariable {R S R\u2098 S\u2098 : Type u} [CommRing R] [CommRing S] [CommRing R\u2098] [CommRing S\u2098]\nvariable (M : Submonoid R)\nvariable [Algebra R S] [Algebra R S\u2098] [Algebra S S\u2098] [Algebra R R\u2098] [Algebra R\u2098 S\u2098]\nvariable [IsScalarTower R R\u2098 S\u2098] [IsScalarTower R S S\u2098]\nvariable [IsLocalization M R\u2098] [IsLocalization (M.map (algebraMap R S)) S\u2098]\n\n-- Porting note: no longer supported\n-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on\n\n/-- This holds in general for epimorphisms. -/\ntheorem of_isLocalization : FormallyUnramified R R\u2098 := by\n constructor\n intro Q _ _ I _ f\u2081 f\u2082 _\n apply AlgHom.coe_ringHom_injective\n refine' IsLocalization.ringHom_ext M _\n ext\n simp\n#align algebra.formally_unramified.of_is_localization Algebra.FormallyUnramified.of_isLocalization\n\n/-- This actually does not need the localization instance, and is stated here again for\nconsistency. See `Algebra.FormallyUnramified.of_comp` instead.\n\n The intended use is for copying proofs between `Formally{Unramified, Smooth, Etale}`\n without the need to change anything (including removing redundant arguments). -/\n-- @[nolint unusedArguments] -- Porting note: removed\ntheorem localization_base [FormallyUnramified R S\u2098] : FormallyUnramified R\u2098 S\u2098 :=\n -- Porting note: added\n let _ := M\n FormallyUnramified.of_comp R R\u2098 S\u2098\n#align algebra.formally_unramified.localization_base Algebra.FormallyUnramified.localization_base\n\ntheorem localization_map [FormallyUnramified R S] :\n FormallyUnramified R\u2098 S\u2098 := by\n haveI : FormallyUnramified S S\u2098 :=\n FormallyUnramified.of_isLocalization (M.map (algebraMap R S))\n haveI : FormallyUnramified R S\u2098 := FormallyUnramified.comp R S S\u2098\n exact FormallyUnramified.localization_base M\n#align algebra.formally_unramified.localization_map Algebra.FormallyUnramified.localization_map\n\nend Localization\n\nend FormallyUnramified\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is unramified if it is formally unramified and of finite type.\n-/\nclass Unramified : Prop where\n formallyUnramified : FormallyUnramified R A := by infer_instance\n finiteType : FiniteType R A := by infer_instance\n\nend\n\nnamespace Unramified\n\nattribute [instance] formallyUnramified finiteType\n\nvariable {R : Type u} [CommRing R]\nvariable {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]\n\n/-- Being unramified is transported via algebra isomorphisms. -/\n", "theoremStatement": "theorem of_equiv [Unramified R A] (e : A \u2243\u2090[R] B) : Unramified R B", "theoremName": "of_equiv", "fileCreated": {"commit": "d991fe8f80", "date": "2024-04-08"}, "theoremCreated": {"commit": "c9ab842e13", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/Unramified/Basic.lean", "positionMetadata": {"lineInFile": 255, "tokenPositionInFile": 9561, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "where\n formallyUnramified := FormallyUnramified.of_equiv e\n finiteType := FiniteType.equiv Unramified.finiteType e", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 116}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport Mathlib.RingTheory.FinitePresentation\nimport Mathlib.RingTheory.Localization.Away.Basic\nimport Mathlib.RingTheory.Localization.Away.AdjoinRoot\nimport Mathlib.RingTheory.QuotientNilpotent\nimport Mathlib.RingTheory.TensorProduct.Basic\n\n/-!\n\n# Unramified morphisms\n\nAn `R`-algebra `A` is formally unramified if for every `R`-algebra,\nevery square-zero ideal `I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists\nat most one lift `A \u2192\u2090[R] B`.\nIt is unramified if it is formally unramified and of finite type.\n\nNote that there are multiple definitions in the literature. The definition we give is equivalent to\nthe one in the Stacks Project https://stacks.math.columbia.edu/tag/00US. Note that in EGA unramified\nis defined as formally unramified and of finite presentation.\n\nWe show that the property extends onto nilpotent ideals, and that it is stable\nunder `R`-algebra homomorphisms and compositions.\n\nWe show that unramified is stable under algebra isomorphisms, composition and\nlocalization at an element.\n\n# TODO\n\n- Show that unramified is stable under base change.\n\n-/\n\n-- Porting note: added to make the syntax work below.\nopen scoped TensorProduct\n\nuniverse u\n\nnamespace Algebra\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal\n`I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists at most one lift `A \u2192\u2090[R] B`. -/\n@[mk_iff]\nclass FormallyUnramified : Prop where\n comp_injective :\n \u2200 \u2983B : Type u\u2984 [CommRing B],\n \u2200 [Algebra R B] (I : Ideal B) (_ : I ^ 2 = \u22a5),\n Function.Injective ((Ideal.Quotient.mk\u2090 R I).comp : (A \u2192\u2090[R] B) \u2192 A \u2192\u2090[R] B \u29f8 I)\n#align algebra.formally_unramified Algebra.FormallyUnramified\n\nend\n\nnamespace FormallyUnramified\n\nsection\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)\n\ntheorem lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]\n [FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g\u2081 g\u2082 : A \u2192\u2090[R] B)\n (h : (Ideal.Quotient.mk\u2090 R I).comp g\u2081 = (Ideal.Quotient.mk\u2090 R I).comp g\u2082) : g\u2081 = g\u2082 := by\n revert g\u2081 g\u2082\n change Function.Injective (Ideal.Quotient.mk\u2090 R I).comp\n revert _RB\n apply Ideal.IsNilpotent.induction_on (R := B) I hI\n \u00b7 intro B _ I hI _; exact FormallyUnramified.comp_injective I hI\n \u00b7 intro B _ I J hIJ h\u2081 h\u2082 _ g\u2081 g\u2082 e\n apply h\u2081\n apply h\u2082\n ext x\n replace e := AlgHom.congr_fun e x\n dsimp only [AlgHom.comp_apply, Ideal.Quotient.mk\u2090_eq_mk] at e \u22a2\n rwa [Ideal.Quotient.eq, \u2190 map_sub, Ideal.mem_quotient_iff_mem hIJ, \u2190 Ideal.Quotient.eq]\n#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique\n\ntheorem ext [FormallyUnramified R A] (hI : IsNilpotent I) {g\u2081 g\u2082 : A \u2192\u2090[R] B}\n (H : \u2200 x, Ideal.Quotient.mk I (g\u2081 x) = Ideal.Quotient.mk I (g\u2082 x)) : g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique I hI g\u2081 g\u2082 (AlgHom.ext H)\n#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext\n\ntheorem lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]\n (f : B \u2192+* C) (hf : IsNilpotent <| RingHom.ker f) (g\u2081 g\u2082 : A \u2192\u2090[R] B)\n (h : f.comp \u2191g\u2081 = f.comp (g\u2082 : A \u2192+* B)) : g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique _ hf _ _\n (by\n ext x\n have := RingHom.congr_fun h x\n simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mk\u2090_eq_mk,\n RingHom.mem_ker, map_sub, sub_eq_zero])\n#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom\n\ntheorem ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B \u2192+* C)\n (hf : IsNilpotent <| RingHom.ker f) (g\u2081 g\u2082 : A \u2192\u2090[R] B) (h : \u2200 x, f (g\u2081 x) = f (g\u2082 x)) :\n g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique_of_ringHom f hf g\u2081 g\u2082 (RingHom.ext h)\n#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'\n\ntheorem lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]\n [Algebra R C] (f : B \u2192\u2090[R] C) (hf : IsNilpotent <| RingHom.ker (f : B \u2192+* C))\n (g\u2081 g\u2082 : A \u2192\u2090[R] B) (h : f.comp g\u2081 = f.comp g\u2082) : g\u2081 = g\u2082 :=\n FormallyUnramified.ext' _ hf g\u2081 g\u2082 (AlgHom.congr_fun h)\n#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'\n\nend\n\nsection OfEquiv\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n\ntheorem of_equiv [FormallyUnramified R A] (e : A \u2243\u2090[R] B) :\n FormallyUnramified R B := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e'\n rw [\u2190 f\u2081.comp_id, \u2190 f\u2082.comp_id, \u2190 e.comp_symm, \u2190 AlgHom.comp_assoc, \u2190 AlgHom.comp_assoc]\n congr 1\n refine' FormallyUnramified.comp_injective I hI _\n rw [\u2190 AlgHom.comp_assoc, e', AlgHom.comp_assoc]\n#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv\n\nend OfEquiv\n\nsection Comp\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [CommSemiring A] [Algebra R A]\nvariable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]\n\ntheorem comp [FormallyUnramified R A] [FormallyUnramified A B] :\n FormallyUnramified R B := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e\n have e' :=\n FormallyUnramified.lift_unique I \u27e82, hI\u27e9 (f\u2081.comp <| IsScalarTower.toAlgHom R A B)\n (f\u2082.comp <| IsScalarTower.toAlgHom R A B) (by rw [\u2190 AlgHom.comp_assoc, e, AlgHom.comp_assoc])\n letI := (f\u2081.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra\n let F\u2081 : B \u2192\u2090[A] C := { f\u2081 with commutes' := fun r => rfl }\n let F\u2082 : B \u2192\u2090[A] C := { f\u2082 with commutes' := AlgHom.congr_fun e'.symm }\n ext1 x\n change F\u2081 x = F\u2082 x\n congr\n exact FormallyUnramified.ext I \u27e82, hI\u27e9 (AlgHom.congr_fun e)\n#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp\n\ntheorem of_comp [FormallyUnramified R B] : FormallyUnramified A B := by\n constructor\n intro Q _ _ I e f\u2081 f\u2082 e'\n letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra\n letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl\n refine' AlgHom.restrictScalars_injective R _\n refine' FormallyUnramified.ext I \u27e82, e\u27e9 _\n intro x\n exact AlgHom.congr_fun e' x\n#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp\n\nend Comp\n\nsection BaseChange\n\nopen scoped TensorProduct\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable (B : Type u) [CommSemiring B] [Algebra R B]\n\ninstance base_change [FormallyUnramified R A] :\n FormallyUnramified B (B \u2297[R] A) := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e\n letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra\n haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl\n ext : 1\n \u00b7 exact Subsingleton.elim _ _\n \u00b7 exact FormallyUnramified.ext I \u27e82, hI\u27e9 fun x => AlgHom.congr_fun e (1 \u2297\u209c x)\n#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change\n\nend BaseChange\n\nsection Localization\n\nvariable {R S R\u2098 S\u2098 : Type u} [CommRing R] [CommRing S] [CommRing R\u2098] [CommRing S\u2098]\nvariable (M : Submonoid R)\nvariable [Algebra R S] [Algebra R S\u2098] [Algebra S S\u2098] [Algebra R R\u2098] [Algebra R\u2098 S\u2098]\nvariable [IsScalarTower R R\u2098 S\u2098] [IsScalarTower R S S\u2098]\nvariable [IsLocalization M R\u2098] [IsLocalization (M.map (algebraMap R S)) S\u2098]\n\n-- Porting note: no longer supported\n-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on\n\n/-- This holds in general for epimorphisms. -/\ntheorem of_isLocalization : FormallyUnramified R R\u2098 := by\n constructor\n intro Q _ _ I _ f\u2081 f\u2082 _\n apply AlgHom.coe_ringHom_injective\n refine' IsLocalization.ringHom_ext M _\n ext\n simp\n#align algebra.formally_unramified.of_is_localization Algebra.FormallyUnramified.of_isLocalization\n\n/-- This actually does not need the localization instance, and is stated here again for\nconsistency. See `Algebra.FormallyUnramified.of_comp` instead.\n\n The intended use is for copying proofs between `Formally{Unramified, Smooth, Etale}`\n without the need to change anything (including removing redundant arguments). -/\n-- @[nolint unusedArguments] -- Porting note: removed\ntheorem localization_base [FormallyUnramified R S\u2098] : FormallyUnramified R\u2098 S\u2098 :=\n -- Porting note: added\n let _ := M\n FormallyUnramified.of_comp R R\u2098 S\u2098\n#align algebra.formally_unramified.localization_base Algebra.FormallyUnramified.localization_base\n\ntheorem localization_map [FormallyUnramified R S] :\n FormallyUnramified R\u2098 S\u2098 := by\n haveI : FormallyUnramified S S\u2098 :=\n FormallyUnramified.of_isLocalization (M.map (algebraMap R S))\n haveI : FormallyUnramified R S\u2098 := FormallyUnramified.comp R S S\u2098\n exact FormallyUnramified.localization_base M\n#align algebra.formally_unramified.localization_map Algebra.FormallyUnramified.localization_map\n\nend Localization\n\nend FormallyUnramified\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is unramified if it is formally unramified and of finite type.\n-/\nclass Unramified : Prop where\n formallyUnramified : FormallyUnramified R A := by infer_instance\n finiteType : FiniteType R A := by infer_instance\n\nend\n\nnamespace Unramified\n\nattribute [instance] formallyUnramified finiteType\n\nvariable {R : Type u} [CommRing R]\nvariable {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]\n\n/-- Being unramified is transported via algebra isomorphisms. -/\ntheorem of_equiv [Unramified R A] (e : A \u2243\u2090[R] B) : Unramified R B where\n formallyUnramified := FormallyUnramified.of_equiv e\n finiteType := FiniteType.equiv Unramified.finiteType e\n\n/-- Localization at an element is unramified. -/\n", "theoremStatement": "theorem of_isLocalization_Away (r : R) [IsLocalization.Away r A] : Unramified R A", "theoremName": "of_isLocalization_Away", "fileCreated": {"commit": "d991fe8f80", "date": "2024-04-08"}, "theoremCreated": {"commit": "c9ab842e13", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/Unramified/Basic.lean", "positionMetadata": {"lineInFile": 260, "tokenPositionInFile": 9795, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "where\n formallyUnramified := Algebra.FormallyUnramified.of_isLocalization (Submonoid.powers r)\n finiteType :=\n haveI : FinitePresentation R A := IsLocalization.Away.finitePresentation r\n inferInstance", "proofType": "term", "proofLengthLines": 5, "proofLengthTokens": 208}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport Mathlib.RingTheory.FinitePresentation\nimport Mathlib.RingTheory.Localization.Away.Basic\nimport Mathlib.RingTheory.Localization.Away.AdjoinRoot\nimport Mathlib.RingTheory.QuotientNilpotent\nimport Mathlib.RingTheory.TensorProduct.Basic\n\n/-!\n\n# Unramified morphisms\n\nAn `R`-algebra `A` is formally unramified if for every `R`-algebra,\nevery square-zero ideal `I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists\nat most one lift `A \u2192\u2090[R] B`.\nIt is unramified if it is formally unramified and of finite type.\n\nNote that there are multiple definitions in the literature. The definition we give is equivalent to\nthe one in the Stacks Project https://stacks.math.columbia.edu/tag/00US. Note that in EGA unramified\nis defined as formally unramified and of finite presentation.\n\nWe show that the property extends onto nilpotent ideals, and that it is stable\nunder `R`-algebra homomorphisms and compositions.\n\nWe show that unramified is stable under algebra isomorphisms, composition and\nlocalization at an element.\n\n# TODO\n\n- Show that unramified is stable under base change.\n\n-/\n\n-- Porting note: added to make the syntax work below.\nopen scoped TensorProduct\n\nuniverse u\n\nnamespace Algebra\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal\n`I : Ideal B` and `f : A \u2192\u2090[R] B \u29f8 I`, there exists at most one lift `A \u2192\u2090[R] B`. -/\n@[mk_iff]\nclass FormallyUnramified : Prop where\n comp_injective :\n \u2200 \u2983B : Type u\u2984 [CommRing B],\n \u2200 [Algebra R B] (I : Ideal B) (_ : I ^ 2 = \u22a5),\n Function.Injective ((Ideal.Quotient.mk\u2090 R I).comp : (A \u2192\u2090[R] B) \u2192 A \u2192\u2090[R] B \u29f8 I)\n#align algebra.formally_unramified Algebra.FormallyUnramified\n\nend\n\nnamespace FormallyUnramified\n\nsection\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)\n\ntheorem lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]\n [FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g\u2081 g\u2082 : A \u2192\u2090[R] B)\n (h : (Ideal.Quotient.mk\u2090 R I).comp g\u2081 = (Ideal.Quotient.mk\u2090 R I).comp g\u2082) : g\u2081 = g\u2082 := by\n revert g\u2081 g\u2082\n change Function.Injective (Ideal.Quotient.mk\u2090 R I).comp\n revert _RB\n apply Ideal.IsNilpotent.induction_on (R := B) I hI\n \u00b7 intro B _ I hI _; exact FormallyUnramified.comp_injective I hI\n \u00b7 intro B _ I J hIJ h\u2081 h\u2082 _ g\u2081 g\u2082 e\n apply h\u2081\n apply h\u2082\n ext x\n replace e := AlgHom.congr_fun e x\n dsimp only [AlgHom.comp_apply, Ideal.Quotient.mk\u2090_eq_mk] at e \u22a2\n rwa [Ideal.Quotient.eq, \u2190 map_sub, Ideal.mem_quotient_iff_mem hIJ, \u2190 Ideal.Quotient.eq]\n#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique\n\ntheorem ext [FormallyUnramified R A] (hI : IsNilpotent I) {g\u2081 g\u2082 : A \u2192\u2090[R] B}\n (H : \u2200 x, Ideal.Quotient.mk I (g\u2081 x) = Ideal.Quotient.mk I (g\u2082 x)) : g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique I hI g\u2081 g\u2082 (AlgHom.ext H)\n#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext\n\ntheorem lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]\n (f : B \u2192+* C) (hf : IsNilpotent <| RingHom.ker f) (g\u2081 g\u2082 : A \u2192\u2090[R] B)\n (h : f.comp \u2191g\u2081 = f.comp (g\u2082 : A \u2192+* B)) : g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique _ hf _ _\n (by\n ext x\n have := RingHom.congr_fun h x\n simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mk\u2090_eq_mk,\n RingHom.mem_ker, map_sub, sub_eq_zero])\n#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom\n\ntheorem ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B \u2192+* C)\n (hf : IsNilpotent <| RingHom.ker f) (g\u2081 g\u2082 : A \u2192\u2090[R] B) (h : \u2200 x, f (g\u2081 x) = f (g\u2082 x)) :\n g\u2081 = g\u2082 :=\n FormallyUnramified.lift_unique_of_ringHom f hf g\u2081 g\u2082 (RingHom.ext h)\n#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'\n\ntheorem lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]\n [Algebra R C] (f : B \u2192\u2090[R] C) (hf : IsNilpotent <| RingHom.ker (f : B \u2192+* C))\n (g\u2081 g\u2082 : A \u2192\u2090[R] B) (h : f.comp g\u2081 = f.comp g\u2082) : g\u2081 = g\u2082 :=\n FormallyUnramified.ext' _ hf g\u2081 g\u2082 (AlgHom.congr_fun h)\n#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'\n\nend\n\nsection OfEquiv\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n\ntheorem of_equiv [FormallyUnramified R A] (e : A \u2243\u2090[R] B) :\n FormallyUnramified R B := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e'\n rw [\u2190 f\u2081.comp_id, \u2190 f\u2082.comp_id, \u2190 e.comp_symm, \u2190 AlgHom.comp_assoc, \u2190 AlgHom.comp_assoc]\n congr 1\n refine' FormallyUnramified.comp_injective I hI _\n rw [\u2190 AlgHom.comp_assoc, e', AlgHom.comp_assoc]\n#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv\n\nend OfEquiv\n\nsection Comp\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [CommSemiring A] [Algebra R A]\nvariable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]\n\ntheorem comp [FormallyUnramified R A] [FormallyUnramified A B] :\n FormallyUnramified R B := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e\n have e' :=\n FormallyUnramified.lift_unique I \u27e82, hI\u27e9 (f\u2081.comp <| IsScalarTower.toAlgHom R A B)\n (f\u2082.comp <| IsScalarTower.toAlgHom R A B) (by rw [\u2190 AlgHom.comp_assoc, e, AlgHom.comp_assoc])\n letI := (f\u2081.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra\n let F\u2081 : B \u2192\u2090[A] C := { f\u2081 with commutes' := fun r => rfl }\n let F\u2082 : B \u2192\u2090[A] C := { f\u2082 with commutes' := AlgHom.congr_fun e'.symm }\n ext1 x\n change F\u2081 x = F\u2082 x\n congr\n exact FormallyUnramified.ext I \u27e82, hI\u27e9 (AlgHom.congr_fun e)\n#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp\n\ntheorem of_comp [FormallyUnramified R B] : FormallyUnramified A B := by\n constructor\n intro Q _ _ I e f\u2081 f\u2082 e'\n letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra\n letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl\n refine' AlgHom.restrictScalars_injective R _\n refine' FormallyUnramified.ext I \u27e82, e\u27e9 _\n intro x\n exact AlgHom.congr_fun e' x\n#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp\n\nend Comp\n\nsection BaseChange\n\nopen scoped TensorProduct\n\nvariable {R : Type u} [CommSemiring R]\nvariable {A : Type u} [Semiring A] [Algebra R A]\nvariable (B : Type u) [CommSemiring B] [Algebra R B]\n\ninstance base_change [FormallyUnramified R A] :\n FormallyUnramified B (B \u2297[R] A) := by\n constructor\n intro C _ _ I hI f\u2081 f\u2082 e\n letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra\n haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl\n ext : 1\n \u00b7 exact Subsingleton.elim _ _\n \u00b7 exact FormallyUnramified.ext I \u27e82, hI\u27e9 fun x => AlgHom.congr_fun e (1 \u2297\u209c x)\n#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change\n\nend BaseChange\n\nsection Localization\n\nvariable {R S R\u2098 S\u2098 : Type u} [CommRing R] [CommRing S] [CommRing R\u2098] [CommRing S\u2098]\nvariable (M : Submonoid R)\nvariable [Algebra R S] [Algebra R S\u2098] [Algebra S S\u2098] [Algebra R R\u2098] [Algebra R\u2098 S\u2098]\nvariable [IsScalarTower R R\u2098 S\u2098] [IsScalarTower R S S\u2098]\nvariable [IsLocalization M R\u2098] [IsLocalization (M.map (algebraMap R S)) S\u2098]\n\n-- Porting note: no longer supported\n-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on\n\n/-- This holds in general for epimorphisms. -/\ntheorem of_isLocalization : FormallyUnramified R R\u2098 := by\n constructor\n intro Q _ _ I _ f\u2081 f\u2082 _\n apply AlgHom.coe_ringHom_injective\n refine' IsLocalization.ringHom_ext M _\n ext\n simp\n#align algebra.formally_unramified.of_is_localization Algebra.FormallyUnramified.of_isLocalization\n\n/-- This actually does not need the localization instance, and is stated here again for\nconsistency. See `Algebra.FormallyUnramified.of_comp` instead.\n\n The intended use is for copying proofs between `Formally{Unramified, Smooth, Etale}`\n without the need to change anything (including removing redundant arguments). -/\n-- @[nolint unusedArguments] -- Porting note: removed\ntheorem localization_base [FormallyUnramified R S\u2098] : FormallyUnramified R\u2098 S\u2098 :=\n -- Porting note: added\n let _ := M\n FormallyUnramified.of_comp R R\u2098 S\u2098\n#align algebra.formally_unramified.localization_base Algebra.FormallyUnramified.localization_base\n\ntheorem localization_map [FormallyUnramified R S] :\n FormallyUnramified R\u2098 S\u2098 := by\n haveI : FormallyUnramified S S\u2098 :=\n FormallyUnramified.of_isLocalization (M.map (algebraMap R S))\n haveI : FormallyUnramified R S\u2098 := FormallyUnramified.comp R S S\u2098\n exact FormallyUnramified.localization_base M\n#align algebra.formally_unramified.localization_map Algebra.FormallyUnramified.localization_map\n\nend Localization\n\nend FormallyUnramified\n\nsection\n\nvariable (R : Type u) [CommSemiring R]\nvariable (A : Type u) [Semiring A] [Algebra R A]\n\n/-- An `R`-algebra `A` is unramified if it is formally unramified and of finite type.\n-/\nclass Unramified : Prop where\n formallyUnramified : FormallyUnramified R A := by infer_instance\n finiteType : FiniteType R A := by infer_instance\n\nend\n\nnamespace Unramified\n\nattribute [instance] formallyUnramified finiteType\n\nvariable {R : Type u} [CommRing R]\nvariable {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]\n\n/-- Being unramified is transported via algebra isomorphisms. -/\ntheorem of_equiv [Unramified R A] (e : A \u2243\u2090[R] B) : Unramified R B where\n formallyUnramified := FormallyUnramified.of_equiv e\n finiteType := FiniteType.equiv Unramified.finiteType e\n\n/-- Localization at an element is unramified. -/\ntheorem of_isLocalization_Away (r : R) [IsLocalization.Away r A] : Unramified R A where\n formallyUnramified := Algebra.FormallyUnramified.of_isLocalization (Submonoid.powers r)\n finiteType :=\n haveI : FinitePresentation R A := IsLocalization.Away.finitePresentation r\n inferInstance\n\nsection Comp\n\nvariable (R A B)\nvariable [Algebra A B] [IsScalarTower R A B]\n\n/-- Unramified is stable under composition. -/\n", "theoremStatement": "theorem comp [Unramified R A] [Unramified A B] : Unramified R B", "theoremName": "comp", "fileCreated": {"commit": "d991fe8f80", "date": "2024-04-08"}, "theoremCreated": {"commit": "c9ab842e13", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/RingTheory/Unramified/Basic.lean", "positionMetadata": {"lineInFile": 272, "tokenPositionInFile": 10211, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "where\n formallyUnramified := FormallyUnramified.comp R A B\n finiteType := FiniteType.trans (S := A) Unramified.finiteType\n Unramified.finiteType", "proofType": "term", "proofLengthLines": 4, "proofLengthTokens": 149}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Floris van Doorn\n-/\nimport Mathlib.CategoryTheory.Limits.Filtered\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts\nimport Mathlib.CategoryTheory.Limits.Shapes.Kernels\nimport Mathlib.CategoryTheory.DiscreteCategory\n\n#align_import category_theory.limits.opposites from \"leanprover-community/mathlib\"@\"ac3ae212f394f508df43e37aa093722fa9b65d31\"\n\n/-!\n# Limits in `C` give colimits in `C\u1d52\u1d56`.\n\nWe also give special cases for (co)products,\n(co)equalizers, and pullbacks / pushouts.\n\n-/\n\n\nuniverse v\u2081 v\u2082 u\u2081 u\u2082\n\nnoncomputable section\n\nopen CategoryTheory\n\nopen CategoryTheory.Functor\n\nopen Opposite\n\nnamespace CategoryTheory.Limits\n\nvariable {C : Type u\u2081} [Category.{v\u2081} C]\nvariable {J : Type u\u2082} [Category.{v\u2082} J]\n\n#align category_theory.limits.is_limit_cocone_op CategoryTheory.Limits.IsColimit.op\n#align category_theory.limits.is_colimit_cone_op CategoryTheory.Limits.IsLimit.op\n#align category_theory.limits.is_limit_cocone_unop CategoryTheory.Limits.IsColimit.unop\n#align category_theory.limits.is_colimit_cone_unop CategoryTheory.Limits.IsLimit.unop\n\n-- 2024-03-26\n@[deprecated] alias isLimitCoconeOp := IsColimit.op\n@[deprecated] alias isColimitConeOp := IsLimit.op\n@[deprecated] alias isLimitCoconeUnop := IsColimit.unop\n@[deprecated] alias isColimitConeUnop := IsLimit.unop\n\n/-- Turn a colimit for `F : J \u2964 C\u1d52\u1d56` into a limit for `F.leftOp : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isLimitConeLeftOpOfCocone (F : J \u2964 C\u1d52\u1d56) {c : Cocone F} (hc : IsColimit c) :\n IsLimit (coneLeftOpOfCocone c)\n where\n lift s := (hc.desc (coconeOfConeLeftOp s)).unop\n fac s j :=\n Quiver.Hom.op_inj <| by\n simp only [coneLeftOpOfCocone_\u03c0_app, op_comp, Quiver.Hom.op_unop, IsColimit.fac,\n coconeOfConeLeftOp_\u03b9_app, op_unop]\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsColimit.fac, coconeOfConeLeftOp_\u03b9_app] using w (op j)\n#align category_theory.limits.is_limit_cone_left_op_of_cocone CategoryTheory.Limits.isLimitConeLeftOpOfCocone\n\n/-- Turn a limit of `F : J \u2964 C\u1d52\u1d56` into a colimit of `F.leftOp : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isColimitCoconeLeftOpOfCone (F : J \u2964 C\u1d52\u1d56) {c : Cone F} (hc : IsLimit c) :\n IsColimit (coconeLeftOpOfCone c)\n where\n desc s := (hc.lift (coneOfCoconeLeftOp s)).unop\n fac s j :=\n Quiver.Hom.op_inj <| by\n simp only [coconeLeftOpOfCone_\u03b9_app, op_comp, Quiver.Hom.op_unop, IsLimit.fac,\n coneOfCoconeLeftOp_\u03c0_app, op_unop]\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsLimit.fac, coneOfCoconeLeftOp_\u03c0_app] using w (op j)\n#align category_theory.limits.is_colimit_cocone_left_op_of_cone CategoryTheory.Limits.isColimitCoconeLeftOpOfCone\n\n/-- Turn a colimit for `F : J\u1d52\u1d56 \u2964 C` into a limit for `F.rightOp : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isLimitConeRightOpOfCocone (F : J\u1d52\u1d56 \u2964 C) {c : Cocone F} (hc : IsColimit c) :\n IsLimit (coneRightOpOfCocone c)\n where\n lift s := (hc.desc (coconeOfConeRightOp s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsColimit.fac] using w (unop j)\n#align category_theory.limits.is_limit_cone_right_op_of_cocone CategoryTheory.Limits.isLimitConeRightOpOfCocone\n\n/-- Turn a limit for `F : J\u1d52\u1d56 \u2964 C` into a colimit for `F.rightOp : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isColimitCoconeRightOpOfCone (F : J\u1d52\u1d56 \u2964 C) {c : Cone F} (hc : IsLimit c) :\n IsColimit (coconeRightOpOfCone c)\n where\n desc s := (hc.lift (coneOfCoconeRightOp s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsLimit.fac] using w (unop j)\n#align category_theory.limits.is_colimit_cocone_right_op_of_cone CategoryTheory.Limits.isColimitCoconeRightOpOfCone\n\n/-- Turn a colimit for `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56` into a limit for `F.unop : J \u2964 C`. -/\n@[simps]\ndef isLimitConeUnopOfCocone (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cocone F} (hc : IsColimit c) :\n IsLimit (coneUnopOfCocone c)\n where\n lift s := (hc.desc (coconeOfConeUnop s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsColimit.fac] using w (unop j)\n#align category_theory.limits.is_limit_cone_unop_of_cocone CategoryTheory.Limits.isLimitConeUnopOfCocone\n\n/-- Turn a limit of `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56` into a colimit of `F.unop : J \u2964 C`. -/\n@[simps]\ndef isColimitCoconeUnopOfCone (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cone F} (hc : IsLimit c) :\n IsColimit (coconeUnopOfCone c)\n where\n desc s := (hc.lift (coneOfCoconeUnop s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsLimit.fac] using w (unop j)\n#align category_theory.limits.is_colimit_cocone_unop_of_cone CategoryTheory.Limits.isColimitCoconeUnopOfCone\n\n/-- Turn a colimit for `F.leftOp : J\u1d52\u1d56 \u2964 C` into a limit for `F : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isLimitConeOfCoconeLeftOp (F : J \u2964 C\u1d52\u1d56) {c : Cocone F.leftOp} (hc : IsColimit c) :\n IsLimit (coneOfCoconeLeftOp c)\n where\n lift s := (hc.desc (coconeLeftOpOfCone s)).op\n fac s j :=\n Quiver.Hom.unop_inj <| by\n simp only [coneOfCoconeLeftOp_\u03c0_app, unop_comp, Quiver.Hom.unop_op, IsColimit.fac,\n coconeLeftOpOfCone_\u03b9_app, unop_op]\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsColimit.fac, coneOfCoconeLeftOp_\u03c0_app] using w (unop j)\n#align category_theory.limits.is_limit_cone_of_cocone_left_op CategoryTheory.Limits.isLimitConeOfCoconeLeftOp\n\n/-- Turn a limit of `F.leftOp : J\u1d52\u1d56 \u2964 C` into a colimit of `F : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isColimitCoconeOfConeLeftOp (F : J \u2964 C\u1d52\u1d56) {c : Cone F.leftOp} (hc : IsLimit c) :\n IsColimit (coconeOfConeLeftOp c)\n where\n desc s := (hc.lift (coneLeftOpOfCocone s)).op\n fac s j :=\n Quiver.Hom.unop_inj <| by\n simp only [coconeOfConeLeftOp_\u03b9_app, unop_comp, Quiver.Hom.unop_op, IsLimit.fac,\n coneLeftOpOfCocone_\u03c0_app, unop_op]\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsLimit.fac, coconeOfConeLeftOp_\u03b9_app] using w (unop j)\n#align category_theory.limits.is_colimit_cocone_of_cone_left_op CategoryTheory.Limits.isColimitCoconeOfConeLeftOp\n\n/-- Turn a colimit for `F.rightOp : J \u2964 C\u1d52\u1d56` into a limit for `F : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isLimitConeOfCoconeRightOp (F : J\u1d52\u1d56 \u2964 C) {c : Cocone F.rightOp} (hc : IsColimit c) :\n IsLimit (coneOfCoconeRightOp c)\n where\n lift s := (hc.desc (coconeRightOpOfCone s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsColimit.fac] using w (op j)\n#align category_theory.limits.is_limit_cone_of_cocone_right_op CategoryTheory.Limits.isLimitConeOfCoconeRightOp\n\n/-- Turn a limit for `F.rightOp : J \u2964 C\u1d52\u1d56` into a limit for `F : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isColimitCoconeOfConeRightOp (F : J\u1d52\u1d56 \u2964 C) {c : Cone F.rightOp} (hc : IsLimit c) :\n IsColimit (coconeOfConeRightOp c)\n where\n desc s := (hc.lift (coneRightOpOfCocone s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsLimit.fac] using w (op j)\n#align category_theory.limits.is_colimit_cocone_of_cone_right_op CategoryTheory.Limits.isColimitCoconeOfConeRightOp\n\n/-- Turn a colimit for `F.unop : J \u2964 C` into a limit for `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isLimitConeOfCoconeUnop (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cocone F.unop} (hc : IsColimit c) :\n IsLimit (coneOfCoconeUnop c)\n where\n lift s := (hc.desc (coconeUnopOfCone s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsColimit.fac] using w (op j)\n#align category_theory.limits.is_limit_cone_of_cocone_unop CategoryTheory.Limits.isLimitConeOfCoconeUnop\n\n/-- Turn a limit for `F.unop : J \u2964 C` into a colimit for `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isColimitConeOfCoconeUnop (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cone F.unop} (hc : IsLimit c) :\n IsColimit (coconeOfConeUnop c)\n where\n desc s := (hc.lift (coneUnopOfCocone s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsLimit.fac] using w (op j)\n#align category_theory.limits.is_colimit_cone_of_cocone_unop CategoryTheory.Limits.isColimitConeOfCoconeUnop\n\n/-- If `F.leftOp : J\u1d52\u1d56 \u2964 C` has a colimit, we can construct a limit for `F : J \u2964 C\u1d52\u1d56`.\n-/\ntheorem hasLimit_of_hasColimit_leftOp (F : J \u2964 C\u1d52\u1d56) [HasColimit F.leftOp] : HasLimit F :=\n HasLimit.mk\n { cone := coneOfCoconeLeftOp (colimit.cocone F.leftOp)\n isLimit := isLimitConeOfCoconeLeftOp _ (colimit.isColimit _) }\n#align category_theory.limits.has_limit_of_has_colimit_left_op CategoryTheory.Limits.hasLimit_of_hasColimit_leftOp\n\ntheorem hasLimit_of_hasColimit_op (F : J \u2964 C) [HasColimit F.op] : HasLimit F :=\n HasLimit.mk\n { cone := (colimit.cocone F.op).unop\n isLimit := (colimit.isColimit _).unop }\n\ntheorem hasLimit_op_of_hasColimit (F : J \u2964 C) [HasColimit F] : HasLimit F.op :=\n HasLimit.mk\n { cone := (colimit.cocone F).op\n isLimit := (colimit.isColimit _).op }\n#align category_theory.limits.has_limit_of_has_colimit_op CategoryTheory.Limits.hasLimit_of_hasColimit_op\n\n/-- If `C` has colimits of shape `J\u1d52\u1d56`, we can construct limits in `C\u1d52\u1d56` of shape `J`.\n-/\ntheorem hasLimitsOfShape_op_of_hasColimitsOfShape [HasColimitsOfShape J\u1d52\u1d56 C] :\n HasLimitsOfShape J C\u1d52\u1d56 :=\n { has_limit := fun F => hasLimit_of_hasColimit_leftOp F }\n#align category_theory.limits.has_limits_of_shape_op_of_has_colimits_of_shape CategoryTheory.Limits.hasLimitsOfShape_op_of_hasColimitsOfShape\n\ntheorem hasLimitsOfShape_of_hasColimitsOfShape_op [HasColimitsOfShape J\u1d52\u1d56 C\u1d52\u1d56] :\n HasLimitsOfShape J C :=\n { has_limit := fun F => hasLimit_of_hasColimit_op F }\n#align category_theory.limits.has_limits_of_shape_of_has_colimits_of_shape_op CategoryTheory.Limits.hasLimitsOfShape_of_hasColimitsOfShape_op\n\nattribute [local instance] hasLimitsOfShape_op_of_hasColimitsOfShape\n\n/-- If `C` has colimits, we can construct limits for `C\u1d52\u1d56`.\n-/\ninstance hasLimits_op_of_hasColimits [HasColimits C] : HasLimits C\u1d52\u1d56 :=\n \u27e8fun _ => inferInstance\u27e9\n#align category_theory.limits.has_limits_op_of_has_colimits CategoryTheory.Limits.hasLimits_op_of_hasColimits\n\ntheorem hasLimits_of_hasColimits_op [HasColimits C\u1d52\u1d56] : HasLimits C :=\n { has_limits_of_shape := fun _ _ => hasLimitsOfShape_of_hasColimitsOfShape_op }\n#align category_theory.limits.has_limits_of_has_colimits_op CategoryTheory.Limits.hasLimits_of_hasColimits_op\n\ninstance has_cofiltered_limits_op_of_has_filtered_colimits [HasFilteredColimitsOfSize.{v\u2082, u\u2082} C] :\n HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56 where\n HasLimitsOfShape _ _ _ := hasLimitsOfShape_op_of_hasColimitsOfShape\n#align category_theory.limits.has_cofiltered_limits_op_of_has_filtered_colimits CategoryTheory.Limits.has_cofiltered_limits_op_of_has_filtered_colimits\n\ntheorem has_cofiltered_limits_of_has_filtered_colimits_op [HasFilteredColimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56] :\n HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C :=\n { HasLimitsOfShape := fun _ _ _ => hasLimitsOfShape_of_hasColimitsOfShape_op }\n#align category_theory.limits.has_cofiltered_limits_of_has_filtered_colimits_op CategoryTheory.Limits.has_cofiltered_limits_of_has_filtered_colimits_op\n\n/-- If `F.leftOp : J\u1d52\u1d56 \u2964 C` has a limit, we can construct a colimit for `F : J \u2964 C\u1d52\u1d56`.\n-/\ntheorem hasColimit_of_hasLimit_leftOp (F : J \u2964 C\u1d52\u1d56) [HasLimit F.leftOp] : HasColimit F :=\n HasColimit.mk\n { cocone := coconeOfConeLeftOp (limit.cone F.leftOp)\n isColimit := isColimitCoconeOfConeLeftOp _ (limit.isLimit _) }\n#align category_theory.limits.has_colimit_of_has_limit_left_op CategoryTheory.Limits.hasColimit_of_hasLimit_leftOp\n\ntheorem hasColimit_of_hasLimit_op (F : J \u2964 C) [HasLimit F.op] : HasColimit F :=\n HasColimit.mk\n { cocone := (limit.cone F.op).unop\n isColimit := (limit.isLimit _).unop }\n#align category_theory.limits.has_colimit_of_has_limit_op CategoryTheory.Limits.hasColimit_of_hasLimit_op\n\ntheorem hasColimit_op_of_hasLimit (F : J \u2964 C) [HasLimit F] : HasColimit F.op :=\n HasColimit.mk\n { cocone := (limit.cone F).op\n isColimit := (limit.isLimit _).op }\n\n/-- If `C` has colimits of shape `J\u1d52\u1d56`, we can construct limits in `C\u1d52\u1d56` of shape `J`.\n-/\ninstance hasColimitsOfShape_op_of_hasLimitsOfShape [HasLimitsOfShape J\u1d52\u1d56 C] :\n HasColimitsOfShape J C\u1d52\u1d56 where has_colimit F := hasColimit_of_hasLimit_leftOp F\n#align category_theory.limits.has_colimits_of_shape_op_of_has_limits_of_shape CategoryTheory.Limits.hasColimitsOfShape_op_of_hasLimitsOfShape\n\ntheorem hasColimitsOfShape_of_hasLimitsOfShape_op [HasLimitsOfShape J\u1d52\u1d56 C\u1d52\u1d56] :\n HasColimitsOfShape J C :=\n { has_colimit := fun F => hasColimit_of_hasLimit_op F }\n#align category_theory.limits.has_colimits_of_shape_of_has_limits_of_shape_op CategoryTheory.Limits.hasColimitsOfShape_of_hasLimitsOfShape_op\n\n/-- If `C` has limits, we can construct colimits for `C\u1d52\u1d56`.\n-/\ninstance hasColimits_op_of_hasLimits [HasLimits C] : HasColimits C\u1d52\u1d56 :=\n \u27e8fun _ => inferInstance\u27e9\n#align category_theory.limits.has_colimits_op_of_has_limits CategoryTheory.Limits.hasColimits_op_of_hasLimits\n\ntheorem hasColimits_of_hasLimits_op [HasLimits C\u1d52\u1d56] : HasColimits C :=\n { has_colimits_of_shape := fun _ _ => hasColimitsOfShape_of_hasLimitsOfShape_op }\n#align category_theory.limits.has_colimits_of_has_limits_op CategoryTheory.Limits.hasColimits_of_hasLimits_op\n\ninstance has_filtered_colimits_op_of_has_cofiltered_limits [HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C] :\n HasFilteredColimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56 where HasColimitsOfShape _ _ _ := inferInstance\n#align category_theory.limits.has_filtered_colimits_op_of_has_cofiltered_limits CategoryTheory.Limits.has_filtered_colimits_op_of_has_cofiltered_limits\n\ntheorem has_filtered_colimits_of_has_cofiltered_limits_op [HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56] :\n HasFilteredColimitsOfSize.{v\u2082, u\u2082} C :=\n { HasColimitsOfShape := fun _ _ _ => hasColimitsOfShape_of_hasLimitsOfShape_op }\n#align category_theory.limits.has_filtered_colimits_of_has_cofiltered_limits_op CategoryTheory.Limits.has_filtered_colimits_of_has_cofiltered_limits_op\n\nvariable (X : Type v\u2082)\n\n/-- If `C` has products indexed by `X`, then `C\u1d52\u1d56` has coproducts indexed by `X`.\n-/\ninstance hasCoproductsOfShape_opposite [HasProductsOfShape X C] : HasCoproductsOfShape X C\u1d52\u1d56 := by\n haveI : HasLimitsOfShape (Discrete X)\u1d52\u1d56 C :=\n hasLimitsOfShape_of_equivalence (Discrete.opposite X).symm\n infer_instance\n#align category_theory.limits.has_coproducts_of_shape_opposite CategoryTheory.Limits.hasCoproductsOfShape_opposite\n\ntheorem hasCoproductsOfShape_of_opposite [HasProductsOfShape X C\u1d52\u1d56] : HasCoproductsOfShape X C :=\n haveI : HasLimitsOfShape (Discrete X)\u1d52\u1d56 C\u1d52\u1d56 :=\n hasLimitsOfShape_of_equivalence (Discrete.opposite X).symm\n hasColimitsOfShape_of_hasLimitsOfShape_op\n#align category_theory.limits.has_coproducts_of_shape_of_opposite CategoryTheory.Limits.hasCoproductsOfShape_of_opposite\n\n/-- If `C` has coproducts indexed by `X`, then `C\u1d52\u1d56` has products indexed by `X`.\n-/\ninstance hasProductsOfShape_opposite [HasCoproductsOfShape X C] : HasProductsOfShape X C\u1d52\u1d56 := by\n haveI : HasColimitsOfShape (Discrete X)\u1d52\u1d56 C :=\n hasColimitsOfShape_of_equivalence (Discrete.opposite X).symm\n infer_instance\n#align category_theory.limits.has_products_of_shape_opposite CategoryTheory.Limits.hasProductsOfShape_opposite\n\ntheorem hasProductsOfShape_of_opposite [HasCoproductsOfShape X C\u1d52\u1d56] : HasProductsOfShape X C :=\n haveI : HasColimitsOfShape (Discrete X)\u1d52\u1d56 C\u1d52\u1d56 :=\n hasColimitsOfShape_of_equivalence (Discrete.opposite X).symm\n hasLimitsOfShape_of_hasColimitsOfShape_op\n#align category_theory.limits.has_products_of_shape_of_opposite CategoryTheory.Limits.hasProductsOfShape_of_opposite\n\ninstance hasProducts_opposite [HasCoproducts.{v\u2082} C] : HasProducts.{v\u2082} C\u1d52\u1d56 := fun _ =>\n inferInstance\n#align category_theory.limits.has_products_opposite CategoryTheory.Limits.hasProducts_opposite\n\ntheorem hasProducts_of_opposite [HasCoproducts.{v\u2082} C\u1d52\u1d56] : HasProducts.{v\u2082} C := fun X =>\n hasProductsOfShape_of_opposite X\n#align category_theory.limits.has_products_of_opposite CategoryTheory.Limits.hasProducts_of_opposite\n\ninstance hasCoproducts_opposite [HasProducts.{v\u2082} C] : HasCoproducts.{v\u2082} C\u1d52\u1d56 := fun _ =>\n inferInstance\n#align category_theory.limits.has_coproducts_opposite CategoryTheory.Limits.hasCoproducts_opposite\n\ntheorem hasCoproducts_of_opposite [HasProducts.{v\u2082} C\u1d52\u1d56] : HasCoproducts.{v\u2082} C := fun X =>\n hasCoproductsOfShape_of_opposite X\n#align category_theory.limits.has_coproducts_of_opposite CategoryTheory.Limits.hasCoproducts_of_opposite\n\ninstance hasFiniteCoproducts_opposite [HasFiniteProducts C] : HasFiniteCoproducts C\u1d52\u1d56 where\n out _ := Limits.hasCoproductsOfShape_opposite _\n#align category_theory.limits.has_finite_coproducts_opposite CategoryTheory.Limits.hasFiniteCoproducts_opposite\n\ntheorem hasFiniteCoproducts_of_opposite [HasFiniteProducts C\u1d52\u1d56] : HasFiniteCoproducts C :=\n { out := fun _ => hasCoproductsOfShape_of_opposite _ }\n#align category_theory.limits.has_finite_coproducts_of_opposite CategoryTheory.Limits.hasFiniteCoproducts_of_opposite\n\ninstance hasFiniteProducts_opposite [HasFiniteCoproducts C] : HasFiniteProducts C\u1d52\u1d56 where\n out _ := inferInstance\n#align category_theory.limits.has_finite_products_opposite CategoryTheory.Limits.hasFiniteProducts_opposite\n\ntheorem hasFiniteProducts_of_opposite [HasFiniteCoproducts C\u1d52\u1d56] : HasFiniteProducts C :=\n { out := fun _ => hasProductsOfShape_of_opposite _ }\n#align category_theory.limits.has_finite_products_of_opposite CategoryTheory.Limits.hasFiniteProducts_of_opposite\n\nsection OppositeCoproducts\n\nvariable {\u03b1 : Type*} {Z : \u03b1 \u2192 C} [HasCoproduct Z]\n\ninstance : HasLimit (Discrete.functor Z).op := hasLimit_op_of_hasColimit (Discrete.functor Z)\n\ninstance : HasLimit ((Discrete.opposite \u03b1).inverse \u22d9 (Discrete.functor Z).op) :=\n hasLimitEquivalenceComp (Discrete.opposite \u03b1).symm\n\ninstance : HasProduct (op <| Z \u00b7) := hasLimitOfIso\n ((Discrete.natIsoFunctor \u226a\u226b Discrete.natIso (fun _ \u21a6 by rfl)) :\n (Discrete.opposite \u03b1).inverse \u22d9 (Discrete.functor Z).op \u2245\n Discrete.functor (op <| Z \u00b7))\n\n/-- A `Cofan` gives a `Fan` in the opposite category. -/\n@[simp]\ndef Cofan.op (c : Cofan Z) : Fan (op <| Z \u00b7) := Fan.mk _ (fun a \u21a6 (c.inj a).op)\n\n/-- If a `Cofan`\u00a0is colimit, then its opposite is limit. -/\ndef Cofan.IsColimit.op {c : Cofan Z} (hc : IsColimit c) : IsLimit c.op := by\n let e : Discrete.functor (Opposite.op <| Z \u00b7) \u2245 (Discrete.opposite \u03b1).inverse \u22d9\n (Discrete.functor Z).op := Discrete.natIso (fun _ \u21a6 Iso.refl _)\n refine IsLimit.ofIsoLimit ((IsLimit.postcomposeInvEquiv e _).2\n (IsLimit.whiskerEquivalence hc.op (Discrete.opposite \u03b1).symm))\n (Cones.ext (Iso.refl _) (fun \u27e8a\u27e9 \u21a6 ?_))\n dsimp\n erw [Category.id_comp, Category.comp_id]\n rfl\n\n/--\nThe canonical isomorphism from the opposite of an abstract coproduct to the corresponding product\nin the opposite category.\n-/\ndef opCoproductIsoProduct' {c : Cofan Z} {f : Fan (op <| Z \u00b7)}\n (hc : IsColimit c) (hf : IsLimit f) : op c.pt \u2245 f.pt :=\n IsLimit.conePointUniqueUpToIso (Cofan.IsColimit.op hc) hf\n\nvariable (Z) in\n/--\nThe canonical isomorphism from the opposite of the coproduct to the product in the opposite\ncategory.\n-/\ndef opCoproductIsoProduct :\n op (\u2210 Z) \u2245 \u220f (op <| Z \u00b7) :=\n opCoproductIsoProduct' (coproductIsCoproduct Z) (productIsProduct (op <| Z \u00b7))\n\ntheorem opCoproductIsoProduct'_inv_comp_inj {c : Cofan Z} {f : Fan (op <| Z \u00b7)}\n (hc : IsColimit c) (hf : IsLimit f) (b : \u03b1) :\n (opCoproductIsoProduct' hc hf).inv \u226b (c.inj b).op = f.proj b :=\n IsLimit.conePointUniqueUpToIso_inv_comp (Cofan.IsColimit.op hc) hf \u27e8b\u27e9\n\n", "theoremStatement": "theorem opCoproductIsoProduct'_comp_self {c c' : Cofan Z} {f : Fan (op <| Z \u00b7)}\n (hc : IsColimit c) (hc' : IsColimit c') (hf : IsLimit f) :\n (opCoproductIsoProduct' hc hf).hom \u226b (opCoproductIsoProduct' hc' hf).inv =\n (hc.coconePointUniqueUpToIso hc').op.inv", "theoremName": "opCoproductIsoProduct'_comp_self", "fileCreated": {"commit": "3fcb15f5aa", "date": "2023-03-11"}, "theoremCreated": {"commit": "423f5783b8", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/CategoryTheory/Limits/Opposites.lean", "positionMetadata": {"lineInFile": 419, "tokenPositionInFile": 20363, "theoremPositionInFile": 54}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n apply Quiver.Hom.unop_inj\n apply hc'.hom_ext\n intro \u27e8j\u27e9\n change c'.inj _ \u226b _ = _\n simp only [unop_op, unop_comp, Discrete.functor_obj, const_obj_obj, Iso.op_inv,\n Quiver.Hom.unop_op, IsColimit.comp_coconePointUniqueUpToIso_inv]\n apply Quiver.Hom.op_inj\n simp only [op_comp, op_unop, Quiver.Hom.op_unop, Category.assoc,\n opCoproductIsoProduct'_inv_comp_inj]\n rw [\u2190 opCoproductIsoProduct'_inv_comp_inj hc hf]\n simp only [Iso.hom_inv_id_assoc]\n rfl", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 465}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Floris van Doorn\n-/\nimport Mathlib.CategoryTheory.Limits.Filtered\nimport Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts\nimport Mathlib.CategoryTheory.Limits.Shapes.Kernels\nimport Mathlib.CategoryTheory.DiscreteCategory\n\n#align_import category_theory.limits.opposites from \"leanprover-community/mathlib\"@\"ac3ae212f394f508df43e37aa093722fa9b65d31\"\n\n/-!\n# Limits in `C` give colimits in `C\u1d52\u1d56`.\n\nWe also give special cases for (co)products,\n(co)equalizers, and pullbacks / pushouts.\n\n-/\n\n\nuniverse v\u2081 v\u2082 u\u2081 u\u2082\n\nnoncomputable section\n\nopen CategoryTheory\n\nopen CategoryTheory.Functor\n\nopen Opposite\n\nnamespace CategoryTheory.Limits\n\nvariable {C : Type u\u2081} [Category.{v\u2081} C]\nvariable {J : Type u\u2082} [Category.{v\u2082} J]\n\n#align category_theory.limits.is_limit_cocone_op CategoryTheory.Limits.IsColimit.op\n#align category_theory.limits.is_colimit_cone_op CategoryTheory.Limits.IsLimit.op\n#align category_theory.limits.is_limit_cocone_unop CategoryTheory.Limits.IsColimit.unop\n#align category_theory.limits.is_colimit_cone_unop CategoryTheory.Limits.IsLimit.unop\n\n-- 2024-03-26\n@[deprecated] alias isLimitCoconeOp := IsColimit.op\n@[deprecated] alias isColimitConeOp := IsLimit.op\n@[deprecated] alias isLimitCoconeUnop := IsColimit.unop\n@[deprecated] alias isColimitConeUnop := IsLimit.unop\n\n/-- Turn a colimit for `F : J \u2964 C\u1d52\u1d56` into a limit for `F.leftOp : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isLimitConeLeftOpOfCocone (F : J \u2964 C\u1d52\u1d56) {c : Cocone F} (hc : IsColimit c) :\n IsLimit (coneLeftOpOfCocone c)\n where\n lift s := (hc.desc (coconeOfConeLeftOp s)).unop\n fac s j :=\n Quiver.Hom.op_inj <| by\n simp only [coneLeftOpOfCocone_\u03c0_app, op_comp, Quiver.Hom.op_unop, IsColimit.fac,\n coconeOfConeLeftOp_\u03b9_app, op_unop]\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsColimit.fac, coconeOfConeLeftOp_\u03b9_app] using w (op j)\n#align category_theory.limits.is_limit_cone_left_op_of_cocone CategoryTheory.Limits.isLimitConeLeftOpOfCocone\n\n/-- Turn a limit of `F : J \u2964 C\u1d52\u1d56` into a colimit of `F.leftOp : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isColimitCoconeLeftOpOfCone (F : J \u2964 C\u1d52\u1d56) {c : Cone F} (hc : IsLimit c) :\n IsColimit (coconeLeftOpOfCone c)\n where\n desc s := (hc.lift (coneOfCoconeLeftOp s)).unop\n fac s j :=\n Quiver.Hom.op_inj <| by\n simp only [coconeLeftOpOfCone_\u03b9_app, op_comp, Quiver.Hom.op_unop, IsLimit.fac,\n coneOfCoconeLeftOp_\u03c0_app, op_unop]\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsLimit.fac, coneOfCoconeLeftOp_\u03c0_app] using w (op j)\n#align category_theory.limits.is_colimit_cocone_left_op_of_cone CategoryTheory.Limits.isColimitCoconeLeftOpOfCone\n\n/-- Turn a colimit for `F : J\u1d52\u1d56 \u2964 C` into a limit for `F.rightOp : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isLimitConeRightOpOfCocone (F : J\u1d52\u1d56 \u2964 C) {c : Cocone F} (hc : IsColimit c) :\n IsLimit (coneRightOpOfCocone c)\n where\n lift s := (hc.desc (coconeOfConeRightOp s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsColimit.fac] using w (unop j)\n#align category_theory.limits.is_limit_cone_right_op_of_cocone CategoryTheory.Limits.isLimitConeRightOpOfCocone\n\n/-- Turn a limit for `F : J\u1d52\u1d56 \u2964 C` into a colimit for `F.rightOp : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isColimitCoconeRightOpOfCone (F : J\u1d52\u1d56 \u2964 C) {c : Cone F} (hc : IsLimit c) :\n IsColimit (coconeRightOpOfCone c)\n where\n desc s := (hc.lift (coneOfCoconeRightOp s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsLimit.fac] using w (unop j)\n#align category_theory.limits.is_colimit_cocone_right_op_of_cone CategoryTheory.Limits.isColimitCoconeRightOpOfCone\n\n/-- Turn a colimit for `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56` into a limit for `F.unop : J \u2964 C`. -/\n@[simps]\ndef isLimitConeUnopOfCocone (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cocone F} (hc : IsColimit c) :\n IsLimit (coneUnopOfCocone c)\n where\n lift s := (hc.desc (coconeOfConeUnop s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsColimit.fac] using w (unop j)\n#align category_theory.limits.is_limit_cone_unop_of_cocone CategoryTheory.Limits.isLimitConeUnopOfCocone\n\n/-- Turn a limit of `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56` into a colimit of `F.unop : J \u2964 C`. -/\n@[simps]\ndef isColimitCoconeUnopOfCone (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cone F} (hc : IsLimit c) :\n IsColimit (coconeUnopOfCone c)\n where\n desc s := (hc.lift (coneOfCoconeUnop s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsLimit.fac] using w (unop j)\n#align category_theory.limits.is_colimit_cocone_unop_of_cone CategoryTheory.Limits.isColimitCoconeUnopOfCone\n\n/-- Turn a colimit for `F.leftOp : J\u1d52\u1d56 \u2964 C` into a limit for `F : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isLimitConeOfCoconeLeftOp (F : J \u2964 C\u1d52\u1d56) {c : Cocone F.leftOp} (hc : IsColimit c) :\n IsLimit (coneOfCoconeLeftOp c)\n where\n lift s := (hc.desc (coconeLeftOpOfCone s)).op\n fac s j :=\n Quiver.Hom.unop_inj <| by\n simp only [coneOfCoconeLeftOp_\u03c0_app, unop_comp, Quiver.Hom.unop_op, IsColimit.fac,\n coconeLeftOpOfCone_\u03b9_app, unop_op]\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsColimit.fac, coneOfCoconeLeftOp_\u03c0_app] using w (unop j)\n#align category_theory.limits.is_limit_cone_of_cocone_left_op CategoryTheory.Limits.isLimitConeOfCoconeLeftOp\n\n/-- Turn a limit of `F.leftOp : J\u1d52\u1d56 \u2964 C` into a colimit of `F : J \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isColimitCoconeOfConeLeftOp (F : J \u2964 C\u1d52\u1d56) {c : Cone F.leftOp} (hc : IsLimit c) :\n IsColimit (coconeOfConeLeftOp c)\n where\n desc s := (hc.lift (coneLeftOpOfCocone s)).op\n fac s j :=\n Quiver.Hom.unop_inj <| by\n simp only [coconeOfConeLeftOp_\u03b9_app, unop_comp, Quiver.Hom.unop_op, IsLimit.fac,\n coneLeftOpOfCocone_\u03c0_app, unop_op]\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsLimit.fac, coconeOfConeLeftOp_\u03b9_app] using w (unop j)\n#align category_theory.limits.is_colimit_cocone_of_cone_left_op CategoryTheory.Limits.isColimitCoconeOfConeLeftOp\n\n/-- Turn a colimit for `F.rightOp : J \u2964 C\u1d52\u1d56` into a limit for `F : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isLimitConeOfCoconeRightOp (F : J\u1d52\u1d56 \u2964 C) {c : Cocone F.rightOp} (hc : IsColimit c) :\n IsLimit (coneOfCoconeRightOp c)\n where\n lift s := (hc.desc (coconeRightOpOfCone s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsColimit.fac] using w (op j)\n#align category_theory.limits.is_limit_cone_of_cocone_right_op CategoryTheory.Limits.isLimitConeOfCoconeRightOp\n\n/-- Turn a limit for `F.rightOp : J \u2964 C\u1d52\u1d56` into a limit for `F : J\u1d52\u1d56 \u2964 C`. -/\n@[simps]\ndef isColimitCoconeOfConeRightOp (F : J\u1d52\u1d56 \u2964 C) {c : Cone F.rightOp} (hc : IsLimit c) :\n IsColimit (coconeOfConeRightOp c)\n where\n desc s := (hc.lift (coneRightOpOfCocone s)).unop\n fac s j := Quiver.Hom.op_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj _)\n simpa only [Quiver.Hom.op_unop, IsLimit.fac] using w (op j)\n#align category_theory.limits.is_colimit_cocone_of_cone_right_op CategoryTheory.Limits.isColimitCoconeOfConeRightOp\n\n/-- Turn a colimit for `F.unop : J \u2964 C` into a limit for `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isLimitConeOfCoconeUnop (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cocone F.unop} (hc : IsColimit c) :\n IsLimit (coneOfCoconeUnop c)\n where\n lift s := (hc.desc (coconeUnopOfCone s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsColimit.fac] using w (op j)\n#align category_theory.limits.is_limit_cone_of_cocone_unop CategoryTheory.Limits.isLimitConeOfCoconeUnop\n\n/-- Turn a limit for `F.unop : J \u2964 C` into a colimit for `F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56`. -/\n@[simps]\ndef isColimitConeOfCoconeUnop (F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56) {c : Cone F.unop} (hc : IsLimit c) :\n IsColimit (coconeOfConeUnop c)\n where\n desc s := (hc.lift (coneUnopOfCocone s)).op\n fac s j := Quiver.Hom.unop_inj (by simp)\n uniq s m w := by\n refine' Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj _)\n simpa only [Quiver.Hom.unop_op, IsLimit.fac] using w (op j)\n#align category_theory.limits.is_colimit_cone_of_cocone_unop CategoryTheory.Limits.isColimitConeOfCoconeUnop\n\n/-- If `F.leftOp : J\u1d52\u1d56 \u2964 C` has a colimit, we can construct a limit for `F : J \u2964 C\u1d52\u1d56`.\n-/\ntheorem hasLimit_of_hasColimit_leftOp (F : J \u2964 C\u1d52\u1d56) [HasColimit F.leftOp] : HasLimit F :=\n HasLimit.mk\n { cone := coneOfCoconeLeftOp (colimit.cocone F.leftOp)\n isLimit := isLimitConeOfCoconeLeftOp _ (colimit.isColimit _) }\n#align category_theory.limits.has_limit_of_has_colimit_left_op CategoryTheory.Limits.hasLimit_of_hasColimit_leftOp\n\ntheorem hasLimit_of_hasColimit_op (F : J \u2964 C) [HasColimit F.op] : HasLimit F :=\n HasLimit.mk\n { cone := (colimit.cocone F.op).unop\n isLimit := (colimit.isColimit _).unop }\n\ntheorem hasLimit_op_of_hasColimit (F : J \u2964 C) [HasColimit F] : HasLimit F.op :=\n HasLimit.mk\n { cone := (colimit.cocone F).op\n isLimit := (colimit.isColimit _).op }\n#align category_theory.limits.has_limit_of_has_colimit_op CategoryTheory.Limits.hasLimit_of_hasColimit_op\n\n/-- If `C` has colimits of shape `J\u1d52\u1d56`, we can construct limits in `C\u1d52\u1d56` of shape `J`.\n-/\ntheorem hasLimitsOfShape_op_of_hasColimitsOfShape [HasColimitsOfShape J\u1d52\u1d56 C] :\n HasLimitsOfShape J C\u1d52\u1d56 :=\n { has_limit := fun F => hasLimit_of_hasColimit_leftOp F }\n#align category_theory.limits.has_limits_of_shape_op_of_has_colimits_of_shape CategoryTheory.Limits.hasLimitsOfShape_op_of_hasColimitsOfShape\n\ntheorem hasLimitsOfShape_of_hasColimitsOfShape_op [HasColimitsOfShape J\u1d52\u1d56 C\u1d52\u1d56] :\n HasLimitsOfShape J C :=\n { has_limit := fun F => hasLimit_of_hasColimit_op F }\n#align category_theory.limits.has_limits_of_shape_of_has_colimits_of_shape_op CategoryTheory.Limits.hasLimitsOfShape_of_hasColimitsOfShape_op\n\nattribute [local instance] hasLimitsOfShape_op_of_hasColimitsOfShape\n\n/-- If `C` has colimits, we can construct limits for `C\u1d52\u1d56`.\n-/\ninstance hasLimits_op_of_hasColimits [HasColimits C] : HasLimits C\u1d52\u1d56 :=\n \u27e8fun _ => inferInstance\u27e9\n#align category_theory.limits.has_limits_op_of_has_colimits CategoryTheory.Limits.hasLimits_op_of_hasColimits\n\ntheorem hasLimits_of_hasColimits_op [HasColimits C\u1d52\u1d56] : HasLimits C :=\n { has_limits_of_shape := fun _ _ => hasLimitsOfShape_of_hasColimitsOfShape_op }\n#align category_theory.limits.has_limits_of_has_colimits_op CategoryTheory.Limits.hasLimits_of_hasColimits_op\n\ninstance has_cofiltered_limits_op_of_has_filtered_colimits [HasFilteredColimitsOfSize.{v\u2082, u\u2082} C] :\n HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56 where\n HasLimitsOfShape _ _ _ := hasLimitsOfShape_op_of_hasColimitsOfShape\n#align category_theory.limits.has_cofiltered_limits_op_of_has_filtered_colimits CategoryTheory.Limits.has_cofiltered_limits_op_of_has_filtered_colimits\n\ntheorem has_cofiltered_limits_of_has_filtered_colimits_op [HasFilteredColimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56] :\n HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C :=\n { HasLimitsOfShape := fun _ _ _ => hasLimitsOfShape_of_hasColimitsOfShape_op }\n#align category_theory.limits.has_cofiltered_limits_of_has_filtered_colimits_op CategoryTheory.Limits.has_cofiltered_limits_of_has_filtered_colimits_op\n\n/-- If `F.leftOp : J\u1d52\u1d56 \u2964 C` has a limit, we can construct a colimit for `F : J \u2964 C\u1d52\u1d56`.\n-/\ntheorem hasColimit_of_hasLimit_leftOp (F : J \u2964 C\u1d52\u1d56) [HasLimit F.leftOp] : HasColimit F :=\n HasColimit.mk\n { cocone := coconeOfConeLeftOp (limit.cone F.leftOp)\n isColimit := isColimitCoconeOfConeLeftOp _ (limit.isLimit _) }\n#align category_theory.limits.has_colimit_of_has_limit_left_op CategoryTheory.Limits.hasColimit_of_hasLimit_leftOp\n\ntheorem hasColimit_of_hasLimit_op (F : J \u2964 C) [HasLimit F.op] : HasColimit F :=\n HasColimit.mk\n { cocone := (limit.cone F.op).unop\n isColimit := (limit.isLimit _).unop }\n#align category_theory.limits.has_colimit_of_has_limit_op CategoryTheory.Limits.hasColimit_of_hasLimit_op\n\ntheorem hasColimit_op_of_hasLimit (F : J \u2964 C) [HasLimit F] : HasColimit F.op :=\n HasColimit.mk\n { cocone := (limit.cone F).op\n isColimit := (limit.isLimit _).op }\n\n/-- If `C` has colimits of shape `J\u1d52\u1d56`, we can construct limits in `C\u1d52\u1d56` of shape `J`.\n-/\ninstance hasColimitsOfShape_op_of_hasLimitsOfShape [HasLimitsOfShape J\u1d52\u1d56 C] :\n HasColimitsOfShape J C\u1d52\u1d56 where has_colimit F := hasColimit_of_hasLimit_leftOp F\n#align category_theory.limits.has_colimits_of_shape_op_of_has_limits_of_shape CategoryTheory.Limits.hasColimitsOfShape_op_of_hasLimitsOfShape\n\ntheorem hasColimitsOfShape_of_hasLimitsOfShape_op [HasLimitsOfShape J\u1d52\u1d56 C\u1d52\u1d56] :\n HasColimitsOfShape J C :=\n { has_colimit := fun F => hasColimit_of_hasLimit_op F }\n#align category_theory.limits.has_colimits_of_shape_of_has_limits_of_shape_op CategoryTheory.Limits.hasColimitsOfShape_of_hasLimitsOfShape_op\n\n/-- If `C` has limits, we can construct colimits for `C\u1d52\u1d56`.\n-/\ninstance hasColimits_op_of_hasLimits [HasLimits C] : HasColimits C\u1d52\u1d56 :=\n \u27e8fun _ => inferInstance\u27e9\n#align category_theory.limits.has_colimits_op_of_has_limits CategoryTheory.Limits.hasColimits_op_of_hasLimits\n\ntheorem hasColimits_of_hasLimits_op [HasLimits C\u1d52\u1d56] : HasColimits C :=\n { has_colimits_of_shape := fun _ _ => hasColimitsOfShape_of_hasLimitsOfShape_op }\n#align category_theory.limits.has_colimits_of_has_limits_op CategoryTheory.Limits.hasColimits_of_hasLimits_op\n\ninstance has_filtered_colimits_op_of_has_cofiltered_limits [HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C] :\n HasFilteredColimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56 where HasColimitsOfShape _ _ _ := inferInstance\n#align category_theory.limits.has_filtered_colimits_op_of_has_cofiltered_limits CategoryTheory.Limits.has_filtered_colimits_op_of_has_cofiltered_limits\n\ntheorem has_filtered_colimits_of_has_cofiltered_limits_op [HasCofilteredLimitsOfSize.{v\u2082, u\u2082} C\u1d52\u1d56] :\n HasFilteredColimitsOfSize.{v\u2082, u\u2082} C :=\n { HasColimitsOfShape := fun _ _ _ => hasColimitsOfShape_of_hasLimitsOfShape_op }\n#align category_theory.limits.has_filtered_colimits_of_has_cofiltered_limits_op CategoryTheory.Limits.has_filtered_colimits_of_has_cofiltered_limits_op\n\nvariable (X : Type v\u2082)\n\n/-- If `C` has products indexed by `X`, then `C\u1d52\u1d56` has coproducts indexed by `X`.\n-/\ninstance hasCoproductsOfShape_opposite [HasProductsOfShape X C] : HasCoproductsOfShape X C\u1d52\u1d56 := by\n haveI : HasLimitsOfShape (Discrete X)\u1d52\u1d56 C :=\n hasLimitsOfShape_of_equivalence (Discrete.opposite X).symm\n infer_instance\n#align category_theory.limits.has_coproducts_of_shape_opposite CategoryTheory.Limits.hasCoproductsOfShape_opposite\n\ntheorem hasCoproductsOfShape_of_opposite [HasProductsOfShape X C\u1d52\u1d56] : HasCoproductsOfShape X C :=\n haveI : HasLimitsOfShape (Discrete X)\u1d52\u1d56 C\u1d52\u1d56 :=\n hasLimitsOfShape_of_equivalence (Discrete.opposite X).symm\n hasColimitsOfShape_of_hasLimitsOfShape_op\n#align category_theory.limits.has_coproducts_of_shape_of_opposite CategoryTheory.Limits.hasCoproductsOfShape_of_opposite\n\n/-- If `C` has coproducts indexed by `X`, then `C\u1d52\u1d56` has products indexed by `X`.\n-/\ninstance hasProductsOfShape_opposite [HasCoproductsOfShape X C] : HasProductsOfShape X C\u1d52\u1d56 := by\n haveI : HasColimitsOfShape (Discrete X)\u1d52\u1d56 C :=\n hasColimitsOfShape_of_equivalence (Discrete.opposite X).symm\n infer_instance\n#align category_theory.limits.has_products_of_shape_opposite CategoryTheory.Limits.hasProductsOfShape_opposite\n\ntheorem hasProductsOfShape_of_opposite [HasCoproductsOfShape X C\u1d52\u1d56] : HasProductsOfShape X C :=\n haveI : HasColimitsOfShape (Discrete X)\u1d52\u1d56 C\u1d52\u1d56 :=\n hasColimitsOfShape_of_equivalence (Discrete.opposite X).symm\n hasLimitsOfShape_of_hasColimitsOfShape_op\n#align category_theory.limits.has_products_of_shape_of_opposite CategoryTheory.Limits.hasProductsOfShape_of_opposite\n\ninstance hasProducts_opposite [HasCoproducts.{v\u2082} C] : HasProducts.{v\u2082} C\u1d52\u1d56 := fun _ =>\n inferInstance\n#align category_theory.limits.has_products_opposite CategoryTheory.Limits.hasProducts_opposite\n\ntheorem hasProducts_of_opposite [HasCoproducts.{v\u2082} C\u1d52\u1d56] : HasProducts.{v\u2082} C := fun X =>\n hasProductsOfShape_of_opposite X\n#align category_theory.limits.has_products_of_opposite CategoryTheory.Limits.hasProducts_of_opposite\n\ninstance hasCoproducts_opposite [HasProducts.{v\u2082} C] : HasCoproducts.{v\u2082} C\u1d52\u1d56 := fun _ =>\n inferInstance\n#align category_theory.limits.has_coproducts_opposite CategoryTheory.Limits.hasCoproducts_opposite\n\ntheorem hasCoproducts_of_opposite [HasProducts.{v\u2082} C\u1d52\u1d56] : HasCoproducts.{v\u2082} C := fun X =>\n hasCoproductsOfShape_of_opposite X\n#align category_theory.limits.has_coproducts_of_opposite CategoryTheory.Limits.hasCoproducts_of_opposite\n\ninstance hasFiniteCoproducts_opposite [HasFiniteProducts C] : HasFiniteCoproducts C\u1d52\u1d56 where\n out _ := Limits.hasCoproductsOfShape_opposite _\n#align category_theory.limits.has_finite_coproducts_opposite CategoryTheory.Limits.hasFiniteCoproducts_opposite\n\ntheorem hasFiniteCoproducts_of_opposite [HasFiniteProducts C\u1d52\u1d56] : HasFiniteCoproducts C :=\n { out := fun _ => hasCoproductsOfShape_of_opposite _ }\n#align category_theory.limits.has_finite_coproducts_of_opposite CategoryTheory.Limits.hasFiniteCoproducts_of_opposite\n\ninstance hasFiniteProducts_opposite [HasFiniteCoproducts C] : HasFiniteProducts C\u1d52\u1d56 where\n out _ := inferInstance\n#align category_theory.limits.has_finite_products_opposite CategoryTheory.Limits.hasFiniteProducts_opposite\n\ntheorem hasFiniteProducts_of_opposite [HasFiniteCoproducts C\u1d52\u1d56] : HasFiniteProducts C :=\n { out := fun _ => hasProductsOfShape_of_opposite _ }\n#align category_theory.limits.has_finite_products_of_opposite CategoryTheory.Limits.hasFiniteProducts_of_opposite\n\nsection OppositeCoproducts\n\nvariable {\u03b1 : Type*} {Z : \u03b1 \u2192 C} [HasCoproduct Z]\n\ninstance : HasLimit (Discrete.functor Z).op := hasLimit_op_of_hasColimit (Discrete.functor Z)\n\ninstance : HasLimit ((Discrete.opposite \u03b1).inverse \u22d9 (Discrete.functor Z).op) :=\n hasLimitEquivalenceComp (Discrete.opposite \u03b1).symm\n\ninstance : HasProduct (op <| Z \u00b7) := hasLimitOfIso\n ((Discrete.natIsoFunctor \u226a\u226b Discrete.natIso (fun _ \u21a6 by rfl)) :\n (Discrete.opposite \u03b1).inverse \u22d9 (Discrete.functor Z).op \u2245\n Discrete.functor (op <| Z \u00b7))\n\n/-- A `Cofan` gives a `Fan` in the opposite category. -/\n@[simp]\ndef Cofan.op (c : Cofan Z) : Fan (op <| Z \u00b7) := Fan.mk _ (fun a \u21a6 (c.inj a).op)\n\n/-- If a `Cofan`\u00a0is colimit, then its opposite is limit. -/\ndef Cofan.IsColimit.op {c : Cofan Z} (hc : IsColimit c) : IsLimit c.op := by\n let e : Discrete.functor (Opposite.op <| Z \u00b7) \u2245 (Discrete.opposite \u03b1).inverse \u22d9\n (Discrete.functor Z).op := Discrete.natIso (fun _ \u21a6 Iso.refl _)\n refine IsLimit.ofIsoLimit ((IsLimit.postcomposeInvEquiv e _).2\n (IsLimit.whiskerEquivalence hc.op (Discrete.opposite \u03b1).symm))\n (Cones.ext (Iso.refl _) (fun \u27e8a\u27e9 \u21a6 ?_))\n dsimp\n erw [Category.id_comp, Category.comp_id]\n rfl\n\n/--\nThe canonical isomorphism from the opposite of an abstract coproduct to the corresponding product\nin the opposite category.\n-/\ndef opCoproductIsoProduct' {c : Cofan Z} {f : Fan (op <| Z \u00b7)}\n (hc : IsColimit c) (hf : IsLimit f) : op c.pt \u2245 f.pt :=\n IsLimit.conePointUniqueUpToIso (Cofan.IsColimit.op hc) hf\n\nvariable (Z) in\n/--\nThe canonical isomorphism from the opposite of the coproduct to the product in the opposite\ncategory.\n-/\ndef opCoproductIsoProduct :\n op (\u2210 Z) \u2245 \u220f (op <| Z \u00b7) :=\n opCoproductIsoProduct' (coproductIsCoproduct Z) (productIsProduct (op <| Z \u00b7))\n\ntheorem opCoproductIsoProduct'_inv_comp_inj {c : Cofan Z} {f : Fan (op <| Z \u00b7)}\n (hc : IsColimit c) (hf : IsLimit f) (b : \u03b1) :\n (opCoproductIsoProduct' hc hf).inv \u226b (c.inj b).op = f.proj b :=\n IsLimit.conePointUniqueUpToIso_inv_comp (Cofan.IsColimit.op hc) hf \u27e8b\u27e9\n\ntheorem opCoproductIsoProduct'_comp_self {c c' : Cofan Z} {f : Fan (op <| Z \u00b7)}\n (hc : IsColimit c) (hc' : IsColimit c') (hf : IsLimit f) :\n (opCoproductIsoProduct' hc hf).hom \u226b (opCoproductIsoProduct' hc' hf).inv =\n (hc.coconePointUniqueUpToIso hc').op.inv := by\n apply Quiver.Hom.unop_inj\n apply hc'.hom_ext\n intro \u27e8j\u27e9\n change c'.inj _ \u226b _ = _\n simp only [unop_op, unop_comp, Discrete.functor_obj, const_obj_obj, Iso.op_inv,\n Quiver.Hom.unop_op, IsColimit.comp_coconePointUniqueUpToIso_inv]\n apply Quiver.Hom.op_inj\n simp only [op_comp, op_unop, Quiver.Hom.op_unop, Category.assoc,\n opCoproductIsoProduct'_inv_comp_inj]\n rw [\u2190 opCoproductIsoProduct'_inv_comp_inj hc hf]\n simp only [Iso.hom_inv_id_assoc]\n rfl\n\nvariable (Z) in\ntheorem opCoproductIsoProduct_inv_comp_\u03b9 (b : \u03b1) :\n (opCoproductIsoProduct Z).inv \u226b (Sigma.\u03b9 Z b).op = Pi.\u03c0 (op <| Z \u00b7) b :=\n opCoproductIsoProduct'_inv_comp_inj _ _ b\n\ntheorem desc_op_comp_opCoproductIsoProduct'_hom {c : Cofan Z} {f : Fan (op <| Z \u00b7)}\n (hc : IsColimit c) (hf : IsLimit f) (c' : Cofan Z) :\n (hc.desc c').op \u226b (opCoproductIsoProduct' hc hf).hom = hf.lift c'.op := by\n refine (Iso.eq_comp_inv _).mp (Quiver.Hom.unop_inj (hc.hom_ext (fun \u27e8j\u27e9 \u21a6 Quiver.Hom.op_inj ?_)))\n simp only [unop_op, Discrete.functor_obj, const_obj_obj, Quiver.Hom.unop_op, IsColimit.fac,\n Cofan.op, unop_comp, op_comp, op_unop, Quiver.Hom.op_unop, Category.assoc]\n erw [opCoproductIsoProduct'_inv_comp_inj, IsLimit.fac]\n rfl\n\ntheorem desc_op_comp_opCoproductIsoProduct_hom {X : C} (\u03c0 : (a : \u03b1) \u2192 Z a \u27f6 X) :\n (Sigma.desc \u03c0).op \u226b (opCoproductIsoProduct Z).hom = Pi.lift (fun a \u21a6 (\u03c0 a).op) := by\n convert desc_op_comp_opCoproductIsoProduct'_hom (coproductIsCoproduct Z)\n (productIsProduct (op <| Z \u00b7)) (Cofan.mk _ \u03c0)\n \u00b7 ext; simp [Sigma.desc, coproductIsCoproduct]\n \u00b7 ext; simp [Pi.lift, productIsProduct]\n\nend OppositeCoproducts\n\nsection OppositeProducts\n\nvariable {\u03b1 : Type*} {Z : \u03b1 \u2192 C} [HasProduct Z]\n\ninstance : HasColimit (Discrete.functor Z).op := hasColimit_op_of_hasLimit (Discrete.functor Z)\n\ninstance : HasColimit ((Discrete.opposite \u03b1).inverse \u22d9 (Discrete.functor Z).op) :=\n hasColimit_equivalence_comp (Discrete.opposite \u03b1).symm\n\ninstance : HasCoproduct (op <| Z \u00b7) := hasColimitOfIso\n ((Discrete.natIsoFunctor \u226a\u226b Discrete.natIso (fun _ \u21a6 by rfl)) :\n (Discrete.opposite \u03b1).inverse \u22d9 (Discrete.functor Z).op \u2245\n Discrete.functor (op <| Z \u00b7)).symm\n\n/-- A `Fan` gives a `Cofan` in the opposite category. -/\n@[simp]\ndef Fan.op (f : Fan Z) : Cofan (op <| Z \u00b7) := Cofan.mk _ (fun a \u21a6 (f.proj a).op)\n\n/-- If a `Fan`\u00a0is limit, then its opposite is colimit. -/\ndef Fan.IsLimit.op {f : Fan Z} (hf : IsLimit f) : IsColimit f.op := by\n let e : Discrete.functor (Opposite.op <| Z \u00b7) \u2245 (Discrete.opposite \u03b1).inverse \u22d9\n (Discrete.functor Z).op := Discrete.natIso (fun _ \u21a6 Iso.refl _)\n refine IsColimit.ofIsoColimit ((IsColimit.precomposeHomEquiv e _).2\n (IsColimit.whiskerEquivalence hf.op (Discrete.opposite \u03b1).symm))\n (Cocones.ext (Iso.refl _) (fun \u27e8a\u27e9 \u21a6 ?_))\n dsimp\n erw [Category.id_comp, Category.comp_id]\n rfl\n\n/--\nThe canonical isomorphism from the opposite of an abstract product to the corresponding coproduct\nin the opposite category.\n-/\ndef opProductIsoCoproduct' {f : Fan Z} {c : Cofan (op <| Z \u00b7)}\n (hf : IsLimit f) (hc : IsColimit c) : op f.pt \u2245 c.pt :=\n IsColimit.coconePointUniqueUpToIso (Fan.IsLimit.op hf) hc\n\nvariable (Z) in\n/--\nThe canonical isomorphism from the opposite of the product to the coproduct in the opposite\ncategory.\n-/\ndef opProductIsoCoproduct :\n op (\u220f Z) \u2245 \u2210 (op <| Z \u00b7) :=\n opProductIsoCoproduct' (productIsProduct Z) (coproductIsCoproduct (op <| Z \u00b7))\n\ntheorem proj_comp_opProductIsoCoproduct'_hom {f : Fan Z} {c : Cofan (op <| Z \u00b7)}\n (hf : IsLimit f) (hc : IsColimit c) (b : \u03b1) :\n (f.proj b).op \u226b (opProductIsoCoproduct' hf hc).hom = c.inj b :=\n IsColimit.comp_coconePointUniqueUpToIso_hom (Fan.IsLimit.op hf) hc \u27e8b\u27e9\n\n", "theoremStatement": "theorem opProductIsoCoproduct'_comp_self {f f' : Fan Z} {c : Cofan (op <| Z \u00b7)}\n (hf : IsLimit f) (hf' : IsLimit f') (hc : IsColimit c) :\n (opProductIsoCoproduct' hf hc).hom \u226b (opProductIsoCoproduct' hf' hc).inv =\n (hf.conePointUniqueUpToIso hf').op.inv", "theoremName": "opProductIsoCoproduct'_comp_self", "fileCreated": {"commit": "3fcb15f5aa", "date": "2023-03-11"}, "theoremCreated": {"commit": "423f5783b8", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/CategoryTheory/Limits/Opposites.lean", "positionMetadata": {"lineInFile": 510, "tokenPositionInFile": 24317, "theoremPositionInFile": 66}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n apply Quiver.Hom.unop_inj\n apply hf.hom_ext\n intro \u27e8j\u27e9\n change _ \u226b f.proj _ = _\n simp only [unop_op, unop_comp, Category.assoc, Discrete.functor_obj, Iso.op_inv,\n Quiver.Hom.unop_op, IsLimit.conePointUniqueUpToIso_inv_comp]\n apply Quiver.Hom.op_inj\n simp only [op_comp, op_unop, Quiver.Hom.op_unop, proj_comp_opProductIsoCoproduct'_hom]\n rw [\u2190 proj_comp_opProductIsoCoproduct'_hom hf' hc]\n simp only [Category.assoc, Iso.hom_inv_id, Category.comp_id]\n rfl", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 472}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2023 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport Mathlib.MeasureTheory.Group.FundamentalDomain\nimport Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar\nimport Mathlib.RingTheory.Localization.Module\n\n#align_import algebra.module.zlattice from \"leanprover-community/mathlib\"@\"a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3\"\n\n/-!\n# \u2124-lattices\n\nLet `E` be a finite dimensional vector space over a `NormedLinearOrderedField` `K` with a solid\nnorm that is also a `FloorRing`, e.g. `\u211d`. A (full) `\u2124`-lattice `L` of `E` is a discrete\nsubgroup of `E` such that `L` spans `E` over `K`.\n\nA `\u2124`-lattice `L` can be defined in two ways:\n* For `b` a basis of `E`, then `L = Submodule.span \u2124 (Set.range b)` is a \u2124-lattice of `E`\n* As an `AddSubgroup E` with the additional properties:\n * `DiscreteTopology L`, that is `L` is discrete\n * `Submodule.span \u211d (L : Set E) = \u22a4`, that is `L` spans `E` over `K`.\n\nResults about the first point of view are in the `Zspan` namespace and results about the second\npoint of view are in the `Zlattice` namespace.\n\n## Main results\n\n* `Zspan.isAddFundamentalDomain`: for a \u2124-lattice `Submodule.span \u2124 (Set.range b)`, proves that\nthe set defined by `Zspan.fundamentalDomain` is a fundamental domain.\n* `Zlattice.module_free`: an AddSubgroup of `E` that is discrete and spans `E` over `K` is a free\n`\u2124`-module\n* `Zlattice.rank`: an AddSubgroup of `E` that is discrete and spans `E` over `K` is a free\n`\u2124`-module of `\u2124`-rank equal to the `K`-rank of `E`\n-/\n\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace Zspan\n\nopen MeasureTheory MeasurableSet Submodule Bornology\n\nvariable {E \u03b9 : Type*}\n\nsection NormedLatticeField\n\nvariable {K : Type*} [NormedLinearOrderedField K]\nvariable [NormedAddCommGroup E] [NormedSpace K E]\nvariable (b : Basis \u03b9 K E)\n\ntheorem span_top : span K (span \u2124 (Set.range b) : Set E) = \u22a4 := by simp [span_span_of_tower]\n\n/-- The fundamental domain of the \u2124-lattice spanned by `b`. See `Zspan.isAddFundamentalDomain`\nfor the proof that it is a fundamental domain. -/\ndef fundamentalDomain : Set E := {m | \u2200 i, b.repr m i \u2208 Set.Ico (0 : K) 1}\n#align zspan.fundamental_domain Zspan.fundamentalDomain\n\n@[simp]\ntheorem mem_fundamentalDomain {m : E} :\n m \u2208 fundamentalDomain b \u2194 \u2200 i, b.repr m i \u2208 Set.Ico (0 : K) 1 := Iff.rfl\n#align zspan.mem_fundamental_domain Zspan.mem_fundamentalDomain\n\ntheorem map_fundamentalDomain {F : Type*} [NormedAddCommGroup F] [NormedSpace K F] (f : E \u2243\u2097[K] F) :\n f '' (fundamentalDomain b) = fundamentalDomain (b.map f) := by\n ext x\n rw [mem_fundamentalDomain, Basis.map_repr, LinearEquiv.trans_apply, \u2190 mem_fundamentalDomain,\n show f.symm x = f.toEquiv.symm x by rfl, \u2190 Set.mem_image_equiv]\n rfl\n\n@[simp]\ntheorem fundamentalDomain_reindex {\u03b9' : Type*} (e : \u03b9 \u2243 \u03b9') :\n fundamentalDomain (b.reindex e) = fundamentalDomain b := by\n ext\n simp_rw [mem_fundamentalDomain, Basis.repr_reindex_apply]\n rw [Equiv.forall_congr' e]\n simp_rw [implies_true]\n\nlemma fundamentalDomain_pi_basisFun [Fintype \u03b9] :\n fundamentalDomain (Pi.basisFun \u211d \u03b9) = Set.pi Set.univ fun _ : \u03b9 \u21a6 Set.Ico (0 : \u211d) 1 := by\n ext; simp\n\nvariable [FloorRing K]\n\nsection Fintype\n\nvariable [Fintype \u03b9]\n\n/-- The map that sends a vector of `E` to the element of the \u2124-lattice spanned by `b` obtained\nby rounding down its coordinates on the basis `b`. -/\ndef floor (m : E) : span \u2124 (Set.range b) := \u2211 i, \u230ab.repr m i\u230b \u2022 b.restrictScalars \u2124 i\n#align zspan.floor Zspan.floor\n\n/-- The map that sends a vector of `E` to the element of the \u2124-lattice spanned by `b` obtained\nby rounding up its coordinates on the basis `b`. -/\ndef ceil (m : E) : span \u2124 (Set.range b) := \u2211 i, \u2308b.repr m i\u2309 \u2022 b.restrictScalars \u2124 i\n#align zspan.ceil Zspan.ceil\n\n@[simp]\ntheorem repr_floor_apply (m : E) (i : \u03b9) : b.repr (floor b m) i = \u230ab.repr m i\u230b := by\n classical simp only [floor, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\n Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\n Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]\n#align zspan.repr_floor_apply Zspan.repr_floor_apply\n\n@[simp]\ntheorem repr_ceil_apply (m : E) (i : \u03b9) : b.repr (ceil b m) i = \u2308b.repr m i\u2309 := by\n classical simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\n Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\n Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]\n#align zspan.repr_ceil_apply Zspan.repr_ceil_apply\n\n@[simp]\ntheorem floor_eq_self_of_mem (m : E) (h : m \u2208 span \u2124 (Set.range b)) : (floor b m : E) = m := by\n apply b.ext_elem\n simp_rw [repr_floor_apply b]\n intro i\n obtain \u27e8z, hz\u27e9 := (b.mem_span_iff_repr_mem \u2124 _).mp h i\n rw [\u2190 hz]\n exact congr_arg (Int.cast : \u2124 \u2192 K) (Int.floor_intCast z)\n#align zspan.floor_eq_self_of_mem Zspan.floor_eq_self_of_mem\n\n@[simp]\ntheorem ceil_eq_self_of_mem (m : E) (h : m \u2208 span \u2124 (Set.range b)) : (ceil b m : E) = m := by\n apply b.ext_elem\n simp_rw [repr_ceil_apply b]\n intro i\n obtain \u27e8z, hz\u27e9 := (b.mem_span_iff_repr_mem \u2124 _).mp h i\n rw [\u2190 hz]\n exact congr_arg (Int.cast : \u2124 \u2192 K) (Int.ceil_intCast z)\n#align zspan.ceil_eq_self_of_mem Zspan.ceil_eq_self_of_mem\n\n/-- The map that sends a vector `E` to the `fundamentalDomain` of the lattice,\nsee `Zspan.fract_mem_fundamentalDomain`, and `fractRestrict` for the map with the codomain\nrestricted to `fundamentalDomain`. -/\ndef fract (m : E) : E := m - floor b m\n#align zspan.fract Zspan.fract\n\ntheorem fract_apply (m : E) : fract b m = m - floor b m := rfl\n#align zspan.fract_apply Zspan.fract_apply\n\n@[simp]\ntheorem repr_fract_apply (m : E) (i : \u03b9) : b.repr (fract b m) i = Int.fract (b.repr m i) := by\n rw [fract, LinearEquiv.map_sub, Finsupp.coe_sub, Pi.sub_apply, repr_floor_apply, Int.fract]\n#align zspan.repr_fract_apply Zspan.repr_fract_apply\n\n@[simp]\ntheorem fract_fract (m : E) : fract b (fract b m) = fract b m :=\n Basis.ext_elem b fun _ => by classical simp only [repr_fract_apply, Int.fract_fract]\n#align zspan.fract_fract Zspan.fract_fract\n\n@[simp]\ntheorem fract_zspan_add (m : E) {v : E} (h : v \u2208 span \u2124 (Set.range b)) :\n fract b (v + m) = fract b m := by\n classical\n refine (Basis.ext_elem_iff b).mpr fun i => ?_\n simp_rw [repr_fract_apply, Int.fract_eq_fract]\n use (b.restrictScalars \u2124).repr \u27e8v, h\u27e9 i\n rw [map_add, Finsupp.coe_add, Pi.add_apply, add_tsub_cancel_right,\n \u2190 eq_intCast (algebraMap \u2124 K) _, Basis.restrictScalars_repr_apply, coe_mk]\n#align zspan.fract_zspan_add Zspan.fract_zspan_add\n\n@[simp]\ntheorem fract_add_zspan (m : E) {v : E} (h : v \u2208 span \u2124 (Set.range b)) :\n fract b (m + v) = fract b m := by rw [add_comm, fract_zspan_add b m h]\n#align zspan.fract_add_zspan Zspan.fract_add_zspan\n\nvariable {b}\n\ntheorem fract_eq_self {x : E} : fract b x = x \u2194 x \u2208 fundamentalDomain b := by\n classical simp only [Basis.ext_elem_iff b, repr_fract_apply, Int.fract_eq_self,\n mem_fundamentalDomain, Set.mem_Ico]\n#align zspan.fract_eq_self Zspan.fract_eq_self\n\nvariable (b)\n\ntheorem fract_mem_fundamentalDomain (x : E) : fract b x \u2208 fundamentalDomain b :=\n fract_eq_self.mp (fract_fract b _)\n#align zspan.fract_mem_fundamental_domain Zspan.fract_mem_fundamentalDomain\n\n/-- The map `fract` with codomain restricted to `fundamentalDomain`. -/\ndef fractRestrict (x : E) : fundamentalDomain b := \u27e8fract b x, fract_mem_fundamentalDomain b x\u27e9\n\ntheorem fractRestrict_surjective : Function.Surjective (fractRestrict b) :=\n fun x => \u27e8\u2191x, Subtype.eq (fract_eq_self.mpr (Subtype.mem x))\u27e9\n\n@[simp]\ntheorem fractRestrict_apply (x : E) : (fractRestrict b x : E) = fract b x := rfl\n\ntheorem fract_eq_fract (m n : E) : fract b m = fract b n \u2194 -m + n \u2208 span \u2124 (Set.range b) := by\n classical\n rw [eq_comm, Basis.ext_elem_iff b]\n simp_rw [repr_fract_apply, Int.fract_eq_fract, eq_comm, Basis.mem_span_iff_repr_mem,\n sub_eq_neg_add, map_add, LinearEquiv.map_neg, Finsupp.coe_add, Finsupp.coe_neg, Pi.add_apply,\n Pi.neg_apply, \u2190 eq_intCast (algebraMap \u2124 K) _, Set.mem_range]\n#align zspan.fract_eq_fract Zspan.fract_eq_fract\n\ntheorem norm_fract_le [HasSolidNorm K] (m : E) : \u2016fract b m\u2016 \u2264 \u2211 i, \u2016b i\u2016 := by\n classical\n calc\n \u2016fract b m\u2016 = \u2016\u2211 i, b.repr (fract b m) i \u2022 b i\u2016 := by rw [b.sum_repr]\n _ = \u2016\u2211 i, Int.fract (b.repr m i) \u2022 b i\u2016 := by simp_rw [repr_fract_apply]\n _ \u2264 \u2211 i, \u2016Int.fract (b.repr m i) \u2022 b i\u2016 := norm_sum_le _ _\n _ = \u2211 i, \u2016Int.fract (b.repr m i)\u2016 * \u2016b i\u2016 := by simp_rw [norm_smul]\n _ \u2264 \u2211 i, \u2016b i\u2016 := Finset.sum_le_sum fun i _ => ?_\n suffices \u2016Int.fract ((b.repr m) i)\u2016 \u2264 1 by\n convert mul_le_mul_of_nonneg_right this (norm_nonneg _ : 0 \u2264 \u2016b i\u2016)\n exact (one_mul _).symm\n rw [(norm_one.symm : 1 = \u2016(1 : K)\u2016)]\n apply norm_le_norm_of_abs_le_abs\n rw [abs_one, Int.abs_fract]\n exact le_of_lt (Int.fract_lt_one _)\n#align zspan.norm_fract_le Zspan.norm_fract_le\n\nsection Unique\n\nvariable [Unique \u03b9]\n\n@[simp]\ntheorem coe_floor_self (k : K) : (floor (Basis.singleton \u03b9 K) k : K) = \u230ak\u230b :=\n Basis.ext_elem _ fun _ => by rw [repr_floor_apply, Basis.singleton_repr, Basis.singleton_repr]\n#align zspan.coe_floor_self Zspan.coe_floor_self\n\n@[simp]\ntheorem coe_fract_self (k : K) : (fract (Basis.singleton \u03b9 K) k : K) = Int.fract k :=\n Basis.ext_elem _ fun _ => by rw [repr_fract_apply, Basis.singleton_repr, Basis.singleton_repr]\n#align zspan.coe_fract_self Zspan.coe_fract_self\n\nend Unique\n\nend Fintype\n\ntheorem fundamentalDomain_isBounded [Finite \u03b9] [HasSolidNorm K] :\n IsBounded (fundamentalDomain b) := by\n cases nonempty_fintype \u03b9\n refine isBounded_iff_forall_norm_le.2 \u27e8\u2211 j, \u2016b j\u2016, fun x hx \u21a6 ?_\u27e9\n rw [\u2190 fract_eq_self.mpr hx]\n apply norm_fract_le\n#align zspan.fundamental_domain_bounded Zspan.fundamentalDomain_isBounded\n\ntheorem vadd_mem_fundamentalDomain [Fintype \u03b9] (y : span \u2124 (Set.range b)) (x : E) :\n y +\u1d65 x \u2208 fundamentalDomain b \u2194 y = -floor b x := by\n rw [Subtype.ext_iff, \u2190 add_right_inj x, NegMemClass.coe_neg, \u2190 sub_eq_add_neg, \u2190 fract_apply,\n \u2190 fract_zspan_add b _ (Subtype.mem y), add_comm, \u2190 vadd_eq_add, \u2190 vadd_def, eq_comm, \u2190\n fract_eq_self]\n#align zspan.vadd_mem_fundamental_domain Zspan.vadd_mem_fundamentalDomain\n\ntheorem exist_unique_vadd_mem_fundamentalDomain [Finite \u03b9] (x : E) :\n \u2203! v : span \u2124 (Set.range b), v +\u1d65 x \u2208 fundamentalDomain b := by\n cases nonempty_fintype \u03b9\n refine \u27e8-floor b x, ?_, fun y h => ?_\u27e9\n \u00b7 exact (vadd_mem_fundamentalDomain b (-floor b x) x).mpr rfl\n \u00b7 exact (vadd_mem_fundamentalDomain b y x).mp h\n#align zspan.exist_unique_vadd_mem_fundamental_domain Zspan.exist_unique_vadd_mem_fundamentalDomain\n\n/-- The map `Zspan.fractRestrict` defines an equiv map between `E \u29f8 span \u2124 (Set.range b)`\nand `Zspan.fundamentalDomain b`. -/\ndef quotientEquiv [Fintype \u03b9] :\n E \u29f8 span \u2124 (Set.range b) \u2243 (fundamentalDomain b) := by\n refine Equiv.ofBijective ?_ \u27e8fun x y => ?_, fun x => ?_\u27e9\n \u00b7 refine fun q => Quotient.liftOn q (fractRestrict b) (fun _ _ h => ?_)\n rw [Subtype.mk.injEq, fractRestrict_apply, fractRestrict_apply, fract_eq_fract]\n exact QuotientAddGroup.leftRel_apply.mp h\n \u00b7 refine Quotient.inductionOn\u2082 x y (fun _ _ hxy => ?_)\n rw [Quotient.liftOn_mk (s := quotientRel (span \u2124 (Set.range b))), fractRestrict,\n Quotient.liftOn_mk (s := quotientRel (span \u2124 (Set.range b))), fractRestrict,\n Subtype.mk.injEq] at hxy\n apply Quotient.sound'\n rwa [QuotientAddGroup.leftRel_apply, mem_toAddSubgroup, \u2190 fract_eq_fract]\n \u00b7 obtain \u27e8a, rfl\u27e9 := fractRestrict_surjective b x\n exact \u27e8Quotient.mk'' a, rfl\u27e9\n\n@[simp]\ntheorem quotientEquiv_apply_mk [Fintype \u03b9] (x : E) :\n quotientEquiv b (Submodule.Quotient.mk x) = fractRestrict b x := rfl\n\n@[simp]\ntheorem quotientEquiv.symm_apply [Fintype \u03b9] (x : fundamentalDomain b) :\n (quotientEquiv b).symm x = Submodule.Quotient.mk \u2191x := by\n rw [Equiv.symm_apply_eq, quotientEquiv_apply_mk b \u2191x, Subtype.ext_iff, fractRestrict_apply]\n exact (fract_eq_self.mpr x.prop).symm\n\nend NormedLatticeField\n\nsection Real\n\ntheorem discreteTopology_pi_basisFun [Finite \u03b9] :\n DiscreteTopology (span \u2124 (Set.range (Pi.basisFun \u211d \u03b9))) := by\n cases nonempty_fintype \u03b9\n refine discreteTopology_iff_isOpen_singleton_zero.mpr \u27e8Metric.ball 0 1, Metric.isOpen_ball, ?_\u27e9\n ext x\n rw [Set.mem_preimage, mem_ball_zero_iff, pi_norm_lt_iff zero_lt_one, Set.mem_singleton_iff]\n simp_rw [\u2190 coe_eq_zero, Function.funext_iff, Pi.zero_apply, Real.norm_eq_abs]\n refine forall_congr' (fun i => ?_)\n rsuffices \u27e8y, hy\u27e9 : \u2203 (y : \u2124), (y : \u211d) = (x : \u03b9 \u2192 \u211d) i\n \u00b7 rw [\u2190 hy, \u2190 Int.cast_abs, \u2190 Int.cast_one, Int.cast_lt, Int.abs_lt_one_iff, Int.cast_eq_zero]\n exact ((Pi.basisFun \u211d \u03b9).mem_span_iff_repr_mem \u2124 x).mp (SetLike.coe_mem x) i\n\nvariable [NormedAddCommGroup E] [NormedSpace \u211d E] (b : Basis \u03b9 \u211d E)\n\ntheorem fundamentalDomain_subset_parallelepiped [Fintype \u03b9] :\n fundamentalDomain b \u2286 parallelepiped b := by\n rw [fundamentalDomain, parallelepiped_basis_eq, Set.setOf_subset_setOf]\n exact fun _ h i \u21a6 Set.Ico_subset_Icc_self (h i)\n\ninstance [Finite \u03b9] : DiscreteTopology (span \u2124 (Set.range b)) := by\n have h : Set.MapsTo b.equivFun (span \u2124 (Set.range b)) (span \u2124 (Set.range (Pi.basisFun \u211d \u03b9))) := by\n intro _ hx\n rwa [SetLike.mem_coe, Basis.mem_span_iff_repr_mem] at hx \u22a2\n convert DiscreteTopology.of_continuous_injective ((continuous_equivFun_basis b).restrict h) ?_\n \u00b7 exact discreteTopology_pi_basisFun\n \u00b7 refine Subtype.map_injective _ (Basis.equivFun b).injective\n\ninstance [Finite \u03b9] : DiscreteTopology (span \u2124 (Set.range b)).toAddSubgroup :=\n inferInstanceAs <| DiscreteTopology (span \u2124 (Set.range b))\n\n@[measurability]\ntheorem fundamentalDomain_measurableSet [MeasurableSpace E] [OpensMeasurableSpace E] [Finite \u03b9] :\n MeasurableSet (fundamentalDomain b) := by\n cases nonempty_fintype \u03b9\n haveI : FiniteDimensional \u211d E := FiniteDimensional.of_fintype_basis b\n let D : Set (\u03b9 \u2192 \u211d) := Set.pi Set.univ fun _ : \u03b9 => Set.Ico (0 : \u211d) 1\n rw [(_ : fundamentalDomain b = b.equivFun.toLinearMap \u207b\u00b9' D)]\n \u00b7 refine measurableSet_preimage (LinearMap.continuous_of_finiteDimensional _).measurable ?_\n exact MeasurableSet.pi Set.countable_univ fun _ _ => measurableSet_Ico\n \u00b7 ext\n simp only [D, fundamentalDomain, Set.mem_Ico, Set.mem_setOf_eq, LinearEquiv.coe_coe,\n Set.mem_preimage, Basis.equivFun_apply, Set.mem_pi, Set.mem_univ, forall_true_left]\n#align zspan.fundamental_domain_measurable_set Zspan.fundamentalDomain_measurableSet\n\n/-- For a \u2124-lattice `Submodule.span \u2124 (Set.range b)`, proves that the set defined\nby `Zspan.fundamentalDomain` is a fundamental domain. -/\nprotected theorem isAddFundamentalDomain [Finite \u03b9] [MeasurableSpace E] [OpensMeasurableSpace E]\n (\u03bc : Measure E) :\n IsAddFundamentalDomain (span \u2124 (Set.range b)).toAddSubgroup (fundamentalDomain b) \u03bc := by\n cases nonempty_fintype \u03b9\n exact IsAddFundamentalDomain.mk' (nullMeasurableSet (fundamentalDomain_measurableSet b))\n fun x => exist_unique_vadd_mem_fundamentalDomain b x\n#align zspan.is_add_fundamental_domain Zspan.isAddFundamentalDomain\n\ntheorem measure_fundamentalDomain_ne_zero [Finite \u03b9] [MeasurableSpace E] [BorelSpace E]\n {\u03bc : Measure E} [Measure.IsAddHaarMeasure \u03bc] :\n \u03bc (fundamentalDomain b) \u2260 0 := by\n convert (Zspan.isAddFundamentalDomain b \u03bc).measure_ne_zero (NeZero.ne \u03bc)\n simp only [mem_toAddSubgroup]\n infer_instance\n\ntheorem measure_fundamentalDomain [Fintype \u03b9] [DecidableEq \u03b9] [MeasurableSpace E] (\u03bc : Measure E)\n [BorelSpace E] [Measure.IsAddHaarMeasure \u03bc] (b\u2080 : Basis \u03b9 \u211d E) :\n \u03bc (fundamentalDomain b) = ENNReal.ofReal |b\u2080.det b| * \u03bc (fundamentalDomain b\u2080) := by\n have : FiniteDimensional \u211d E := FiniteDimensional.of_fintype_basis b\n convert \u03bc.addHaar_preimage_linearEquiv (b.equiv b\u2080 (Equiv.refl \u03b9)) (fundamentalDomain b\u2080)\n \u00b7 rw [Set.eq_preimage_iff_image_eq (LinearEquiv.bijective _), map_fundamentalDomain,\n Basis.map_equiv, Equiv.refl_symm, Basis.reindex_refl]\n \u00b7 rw [\u2190 LinearMap.det_toMatrix b\u2080, Basis.equiv_symm, Equiv.refl_symm, Basis.det_apply]\n congr\n ext\n simp [Basis.toMatrix_apply, LinearMap.toMatrix_apply, LinearEquiv.coe_coe, Basis.equiv_apply]\n\n@[simp]\ntheorem volume_fundamentalDomain [Fintype \u03b9] [DecidableEq \u03b9] (b : Basis \u03b9 \u211d (\u03b9 \u2192 \u211d)) :\n volume (fundamentalDomain b) = ENNReal.ofReal |(Matrix.of b).det| := by\n rw [measure_fundamentalDomain b volume (b\u2080 := Pi.basisFun \u211d \u03b9), fundamentalDomain_pi_basisFun,\n volume_pi, Measure.pi_pi, Real.volume_Ico, sub_zero, ENNReal.ofReal_one, Finset.prod_const_one,\n mul_one, \u2190 Matrix.det_transpose]\n rfl\n\ntheorem fundamentalDomain_ae_parallelepiped [Fintype \u03b9] [MeasurableSpace E] (\u03bc : Measure E)\n [BorelSpace E] [Measure.IsAddHaarMeasure \u03bc] :\n fundamentalDomain b =\u1d50[\u03bc] parallelepiped b := by\n classical\n have : FiniteDimensional \u211d E := FiniteDimensional.of_fintype_basis b\n rw [\u2190 measure_symmDiff_eq_zero_iff, symmDiff_of_le (fundamentalDomain_subset_parallelepiped b)]\n suffices (parallelepiped b \\ fundamentalDomain b) \u2286 \u22c3 i,\n AffineSubspace.mk' (b i) (span \u211d (b '' (Set.univ \\ {i}))) by\n refine measure_mono_null this\n (measure_iUnion_null_iff.mpr fun i \u21a6 Measure.addHaar_affineSubspace \u03bc _ ?_)\n refine (ne_of_mem_of_not_mem' (AffineSubspace.mem_top _ _ 0)\n (AffineSubspace.mem_mk'_iff_vsub_mem.not.mpr ?_)).symm\n simp_rw [vsub_eq_sub, zero_sub, neg_mem_iff]\n exact linearIndependent_iff_not_mem_span.mp b.linearIndependent i\n intro x hx\n simp_rw [parallelepiped_basis_eq, Set.mem_Icc, Set.mem_diff, Set.mem_setOf_eq,\n mem_fundamentalDomain, Set.mem_Ico, not_forall, not_and, not_lt] at hx\n obtain \u27e8i, hi\u27e9 := hx.2\n have : b.repr x i = 1 := le_antisymm (hx.1 i).2 (hi (hx.1 i).1)\n rw [\u2190 b.sum_repr x, \u2190 Finset.sum_erase_add _ _ (Finset.mem_univ i), this, one_smul, \u2190 vadd_eq_add]\n refine Set.mem_iUnion.mpr \u27e8i, AffineSubspace.vadd_mem_mk' _\n (sum_smul_mem _ _ (fun i hi \u21a6 Submodule.subset_span ?_))\u27e9\n exact \u27e8i, Set.mem_diff_singleton.mpr \u27e8trivial, Finset.ne_of_mem_erase hi\u27e9, rfl\u27e9\n\nend Real\n\nend Zspan\n\nsection Zlattice\n\nopen Submodule FiniteDimensional\n\n-- TODO: generalize this class to other rings than `\u2124`\n/-- An `L : Addsubgroup E` where `E` is a vector space over a normed field `K` is a `\u2124`-lattice if\nit is discrete and spans `E` over `K`. -/\nclass IsZlattice (K : Type*) [NormedField K] {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]\n (L : AddSubgroup E) [DiscreteTopology L] : Prop where\n /-- `L` spans the full space `E` over `K`. -/\n span_top : span K (L : Set E) = \u22a4\n\ntheorem _root_.Zspan.isZlattice {E \u03b9 : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n [Finite \u03b9] (b : Basis \u03b9 \u211d E) :\n IsZlattice \u211d (span \u2124 (Set.range b)).toAddSubgroup where\n span_top := Zspan.span_top b\n\nvariable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]\nvariable [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology L]\n\ntheorem Zlattice.FG [hs : IsZlattice K L] : AddSubgroup.FG L := by\n suffices (AddSubgroup.toIntSubmodule L).FG by exact (fg_iff_add_subgroup_fg _).mp this\n obtain \u27e8s, \u27e8h_incl, \u27e8h_span, h_lind\u27e9\u27e9\u27e9 := exists_linearIndependent K (L : Set E)\n -- Let `s` be a maximal `K`-linear independent family of elements of `L`. We show that\n -- `L` is finitely generated (as a \u2124-module) because it fits in the exact sequence\n -- `0 \u2192 span \u2124 s \u2192 L \u2192 L \u29f8 span \u2124 s \u2192 0` with `span \u2124 s` and `L \u29f8 span \u2124 s` finitely generated.\n refine fg_of_fg_map_of_fg_inf_ker (span \u2124 s).mkQ ?_ ?_\n \u00b7 -- Let `b` be the `K`-basis of `E` formed by the vectors in `s`. The elements of\n -- `L \u29f8 span \u2124 s = L \u29f8 span \u2124 b` are in bijection with elements of `L \u2229 fundamentalDomain b`\n -- so there are finitely many since `fundamentalDomain b` is bounded.\n refine fg_def.mpr \u27e8map (span \u2124 s).mkQ (AddSubgroup.toIntSubmodule L), ?_, span_eq _\u27e9\n let b := Basis.mk h_lind (by\n rw [\u2190 hs.span_top, \u2190 h_span]\n exact span_mono (by simp only [Subtype.range_coe_subtype, Set.setOf_mem_eq, subset_rfl]))\n rw [show span \u2124 s = span \u2124 (Set.range b) by simp [b, Basis.coe_mk, Subtype.range_coe_subtype]]\n have : Fintype s := h_lind.setFinite.fintype\n refine Set.Finite.of_finite_image (f := ((\u2191) : _ \u2192 E) \u2218 Zspan.quotientEquiv b) ?_\n (Function.Injective.injOn (Subtype.coe_injective.comp (Zspan.quotientEquiv b).injective) _)\n have : Set.Finite ((Zspan.fundamentalDomain b) \u2229 L) :=\n Metric.finite_isBounded_inter_isClosed (Zspan.fundamentalDomain_isBounded b) inferInstance\n refine Set.Finite.subset this ?_\n rintro _ \u27e8_, \u27e8\u27e8x, \u27e8h_mem, rfl\u27e9\u27e9, rfl\u27e9\u27e9\n rw [Function.comp_apply, mkQ_apply, Zspan.quotientEquiv_apply_mk, Zspan.fractRestrict_apply]\n refine \u27e8?_, ?_\u27e9\n \u00b7 exact Zspan.fract_mem_fundamentalDomain b x\n \u00b7 rw [Zspan.fract, SetLike.mem_coe, sub_eq_add_neg]\n refine AddSubgroup.add_mem _ h_mem\n (neg_mem (Set.mem_of_subset_of_mem ?_ (Subtype.mem (Zspan.floor b x))))\n rw [show (L : Set E) = AddSubgroup.toIntSubmodule L by rfl]\n rw [SetLike.coe_subset_coe, Basis.coe_mk, Subtype.range_coe_subtype, Set.setOf_mem_eq]\n exact span_le.mpr h_incl\n \u00b7 -- `span \u2124 s` is finitely generated because `s` is finite\n rw [ker_mkQ, inf_of_le_right (span_le.mpr h_incl)]\n exact fg_span (LinearIndependent.setFinite h_lind)\n\ntheorem Zlattice.module_finite [IsZlattice K L] : Module.Finite \u2124 L :=\n Module.Finite.iff_addGroup_fg.mpr ((AddGroup.fg_iff_addSubgroup_fg L).mpr (FG K L))\n\ninstance instModuleFinite_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E]\n [NormedSpace \u211d E] [FiniteDimensional \u211d E] (L : AddSubgroup E) [DiscreteTopology L] :\n Module.Finite \u2124 L := by\n let f := (span \u211d (L : Set E)).subtype\n let L\u2080 := (AddSubgroup.toIntSubmodule L).comap (f.restrictScalars \u2124)\n have h_img : f '' L\u2080 = L := by\n rw [\u2190 LinearMap.coe_restrictScalars \u2124 f, \u2190 Submodule.map_coe (f.restrictScalars \u2124),\n Submodule.map_comap_eq_self, AddSubgroup.coe_toIntSubmodule]\n exact fun x hx \u21a6 LinearMap.mem_range.mpr \u27e8\u27e8x, Submodule.subset_span hx\u27e9, rfl\u27e9\n suffices Module.Finite \u2124 L\u2080 by\n have : L\u2080.map (f.restrictScalars \u2124) = (AddSubgroup.toIntSubmodule L) :=\n SetLike.ext'_iff.mpr h_img\n convert this \u25b8 Module.Finite.map L\u2080 (f.restrictScalars \u2124)\n have : DiscreteTopology L\u2080.toAddSubgroup := by\n refine DiscreteTopology.preimage_of_continuous_injective (L : Set E) ?_ (injective_subtype _)\n exact LinearMap.continuous_of_finiteDimensional f\n have : IsZlattice \u211d L\u2080.toAddSubgroup := \u27e8by\n rw [\u2190 (Submodule.map_injective_of_injective (injective_subtype _)).eq_iff, Submodule.map_span,\n Submodule.map_top, range_subtype, coe_toAddSubgroup, h_img]\u27e9\n exact Zlattice.module_finite \u211d L\u2080.toAddSubgroup\n\ntheorem Zlattice.module_free [IsZlattice K L] : Module.Free \u2124 L := by\n have : Module.Finite \u2124 L := module_finite K L\n have : Module \u211a E := Module.compHom E (algebraMap \u211a K)\n have : NoZeroSMulDivisors \u2124 E := RatModule.noZeroSMulDivisors\n have : NoZeroSMulDivisors \u2124 L := by\n change NoZeroSMulDivisors \u2124 (AddSubgroup.toIntSubmodule L)\n exact noZeroSMulDivisors _\n infer_instance\n\ninstance instModuleFree_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E]\n [NormedSpace \u211d E] [FiniteDimensional \u211d E] (L : AddSubgroup E) [DiscreteTopology L] :\n Module.Free \u2124 L := by\n have : Module \u211a E := Module.compHom E (algebraMap \u211a \u211d)\n have : NoZeroSMulDivisors \u2124 E := RatModule.noZeroSMulDivisors\n have : NoZeroSMulDivisors \u2124 L := by\n change NoZeroSMulDivisors \u2124 (AddSubgroup.toIntSubmodule L)\n exact noZeroSMulDivisors _\n infer_instance\n\ntheorem Zlattice.rank [hs : IsZlattice K L] : finrank \u2124 L = finrank K E := by\n classical\n have : Module.Finite \u2124 L := module_finite K L\n have : Module.Free \u2124 L := module_free K L\n have : Module \u211a E := Module.compHom E (algebraMap \u211a K)\n let b\u2080 := Module.Free.chooseBasis \u2124 L\n -- Let `b` be a `\u2124`-basis of `L` formed of vectors of `E`\n let b := Subtype.val \u2218 b\u2080\n have : LinearIndependent \u2124 b :=\n LinearIndependent.map' b\u2080.linearIndependent (L.toIntSubmodule.subtype) (ker_subtype _)\n -- We prove some assertions that will be useful later on\n have h_spanL : span \u2124 (Set.range b) = AddSubgroup.toIntSubmodule L := by\n convert congrArg (map (Submodule.subtype (AddSubgroup.toIntSubmodule L))) b\u2080.span_eq\n \u00b7 rw [map_span, Set.range_comp]\n rfl\n \u00b7 exact (map_subtype_top _).symm\n have h_spanE : span K (Set.range b) = \u22a4 := by\n rw [\u2190 span_span_of_tower (R := \u2124), h_spanL]\n exact hs.span_top\n have h_card : Fintype.card (Module.Free.ChooseBasisIndex \u2124 L) =\n (Set.range b).toFinset.card := by\n rw [Set.toFinset_range, Finset.univ.card_image_of_injective]\n rfl\n exact Subtype.coe_injective.comp (Basis.injective _)\n rw [finrank_eq_card_chooseBasisIndex]\n -- We prove that `finrank \u2124 L \u2264 finrank K E` and `finrank K E \u2264 finrank \u2124 L`\n refine le_antisymm ?_ ?_\n \u00b7 -- To prove that `finrank \u2124 L \u2264 finrank K E`, we proceed by contradiction and prove that, in\n -- this case, there is a \u2124-relation between the vectors of `b`\n obtain \u27e8t, \u27e8ht_inc, \u27e8ht_span, ht_lin\u27e9\u27e9\u27e9 := exists_linearIndependent K (Set.range b)\n -- `e` is a `K`-basis of `E` formed of vectors of `b`\n let e : Basis t K E := Basis.mk ht_lin (by simp [ht_span, h_spanE])\n have : Fintype t := Set.Finite.fintype ((Set.range b).toFinite.subset ht_inc)\n have h : LinearIndependent \u2124 (fun x : (Set.range b) => (x : E)) := by\n rwa [linearIndependent_subtype_range (Subtype.coe_injective.comp b\u2080.injective)]\n contrapose! h\n -- Since `finrank \u2124 L > finrank K E`, there exists a vector `v \u2208 b` with `v \u2209 e`\n obtain \u27e8v, hv\u27e9 : (Set.range b \\ Set.range e).Nonempty := by\n rw [Basis.coe_mk, Subtype.range_coe_subtype, Set.setOf_mem_eq, \u2190 Set.toFinset_nonempty]\n contrapose h\n rw [Finset.not_nonempty_iff_eq_empty, Set.toFinset_diff,\n Finset.sdiff_eq_empty_iff_subset] at h\n replace h := Finset.card_le_card h\n rwa [not_lt, h_card, \u2190 topEquiv.finrank_eq, \u2190 h_spanE, \u2190 ht_span,\n finrank_span_set_eq_card ht_lin]\n -- Assume that `e \u222a {v}` is not `\u2124`-linear independent then we get the contradiction\n suffices \u00ac LinearIndependent \u2124 (fun x : \u21a5(insert v (Set.range e)) => (x : E)) by\n contrapose! this\n refine LinearIndependent.mono ?_ this\n exact Set.insert_subset (Set.mem_of_mem_diff hv) (by simp [e, ht_inc])\n -- We prove finally that `e \u222a {v}` is not \u2124-linear independent or, equivalently,\n -- not \u211a-linear independent by showing that `v \u2208 span \u211a e`.\n rw [LinearIndependent.iff_fractionRing \u2124 \u211a,\n linearIndependent_insert (Set.not_mem_of_mem_diff hv), not_and, not_not]\n intro _\n -- But that follows from the fact that there exist `n, m : \u2115`, `n \u2260 m`\n -- such that `(n - m) \u2022 v \u2208 span \u2124 e` which is true since `n \u21a6 Zspan.fract e (n \u2022 v)`\n -- takes value into the finite set `fundamentalDomain e \u2229 L`\n have h_mapsto : Set.MapsTo (fun n : \u2124 => Zspan.fract e (n \u2022 v)) Set.univ\n (Metric.closedBall 0 (\u2211 i, \u2016e i\u2016) \u2229 (L : Set E)) := by\n rw [Set.mapsTo_inter, Set.mapsTo_univ_iff, Set.mapsTo_univ_iff]\n refine \u27e8fun _ \u21a6 mem_closedBall_zero_iff.mpr (Zspan.norm_fract_le e _), fun _ => ?_\u27e9\n \u00b7 change _ \u2208 AddSubgroup.toIntSubmodule L\n rw [\u2190 h_spanL]\n refine sub_mem ?_ ?_\n \u00b7 exact zsmul_mem (subset_span (Set.diff_subset _ _ hv)) _\n \u00b7 exact span_mono (by simp [e, ht_inc]) (coe_mem _)\n have h_finite : Set.Finite (Metric.closedBall 0 (\u2211 i, \u2016e i\u2016) \u2229 (L : Set E)) :=\n Metric.finite_isBounded_inter_isClosed Metric.isBounded_closedBall inferInstance\n obtain \u27e8n, -, m, -, h_neq, h_eq\u27e9 := Set.Infinite.exists_ne_map_eq_of_mapsTo\n Set.infinite_univ h_mapsto h_finite\n have h_nz : (-n + m : \u211a) \u2260 0 := by\n rwa [Ne, add_eq_zero_iff_eq_neg.not, neg_inj, Rat.coe_int_inj, \u2190 Ne]\n apply (smul_mem_iff _ h_nz).mp\n refine span_subset_span \u2124 \u211a _ ?_\n rwa [add_smul, neg_smul, SetLike.mem_coe, \u2190 Zspan.fract_eq_fract, \u2190 zsmul_eq_smul_cast \u211a,\n \u2190 zsmul_eq_smul_cast \u211a]\n \u00b7 -- To prove that `finrank K E \u2264 finrank \u2124 L`, we use the fact `b` generates `E` over `K`\n -- and thus `finrank K E \u2264 card b = finrank \u2124 L`\n rw [\u2190 topEquiv.finrank_eq, \u2190 h_spanE]\n convert finrank_span_le_card (R := K) (Set.range b)\n\nopen Module\n\nvariable {\u03b9 : Type*} [hs : IsZlattice K L] (b : Basis \u03b9 \u2124 L)\n/-- Any `\u2124`-basis of `L` is also a `K`-basis of `E`. -/\ndef Basis.ofZlatticeBasis :\n Basis \u03b9 K E := by\n have : Finite \u2124 L := Zlattice.module_finite K L\n have : Free \u2124 L := Zlattice.module_free K L\n let e := Basis.indexEquiv (Free.chooseBasis \u2124 L) b\n have : Fintype \u03b9 := Fintype.ofEquiv _ e\n refine basisOfTopLeSpanOfCardEqFinrank (L.subtype.toIntLinearMap \u2218 b) ?_ ?_\n \u00b7 rw [\u2190 span_span_of_tower \u2124, Set.range_comp, \u2190 map_span, Basis.span_eq, Submodule.map_top,\n top_le_iff, AddMonoidHom.coe_toIntLinearMap_range, AddSubgroup.subtype_range,\n AddSubgroup.coe_toIntSubmodule, hs.span_top]\n \u00b7 rw [\u2190 Fintype.card_congr e, \u2190 finrank_eq_card_chooseBasisIndex, Zlattice.rank K L]\n\n@[simp]\ntheorem Basis.ofZlatticeBasis_apply (i : \u03b9) :\n b.ofZlatticeBasis K L i = b i := by simp [Basis.ofZlatticeBasis]\n\n@[simp]\ntheorem Basis.ofZlatticeBasis_repr_apply (x : L) (i : \u03b9) :\n (b.ofZlatticeBasis K L).repr x i = b.repr x i := by\n suffices ((b.ofZlatticeBasis K L).repr.toLinearMap.restrictScalars \u2124) \u2218\u2097 L.subtype.toIntLinearMap\n = Finsupp.mapRange.linearMap (Algebra.linearMap \u2124 K) \u2218\u2097 b.repr.toLinearMap by\n exact DFunLike.congr_fun (LinearMap.congr_fun this x) i\n refine Basis.ext b fun i \u21a6 ?_\n simp_rw [LinearMap.coe_comp, Function.comp_apply, LinearMap.coe_restrictScalars,\n LinearEquiv.coe_coe, AddMonoidHom.coe_toIntLinearMap, AddSubgroup.coeSubtype,\n \u2190 b.ofZlatticeBasis_apply K, repr_self, Finsupp.mapRange.linearMap_apply,\n Finsupp.mapRange_single, Algebra.linearMap_apply, map_one]\n\ntheorem Basis.ofZlatticeBasis_span :\n (span \u2124 (Set.range (b.ofZlatticeBasis K))).toAddSubgroup = L := by\n calc (span \u2124 (Set.range (b.ofZlatticeBasis K))).toAddSubgroup\n _ = (span \u2124 (L.subtype.toIntLinearMap '' (Set.range b))).toAddSubgroup := by congr; ext; simp\n _ = (map L.subtype.toIntLinearMap (span \u2124 (Set.range b))).toAddSubgroup := by\n rw [Submodule.map_span]\n _ = L := by simp [b.span_eq]\n\n", "theoremStatement": "theorem Zlattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n [FiniteDimensional \u211d E] {L : AddSubgroup E} [DiscreteTopology L] [IsZlattice \u211d L] [Finite \u03b9]\n (b : Basis \u03b9 \u2124 L) [MeasurableSpace E] [OpensMeasurableSpace E] (\u03bc : MeasureTheory.Measure E) :\n MeasureTheory.IsAddFundamentalDomain L (Zspan.fundamentalDomain (b.ofZlatticeBasis \u211d)) \u03bc", "theoremName": "Zlattice.isAddFundamentalDomain", "fileCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "theoremCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/Zlattice/Basic.lean", "positionMetadata": {"lineInFile": 621, "tokenPositionInFile": 30285, "theoremPositionInFile": 53}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n convert Zspan.isAddFundamentalDomain (b.ofZlatticeBasis \u211d) \u03bc\n all_goals exact (b.ofZlatticeBasis_span \u211d).symm", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 115}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2023 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport Mathlib.MeasureTheory.Group.FundamentalDomain\nimport Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar\nimport Mathlib.RingTheory.Localization.Module\n\n#align_import algebra.module.zlattice from \"leanprover-community/mathlib\"@\"a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3\"\n\n/-!\n# \u2124-lattices\n\nLet `E` be a finite dimensional vector space over a `NormedLinearOrderedField` `K` with a solid\nnorm that is also a `FloorRing`, e.g. `\u211d`. A (full) `\u2124`-lattice `L` of `E` is a discrete\nsubgroup of `E` such that `L` spans `E` over `K`.\n\nA `\u2124`-lattice `L` can be defined in two ways:\n* For `b` a basis of `E`, then `L = Submodule.span \u2124 (Set.range b)` is a \u2124-lattice of `E`\n* As an `AddSubgroup E` with the additional properties:\n * `DiscreteTopology L`, that is `L` is discrete\n * `Submodule.span \u211d (L : Set E) = \u22a4`, that is `L` spans `E` over `K`.\n\nResults about the first point of view are in the `Zspan` namespace and results about the second\npoint of view are in the `Zlattice` namespace.\n\n## Main results\n\n* `Zspan.isAddFundamentalDomain`: for a \u2124-lattice `Submodule.span \u2124 (Set.range b)`, proves that\nthe set defined by `Zspan.fundamentalDomain` is a fundamental domain.\n* `Zlattice.module_free`: an AddSubgroup of `E` that is discrete and spans `E` over `K` is a free\n`\u2124`-module\n* `Zlattice.rank`: an AddSubgroup of `E` that is discrete and spans `E` over `K` is a free\n`\u2124`-module of `\u2124`-rank equal to the `K`-rank of `E`\n-/\n\n\nopen scoped BigOperators\n\nnoncomputable section\n\nnamespace Zspan\n\nopen MeasureTheory MeasurableSet Submodule Bornology\n\nvariable {E \u03b9 : Type*}\n\nsection NormedLatticeField\n\nvariable {K : Type*} [NormedLinearOrderedField K]\nvariable [NormedAddCommGroup E] [NormedSpace K E]\nvariable (b : Basis \u03b9 K E)\n\ntheorem span_top : span K (span \u2124 (Set.range b) : Set E) = \u22a4 := by simp [span_span_of_tower]\n\n/-- The fundamental domain of the \u2124-lattice spanned by `b`. See `Zspan.isAddFundamentalDomain`\nfor the proof that it is a fundamental domain. -/\ndef fundamentalDomain : Set E := {m | \u2200 i, b.repr m i \u2208 Set.Ico (0 : K) 1}\n#align zspan.fundamental_domain Zspan.fundamentalDomain\n\n@[simp]\ntheorem mem_fundamentalDomain {m : E} :\n m \u2208 fundamentalDomain b \u2194 \u2200 i, b.repr m i \u2208 Set.Ico (0 : K) 1 := Iff.rfl\n#align zspan.mem_fundamental_domain Zspan.mem_fundamentalDomain\n\ntheorem map_fundamentalDomain {F : Type*} [NormedAddCommGroup F] [NormedSpace K F] (f : E \u2243\u2097[K] F) :\n f '' (fundamentalDomain b) = fundamentalDomain (b.map f) := by\n ext x\n rw [mem_fundamentalDomain, Basis.map_repr, LinearEquiv.trans_apply, \u2190 mem_fundamentalDomain,\n show f.symm x = f.toEquiv.symm x by rfl, \u2190 Set.mem_image_equiv]\n rfl\n\n@[simp]\ntheorem fundamentalDomain_reindex {\u03b9' : Type*} (e : \u03b9 \u2243 \u03b9') :\n fundamentalDomain (b.reindex e) = fundamentalDomain b := by\n ext\n simp_rw [mem_fundamentalDomain, Basis.repr_reindex_apply]\n rw [Equiv.forall_congr' e]\n simp_rw [implies_true]\n\nlemma fundamentalDomain_pi_basisFun [Fintype \u03b9] :\n fundamentalDomain (Pi.basisFun \u211d \u03b9) = Set.pi Set.univ fun _ : \u03b9 \u21a6 Set.Ico (0 : \u211d) 1 := by\n ext; simp\n\nvariable [FloorRing K]\n\nsection Fintype\n\nvariable [Fintype \u03b9]\n\n/-- The map that sends a vector of `E` to the element of the \u2124-lattice spanned by `b` obtained\nby rounding down its coordinates on the basis `b`. -/\ndef floor (m : E) : span \u2124 (Set.range b) := \u2211 i, \u230ab.repr m i\u230b \u2022 b.restrictScalars \u2124 i\n#align zspan.floor Zspan.floor\n\n/-- The map that sends a vector of `E` to the element of the \u2124-lattice spanned by `b` obtained\nby rounding up its coordinates on the basis `b`. -/\ndef ceil (m : E) : span \u2124 (Set.range b) := \u2211 i, \u2308b.repr m i\u2309 \u2022 b.restrictScalars \u2124 i\n#align zspan.ceil Zspan.ceil\n\n@[simp]\ntheorem repr_floor_apply (m : E) (i : \u03b9) : b.repr (floor b m) i = \u230ab.repr m i\u230b := by\n classical simp only [floor, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\n Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\n Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]\n#align zspan.repr_floor_apply Zspan.repr_floor_apply\n\n@[simp]\ntheorem repr_ceil_apply (m : E) (i : \u03b9) : b.repr (ceil b m) i = \u2308b.repr m i\u2309 := by\n classical simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\n Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\n Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]\n#align zspan.repr_ceil_apply Zspan.repr_ceil_apply\n\n@[simp]\ntheorem floor_eq_self_of_mem (m : E) (h : m \u2208 span \u2124 (Set.range b)) : (floor b m : E) = m := by\n apply b.ext_elem\n simp_rw [repr_floor_apply b]\n intro i\n obtain \u27e8z, hz\u27e9 := (b.mem_span_iff_repr_mem \u2124 _).mp h i\n rw [\u2190 hz]\n exact congr_arg (Int.cast : \u2124 \u2192 K) (Int.floor_intCast z)\n#align zspan.floor_eq_self_of_mem Zspan.floor_eq_self_of_mem\n\n@[simp]\ntheorem ceil_eq_self_of_mem (m : E) (h : m \u2208 span \u2124 (Set.range b)) : (ceil b m : E) = m := by\n apply b.ext_elem\n simp_rw [repr_ceil_apply b]\n intro i\n obtain \u27e8z, hz\u27e9 := (b.mem_span_iff_repr_mem \u2124 _).mp h i\n rw [\u2190 hz]\n exact congr_arg (Int.cast : \u2124 \u2192 K) (Int.ceil_intCast z)\n#align zspan.ceil_eq_self_of_mem Zspan.ceil_eq_self_of_mem\n\n/-- The map that sends a vector `E` to the `fundamentalDomain` of the lattice,\nsee `Zspan.fract_mem_fundamentalDomain`, and `fractRestrict` for the map with the codomain\nrestricted to `fundamentalDomain`. -/\ndef fract (m : E) : E := m - floor b m\n#align zspan.fract Zspan.fract\n\ntheorem fract_apply (m : E) : fract b m = m - floor b m := rfl\n#align zspan.fract_apply Zspan.fract_apply\n\n@[simp]\ntheorem repr_fract_apply (m : E) (i : \u03b9) : b.repr (fract b m) i = Int.fract (b.repr m i) := by\n rw [fract, LinearEquiv.map_sub, Finsupp.coe_sub, Pi.sub_apply, repr_floor_apply, Int.fract]\n#align zspan.repr_fract_apply Zspan.repr_fract_apply\n\n@[simp]\ntheorem fract_fract (m : E) : fract b (fract b m) = fract b m :=\n Basis.ext_elem b fun _ => by classical simp only [repr_fract_apply, Int.fract_fract]\n#align zspan.fract_fract Zspan.fract_fract\n\n@[simp]\ntheorem fract_zspan_add (m : E) {v : E} (h : v \u2208 span \u2124 (Set.range b)) :\n fract b (v + m) = fract b m := by\n classical\n refine (Basis.ext_elem_iff b).mpr fun i => ?_\n simp_rw [repr_fract_apply, Int.fract_eq_fract]\n use (b.restrictScalars \u2124).repr \u27e8v, h\u27e9 i\n rw [map_add, Finsupp.coe_add, Pi.add_apply, add_tsub_cancel_right,\n \u2190 eq_intCast (algebraMap \u2124 K) _, Basis.restrictScalars_repr_apply, coe_mk]\n#align zspan.fract_zspan_add Zspan.fract_zspan_add\n\n@[simp]\ntheorem fract_add_zspan (m : E) {v : E} (h : v \u2208 span \u2124 (Set.range b)) :\n fract b (m + v) = fract b m := by rw [add_comm, fract_zspan_add b m h]\n#align zspan.fract_add_zspan Zspan.fract_add_zspan\n\nvariable {b}\n\ntheorem fract_eq_self {x : E} : fract b x = x \u2194 x \u2208 fundamentalDomain b := by\n classical simp only [Basis.ext_elem_iff b, repr_fract_apply, Int.fract_eq_self,\n mem_fundamentalDomain, Set.mem_Ico]\n#align zspan.fract_eq_self Zspan.fract_eq_self\n\nvariable (b)\n\ntheorem fract_mem_fundamentalDomain (x : E) : fract b x \u2208 fundamentalDomain b :=\n fract_eq_self.mp (fract_fract b _)\n#align zspan.fract_mem_fundamental_domain Zspan.fract_mem_fundamentalDomain\n\n/-- The map `fract` with codomain restricted to `fundamentalDomain`. -/\ndef fractRestrict (x : E) : fundamentalDomain b := \u27e8fract b x, fract_mem_fundamentalDomain b x\u27e9\n\ntheorem fractRestrict_surjective : Function.Surjective (fractRestrict b) :=\n fun x => \u27e8\u2191x, Subtype.eq (fract_eq_self.mpr (Subtype.mem x))\u27e9\n\n@[simp]\ntheorem fractRestrict_apply (x : E) : (fractRestrict b x : E) = fract b x := rfl\n\ntheorem fract_eq_fract (m n : E) : fract b m = fract b n \u2194 -m + n \u2208 span \u2124 (Set.range b) := by\n classical\n rw [eq_comm, Basis.ext_elem_iff b]\n simp_rw [repr_fract_apply, Int.fract_eq_fract, eq_comm, Basis.mem_span_iff_repr_mem,\n sub_eq_neg_add, map_add, LinearEquiv.map_neg, Finsupp.coe_add, Finsupp.coe_neg, Pi.add_apply,\n Pi.neg_apply, \u2190 eq_intCast (algebraMap \u2124 K) _, Set.mem_range]\n#align zspan.fract_eq_fract Zspan.fract_eq_fract\n\ntheorem norm_fract_le [HasSolidNorm K] (m : E) : \u2016fract b m\u2016 \u2264 \u2211 i, \u2016b i\u2016 := by\n classical\n calc\n \u2016fract b m\u2016 = \u2016\u2211 i, b.repr (fract b m) i \u2022 b i\u2016 := by rw [b.sum_repr]\n _ = \u2016\u2211 i, Int.fract (b.repr m i) \u2022 b i\u2016 := by simp_rw [repr_fract_apply]\n _ \u2264 \u2211 i, \u2016Int.fract (b.repr m i) \u2022 b i\u2016 := norm_sum_le _ _\n _ = \u2211 i, \u2016Int.fract (b.repr m i)\u2016 * \u2016b i\u2016 := by simp_rw [norm_smul]\n _ \u2264 \u2211 i, \u2016b i\u2016 := Finset.sum_le_sum fun i _ => ?_\n suffices \u2016Int.fract ((b.repr m) i)\u2016 \u2264 1 by\n convert mul_le_mul_of_nonneg_right this (norm_nonneg _ : 0 \u2264 \u2016b i\u2016)\n exact (one_mul _).symm\n rw [(norm_one.symm : 1 = \u2016(1 : K)\u2016)]\n apply norm_le_norm_of_abs_le_abs\n rw [abs_one, Int.abs_fract]\n exact le_of_lt (Int.fract_lt_one _)\n#align zspan.norm_fract_le Zspan.norm_fract_le\n\nsection Unique\n\nvariable [Unique \u03b9]\n\n@[simp]\ntheorem coe_floor_self (k : K) : (floor (Basis.singleton \u03b9 K) k : K) = \u230ak\u230b :=\n Basis.ext_elem _ fun _ => by rw [repr_floor_apply, Basis.singleton_repr, Basis.singleton_repr]\n#align zspan.coe_floor_self Zspan.coe_floor_self\n\n@[simp]\ntheorem coe_fract_self (k : K) : (fract (Basis.singleton \u03b9 K) k : K) = Int.fract k :=\n Basis.ext_elem _ fun _ => by rw [repr_fract_apply, Basis.singleton_repr, Basis.singleton_repr]\n#align zspan.coe_fract_self Zspan.coe_fract_self\n\nend Unique\n\nend Fintype\n\ntheorem fundamentalDomain_isBounded [Finite \u03b9] [HasSolidNorm K] :\n IsBounded (fundamentalDomain b) := by\n cases nonempty_fintype \u03b9\n refine isBounded_iff_forall_norm_le.2 \u27e8\u2211 j, \u2016b j\u2016, fun x hx \u21a6 ?_\u27e9\n rw [\u2190 fract_eq_self.mpr hx]\n apply norm_fract_le\n#align zspan.fundamental_domain_bounded Zspan.fundamentalDomain_isBounded\n\ntheorem vadd_mem_fundamentalDomain [Fintype \u03b9] (y : span \u2124 (Set.range b)) (x : E) :\n y +\u1d65 x \u2208 fundamentalDomain b \u2194 y = -floor b x := by\n rw [Subtype.ext_iff, \u2190 add_right_inj x, NegMemClass.coe_neg, \u2190 sub_eq_add_neg, \u2190 fract_apply,\n \u2190 fract_zspan_add b _ (Subtype.mem y), add_comm, \u2190 vadd_eq_add, \u2190 vadd_def, eq_comm, \u2190\n fract_eq_self]\n#align zspan.vadd_mem_fundamental_domain Zspan.vadd_mem_fundamentalDomain\n\ntheorem exist_unique_vadd_mem_fundamentalDomain [Finite \u03b9] (x : E) :\n \u2203! v : span \u2124 (Set.range b), v +\u1d65 x \u2208 fundamentalDomain b := by\n cases nonempty_fintype \u03b9\n refine \u27e8-floor b x, ?_, fun y h => ?_\u27e9\n \u00b7 exact (vadd_mem_fundamentalDomain b (-floor b x) x).mpr rfl\n \u00b7 exact (vadd_mem_fundamentalDomain b y x).mp h\n#align zspan.exist_unique_vadd_mem_fundamental_domain Zspan.exist_unique_vadd_mem_fundamentalDomain\n\n/-- The map `Zspan.fractRestrict` defines an equiv map between `E \u29f8 span \u2124 (Set.range b)`\nand `Zspan.fundamentalDomain b`. -/\ndef quotientEquiv [Fintype \u03b9] :\n E \u29f8 span \u2124 (Set.range b) \u2243 (fundamentalDomain b) := by\n refine Equiv.ofBijective ?_ \u27e8fun x y => ?_, fun x => ?_\u27e9\n \u00b7 refine fun q => Quotient.liftOn q (fractRestrict b) (fun _ _ h => ?_)\n rw [Subtype.mk.injEq, fractRestrict_apply, fractRestrict_apply, fract_eq_fract]\n exact QuotientAddGroup.leftRel_apply.mp h\n \u00b7 refine Quotient.inductionOn\u2082 x y (fun _ _ hxy => ?_)\n rw [Quotient.liftOn_mk (s := quotientRel (span \u2124 (Set.range b))), fractRestrict,\n Quotient.liftOn_mk (s := quotientRel (span \u2124 (Set.range b))), fractRestrict,\n Subtype.mk.injEq] at hxy\n apply Quotient.sound'\n rwa [QuotientAddGroup.leftRel_apply, mem_toAddSubgroup, \u2190 fract_eq_fract]\n \u00b7 obtain \u27e8a, rfl\u27e9 := fractRestrict_surjective b x\n exact \u27e8Quotient.mk'' a, rfl\u27e9\n\n@[simp]\ntheorem quotientEquiv_apply_mk [Fintype \u03b9] (x : E) :\n quotientEquiv b (Submodule.Quotient.mk x) = fractRestrict b x := rfl\n\n@[simp]\ntheorem quotientEquiv.symm_apply [Fintype \u03b9] (x : fundamentalDomain b) :\n (quotientEquiv b).symm x = Submodule.Quotient.mk \u2191x := by\n rw [Equiv.symm_apply_eq, quotientEquiv_apply_mk b \u2191x, Subtype.ext_iff, fractRestrict_apply]\n exact (fract_eq_self.mpr x.prop).symm\n\nend NormedLatticeField\n\nsection Real\n\ntheorem discreteTopology_pi_basisFun [Finite \u03b9] :\n DiscreteTopology (span \u2124 (Set.range (Pi.basisFun \u211d \u03b9))) := by\n cases nonempty_fintype \u03b9\n refine discreteTopology_iff_isOpen_singleton_zero.mpr \u27e8Metric.ball 0 1, Metric.isOpen_ball, ?_\u27e9\n ext x\n rw [Set.mem_preimage, mem_ball_zero_iff, pi_norm_lt_iff zero_lt_one, Set.mem_singleton_iff]\n simp_rw [\u2190 coe_eq_zero, Function.funext_iff, Pi.zero_apply, Real.norm_eq_abs]\n refine forall_congr' (fun i => ?_)\n rsuffices \u27e8y, hy\u27e9 : \u2203 (y : \u2124), (y : \u211d) = (x : \u03b9 \u2192 \u211d) i\n \u00b7 rw [\u2190 hy, \u2190 Int.cast_abs, \u2190 Int.cast_one, Int.cast_lt, Int.abs_lt_one_iff, Int.cast_eq_zero]\n exact ((Pi.basisFun \u211d \u03b9).mem_span_iff_repr_mem \u2124 x).mp (SetLike.coe_mem x) i\n\nvariable [NormedAddCommGroup E] [NormedSpace \u211d E] (b : Basis \u03b9 \u211d E)\n\ntheorem fundamentalDomain_subset_parallelepiped [Fintype \u03b9] :\n fundamentalDomain b \u2286 parallelepiped b := by\n rw [fundamentalDomain, parallelepiped_basis_eq, Set.setOf_subset_setOf]\n exact fun _ h i \u21a6 Set.Ico_subset_Icc_self (h i)\n\ninstance [Finite \u03b9] : DiscreteTopology (span \u2124 (Set.range b)) := by\n have h : Set.MapsTo b.equivFun (span \u2124 (Set.range b)) (span \u2124 (Set.range (Pi.basisFun \u211d \u03b9))) := by\n intro _ hx\n rwa [SetLike.mem_coe, Basis.mem_span_iff_repr_mem] at hx \u22a2\n convert DiscreteTopology.of_continuous_injective ((continuous_equivFun_basis b).restrict h) ?_\n \u00b7 exact discreteTopology_pi_basisFun\n \u00b7 refine Subtype.map_injective _ (Basis.equivFun b).injective\n\ninstance [Finite \u03b9] : DiscreteTopology (span \u2124 (Set.range b)).toAddSubgroup :=\n inferInstanceAs <| DiscreteTopology (span \u2124 (Set.range b))\n\n@[measurability]\ntheorem fundamentalDomain_measurableSet [MeasurableSpace E] [OpensMeasurableSpace E] [Finite \u03b9] :\n MeasurableSet (fundamentalDomain b) := by\n cases nonempty_fintype \u03b9\n haveI : FiniteDimensional \u211d E := FiniteDimensional.of_fintype_basis b\n let D : Set (\u03b9 \u2192 \u211d) := Set.pi Set.univ fun _ : \u03b9 => Set.Ico (0 : \u211d) 1\n rw [(_ : fundamentalDomain b = b.equivFun.toLinearMap \u207b\u00b9' D)]\n \u00b7 refine measurableSet_preimage (LinearMap.continuous_of_finiteDimensional _).measurable ?_\n exact MeasurableSet.pi Set.countable_univ fun _ _ => measurableSet_Ico\n \u00b7 ext\n simp only [D, fundamentalDomain, Set.mem_Ico, Set.mem_setOf_eq, LinearEquiv.coe_coe,\n Set.mem_preimage, Basis.equivFun_apply, Set.mem_pi, Set.mem_univ, forall_true_left]\n#align zspan.fundamental_domain_measurable_set Zspan.fundamentalDomain_measurableSet\n\n/-- For a \u2124-lattice `Submodule.span \u2124 (Set.range b)`, proves that the set defined\nby `Zspan.fundamentalDomain` is a fundamental domain. -/\nprotected theorem isAddFundamentalDomain [Finite \u03b9] [MeasurableSpace E] [OpensMeasurableSpace E]\n (\u03bc : Measure E) :\n IsAddFundamentalDomain (span \u2124 (Set.range b)).toAddSubgroup (fundamentalDomain b) \u03bc := by\n cases nonempty_fintype \u03b9\n exact IsAddFundamentalDomain.mk' (nullMeasurableSet (fundamentalDomain_measurableSet b))\n fun x => exist_unique_vadd_mem_fundamentalDomain b x\n#align zspan.is_add_fundamental_domain Zspan.isAddFundamentalDomain\n\ntheorem measure_fundamentalDomain_ne_zero [Finite \u03b9] [MeasurableSpace E] [BorelSpace E]\n {\u03bc : Measure E} [Measure.IsAddHaarMeasure \u03bc] :\n \u03bc (fundamentalDomain b) \u2260 0 := by\n convert (Zspan.isAddFundamentalDomain b \u03bc).measure_ne_zero (NeZero.ne \u03bc)\n simp only [mem_toAddSubgroup]\n infer_instance\n\ntheorem measure_fundamentalDomain [Fintype \u03b9] [DecidableEq \u03b9] [MeasurableSpace E] (\u03bc : Measure E)\n [BorelSpace E] [Measure.IsAddHaarMeasure \u03bc] (b\u2080 : Basis \u03b9 \u211d E) :\n \u03bc (fundamentalDomain b) = ENNReal.ofReal |b\u2080.det b| * \u03bc (fundamentalDomain b\u2080) := by\n have : FiniteDimensional \u211d E := FiniteDimensional.of_fintype_basis b\n convert \u03bc.addHaar_preimage_linearEquiv (b.equiv b\u2080 (Equiv.refl \u03b9)) (fundamentalDomain b\u2080)\n \u00b7 rw [Set.eq_preimage_iff_image_eq (LinearEquiv.bijective _), map_fundamentalDomain,\n Basis.map_equiv, Equiv.refl_symm, Basis.reindex_refl]\n \u00b7 rw [\u2190 LinearMap.det_toMatrix b\u2080, Basis.equiv_symm, Equiv.refl_symm, Basis.det_apply]\n congr\n ext\n simp [Basis.toMatrix_apply, LinearMap.toMatrix_apply, LinearEquiv.coe_coe, Basis.equiv_apply]\n\n@[simp]\ntheorem volume_fundamentalDomain [Fintype \u03b9] [DecidableEq \u03b9] (b : Basis \u03b9 \u211d (\u03b9 \u2192 \u211d)) :\n volume (fundamentalDomain b) = ENNReal.ofReal |(Matrix.of b).det| := by\n rw [measure_fundamentalDomain b volume (b\u2080 := Pi.basisFun \u211d \u03b9), fundamentalDomain_pi_basisFun,\n volume_pi, Measure.pi_pi, Real.volume_Ico, sub_zero, ENNReal.ofReal_one, Finset.prod_const_one,\n mul_one, \u2190 Matrix.det_transpose]\n rfl\n\ntheorem fundamentalDomain_ae_parallelepiped [Fintype \u03b9] [MeasurableSpace E] (\u03bc : Measure E)\n [BorelSpace E] [Measure.IsAddHaarMeasure \u03bc] :\n fundamentalDomain b =\u1d50[\u03bc] parallelepiped b := by\n classical\n have : FiniteDimensional \u211d E := FiniteDimensional.of_fintype_basis b\n rw [\u2190 measure_symmDiff_eq_zero_iff, symmDiff_of_le (fundamentalDomain_subset_parallelepiped b)]\n suffices (parallelepiped b \\ fundamentalDomain b) \u2286 \u22c3 i,\n AffineSubspace.mk' (b i) (span \u211d (b '' (Set.univ \\ {i}))) by\n refine measure_mono_null this\n (measure_iUnion_null_iff.mpr fun i \u21a6 Measure.addHaar_affineSubspace \u03bc _ ?_)\n refine (ne_of_mem_of_not_mem' (AffineSubspace.mem_top _ _ 0)\n (AffineSubspace.mem_mk'_iff_vsub_mem.not.mpr ?_)).symm\n simp_rw [vsub_eq_sub, zero_sub, neg_mem_iff]\n exact linearIndependent_iff_not_mem_span.mp b.linearIndependent i\n intro x hx\n simp_rw [parallelepiped_basis_eq, Set.mem_Icc, Set.mem_diff, Set.mem_setOf_eq,\n mem_fundamentalDomain, Set.mem_Ico, not_forall, not_and, not_lt] at hx\n obtain \u27e8i, hi\u27e9 := hx.2\n have : b.repr x i = 1 := le_antisymm (hx.1 i).2 (hi (hx.1 i).1)\n rw [\u2190 b.sum_repr x, \u2190 Finset.sum_erase_add _ _ (Finset.mem_univ i), this, one_smul, \u2190 vadd_eq_add]\n refine Set.mem_iUnion.mpr \u27e8i, AffineSubspace.vadd_mem_mk' _\n (sum_smul_mem _ _ (fun i hi \u21a6 Submodule.subset_span ?_))\u27e9\n exact \u27e8i, Set.mem_diff_singleton.mpr \u27e8trivial, Finset.ne_of_mem_erase hi\u27e9, rfl\u27e9\n\nend Real\n\nend Zspan\n\nsection Zlattice\n\nopen Submodule FiniteDimensional\n\n-- TODO: generalize this class to other rings than `\u2124`\n/-- An `L : Addsubgroup E` where `E` is a vector space over a normed field `K` is a `\u2124`-lattice if\nit is discrete and spans `E` over `K`. -/\nclass IsZlattice (K : Type*) [NormedField K] {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]\n (L : AddSubgroup E) [DiscreteTopology L] : Prop where\n /-- `L` spans the full space `E` over `K`. -/\n span_top : span K (L : Set E) = \u22a4\n\ntheorem _root_.Zspan.isZlattice {E \u03b9 : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n [Finite \u03b9] (b : Basis \u03b9 \u211d E) :\n IsZlattice \u211d (span \u2124 (Set.range b)).toAddSubgroup where\n span_top := Zspan.span_top b\n\nvariable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]\nvariable [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology L]\n\ntheorem Zlattice.FG [hs : IsZlattice K L] : AddSubgroup.FG L := by\n suffices (AddSubgroup.toIntSubmodule L).FG by exact (fg_iff_add_subgroup_fg _).mp this\n obtain \u27e8s, \u27e8h_incl, \u27e8h_span, h_lind\u27e9\u27e9\u27e9 := exists_linearIndependent K (L : Set E)\n -- Let `s` be a maximal `K`-linear independent family of elements of `L`. We show that\n -- `L` is finitely generated (as a \u2124-module) because it fits in the exact sequence\n -- `0 \u2192 span \u2124 s \u2192 L \u2192 L \u29f8 span \u2124 s \u2192 0` with `span \u2124 s` and `L \u29f8 span \u2124 s` finitely generated.\n refine fg_of_fg_map_of_fg_inf_ker (span \u2124 s).mkQ ?_ ?_\n \u00b7 -- Let `b` be the `K`-basis of `E` formed by the vectors in `s`. The elements of\n -- `L \u29f8 span \u2124 s = L \u29f8 span \u2124 b` are in bijection with elements of `L \u2229 fundamentalDomain b`\n -- so there are finitely many since `fundamentalDomain b` is bounded.\n refine fg_def.mpr \u27e8map (span \u2124 s).mkQ (AddSubgroup.toIntSubmodule L), ?_, span_eq _\u27e9\n let b := Basis.mk h_lind (by\n rw [\u2190 hs.span_top, \u2190 h_span]\n exact span_mono (by simp only [Subtype.range_coe_subtype, Set.setOf_mem_eq, subset_rfl]))\n rw [show span \u2124 s = span \u2124 (Set.range b) by simp [b, Basis.coe_mk, Subtype.range_coe_subtype]]\n have : Fintype s := h_lind.setFinite.fintype\n refine Set.Finite.of_finite_image (f := ((\u2191) : _ \u2192 E) \u2218 Zspan.quotientEquiv b) ?_\n (Function.Injective.injOn (Subtype.coe_injective.comp (Zspan.quotientEquiv b).injective) _)\n have : Set.Finite ((Zspan.fundamentalDomain b) \u2229 L) :=\n Metric.finite_isBounded_inter_isClosed (Zspan.fundamentalDomain_isBounded b) inferInstance\n refine Set.Finite.subset this ?_\n rintro _ \u27e8_, \u27e8\u27e8x, \u27e8h_mem, rfl\u27e9\u27e9, rfl\u27e9\u27e9\n rw [Function.comp_apply, mkQ_apply, Zspan.quotientEquiv_apply_mk, Zspan.fractRestrict_apply]\n refine \u27e8?_, ?_\u27e9\n \u00b7 exact Zspan.fract_mem_fundamentalDomain b x\n \u00b7 rw [Zspan.fract, SetLike.mem_coe, sub_eq_add_neg]\n refine AddSubgroup.add_mem _ h_mem\n (neg_mem (Set.mem_of_subset_of_mem ?_ (Subtype.mem (Zspan.floor b x))))\n rw [show (L : Set E) = AddSubgroup.toIntSubmodule L by rfl]\n rw [SetLike.coe_subset_coe, Basis.coe_mk, Subtype.range_coe_subtype, Set.setOf_mem_eq]\n exact span_le.mpr h_incl\n \u00b7 -- `span \u2124 s` is finitely generated because `s` is finite\n rw [ker_mkQ, inf_of_le_right (span_le.mpr h_incl)]\n exact fg_span (LinearIndependent.setFinite h_lind)\n\ntheorem Zlattice.module_finite [IsZlattice K L] : Module.Finite \u2124 L :=\n Module.Finite.iff_addGroup_fg.mpr ((AddGroup.fg_iff_addSubgroup_fg L).mpr (FG K L))\n\ninstance instModuleFinite_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E]\n [NormedSpace \u211d E] [FiniteDimensional \u211d E] (L : AddSubgroup E) [DiscreteTopology L] :\n Module.Finite \u2124 L := by\n let f := (span \u211d (L : Set E)).subtype\n let L\u2080 := (AddSubgroup.toIntSubmodule L).comap (f.restrictScalars \u2124)\n have h_img : f '' L\u2080 = L := by\n rw [\u2190 LinearMap.coe_restrictScalars \u2124 f, \u2190 Submodule.map_coe (f.restrictScalars \u2124),\n Submodule.map_comap_eq_self, AddSubgroup.coe_toIntSubmodule]\n exact fun x hx \u21a6 LinearMap.mem_range.mpr \u27e8\u27e8x, Submodule.subset_span hx\u27e9, rfl\u27e9\n suffices Module.Finite \u2124 L\u2080 by\n have : L\u2080.map (f.restrictScalars \u2124) = (AddSubgroup.toIntSubmodule L) :=\n SetLike.ext'_iff.mpr h_img\n convert this \u25b8 Module.Finite.map L\u2080 (f.restrictScalars \u2124)\n have : DiscreteTopology L\u2080.toAddSubgroup := by\n refine DiscreteTopology.preimage_of_continuous_injective (L : Set E) ?_ (injective_subtype _)\n exact LinearMap.continuous_of_finiteDimensional f\n have : IsZlattice \u211d L\u2080.toAddSubgroup := \u27e8by\n rw [\u2190 (Submodule.map_injective_of_injective (injective_subtype _)).eq_iff, Submodule.map_span,\n Submodule.map_top, range_subtype, coe_toAddSubgroup, h_img]\u27e9\n exact Zlattice.module_finite \u211d L\u2080.toAddSubgroup\n\ntheorem Zlattice.module_free [IsZlattice K L] : Module.Free \u2124 L := by\n have : Module.Finite \u2124 L := module_finite K L\n have : Module \u211a E := Module.compHom E (algebraMap \u211a K)\n have : NoZeroSMulDivisors \u2124 E := RatModule.noZeroSMulDivisors\n have : NoZeroSMulDivisors \u2124 L := by\n change NoZeroSMulDivisors \u2124 (AddSubgroup.toIntSubmodule L)\n exact noZeroSMulDivisors _\n infer_instance\n\ninstance instModuleFree_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E]\n [NormedSpace \u211d E] [FiniteDimensional \u211d E] (L : AddSubgroup E) [DiscreteTopology L] :\n Module.Free \u2124 L := by\n have : Module \u211a E := Module.compHom E (algebraMap \u211a \u211d)\n have : NoZeroSMulDivisors \u2124 E := RatModule.noZeroSMulDivisors\n have : NoZeroSMulDivisors \u2124 L := by\n change NoZeroSMulDivisors \u2124 (AddSubgroup.toIntSubmodule L)\n exact noZeroSMulDivisors _\n infer_instance\n\ntheorem Zlattice.rank [hs : IsZlattice K L] : finrank \u2124 L = finrank K E := by\n classical\n have : Module.Finite \u2124 L := module_finite K L\n have : Module.Free \u2124 L := module_free K L\n have : Module \u211a E := Module.compHom E (algebraMap \u211a K)\n let b\u2080 := Module.Free.chooseBasis \u2124 L\n -- Let `b` be a `\u2124`-basis of `L` formed of vectors of `E`\n let b := Subtype.val \u2218 b\u2080\n have : LinearIndependent \u2124 b :=\n LinearIndependent.map' b\u2080.linearIndependent (L.toIntSubmodule.subtype) (ker_subtype _)\n -- We prove some assertions that will be useful later on\n have h_spanL : span \u2124 (Set.range b) = AddSubgroup.toIntSubmodule L := by\n convert congrArg (map (Submodule.subtype (AddSubgroup.toIntSubmodule L))) b\u2080.span_eq\n \u00b7 rw [map_span, Set.range_comp]\n rfl\n \u00b7 exact (map_subtype_top _).symm\n have h_spanE : span K (Set.range b) = \u22a4 := by\n rw [\u2190 span_span_of_tower (R := \u2124), h_spanL]\n exact hs.span_top\n have h_card : Fintype.card (Module.Free.ChooseBasisIndex \u2124 L) =\n (Set.range b).toFinset.card := by\n rw [Set.toFinset_range, Finset.univ.card_image_of_injective]\n rfl\n exact Subtype.coe_injective.comp (Basis.injective _)\n rw [finrank_eq_card_chooseBasisIndex]\n -- We prove that `finrank \u2124 L \u2264 finrank K E` and `finrank K E \u2264 finrank \u2124 L`\n refine le_antisymm ?_ ?_\n \u00b7 -- To prove that `finrank \u2124 L \u2264 finrank K E`, we proceed by contradiction and prove that, in\n -- this case, there is a \u2124-relation between the vectors of `b`\n obtain \u27e8t, \u27e8ht_inc, \u27e8ht_span, ht_lin\u27e9\u27e9\u27e9 := exists_linearIndependent K (Set.range b)\n -- `e` is a `K`-basis of `E` formed of vectors of `b`\n let e : Basis t K E := Basis.mk ht_lin (by simp [ht_span, h_spanE])\n have : Fintype t := Set.Finite.fintype ((Set.range b).toFinite.subset ht_inc)\n have h : LinearIndependent \u2124 (fun x : (Set.range b) => (x : E)) := by\n rwa [linearIndependent_subtype_range (Subtype.coe_injective.comp b\u2080.injective)]\n contrapose! h\n -- Since `finrank \u2124 L > finrank K E`, there exists a vector `v \u2208 b` with `v \u2209 e`\n obtain \u27e8v, hv\u27e9 : (Set.range b \\ Set.range e).Nonempty := by\n rw [Basis.coe_mk, Subtype.range_coe_subtype, Set.setOf_mem_eq, \u2190 Set.toFinset_nonempty]\n contrapose h\n rw [Finset.not_nonempty_iff_eq_empty, Set.toFinset_diff,\n Finset.sdiff_eq_empty_iff_subset] at h\n replace h := Finset.card_le_card h\n rwa [not_lt, h_card, \u2190 topEquiv.finrank_eq, \u2190 h_spanE, \u2190 ht_span,\n finrank_span_set_eq_card ht_lin]\n -- Assume that `e \u222a {v}` is not `\u2124`-linear independent then we get the contradiction\n suffices \u00ac LinearIndependent \u2124 (fun x : \u21a5(insert v (Set.range e)) => (x : E)) by\n contrapose! this\n refine LinearIndependent.mono ?_ this\n exact Set.insert_subset (Set.mem_of_mem_diff hv) (by simp [e, ht_inc])\n -- We prove finally that `e \u222a {v}` is not \u2124-linear independent or, equivalently,\n -- not \u211a-linear independent by showing that `v \u2208 span \u211a e`.\n rw [LinearIndependent.iff_fractionRing \u2124 \u211a,\n linearIndependent_insert (Set.not_mem_of_mem_diff hv), not_and, not_not]\n intro _\n -- But that follows from the fact that there exist `n, m : \u2115`, `n \u2260 m`\n -- such that `(n - m) \u2022 v \u2208 span \u2124 e` which is true since `n \u21a6 Zspan.fract e (n \u2022 v)`\n -- takes value into the finite set `fundamentalDomain e \u2229 L`\n have h_mapsto : Set.MapsTo (fun n : \u2124 => Zspan.fract e (n \u2022 v)) Set.univ\n (Metric.closedBall 0 (\u2211 i, \u2016e i\u2016) \u2229 (L : Set E)) := by\n rw [Set.mapsTo_inter, Set.mapsTo_univ_iff, Set.mapsTo_univ_iff]\n refine \u27e8fun _ \u21a6 mem_closedBall_zero_iff.mpr (Zspan.norm_fract_le e _), fun _ => ?_\u27e9\n \u00b7 change _ \u2208 AddSubgroup.toIntSubmodule L\n rw [\u2190 h_spanL]\n refine sub_mem ?_ ?_\n \u00b7 exact zsmul_mem (subset_span (Set.diff_subset _ _ hv)) _\n \u00b7 exact span_mono (by simp [e, ht_inc]) (coe_mem _)\n have h_finite : Set.Finite (Metric.closedBall 0 (\u2211 i, \u2016e i\u2016) \u2229 (L : Set E)) :=\n Metric.finite_isBounded_inter_isClosed Metric.isBounded_closedBall inferInstance\n obtain \u27e8n, -, m, -, h_neq, h_eq\u27e9 := Set.Infinite.exists_ne_map_eq_of_mapsTo\n Set.infinite_univ h_mapsto h_finite\n have h_nz : (-n + m : \u211a) \u2260 0 := by\n rwa [Ne, add_eq_zero_iff_eq_neg.not, neg_inj, Rat.coe_int_inj, \u2190 Ne]\n apply (smul_mem_iff _ h_nz).mp\n refine span_subset_span \u2124 \u211a _ ?_\n rwa [add_smul, neg_smul, SetLike.mem_coe, \u2190 Zspan.fract_eq_fract, \u2190 zsmul_eq_smul_cast \u211a,\n \u2190 zsmul_eq_smul_cast \u211a]\n \u00b7 -- To prove that `finrank K E \u2264 finrank \u2124 L`, we use the fact `b` generates `E` over `K`\n -- and thus `finrank K E \u2264 card b = finrank \u2124 L`\n rw [\u2190 topEquiv.finrank_eq, \u2190 h_spanE]\n convert finrank_span_le_card (R := K) (Set.range b)\n\nopen Module\n\nvariable {\u03b9 : Type*} [hs : IsZlattice K L] (b : Basis \u03b9 \u2124 L)\n/-- Any `\u2124`-basis of `L` is also a `K`-basis of `E`. -/\ndef Basis.ofZlatticeBasis :\n Basis \u03b9 K E := by\n have : Finite \u2124 L := Zlattice.module_finite K L\n have : Free \u2124 L := Zlattice.module_free K L\n let e := Basis.indexEquiv (Free.chooseBasis \u2124 L) b\n have : Fintype \u03b9 := Fintype.ofEquiv _ e\n refine basisOfTopLeSpanOfCardEqFinrank (L.subtype.toIntLinearMap \u2218 b) ?_ ?_\n \u00b7 rw [\u2190 span_span_of_tower \u2124, Set.range_comp, \u2190 map_span, Basis.span_eq, Submodule.map_top,\n top_le_iff, AddMonoidHom.coe_toIntLinearMap_range, AddSubgroup.subtype_range,\n AddSubgroup.coe_toIntSubmodule, hs.span_top]\n \u00b7 rw [\u2190 Fintype.card_congr e, \u2190 finrank_eq_card_chooseBasisIndex, Zlattice.rank K L]\n\n@[simp]\ntheorem Basis.ofZlatticeBasis_apply (i : \u03b9) :\n b.ofZlatticeBasis K L i = b i := by simp [Basis.ofZlatticeBasis]\n\n@[simp]\ntheorem Basis.ofZlatticeBasis_repr_apply (x : L) (i : \u03b9) :\n (b.ofZlatticeBasis K L).repr x i = b.repr x i := by\n suffices ((b.ofZlatticeBasis K L).repr.toLinearMap.restrictScalars \u2124) \u2218\u2097 L.subtype.toIntLinearMap\n = Finsupp.mapRange.linearMap (Algebra.linearMap \u2124 K) \u2218\u2097 b.repr.toLinearMap by\n exact DFunLike.congr_fun (LinearMap.congr_fun this x) i\n refine Basis.ext b fun i \u21a6 ?_\n simp_rw [LinearMap.coe_comp, Function.comp_apply, LinearMap.coe_restrictScalars,\n LinearEquiv.coe_coe, AddMonoidHom.coe_toIntLinearMap, AddSubgroup.coeSubtype,\n \u2190 b.ofZlatticeBasis_apply K, repr_self, Finsupp.mapRange.linearMap_apply,\n Finsupp.mapRange_single, Algebra.linearMap_apply, map_one]\n\ntheorem Basis.ofZlatticeBasis_span :\n (span \u2124 (Set.range (b.ofZlatticeBasis K))).toAddSubgroup = L := by\n calc (span \u2124 (Set.range (b.ofZlatticeBasis K))).toAddSubgroup\n _ = (span \u2124 (L.subtype.toIntLinearMap '' (Set.range b))).toAddSubgroup := by congr; ext; simp\n _ = (map L.subtype.toIntLinearMap (span \u2124 (Set.range b))).toAddSubgroup := by\n rw [Submodule.map_span]\n _ = L := by simp [b.span_eq]\n\ntheorem Zlattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n [FiniteDimensional \u211d E] {L : AddSubgroup E} [DiscreteTopology L] [IsZlattice \u211d L] [Finite \u03b9]\n (b : Basis \u03b9 \u2124 L) [MeasurableSpace E] [OpensMeasurableSpace E] (\u03bc : MeasureTheory.Measure E) :\n MeasureTheory.IsAddFundamentalDomain L (Zspan.fundamentalDomain (b.ofZlatticeBasis \u211d)) \u03bc := by\n convert Zspan.isAddFundamentalDomain (b.ofZlatticeBasis \u211d) \u03bc\n all_goals exact (b.ofZlatticeBasis_span \u211d).symm\n\n", "theoremStatement": "instance instCountable_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n [FiniteDimensional \u211d E] (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice \u211d L] :\n Countable L", "theoremName": null, "fileCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "theoremCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/Zlattice/Basic.lean", "positionMetadata": {"lineInFile": 628, "tokenPositionInFile": 30787, "theoremPositionInFile": 54}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rw [\u2190 (Module.Free.chooseBasis \u2124 L).ofZlatticeBasis_span \u211d]\n change Countable (span \u2124 (Set.range (Basis.ofZlatticeBasis \u211d L _)))\n infer_instance", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 151}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\nimport Mathlib.Algebra.Module.Zlattice.Basic\n\n/-!\n# Covolume of \u2124-lattices\n\nLet `E` be a finite dimensional real vector space with an inner product.\n\nLet `L` be a `\u2124`-lattice `L` defined as a discrete `AddSubgroup E` that spans `E` over `\u211d`.\n\n## Main definitions and results\n\n* `Zlattice.covolume`: the covolume of `L` defined as the volume of an arbitrary fundamental\ndomain of `L`.\n\n* `Zlattice.covolume_eq_measure_fundamentalDomain`: the covolume of `L` does not depend on the\nchoice of the fundamental domain of `L`.\n\n* `Zlattice.covolume_eq_det`: if `L` is a lattice in `\u211d^n`, then its covolume is the absolute\nvalue of the determinant of any `\u2124`-basis of `L`.\n\n-/\n\nnoncomputable section\n\nnamespace Zlattice\n\nopen Submodule MeasureTheory FiniteDimensional MeasureTheory Module\n\nsection General\n\nvariable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]\nvariable [ProperSpace E] [MeasurableSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice K L]\n\n/-- The covolume of a `\u2124`-lattice is the volume of some fundamental domain; see\n`Zlattice.covolume_eq_volume` for the proof that the volume does not depend on the choice of\nthe fundamental domain. -/\ndef covolume (\u03bc : Measure E := by volume_tac) : \u211d := (addCovolume L E \u03bc).toReal\n\nend General\n\nsection Real\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [FiniteDimensional \u211d E]\nvariable [MeasurableSpace E] [BorelSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice \u211d L]\nvariable (\u03bc : Measure E := by volume_tac) [Measure.IsAddHaarMeasure \u03bc]\n\n", "theoremStatement": "theorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalDomain L F \u03bc) :\n covolume L \u03bc = (\u03bc F).toReal", "theoremName": "covolume_eq_measure_fundamentalDomain", "fileCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "theoremCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/Zlattice/Covolume.lean", "positionMetadata": {"lineInFile": 55, "tokenPositionInFile": 1809, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "congr_arg ENNReal.toReal (h.covolume_eq_volume \u03bc)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 49}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\nimport Mathlib.Algebra.Module.Zlattice.Basic\n\n/-!\n# Covolume of \u2124-lattices\n\nLet `E` be a finite dimensional real vector space with an inner product.\n\nLet `L` be a `\u2124`-lattice `L` defined as a discrete `AddSubgroup E` that spans `E` over `\u211d`.\n\n## Main definitions and results\n\n* `Zlattice.covolume`: the covolume of `L` defined as the volume of an arbitrary fundamental\ndomain of `L`.\n\n* `Zlattice.covolume_eq_measure_fundamentalDomain`: the covolume of `L` does not depend on the\nchoice of the fundamental domain of `L`.\n\n* `Zlattice.covolume_eq_det`: if `L` is a lattice in `\u211d^n`, then its covolume is the absolute\nvalue of the determinant of any `\u2124`-basis of `L`.\n\n-/\n\nnoncomputable section\n\nnamespace Zlattice\n\nopen Submodule MeasureTheory FiniteDimensional MeasureTheory Module\n\nsection General\n\nvariable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]\nvariable [ProperSpace E] [MeasurableSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice K L]\n\n/-- The covolume of a `\u2124`-lattice is the volume of some fundamental domain; see\n`Zlattice.covolume_eq_volume` for the proof that the volume does not depend on the choice of\nthe fundamental domain. -/\ndef covolume (\u03bc : Measure E := by volume_tac) : \u211d := (addCovolume L E \u03bc).toReal\n\nend General\n\nsection Real\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [FiniteDimensional \u211d E]\nvariable [MeasurableSpace E] [BorelSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice \u211d L]\nvariable (\u03bc : Measure E := by volume_tac) [Measure.IsAddHaarMeasure \u03bc]\n\ntheorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalDomain L F \u03bc) :\n covolume L \u03bc = (\u03bc F).toReal := congr_arg ENNReal.toReal (h.covolume_eq_volume \u03bc)\n\n", "theoremStatement": "theorem covolume_ne_zero : covolume L \u03bc \u2260 0", "theoremName": "covolume_ne_zero", "fileCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "theoremCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/Zlattice/Covolume.lean", "positionMetadata": {"lineInFile": 58, "tokenPositionInFile": 1990, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rw [covolume_eq_measure_fundamentalDomain L \u03bc (isAddFundamentalDomain (Free.chooseBasis \u2124 L) \u03bc),\n ENNReal.toReal_ne_zero]\n refine \u27e8Zspan.measure_fundamentalDomain_ne_zero _, ne_of_lt ?_\u27e9\n exact Bornology.IsBounded.measure_lt_top (Zspan.fundamentalDomain_isBounded _)", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 276}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\nimport Mathlib.Algebra.Module.Zlattice.Basic\n\n/-!\n# Covolume of \u2124-lattices\n\nLet `E` be a finite dimensional real vector space with an inner product.\n\nLet `L` be a `\u2124`-lattice `L` defined as a discrete `AddSubgroup E` that spans `E` over `\u211d`.\n\n## Main definitions and results\n\n* `Zlattice.covolume`: the covolume of `L` defined as the volume of an arbitrary fundamental\ndomain of `L`.\n\n* `Zlattice.covolume_eq_measure_fundamentalDomain`: the covolume of `L` does not depend on the\nchoice of the fundamental domain of `L`.\n\n* `Zlattice.covolume_eq_det`: if `L` is a lattice in `\u211d^n`, then its covolume is the absolute\nvalue of the determinant of any `\u2124`-basis of `L`.\n\n-/\n\nnoncomputable section\n\nnamespace Zlattice\n\nopen Submodule MeasureTheory FiniteDimensional MeasureTheory Module\n\nsection General\n\nvariable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]\nvariable [ProperSpace E] [MeasurableSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice K L]\n\n/-- The covolume of a `\u2124`-lattice is the volume of some fundamental domain; see\n`Zlattice.covolume_eq_volume` for the proof that the volume does not depend on the choice of\nthe fundamental domain. -/\ndef covolume (\u03bc : Measure E := by volume_tac) : \u211d := (addCovolume L E \u03bc).toReal\n\nend General\n\nsection Real\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [FiniteDimensional \u211d E]\nvariable [MeasurableSpace E] [BorelSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice \u211d L]\nvariable (\u03bc : Measure E := by volume_tac) [Measure.IsAddHaarMeasure \u03bc]\n\ntheorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalDomain L F \u03bc) :\n covolume L \u03bc = (\u03bc F).toReal := congr_arg ENNReal.toReal (h.covolume_eq_volume \u03bc)\n\ntheorem covolume_ne_zero : covolume L \u03bc \u2260 0 := by\n rw [covolume_eq_measure_fundamentalDomain L \u03bc (isAddFundamentalDomain (Free.chooseBasis \u2124 L) \u03bc),\n ENNReal.toReal_ne_zero]\n refine \u27e8Zspan.measure_fundamentalDomain_ne_zero _, ne_of_lt ?_\u27e9\n exact Bornology.IsBounded.measure_lt_top (Zspan.fundamentalDomain_isBounded _)\n\n", "theoremStatement": "theorem covolume_pos : 0 < covolume L \u03bc", "theoremName": "covolume_pos", "fileCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "theoremCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/Zlattice/Covolume.lean", "positionMetadata": {"lineInFile": 64, "tokenPositionInFile": 2315, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "lt_of_le_of_ne ENNReal.toReal_nonneg (covolume_ne_zero L \u03bc).symm", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 64}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\nimport Mathlib.Algebra.Module.Zlattice.Basic\n\n/-!\n# Covolume of \u2124-lattices\n\nLet `E` be a finite dimensional real vector space with an inner product.\n\nLet `L` be a `\u2124`-lattice `L` defined as a discrete `AddSubgroup E` that spans `E` over `\u211d`.\n\n## Main definitions and results\n\n* `Zlattice.covolume`: the covolume of `L` defined as the volume of an arbitrary fundamental\ndomain of `L`.\n\n* `Zlattice.covolume_eq_measure_fundamentalDomain`: the covolume of `L` does not depend on the\nchoice of the fundamental domain of `L`.\n\n* `Zlattice.covolume_eq_det`: if `L` is a lattice in `\u211d^n`, then its covolume is the absolute\nvalue of the determinant of any `\u2124`-basis of `L`.\n\n-/\n\nnoncomputable section\n\nnamespace Zlattice\n\nopen Submodule MeasureTheory FiniteDimensional MeasureTheory Module\n\nsection General\n\nvariable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]\nvariable [ProperSpace E] [MeasurableSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice K L]\n\n/-- The covolume of a `\u2124`-lattice is the volume of some fundamental domain; see\n`Zlattice.covolume_eq_volume` for the proof that the volume does not depend on the choice of\nthe fundamental domain. -/\ndef covolume (\u03bc : Measure E := by volume_tac) : \u211d := (addCovolume L E \u03bc).toReal\n\nend General\n\nsection Real\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [FiniteDimensional \u211d E]\nvariable [MeasurableSpace E] [BorelSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice \u211d L]\nvariable (\u03bc : Measure E := by volume_tac) [Measure.IsAddHaarMeasure \u03bc]\n\ntheorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalDomain L F \u03bc) :\n covolume L \u03bc = (\u03bc F).toReal := congr_arg ENNReal.toReal (h.covolume_eq_volume \u03bc)\n\ntheorem covolume_ne_zero : covolume L \u03bc \u2260 0 := by\n rw [covolume_eq_measure_fundamentalDomain L \u03bc (isAddFundamentalDomain (Free.chooseBasis \u2124 L) \u03bc),\n ENNReal.toReal_ne_zero]\n refine \u27e8Zspan.measure_fundamentalDomain_ne_zero _, ne_of_lt ?_\u27e9\n exact Bornology.IsBounded.measure_lt_top (Zspan.fundamentalDomain_isBounded _)\n\ntheorem covolume_pos : 0 < covolume L \u03bc :=\n lt_of_le_of_ne ENNReal.toReal_nonneg (covolume_ne_zero L \u03bc).symm\n\n", "theoremStatement": "theorem covolume_eq_det_mul_measure {\u03b9 : Type*} [Fintype \u03b9] [DecidableEq \u03b9] (b : Basis \u03b9 \u2124 L)\n (b\u2080 : Basis \u03b9 \u211d E) :\n covolume L \u03bc = |b\u2080.det ((\u2191) \u2218 b)| * (\u03bc (Zspan.fundamentalDomain b\u2080)).toReal", "theoremName": "covolume_eq_det_mul_measure", "fileCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "theoremCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/Zlattice/Covolume.lean", "positionMetadata": {"lineInFile": 67, "tokenPositionInFile": 2426, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rw [covolume_eq_measure_fundamentalDomain L \u03bc (isAddFundamentalDomain b \u03bc),\n Zspan.measure_fundamentalDomain _ _ b\u2080, measure_congr\n (Zspan.fundamentalDomain_ae_parallelepiped b\u2080 \u03bc), ENNReal.toReal_mul, ENNReal.toReal_ofReal\n (by positivity)]\n congr\n ext\n exact b.ofZlatticeBasis_apply \u211d L _", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 307}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\nimport Mathlib.Algebra.Module.Zlattice.Basic\n\n/-!\n# Covolume of \u2124-lattices\n\nLet `E` be a finite dimensional real vector space with an inner product.\n\nLet `L` be a `\u2124`-lattice `L` defined as a discrete `AddSubgroup E` that spans `E` over `\u211d`.\n\n## Main definitions and results\n\n* `Zlattice.covolume`: the covolume of `L` defined as the volume of an arbitrary fundamental\ndomain of `L`.\n\n* `Zlattice.covolume_eq_measure_fundamentalDomain`: the covolume of `L` does not depend on the\nchoice of the fundamental domain of `L`.\n\n* `Zlattice.covolume_eq_det`: if `L` is a lattice in `\u211d^n`, then its covolume is the absolute\nvalue of the determinant of any `\u2124`-basis of `L`.\n\n-/\n\nnoncomputable section\n\nnamespace Zlattice\n\nopen Submodule MeasureTheory FiniteDimensional MeasureTheory Module\n\nsection General\n\nvariable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]\nvariable [ProperSpace E] [MeasurableSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice K L]\n\n/-- The covolume of a `\u2124`-lattice is the volume of some fundamental domain; see\n`Zlattice.covolume_eq_volume` for the proof that the volume does not depend on the choice of\nthe fundamental domain. -/\ndef covolume (\u03bc : Measure E := by volume_tac) : \u211d := (addCovolume L E \u03bc).toReal\n\nend General\n\nsection Real\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [FiniteDimensional \u211d E]\nvariable [MeasurableSpace E] [BorelSpace E]\nvariable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice \u211d L]\nvariable (\u03bc : Measure E := by volume_tac) [Measure.IsAddHaarMeasure \u03bc]\n\ntheorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalDomain L F \u03bc) :\n covolume L \u03bc = (\u03bc F).toReal := congr_arg ENNReal.toReal (h.covolume_eq_volume \u03bc)\n\ntheorem covolume_ne_zero : covolume L \u03bc \u2260 0 := by\n rw [covolume_eq_measure_fundamentalDomain L \u03bc (isAddFundamentalDomain (Free.chooseBasis \u2124 L) \u03bc),\n ENNReal.toReal_ne_zero]\n refine \u27e8Zspan.measure_fundamentalDomain_ne_zero _, ne_of_lt ?_\u27e9\n exact Bornology.IsBounded.measure_lt_top (Zspan.fundamentalDomain_isBounded _)\n\ntheorem covolume_pos : 0 < covolume L \u03bc :=\n lt_of_le_of_ne ENNReal.toReal_nonneg (covolume_ne_zero L \u03bc).symm\n\ntheorem covolume_eq_det_mul_measure {\u03b9 : Type*} [Fintype \u03b9] [DecidableEq \u03b9] (b : Basis \u03b9 \u2124 L)\n (b\u2080 : Basis \u03b9 \u211d E) :\n covolume L \u03bc = |b\u2080.det ((\u2191) \u2218 b)| * (\u03bc (Zspan.fundamentalDomain b\u2080)).toReal := by\n rw [covolume_eq_measure_fundamentalDomain L \u03bc (isAddFundamentalDomain b \u03bc),\n Zspan.measure_fundamentalDomain _ _ b\u2080, measure_congr\n (Zspan.fundamentalDomain_ae_parallelepiped b\u2080 \u03bc), ENNReal.toReal_mul, ENNReal.toReal_ofReal\n (by positivity)]\n congr\n ext\n exact b.ofZlatticeBasis_apply \u211d L _\n\n", "theoremStatement": "theorem covolume_eq_det {\u03b9 : Type*} [Fintype \u03b9] [DecidableEq \u03b9] (L : AddSubgroup (\u03b9 \u2192 \u211d))\n [DiscreteTopology L] [IsZlattice \u211d L] (b : Basis \u03b9 \u2124 L) :\n covolume L = |(Matrix.of ((\u2191) \u2218 b)).det|", "theoremName": "covolume_eq_det", "fileCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "theoremCreated": {"commit": "12c27420d4", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/Zlattice/Covolume.lean", "positionMetadata": {"lineInFile": 78, "tokenPositionInFile": 2937, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n rw [covolume_eq_measure_fundamentalDomain L volume (isAddFundamentalDomain b volume),\n Zspan.volume_fundamentalDomain, ENNReal.toReal_ofReal (by positivity)]\n congr\n ext1\n exact b.ofZlatticeBasis_apply \u211d L _", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 218}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.Localization.CalculusOfFractions\n\n/-!\n# Lemmas on fractions\n\nLet `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,\nwe introduce structures like `W.LeftFraction\u2082 X Y` which consists of two\nleft fractions with the \"same denominator\" which shall be important in\nthe construction of the preadditive structure on the localized category\nwhen `C` is preadditive and `W` has a left calculus of fractions.\n\nWhen `W` has a left calculus of fractions, we generalize the lemmas\n`RightFraction.exists_leftFraction` as `RightFraction\u2082.exists_leftFraction\u2082`,\n`Localization.exists_leftFraction` as `Localization.exists_leftFraction\u2082` and\n`Localization.exists_leftFraction\u2083`, and\n`LeftFraction.map_eq_iff` as `LeftFraction\u2082.map_eq_iff`.\n\n## Implementation note\n\nThe lemmas in this file are phrased with data that is bundled into structures like\n`LeftFraction\u2082` or `LeftFraction\u2083`. It could have been possible to phrase them\nwith \"unbundled data\". However, this would require introducing 4 or 5 variables instead\nof one. It is also very convenient to use dot notation.\nMany definitions have been made reducible so as to ease rewrites when this API is used.\n\n-/\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D] (L : C \u2964 D) (W : MorphismProperty C)\n [L.IsLocalization W]\n\nnamespace MorphismProperty\n\n/-- This structure contains the data of two left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2082 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of three left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2083 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the numerator of the third left fraction -/\n f'' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of two right fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure RightFraction\u2082 (X Y : C) where\n /-- the auxiliary object of right fractions -/\n {X' : C}\n /-- the denominator of the right fractions -/\n s : X' \u27f6 X\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n /-- the numerator of the first right fraction -/\n f : X' \u27f6 Y\n /-- the numerator of the second right fraction -/\n f' : X' \u27f6 Y\n\nvariable {W}\n\n/-- The equivalence relation on tuples of left fractions with the same denominator\nfor a morphism property `W`. The fact it is an equivalence relation is not\nformalized, but it would follow easily from `LeftFraction\u2082.map_eq_iff`. -/\ndef LeftFraction\u2082Rel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction\u2082 X Y) : Prop :=\n \u2203 (Z : C) (t\u2081 : z\u2081.Y' \u27f6 Z) (t\u2082 : z\u2082.Y' \u27f6 Z) (_ : z\u2081.s \u226b t\u2081 = z\u2082.s \u226b t\u2082)\n (_ : z\u2081.f \u226b t\u2081 = z\u2082.f \u226b t\u2082) (_ : z\u2081.f' \u226b t\u2081 = z\u2082.f' \u226b t\u2082), W (z\u2081.s \u226b t\u2081)\n\nnamespace LeftFraction\u2082\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2082 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The exchange of the two fractions. -/\n@[reducible]\ndef symm : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2082\n\nnamespace LeftFraction\u2083\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2083 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The third left fraction. -/\n@[reducible]\ndef thd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the first fraction. -/\n@[reducible]\ndef forgetFst : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the second fraction. -/\n@[reducible]\ndef forgetSnd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the third fraction. -/\n@[reducible]\ndef forgetThd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2083\n\nnamespace LeftFraction\u2082Rel\n\nvariable {X Y : C} {z\u2081 z\u2082 : W.LeftFraction\u2082 X Y} (h : LeftFraction\u2082Rel z\u2081 z\u2082)\n\n", "theoremStatement": "lemma fst : LeftFractionRel z\u2081.fst z\u2082.fst", "theoremName": "fst", "fileCreated": {"commit": "429161409e", "date": "2024-04-18"}, "theoremCreated": {"commit": "429161409e", "date": "2024-04-18"}, "file": "mathlib4/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean", "positionMetadata": {"lineInFile": 185, "tokenPositionInFile": 5121, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, _, ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 79}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.Localization.CalculusOfFractions\n\n/-!\n# Lemmas on fractions\n\nLet `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,\nwe introduce structures like `W.LeftFraction\u2082 X Y` which consists of two\nleft fractions with the \"same denominator\" which shall be important in\nthe construction of the preadditive structure on the localized category\nwhen `C` is preadditive and `W` has a left calculus of fractions.\n\nWhen `W` has a left calculus of fractions, we generalize the lemmas\n`RightFraction.exists_leftFraction` as `RightFraction\u2082.exists_leftFraction\u2082`,\n`Localization.exists_leftFraction` as `Localization.exists_leftFraction\u2082` and\n`Localization.exists_leftFraction\u2083`, and\n`LeftFraction.map_eq_iff` as `LeftFraction\u2082.map_eq_iff`.\n\n## Implementation note\n\nThe lemmas in this file are phrased with data that is bundled into structures like\n`LeftFraction\u2082` or `LeftFraction\u2083`. It could have been possible to phrase them\nwith \"unbundled data\". However, this would require introducing 4 or 5 variables instead\nof one. It is also very convenient to use dot notation.\nMany definitions have been made reducible so as to ease rewrites when this API is used.\n\n-/\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D] (L : C \u2964 D) (W : MorphismProperty C)\n [L.IsLocalization W]\n\nnamespace MorphismProperty\n\n/-- This structure contains the data of two left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2082 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of three left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2083 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the numerator of the third left fraction -/\n f'' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of two right fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure RightFraction\u2082 (X Y : C) where\n /-- the auxiliary object of right fractions -/\n {X' : C}\n /-- the denominator of the right fractions -/\n s : X' \u27f6 X\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n /-- the numerator of the first right fraction -/\n f : X' \u27f6 Y\n /-- the numerator of the second right fraction -/\n f' : X' \u27f6 Y\n\nvariable {W}\n\n/-- The equivalence relation on tuples of left fractions with the same denominator\nfor a morphism property `W`. The fact it is an equivalence relation is not\nformalized, but it would follow easily from `LeftFraction\u2082.map_eq_iff`. -/\ndef LeftFraction\u2082Rel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction\u2082 X Y) : Prop :=\n \u2203 (Z : C) (t\u2081 : z\u2081.Y' \u27f6 Z) (t\u2082 : z\u2082.Y' \u27f6 Z) (_ : z\u2081.s \u226b t\u2081 = z\u2082.s \u226b t\u2082)\n (_ : z\u2081.f \u226b t\u2081 = z\u2082.f \u226b t\u2082) (_ : z\u2081.f' \u226b t\u2081 = z\u2082.f' \u226b t\u2082), W (z\u2081.s \u226b t\u2081)\n\nnamespace LeftFraction\u2082\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2082 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The exchange of the two fractions. -/\n@[reducible]\ndef symm : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2082\n\nnamespace LeftFraction\u2083\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2083 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The third left fraction. -/\n@[reducible]\ndef thd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the first fraction. -/\n@[reducible]\ndef forgetFst : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the second fraction. -/\n@[reducible]\ndef forgetSnd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the third fraction. -/\n@[reducible]\ndef forgetThd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2083\n\nnamespace LeftFraction\u2082Rel\n\nvariable {X Y : C} {z\u2081 z\u2082 : W.LeftFraction\u2082 X Y} (h : LeftFraction\u2082Rel z\u2081 z\u2082)\n\nlemma fst : LeftFractionRel z\u2081.fst z\u2082.fst := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, _, ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9\n\n", "theoremStatement": "lemma snd : LeftFractionRel z\u2081.snd z\u2082.snd", "theoremName": "snd", "fileCreated": {"commit": "429161409e", "date": "2024-04-18"}, "theoremCreated": {"commit": "429161409e", "date": "2024-04-18"}, "file": "mathlib4/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean", "positionMetadata": {"lineInFile": 189, "tokenPositionInFile": 5247, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n obtain \u27e8Z, t\u2081, t\u2082, hst, _, hft', ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft', ht\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 81}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.Localization.CalculusOfFractions\n\n/-!\n# Lemmas on fractions\n\nLet `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,\nwe introduce structures like `W.LeftFraction\u2082 X Y` which consists of two\nleft fractions with the \"same denominator\" which shall be important in\nthe construction of the preadditive structure on the localized category\nwhen `C` is preadditive and `W` has a left calculus of fractions.\n\nWhen `W` has a left calculus of fractions, we generalize the lemmas\n`RightFraction.exists_leftFraction` as `RightFraction\u2082.exists_leftFraction\u2082`,\n`Localization.exists_leftFraction` as `Localization.exists_leftFraction\u2082` and\n`Localization.exists_leftFraction\u2083`, and\n`LeftFraction.map_eq_iff` as `LeftFraction\u2082.map_eq_iff`.\n\n## Implementation note\n\nThe lemmas in this file are phrased with data that is bundled into structures like\n`LeftFraction\u2082` or `LeftFraction\u2083`. It could have been possible to phrase them\nwith \"unbundled data\". However, this would require introducing 4 or 5 variables instead\nof one. It is also very convenient to use dot notation.\nMany definitions have been made reducible so as to ease rewrites when this API is used.\n\n-/\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D] (L : C \u2964 D) (W : MorphismProperty C)\n [L.IsLocalization W]\n\nnamespace MorphismProperty\n\n/-- This structure contains the data of two left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2082 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of three left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2083 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the numerator of the third left fraction -/\n f'' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of two right fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure RightFraction\u2082 (X Y : C) where\n /-- the auxiliary object of right fractions -/\n {X' : C}\n /-- the denominator of the right fractions -/\n s : X' \u27f6 X\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n /-- the numerator of the first right fraction -/\n f : X' \u27f6 Y\n /-- the numerator of the second right fraction -/\n f' : X' \u27f6 Y\n\nvariable {W}\n\n/-- The equivalence relation on tuples of left fractions with the same denominator\nfor a morphism property `W`. The fact it is an equivalence relation is not\nformalized, but it would follow easily from `LeftFraction\u2082.map_eq_iff`. -/\ndef LeftFraction\u2082Rel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction\u2082 X Y) : Prop :=\n \u2203 (Z : C) (t\u2081 : z\u2081.Y' \u27f6 Z) (t\u2082 : z\u2082.Y' \u27f6 Z) (_ : z\u2081.s \u226b t\u2081 = z\u2082.s \u226b t\u2082)\n (_ : z\u2081.f \u226b t\u2081 = z\u2082.f \u226b t\u2082) (_ : z\u2081.f' \u226b t\u2081 = z\u2082.f' \u226b t\u2082), W (z\u2081.s \u226b t\u2081)\n\nnamespace LeftFraction\u2082\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2082 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The exchange of the two fractions. -/\n@[reducible]\ndef symm : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2082\n\nnamespace LeftFraction\u2083\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2083 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The third left fraction. -/\n@[reducible]\ndef thd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the first fraction. -/\n@[reducible]\ndef forgetFst : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the second fraction. -/\n@[reducible]\ndef forgetSnd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the third fraction. -/\n@[reducible]\ndef forgetThd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2083\n\nnamespace LeftFraction\u2082Rel\n\nvariable {X Y : C} {z\u2081 z\u2082 : W.LeftFraction\u2082 X Y} (h : LeftFraction\u2082Rel z\u2081 z\u2082)\n\nlemma fst : LeftFractionRel z\u2081.fst z\u2082.fst := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, _, ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9\n\nlemma snd : LeftFractionRel z\u2081.snd z\u2082.snd := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, _, hft', ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft', ht\u27e9\n\nend LeftFraction\u2082Rel\n\nnamespace LeftFraction\u2082\n\nvariable (W)\nvariable [W.HasLeftCalculusOfFractions]\n\n", "theoremStatement": "lemma map_eq_iff {X Y : C} (\u03c6 \u03c8 : W.LeftFraction\u2082 X Y) :\n (\u03c6.fst.map L (Localization.inverts _ _) = \u03c8.fst.map L (Localization.inverts _ _) \u2227\n \u03c6.snd.map L (Localization.inverts _ _) = \u03c8.snd.map L (Localization.inverts _ _)) \u2194\n LeftFraction\u2082Rel \u03c6 \u03c8", "theoremName": "map_eq_iff", "fileCreated": {"commit": "429161409e", "date": "2024-04-18"}, "theoremCreated": {"commit": "429161409e", "date": "2024-04-18"}, "file": "mathlib4/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean", "positionMetadata": {"lineInFile": 200, "tokenPositionInFile": 5476, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n simp only [LeftFraction.map_eq_iff L W]\n constructor\n \u00b7 intro \u27e8h, h'\u27e9\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9 := h\n obtain \u27e8Z', t\u2081', t\u2082', hst', hft', ht'\u27e9 := h'\n dsimp at t\u2081 t\u2082 t\u2081' t\u2082' hst hft hst' hft' ht ht'\n have \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ ht (\u03c6.s \u226b t\u2081')).exists_leftFraction\n simp only [Category.assoc] at h\u03b1\n obtain \u27e8Z'', u, hu, fac\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c6.hs h\u03b1\n have h\u03b1' : \u03c8.s \u226b t\u2082 \u226b \u03b1.f \u226b u = \u03c8.s \u226b t\u2082' \u226b \u03b1.s \u226b u := by\n rw [\u2190 reassoc_of% hst, \u2190 reassoc_of% h\u03b1, \u2190 reassoc_of% hst']\n obtain \u27e8Z''', u', hu', fac'\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c8.hs h\u03b1'\n simp only [Category.assoc] at fac fac'\n refine' \u27e8Z''', t\u2081' \u226b \u03b1.s \u226b u \u226b u', t\u2082' \u226b \u03b1.s \u226b u \u226b u', _, _, _, _\u27e9\n \u00b7 rw [reassoc_of% hst']\n \u00b7 rw [reassoc_of% fac, reassoc_of% hft, fac']\n \u00b7 rw [reassoc_of% hft']\n \u00b7 rw [\u2190 Category.assoc]\n exact W.comp_mem _ _ ht' (W.comp_mem _ _ \u03b1.hs (W.comp_mem _ _ hu hu'))\n \u00b7 intro h\n exact \u27e8h.fst, h.snd\u27e9", "proofType": "tactic", "proofLengthLines": 22, "proofLengthTokens": 982}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.Localization.CalculusOfFractions\n\n/-!\n# Lemmas on fractions\n\nLet `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,\nwe introduce structures like `W.LeftFraction\u2082 X Y` which consists of two\nleft fractions with the \"same denominator\" which shall be important in\nthe construction of the preadditive structure on the localized category\nwhen `C` is preadditive and `W` has a left calculus of fractions.\n\nWhen `W` has a left calculus of fractions, we generalize the lemmas\n`RightFraction.exists_leftFraction` as `RightFraction\u2082.exists_leftFraction\u2082`,\n`Localization.exists_leftFraction` as `Localization.exists_leftFraction\u2082` and\n`Localization.exists_leftFraction\u2083`, and\n`LeftFraction.map_eq_iff` as `LeftFraction\u2082.map_eq_iff`.\n\n## Implementation note\n\nThe lemmas in this file are phrased with data that is bundled into structures like\n`LeftFraction\u2082` or `LeftFraction\u2083`. It could have been possible to phrase them\nwith \"unbundled data\". However, this would require introducing 4 or 5 variables instead\nof one. It is also very convenient to use dot notation.\nMany definitions have been made reducible so as to ease rewrites when this API is used.\n\n-/\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D] (L : C \u2964 D) (W : MorphismProperty C)\n [L.IsLocalization W]\n\nnamespace MorphismProperty\n\n/-- This structure contains the data of two left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2082 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of three left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2083 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the numerator of the third left fraction -/\n f'' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of two right fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure RightFraction\u2082 (X Y : C) where\n /-- the auxiliary object of right fractions -/\n {X' : C}\n /-- the denominator of the right fractions -/\n s : X' \u27f6 X\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n /-- the numerator of the first right fraction -/\n f : X' \u27f6 Y\n /-- the numerator of the second right fraction -/\n f' : X' \u27f6 Y\n\nvariable {W}\n\n/-- The equivalence relation on tuples of left fractions with the same denominator\nfor a morphism property `W`. The fact it is an equivalence relation is not\nformalized, but it would follow easily from `LeftFraction\u2082.map_eq_iff`. -/\ndef LeftFraction\u2082Rel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction\u2082 X Y) : Prop :=\n \u2203 (Z : C) (t\u2081 : z\u2081.Y' \u27f6 Z) (t\u2082 : z\u2082.Y' \u27f6 Z) (_ : z\u2081.s \u226b t\u2081 = z\u2082.s \u226b t\u2082)\n (_ : z\u2081.f \u226b t\u2081 = z\u2082.f \u226b t\u2082) (_ : z\u2081.f' \u226b t\u2081 = z\u2082.f' \u226b t\u2082), W (z\u2081.s \u226b t\u2081)\n\nnamespace LeftFraction\u2082\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2082 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The exchange of the two fractions. -/\n@[reducible]\ndef symm : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2082\n\nnamespace LeftFraction\u2083\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2083 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The third left fraction. -/\n@[reducible]\ndef thd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the first fraction. -/\n@[reducible]\ndef forgetFst : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the second fraction. -/\n@[reducible]\ndef forgetSnd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the third fraction. -/\n@[reducible]\ndef forgetThd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2083\n\nnamespace LeftFraction\u2082Rel\n\nvariable {X Y : C} {z\u2081 z\u2082 : W.LeftFraction\u2082 X Y} (h : LeftFraction\u2082Rel z\u2081 z\u2082)\n\nlemma fst : LeftFractionRel z\u2081.fst z\u2082.fst := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, _, ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9\n\nlemma snd : LeftFractionRel z\u2081.snd z\u2082.snd := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, _, hft', ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft', ht\u27e9\n\nend LeftFraction\u2082Rel\n\nnamespace LeftFraction\u2082\n\nvariable (W)\nvariable [W.HasLeftCalculusOfFractions]\n\nlemma map_eq_iff {X Y : C} (\u03c6 \u03c8 : W.LeftFraction\u2082 X Y) :\n (\u03c6.fst.map L (Localization.inverts _ _) = \u03c8.fst.map L (Localization.inverts _ _) \u2227\n \u03c6.snd.map L (Localization.inverts _ _) = \u03c8.snd.map L (Localization.inverts _ _)) \u2194\n LeftFraction\u2082Rel \u03c6 \u03c8 := by\n simp only [LeftFraction.map_eq_iff L W]\n constructor\n \u00b7 intro \u27e8h, h'\u27e9\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9 := h\n obtain \u27e8Z', t\u2081', t\u2082', hst', hft', ht'\u27e9 := h'\n dsimp at t\u2081 t\u2082 t\u2081' t\u2082' hst hft hst' hft' ht ht'\n have \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ ht (\u03c6.s \u226b t\u2081')).exists_leftFraction\n simp only [Category.assoc] at h\u03b1\n obtain \u27e8Z'', u, hu, fac\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c6.hs h\u03b1\n have h\u03b1' : \u03c8.s \u226b t\u2082 \u226b \u03b1.f \u226b u = \u03c8.s \u226b t\u2082' \u226b \u03b1.s \u226b u := by\n rw [\u2190 reassoc_of% hst, \u2190 reassoc_of% h\u03b1, \u2190 reassoc_of% hst']\n obtain \u27e8Z''', u', hu', fac'\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c8.hs h\u03b1'\n simp only [Category.assoc] at fac fac'\n refine' \u27e8Z''', t\u2081' \u226b \u03b1.s \u226b u \u226b u', t\u2082' \u226b \u03b1.s \u226b u \u226b u', _, _, _, _\u27e9\n \u00b7 rw [reassoc_of% hst']\n \u00b7 rw [reassoc_of% fac, reassoc_of% hft, fac']\n \u00b7 rw [reassoc_of% hft']\n \u00b7 rw [\u2190 Category.assoc]\n exact W.comp_mem _ _ ht' (W.comp_mem _ _ \u03b1.hs (W.comp_mem _ _ hu hu'))\n \u00b7 intro h\n exact \u27e8h.fst, h.snd\u27e9\n\nend LeftFraction\u2082\n\nnamespace RightFraction\u2082\n\nvariable {X Y : C}\nvariable (\u03c6 : W.RightFraction\u2082 X Y)\n\n/-- The first right fraction. -/\n@[reducible]\ndef fst : W.RightFraction X Y where\n X' := \u03c6.X'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second right fraction. -/\n@[reducible]\ndef snd : W.RightFraction X Y where\n X' := \u03c6.X'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n", "theoremStatement": "lemma exists_leftFraction\u2082 [W.HasLeftCalculusOfFractions] :\n \u2203 (\u03c8 : W.LeftFraction\u2082 X Y), \u03c6.f \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f \u2227\n \u03c6.f' \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f'", "theoremName": "exists_leftFraction\u2082", "fileCreated": {"commit": "429161409e", "date": "2024-04-18"}, "theoremCreated": {"commit": "429161409e", "date": "2024-04-18"}, "file": "mathlib4/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean", "positionMetadata": {"lineInFile": 249, "tokenPositionInFile": 7086, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n obtain \u27e8\u03c8\u2081, h\u03c8\u2081\u27e9 := \u03c6.fst.exists_leftFraction\n obtain \u27e8\u03c8\u2082, h\u03c8\u2082\u27e9 := \u03c6.snd.exists_leftFraction\n obtain \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ \u03c8\u2081.hs \u03c8\u2082.s).exists_leftFraction\n dsimp at h\u03c8\u2081 h\u03c8\u2082 h\u03b1\n refine' \u27e8LeftFraction\u2082.mk (\u03c8\u2081.f \u226b \u03b1.f) (\u03c8\u2082.f \u226b \u03b1.s) (\u03c8\u2082.s \u226b \u03b1.s)\n (W.comp_mem _ _ \u03c8\u2082.hs \u03b1.hs), _, _\u27e9\n \u00b7 dsimp\n rw [h\u03b1, reassoc_of% h\u03c8\u2081]\n \u00b7 rw [reassoc_of% h\u03c8\u2082]", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 364}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.Localization.CalculusOfFractions\n\n/-!\n# Lemmas on fractions\n\nLet `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,\nwe introduce structures like `W.LeftFraction\u2082 X Y` which consists of two\nleft fractions with the \"same denominator\" which shall be important in\nthe construction of the preadditive structure on the localized category\nwhen `C` is preadditive and `W` has a left calculus of fractions.\n\nWhen `W` has a left calculus of fractions, we generalize the lemmas\n`RightFraction.exists_leftFraction` as `RightFraction\u2082.exists_leftFraction\u2082`,\n`Localization.exists_leftFraction` as `Localization.exists_leftFraction\u2082` and\n`Localization.exists_leftFraction\u2083`, and\n`LeftFraction.map_eq_iff` as `LeftFraction\u2082.map_eq_iff`.\n\n## Implementation note\n\nThe lemmas in this file are phrased with data that is bundled into structures like\n`LeftFraction\u2082` or `LeftFraction\u2083`. It could have been possible to phrase them\nwith \"unbundled data\". However, this would require introducing 4 or 5 variables instead\nof one. It is also very convenient to use dot notation.\nMany definitions have been made reducible so as to ease rewrites when this API is used.\n\n-/\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D] (L : C \u2964 D) (W : MorphismProperty C)\n [L.IsLocalization W]\n\nnamespace MorphismProperty\n\n/-- This structure contains the data of two left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2082 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of three left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2083 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the numerator of the third left fraction -/\n f'' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of two right fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure RightFraction\u2082 (X Y : C) where\n /-- the auxiliary object of right fractions -/\n {X' : C}\n /-- the denominator of the right fractions -/\n s : X' \u27f6 X\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n /-- the numerator of the first right fraction -/\n f : X' \u27f6 Y\n /-- the numerator of the second right fraction -/\n f' : X' \u27f6 Y\n\nvariable {W}\n\n/-- The equivalence relation on tuples of left fractions with the same denominator\nfor a morphism property `W`. The fact it is an equivalence relation is not\nformalized, but it would follow easily from `LeftFraction\u2082.map_eq_iff`. -/\ndef LeftFraction\u2082Rel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction\u2082 X Y) : Prop :=\n \u2203 (Z : C) (t\u2081 : z\u2081.Y' \u27f6 Z) (t\u2082 : z\u2082.Y' \u27f6 Z) (_ : z\u2081.s \u226b t\u2081 = z\u2082.s \u226b t\u2082)\n (_ : z\u2081.f \u226b t\u2081 = z\u2082.f \u226b t\u2082) (_ : z\u2081.f' \u226b t\u2081 = z\u2082.f' \u226b t\u2082), W (z\u2081.s \u226b t\u2081)\n\nnamespace LeftFraction\u2082\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2082 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The exchange of the two fractions. -/\n@[reducible]\ndef symm : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2082\n\nnamespace LeftFraction\u2083\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2083 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The third left fraction. -/\n@[reducible]\ndef thd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the first fraction. -/\n@[reducible]\ndef forgetFst : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the second fraction. -/\n@[reducible]\ndef forgetSnd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the third fraction. -/\n@[reducible]\ndef forgetThd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2083\n\nnamespace LeftFraction\u2082Rel\n\nvariable {X Y : C} {z\u2081 z\u2082 : W.LeftFraction\u2082 X Y} (h : LeftFraction\u2082Rel z\u2081 z\u2082)\n\nlemma fst : LeftFractionRel z\u2081.fst z\u2082.fst := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, _, ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9\n\nlemma snd : LeftFractionRel z\u2081.snd z\u2082.snd := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, _, hft', ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft', ht\u27e9\n\nend LeftFraction\u2082Rel\n\nnamespace LeftFraction\u2082\n\nvariable (W)\nvariable [W.HasLeftCalculusOfFractions]\n\nlemma map_eq_iff {X Y : C} (\u03c6 \u03c8 : W.LeftFraction\u2082 X Y) :\n (\u03c6.fst.map L (Localization.inverts _ _) = \u03c8.fst.map L (Localization.inverts _ _) \u2227\n \u03c6.snd.map L (Localization.inverts _ _) = \u03c8.snd.map L (Localization.inverts _ _)) \u2194\n LeftFraction\u2082Rel \u03c6 \u03c8 := by\n simp only [LeftFraction.map_eq_iff L W]\n constructor\n \u00b7 intro \u27e8h, h'\u27e9\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9 := h\n obtain \u27e8Z', t\u2081', t\u2082', hst', hft', ht'\u27e9 := h'\n dsimp at t\u2081 t\u2082 t\u2081' t\u2082' hst hft hst' hft' ht ht'\n have \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ ht (\u03c6.s \u226b t\u2081')).exists_leftFraction\n simp only [Category.assoc] at h\u03b1\n obtain \u27e8Z'', u, hu, fac\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c6.hs h\u03b1\n have h\u03b1' : \u03c8.s \u226b t\u2082 \u226b \u03b1.f \u226b u = \u03c8.s \u226b t\u2082' \u226b \u03b1.s \u226b u := by\n rw [\u2190 reassoc_of% hst, \u2190 reassoc_of% h\u03b1, \u2190 reassoc_of% hst']\n obtain \u27e8Z''', u', hu', fac'\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c8.hs h\u03b1'\n simp only [Category.assoc] at fac fac'\n refine' \u27e8Z''', t\u2081' \u226b \u03b1.s \u226b u \u226b u', t\u2082' \u226b \u03b1.s \u226b u \u226b u', _, _, _, _\u27e9\n \u00b7 rw [reassoc_of% hst']\n \u00b7 rw [reassoc_of% fac, reassoc_of% hft, fac']\n \u00b7 rw [reassoc_of% hft']\n \u00b7 rw [\u2190 Category.assoc]\n exact W.comp_mem _ _ ht' (W.comp_mem _ _ \u03b1.hs (W.comp_mem _ _ hu hu'))\n \u00b7 intro h\n exact \u27e8h.fst, h.snd\u27e9\n\nend LeftFraction\u2082\n\nnamespace RightFraction\u2082\n\nvariable {X Y : C}\nvariable (\u03c6 : W.RightFraction\u2082 X Y)\n\n/-- The first right fraction. -/\n@[reducible]\ndef fst : W.RightFraction X Y where\n X' := \u03c6.X'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second right fraction. -/\n@[reducible]\ndef snd : W.RightFraction X Y where\n X' := \u03c6.X'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nlemma exists_leftFraction\u2082 [W.HasLeftCalculusOfFractions] :\n \u2203 (\u03c8 : W.LeftFraction\u2082 X Y), \u03c6.f \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f \u2227\n \u03c6.f' \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f' := by\n obtain \u27e8\u03c8\u2081, h\u03c8\u2081\u27e9 := \u03c6.fst.exists_leftFraction\n obtain \u27e8\u03c8\u2082, h\u03c8\u2082\u27e9 := \u03c6.snd.exists_leftFraction\n obtain \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ \u03c8\u2081.hs \u03c8\u2082.s).exists_leftFraction\n dsimp at h\u03c8\u2081 h\u03c8\u2082 h\u03b1\n refine' \u27e8LeftFraction\u2082.mk (\u03c8\u2081.f \u226b \u03b1.f) (\u03c8\u2082.f \u226b \u03b1.s) (\u03c8\u2082.s \u226b \u03b1.s)\n (W.comp_mem _ _ \u03c8\u2082.hs \u03b1.hs), _, _\u27e9\n \u00b7 dsimp\n rw [h\u03b1, reassoc_of% h\u03c8\u2081]\n \u00b7 rw [reassoc_of% h\u03c8\u2082]\n\nend RightFraction\u2082\n\nend MorphismProperty\n\nnamespace Localization\n\nvariable [W.HasLeftCalculusOfFractions]\n\nopen MorphismProperty\n\n", "theoremStatement": "lemma exists_leftFraction\u2082 {X Y : C} (f f' : L.obj X \u27f6 L.obj Y) :\n \u2203 (\u03c6 : W.LeftFraction\u2082 X Y), f = \u03c6.fst.map L (inverts L W) \u2227\n f' = \u03c6.snd.map L (inverts L W)", "theoremName": "exists_leftFraction\u2082", "fileCreated": {"commit": "429161409e", "date": "2024-04-18"}, "theoremCreated": {"commit": "429161409e", "date": "2024-04-18"}, "file": "mathlib4/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean", "positionMetadata": {"lineInFile": 272, "tokenPositionInFile": 7732, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n have \u27e8\u03c6, h\u03c6\u27e9 := exists_leftFraction L W f\n have \u27e8\u03c6', h\u03c6'\u27e9 := exists_leftFraction L W f'\n obtain \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ \u03c6.hs \u03c6'.s).exists_leftFraction\n let \u03c8 : W.LeftFraction\u2082 X Y :=\n { Y' := \u03b1.Y'\n f := \u03c6.f \u226b \u03b1.f\n f' := \u03c6'.f \u226b \u03b1.s\n s := \u03c6'.s \u226b \u03b1.s\n hs := W.comp_mem _ _ \u03c6'.hs \u03b1.hs }\n have := inverts L W _ \u03c6'.hs\n have := inverts L W _ \u03b1.hs\n have : IsIso (L.map (\u03c6'.s \u226b \u03b1.s)) := by\n rw [L.map_comp]\n infer_instance\n refine' \u27e8\u03c8, _, _\u27e9\n \u00b7 rw [\u2190 cancel_mono (L.map (\u03c6'.s \u226b \u03b1.s)), LeftFraction.map_comp_map_s,\n h\u03b1, L.map_comp, h\u03c6, LeftFraction.map_comp_map_s_assoc,\n L.map_comp]\n \u00b7 rw [\u2190 cancel_mono (L.map (\u03c6'.s \u226b \u03b1.s)), h\u03c6']\n nth_rw 1 [L.map_comp]\n rw [LeftFraction.map_comp_map_s_assoc, LeftFraction.map_comp_map_s,\n L.map_comp]", "proofType": "tactic", "proofLengthLines": 23, "proofLengthTokens": 797}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.Localization.CalculusOfFractions\n\n/-!\n# Lemmas on fractions\n\nLet `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,\nwe introduce structures like `W.LeftFraction\u2082 X Y` which consists of two\nleft fractions with the \"same denominator\" which shall be important in\nthe construction of the preadditive structure on the localized category\nwhen `C` is preadditive and `W` has a left calculus of fractions.\n\nWhen `W` has a left calculus of fractions, we generalize the lemmas\n`RightFraction.exists_leftFraction` as `RightFraction\u2082.exists_leftFraction\u2082`,\n`Localization.exists_leftFraction` as `Localization.exists_leftFraction\u2082` and\n`Localization.exists_leftFraction\u2083`, and\n`LeftFraction.map_eq_iff` as `LeftFraction\u2082.map_eq_iff`.\n\n## Implementation note\n\nThe lemmas in this file are phrased with data that is bundled into structures like\n`LeftFraction\u2082` or `LeftFraction\u2083`. It could have been possible to phrase them\nwith \"unbundled data\". However, this would require introducing 4 or 5 variables instead\nof one. It is also very convenient to use dot notation.\nMany definitions have been made reducible so as to ease rewrites when this API is used.\n\n-/\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D] (L : C \u2964 D) (W : MorphismProperty C)\n [L.IsLocalization W]\n\nnamespace MorphismProperty\n\n/-- This structure contains the data of two left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2082 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of three left fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure LeftFraction\u2083 (X Y : C) where\n /-- the auxiliary object of left fractions -/\n {Y' : C}\n /-- the numerator of the first left fraction -/\n f : X \u27f6 Y'\n /-- the numerator of the second left fraction -/\n f' : X \u27f6 Y'\n /-- the numerator of the third left fraction -/\n f'' : X \u27f6 Y'\n /-- the denominator of the left fractions -/\n s : Y \u27f6 Y'\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n\n/-- This structure contains the data of two right fractions for\n`W : MorphismProperty C` that have the same \"denominator\". -/\nstructure RightFraction\u2082 (X Y : C) where\n /-- the auxiliary object of right fractions -/\n {X' : C}\n /-- the denominator of the right fractions -/\n s : X' \u27f6 X\n /-- the condition that the denominator belongs to the given morphism property -/\n hs : W s\n /-- the numerator of the first right fraction -/\n f : X' \u27f6 Y\n /-- the numerator of the second right fraction -/\n f' : X' \u27f6 Y\n\nvariable {W}\n\n/-- The equivalence relation on tuples of left fractions with the same denominator\nfor a morphism property `W`. The fact it is an equivalence relation is not\nformalized, but it would follow easily from `LeftFraction\u2082.map_eq_iff`. -/\ndef LeftFraction\u2082Rel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction\u2082 X Y) : Prop :=\n \u2203 (Z : C) (t\u2081 : z\u2081.Y' \u27f6 Z) (t\u2082 : z\u2082.Y' \u27f6 Z) (_ : z\u2081.s \u226b t\u2081 = z\u2082.s \u226b t\u2082)\n (_ : z\u2081.f \u226b t\u2081 = z\u2082.f \u226b t\u2082) (_ : z\u2081.f' \u226b t\u2081 = z\u2082.f' \u226b t\u2082), W (z\u2081.s \u226b t\u2081)\n\nnamespace LeftFraction\u2082\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2082 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The exchange of the two fractions. -/\n@[reducible]\ndef symm : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2082\n\nnamespace LeftFraction\u2083\n\nvariable {X Y : C} (\u03c6 : W.LeftFraction\u2083 X Y)\n\n/-- The first left fraction. -/\n@[reducible]\ndef fst : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second left fraction. -/\n@[reducible]\ndef snd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The third left fraction. -/\n@[reducible]\ndef thd : W.LeftFraction X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the first fraction. -/\n@[reducible]\ndef forgetFst : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f'\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the second fraction. -/\n@[reducible]\ndef forgetSnd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f''\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- Forgets the third fraction. -/\n@[reducible]\ndef forgetThd : W.LeftFraction\u2082 X Y where\n Y' := \u03c6.Y'\n f := \u03c6.f\n f' := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nend LeftFraction\u2083\n\nnamespace LeftFraction\u2082Rel\n\nvariable {X Y : C} {z\u2081 z\u2082 : W.LeftFraction\u2082 X Y} (h : LeftFraction\u2082Rel z\u2081 z\u2082)\n\nlemma fst : LeftFractionRel z\u2081.fst z\u2082.fst := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, _, ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9\n\nlemma snd : LeftFractionRel z\u2081.snd z\u2082.snd := by\n obtain \u27e8Z, t\u2081, t\u2082, hst, _, hft', ht\u27e9 := h\n exact \u27e8Z, t\u2081, t\u2082, hst, hft', ht\u27e9\n\nend LeftFraction\u2082Rel\n\nnamespace LeftFraction\u2082\n\nvariable (W)\nvariable [W.HasLeftCalculusOfFractions]\n\nlemma map_eq_iff {X Y : C} (\u03c6 \u03c8 : W.LeftFraction\u2082 X Y) :\n (\u03c6.fst.map L (Localization.inverts _ _) = \u03c8.fst.map L (Localization.inverts _ _) \u2227\n \u03c6.snd.map L (Localization.inverts _ _) = \u03c8.snd.map L (Localization.inverts _ _)) \u2194\n LeftFraction\u2082Rel \u03c6 \u03c8 := by\n simp only [LeftFraction.map_eq_iff L W]\n constructor\n \u00b7 intro \u27e8h, h'\u27e9\n obtain \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9 := h\n obtain \u27e8Z', t\u2081', t\u2082', hst', hft', ht'\u27e9 := h'\n dsimp at t\u2081 t\u2082 t\u2081' t\u2082' hst hft hst' hft' ht ht'\n have \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ ht (\u03c6.s \u226b t\u2081')).exists_leftFraction\n simp only [Category.assoc] at h\u03b1\n obtain \u27e8Z'', u, hu, fac\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c6.hs h\u03b1\n have h\u03b1' : \u03c8.s \u226b t\u2082 \u226b \u03b1.f \u226b u = \u03c8.s \u226b t\u2082' \u226b \u03b1.s \u226b u := by\n rw [\u2190 reassoc_of% hst, \u2190 reassoc_of% h\u03b1, \u2190 reassoc_of% hst']\n obtain \u27e8Z''', u', hu', fac'\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ \u03c8.hs h\u03b1'\n simp only [Category.assoc] at fac fac'\n refine' \u27e8Z''', t\u2081' \u226b \u03b1.s \u226b u \u226b u', t\u2082' \u226b \u03b1.s \u226b u \u226b u', _, _, _, _\u27e9\n \u00b7 rw [reassoc_of% hst']\n \u00b7 rw [reassoc_of% fac, reassoc_of% hft, fac']\n \u00b7 rw [reassoc_of% hft']\n \u00b7 rw [\u2190 Category.assoc]\n exact W.comp_mem _ _ ht' (W.comp_mem _ _ \u03b1.hs (W.comp_mem _ _ hu hu'))\n \u00b7 intro h\n exact \u27e8h.fst, h.snd\u27e9\n\nend LeftFraction\u2082\n\nnamespace RightFraction\u2082\n\nvariable {X Y : C}\nvariable (\u03c6 : W.RightFraction\u2082 X Y)\n\n/-- The first right fraction. -/\n@[reducible]\ndef fst : W.RightFraction X Y where\n X' := \u03c6.X'\n f := \u03c6.f\n s := \u03c6.s\n hs := \u03c6.hs\n\n/-- The second right fraction. -/\n@[reducible]\ndef snd : W.RightFraction X Y where\n X' := \u03c6.X'\n f := \u03c6.f'\n s := \u03c6.s\n hs := \u03c6.hs\n\nlemma exists_leftFraction\u2082 [W.HasLeftCalculusOfFractions] :\n \u2203 (\u03c8 : W.LeftFraction\u2082 X Y), \u03c6.f \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f \u2227\n \u03c6.f' \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f' := by\n obtain \u27e8\u03c8\u2081, h\u03c8\u2081\u27e9 := \u03c6.fst.exists_leftFraction\n obtain \u27e8\u03c8\u2082, h\u03c8\u2082\u27e9 := \u03c6.snd.exists_leftFraction\n obtain \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ \u03c8\u2081.hs \u03c8\u2082.s).exists_leftFraction\n dsimp at h\u03c8\u2081 h\u03c8\u2082 h\u03b1\n refine' \u27e8LeftFraction\u2082.mk (\u03c8\u2081.f \u226b \u03b1.f) (\u03c8\u2082.f \u226b \u03b1.s) (\u03c8\u2082.s \u226b \u03b1.s)\n (W.comp_mem _ _ \u03c8\u2082.hs \u03b1.hs), _, _\u27e9\n \u00b7 dsimp\n rw [h\u03b1, reassoc_of% h\u03c8\u2081]\n \u00b7 rw [reassoc_of% h\u03c8\u2082]\n\nend RightFraction\u2082\n\nend MorphismProperty\n\nnamespace Localization\n\nvariable [W.HasLeftCalculusOfFractions]\n\nopen MorphismProperty\n\nlemma exists_leftFraction\u2082 {X Y : C} (f f' : L.obj X \u27f6 L.obj Y) :\n \u2203 (\u03c6 : W.LeftFraction\u2082 X Y), f = \u03c6.fst.map L (inverts L W) \u2227\n f' = \u03c6.snd.map L (inverts L W) := by\n have \u27e8\u03c6, h\u03c6\u27e9 := exists_leftFraction L W f\n have \u27e8\u03c6', h\u03c6'\u27e9 := exists_leftFraction L W f'\n obtain \u27e8\u03b1, h\u03b1\u27e9 := (RightFraction.mk _ \u03c6.hs \u03c6'.s).exists_leftFraction\n let \u03c8 : W.LeftFraction\u2082 X Y :=\n { Y' := \u03b1.Y'\n f := \u03c6.f \u226b \u03b1.f\n f' := \u03c6'.f \u226b \u03b1.s\n s := \u03c6'.s \u226b \u03b1.s\n hs := W.comp_mem _ _ \u03c6'.hs \u03b1.hs }\n have := inverts L W _ \u03c6'.hs\n have := inverts L W _ \u03b1.hs\n have : IsIso (L.map (\u03c6'.s \u226b \u03b1.s)) := by\n rw [L.map_comp]\n infer_instance\n refine' \u27e8\u03c8, _, _\u27e9\n \u00b7 rw [\u2190 cancel_mono (L.map (\u03c6'.s \u226b \u03b1.s)), LeftFraction.map_comp_map_s,\n h\u03b1, L.map_comp, h\u03c6, LeftFraction.map_comp_map_s_assoc,\n L.map_comp]\n \u00b7 rw [\u2190 cancel_mono (L.map (\u03c6'.s \u226b \u03b1.s)), h\u03c6']\n nth_rw 1 [L.map_comp]\n rw [LeftFraction.map_comp_map_s_assoc, LeftFraction.map_comp_map_s,\n L.map_comp]\n\n", "theoremStatement": "lemma exists_leftFraction\u2083 {X Y : C} (f f' f'' : L.obj X \u27f6 L.obj Y) :\n \u2203 (\u03c6 : W.LeftFraction\u2083 X Y), f = \u03c6.fst.map L (inverts L W) \u2227\n f' = \u03c6.snd.map L (inverts L W) \u2227\n f'' = \u03c6.thd.map L (inverts L W)", "theoremName": "exists_leftFraction\u2083", "fileCreated": {"commit": "429161409e", "date": "2024-04-18"}, "theoremCreated": {"commit": "429161409e", "date": "2024-04-18"}, "file": "mathlib4/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean", "positionMetadata": {"lineInFile": 298, "tokenPositionInFile": 8702, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n obtain \u27e8\u03b1, h\u03b1, h\u03b1'\u27e9 := exists_leftFraction\u2082 L W f f'\n have \u27e8\u03b2, h\u03b2\u27e9 := exists_leftFraction L W f''\n obtain \u27e8\u03b3, h\u03b3\u27e9 := (RightFraction.mk _ \u03b1.hs \u03b2.s).exists_leftFraction\n dsimp at h\u03b3\n let \u03c8 : W.LeftFraction\u2083 X Y :=\n { Y' := \u03b3.Y'\n f := \u03b1.f \u226b \u03b3.f\n f' := \u03b1.f' \u226b \u03b3.f\n f'' := \u03b2.f \u226b \u03b3.s\n s := \u03b2.s \u226b \u03b3.s\n hs := W.comp_mem _ _ \u03b2.hs \u03b3.hs }\n have := inverts L W _ \u03b2.hs\n have := inverts L W _ \u03b3.hs\n have : IsIso (L.map (\u03b2.s \u226b \u03b3.s)) := by\n rw [L.map_comp]\n infer_instance\n refine' \u27e8\u03c8, _, _, _\u27e9\n \u00b7 rw [\u2190 cancel_mono (L.map (\u03b2.s \u226b \u03b3.s)), LeftFraction.map_comp_map_s, h\u03b1, h\u03b3,\n L.map_comp, LeftFraction.map_comp_map_s_assoc, L.map_comp]\n \u00b7 rw [\u2190 cancel_mono (L.map (\u03b2.s \u226b \u03b3.s)), LeftFraction.map_comp_map_s, h\u03b1', h\u03b3,\n L.map_comp, LeftFraction.map_comp_map_s_assoc, L.map_comp]\n \u00b7 rw [\u2190 cancel_mono (L.map (\u03b2.s \u226b \u03b3.s)), h\u03b2]\n nth_rw 1 [L.map_comp]\n rw [LeftFraction.map_comp_map_s_assoc, LeftFraction.map_comp_map_s, L.map_comp]", "proofType": "tactic", "proofLengthLines": 25, "proofLengthTokens": 973}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.Algebra.MvPolynomial.Monad\nimport Mathlib.LinearAlgebra.Charpoly.ToMatrix\nimport Mathlib.LinearAlgebra.Dimension.Finrank\nimport Mathlib.LinearAlgebra.Matrix.Charpoly.Univ\n\n/-!\n# Characteristic polynomials of linear families of endomorphisms\n\nThe coefficients of the characteristic polynomials of a linear family of endomorphisms\nare homogeneous polynomials in the parameters.\nThis result is used in Lie theory\nto establish the existence of regular elements and Cartan subalgebras,\nand ultimately a well-defined notion of rank for Lie algebras.\n\nIn this file we prove this result about characteristic polynomials.\nLet `L` and `M` be modules over a nontrivial commutative ring `R`,\nand let `\u03c6 : L \u2192\u2097[R] Module.End R M` be a linear map.\nLet `b` be a basis of `L`, indexed by `\u03b9`.\nThen we define a multivariate polynomial with variables indexed by `\u03b9`\nthat evaluates on elements `x` of `L` to the characteristic polynomial of `\u03c6 x`.\n\n## Main declarations\n\n* `Matrix.toMvPolynomial M i`: the family of multivariate polynomials that evaluates on `c : n \u2192 R`\n to the dot product of the `i`-th row of `M` with `c`.\n `Matrix.toMvPolynomial M i` is the sum of the monomials `C (M i j) * X j`.\n* `LinearMap.toMvPolynomial b\u2081 b\u2082 f`: a version of `Matrix.toMvPolynomial` for linear maps `f`\n with respect to bases `b\u2081` and `b\u2082` of the domain and codomain.\n* `LinearMap.polyCharpoly`: the multivariate polynomial that evaluates on elements `x` of `L`\n to the characteristic polynomial of `\u03c6 x`.\n* `LinearMap.polyCharpoly_map_eq_charpoly`: the evaluation of `polyCharpoly` on elements `x` of `L`\n is the characteristic polynomial of `\u03c6 x`.\n* `LinearMap.polyCharpoly_coeff_isHomogeneous`: the coefficients of `polyCharpoly`\n are homogeneous polynomials in the parameters.\n\n## Implementation details\n\nWe show that `LinearMap.polyCharpoly` does not depend on the choice of basis of the target module.\nThis is done via `LinearMap.polyCharpoly_eq_polyCharpolyAux`\nand `LinearMap.polyCharpolyAux_basisIndep`.\nThe latter is proven by considering\nthe base change of the `R`-linear map `\u03c6 : L \u2192\u2097[R] End R M`\nto the multivariate polynomial ring `MvPolynomial \u03b9 R`,\nand showing that `polyCharpolyAux \u03c6` is equal to the characteristic polynomial of this base change.\nThe proof concludes because characteristic polynomials are independent of the chosen basis.\n\n-/\n\nopen scoped BigOperators Matrix\n\nnamespace Matrix\n\nvariable {m n o R S : Type*}\nvariable [Fintype n] [Fintype o] [CommSemiring R] [CommSemiring S]\n\nopen MvPolynomial\n\n/-- Let `M` be an `(m \u00d7 n)`-matrix over `R`.\nThen `Matrix.toMvPolynomial M` is the family (indexed by `i : m`)\nof multivariate polynomials in `n` variables over `R` that evaluates on `c : n \u2192 R`\nto the dot product of the `i`-th row of `M` with `c`:\n`Matrix.toMvPolynomial M i` is the sum of the monomials `C (M i j) * X j`. -/\nnoncomputable\ndef toMvPolynomial (M : Matrix m n R) (i : m) : MvPolynomial n R :=\n \u2211 j, monomial (.single j 1) (M i j)\n\n", "theoremStatement": "lemma toMvPolynomial_eval_eq_apply (M : Matrix m n R) (i : m) (c : n \u2192 R) :\n eval c (M.toMvPolynomial i) = (M *\u1d65 c) i", "theoremName": "toMvPolynomial_eval_eq_apply", "fileCreated": {"commit": "48970a4b9d", "date": "2024-04-17"}, "theoremCreated": {"commit": "48970a4b9d", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/LinearMap/Polynomial.lean", "positionMetadata": {"lineInFile": 72, "tokenPositionInFile": 3127, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n simp only [toMvPolynomial, map_sum, eval_monomial, pow_zero, Finsupp.prod_single_index, pow_one,\n mulVec, dotProduct]", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 125}, "split": "mathlib"} +{"srcContext": "/-\nCopyright (c) 2024 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.Algebra.MvPolynomial.Monad\nimport Mathlib.LinearAlgebra.Charpoly.ToMatrix\nimport Mathlib.LinearAlgebra.Dimension.Finrank\nimport Mathlib.LinearAlgebra.Matrix.Charpoly.Univ\n\n/-!\n# Characteristic polynomials of linear families of endomorphisms\n\nThe coefficients of the characteristic polynomials of a linear family of endomorphisms\nare homogeneous polynomials in the parameters.\nThis result is used in Lie theory\nto establish the existence of regular elements and Cartan subalgebras,\nand ultimately a well-defined notion of rank for Lie algebras.\n\nIn this file we prove this result about characteristic polynomials.\nLet `L` and `M` be modules over a nontrivial commutative ring `R`,\nand let `\u03c6 : L \u2192\u2097[R] Module.End R M` be a linear map.\nLet `b` be a basis of `L`, indexed by `\u03b9`.\nThen we define a multivariate polynomial with variables indexed by `\u03b9`\nthat evaluates on elements `x` of `L` to the characteristic polynomial of `\u03c6 x`.\n\n## Main declarations\n\n* `Matrix.toMvPolynomial M i`: the family of multivariate polynomials that evaluates on `c : n \u2192 R`\n to the dot product of the `i`-th row of `M` with `c`.\n `Matrix.toMvPolynomial M i` is the sum of the monomials `C (M i j) * X j`.\n* `LinearMap.toMvPolynomial b\u2081 b\u2082 f`: a version of `Matrix.toMvPolynomial` for linear maps `f`\n with respect to bases `b\u2081` and `b\u2082` of the domain and codomain.\n* `LinearMap.polyCharpoly`: the multivariate polynomial that evaluates on elements `x` of `L`\n to the characteristic polynomial of `\u03c6 x`.\n* `LinearMap.polyCharpoly_map_eq_charpoly`: the evaluation of `polyCharpoly` on elements `x` of `L`\n is the characteristic polynomial of `\u03c6 x`.\n* `LinearMap.polyCharpoly_coeff_isHomogeneous`: the coefficients of `polyCharpoly`\n are homogeneous polynomials in the parameters.\n\n## Implementation details\n\nWe show that `LinearMap.polyCharpoly` does not depend on the choice of basis of the target module.\nThis is done via `LinearMap.polyCharpoly_eq_polyCharpolyAux`\nand `LinearMap.polyCharpolyAux_basisIndep`.\nThe latter is proven by considering\nthe base change of the `R`-linear map `\u03c6 : L \u2192\u2097[R] End R M`\nto the multivariate polynomial ring `MvPolynomial \u03b9 R`,\nand showing that `polyCharpolyAux \u03c6` is equal to the characteristic polynomial of this base change.\nThe proof concludes because characteristic polynomials are independent of the chosen basis.\n\n-/\n\nopen scoped BigOperators Matrix\n\nnamespace Matrix\n\nvariable {m n o R S : Type*}\nvariable [Fintype n] [Fintype o] [CommSemiring R] [CommSemiring S]\n\nopen MvPolynomial\n\n/-- Let `M` be an `(m \u00d7 n)`-matrix over `R`.\nThen `Matrix.toMvPolynomial M` is the family (indexed by `i : m`)\nof multivariate polynomials in `n` variables over `R` that evaluates on `c : n \u2192 R`\nto the dot product of the `i`-th row of `M` with `c`:\n`Matrix.toMvPolynomial M i` is the sum of the monomials `C (M i j) * X j`. -/\nnoncomputable\ndef toMvPolynomial (M : Matrix m n R) (i : m) : MvPolynomial n R :=\n \u2211 j, monomial (.single j 1) (M i j)\n\nlemma toMvPolynomial_eval_eq_apply (M : Matrix m n R) (i : m) (c : n \u2192 R) :\n eval c (M.toMvPolynomial i) = (M *\u1d65 c) i := by\n simp only [toMvPolynomial, map_sum, eval_monomial, pow_zero, Finsupp.prod_single_index, pow_one,\n mulVec, dotProduct]\n\n", "theoremStatement": "lemma toMvPolynomial_map (f : R \u2192+* S) (M : Matrix m n R) (i : m) :\n (M.map f).toMvPolynomial i = MvPolynomial.map f (M.toMvPolynomial i)", "theoremName": "toMvPolynomial_map", "fileCreated": {"commit": "48970a4b9d", "date": "2024-04-17"}, "theoremCreated": {"commit": "48970a4b9d", "date": "2024-04-17"}, "file": "mathlib4/Mathlib/Algebra/Module/LinearMap/Polynomial.lean", "positionMetadata": {"lineInFile": 77, "tokenPositionInFile": 3378, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "repositoryPremises": true}, "proofMetadata": {"hasProof": true, "proof": "by\n simp only [toMvPolynomial, map_apply, map_sum, map_monomial]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 65}, "split": "mathlib"} diff --git a/minictx/pfr.jsonl b/minictx/pfr.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..bb17b9cd0ffd44a91825ba493bf370ad673367ca --- /dev/null +++ b/minictx/pfr.jsonl @@ -0,0 +1,51 @@ +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\n", "theoremStatement": "lemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B ", "theoremName": "IsShift.sub_self_congr", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 33, "tokenPositionInFile": 881, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 36, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 66}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\n", "theoremStatement": "lemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B ", "theoremName": "IsShift.card_congr", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 36, "tokenPositionInFile": 1008, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 25, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by rintro \u27e8x, rfl\u27e9; simp", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 27}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n", "theoremStatement": "/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' ", "theoremName": "wlog_notInCoset", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 42, "tokenPositionInFile": 1294, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 121, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]", "proofType": "tactic", "proofLengthLines": 24, "proofLengthTokens": 1358}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n", "theoremStatement": "/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] ", "theoremName": "torsion_free_doubling", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "1226201", "date": "2024-01-02"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 81, "tokenPositionInFile": 3354, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 19, "numPremises": 426, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]", "proofType": "tactic", "proofLengthLines": 117, "proofLengthTokens": 8125}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n", "theoremStatement": "/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] ", "theoremName": "torsion_dist_shrinking", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 203, "tokenPositionInFile": 11772, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 125, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 492}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n", "theoremStatement": "/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) ", "theoremName": "app_ent_PFR'", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "9692d13", "date": "2024-01-11"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 232, "tokenPositionInFile": 13219, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 15, "numPremises": 282, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9", "proofType": "tactic", "proofLengthLines": 36, "proofLengthTokens": 1584}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n", "theoremStatement": "lemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) ", "theoremName": "app_ent_PFR", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "9692d13", "date": "2024-01-11"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 286, "tokenPositionInFile": 15840, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 60, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 96}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n", "theoremStatement": "/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] ", "theoremName": "PFR_projection'", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "9692d13", "date": "2024-01-11"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 294, "tokenPositionInFile": 16290, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 435, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9", "proofType": "tactic", "proofLengthLines": 93, "proofLengthTokens": 4973}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n", "theoremStatement": "/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] ", "theoremName": "PFR_projection", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "9692d13", "date": "2024-01-11"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 398, "tokenPositionInFile": 22003, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 281, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 343}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\n", "theoremStatement": "lemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d ", "theoremName": "four_logs", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 420, "tokenPositionInFile": 23051, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 46, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 62}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\n", "theoremStatement": "lemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 ", "theoremName": "sum_prob_preimage", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 424, "tokenPositionInFile": 23253, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 107, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 530}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n", "theoremStatement": "/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) ", "theoremName": "single_fibres", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 438, "tokenPositionInFile": 24019, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 25, "numPremises": 476, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]", "proofType": "tactic", "proofLengthLines": 131, "proofLengthTokens": 7867}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\n", "theoremStatement": "lemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S ", "theoremName": "exists_coset_cover", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 595, "tokenPositionInFile": 33499, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 37, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 103}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\n", "theoremStatement": "lemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S ", "theoremName": "dimension_le_of_coset_cover", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 604, "tokenPositionInFile": 33913, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 59}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\n", "theoremStatement": "lemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G ", "theoremName": "dimension_le_rank", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 610, "tokenPositionInFile": 34124, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 43, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 121}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n", "theoremStatement": "/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) ", "theoremName": "Finsupp.mapRange_surjective", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 622, "tokenPositionInFile": 34560, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 40, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 461}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n", "theoremStatement": "/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G ", "theoremName": "torsion_free", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "f0dd05d", "date": "2023-12-27"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 640, "tokenPositionInFile": 35316, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 223, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 262}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n", "theoremStatement": "/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) ", "theoremName": "weak_PFR_quotient_prelim", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "f0dd05d", "date": "2023-12-27"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 649, "tokenPositionInFile": 35684, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 280, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9", "proofType": "tactic", "proofLengthLines": 87, "proofLengthTokens": 4047}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n", "theoremStatement": "/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x ", "theoremName": "third_iso", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 743, "tokenPositionInFile": 40069, "theoremPositionInFile": 21}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 97, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1", "proofType": "tactic", "proofLengthLines": 19, "proofLengthTokens": 648}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\n", "theoremStatement": "lemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n z \u2208 A ", "theoremName": "single", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 769, "tokenPositionInFile": 41187, "theoremPositionInFile": 22}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 98, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n contrapose! hz\n have : Disjoint {z} A := by simp [hz]\n replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n simp [hA] at this\n have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n simp [this] at h\n assumption", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 266}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n z \u2208 A := by\n contrapose! hz\n have : Disjoint {z} A := by simp [hz]\n replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n simp [hA] at this\n have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n simp [this] at h\n assumption\n\n", "theoremStatement": "/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n 40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) ", "theoremName": "weak_PFR_asymm_prelim", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 780, "tokenPositionInFile": 41648, "theoremPositionInFile": 23}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 9, "repositoryPremises": true, "numRepositoryPremises": 29, "numPremises": 490, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : Nonempty A := hnA.to_subtype\n have : Nonempty B := hnB.to_subtype\n obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n let _mH : MeasurableSpace H := \u22a4\n have h_fintype : Fintype H := Fintype.ofFinite H\n have h_torsionfree := torsion_free (G := G)\n\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n exists_isUniform_measureSpace' B\n\n rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n let N := AddSubgroup.comap \u03c6 H'\n set \u03c6' := QuotientAddGroup.mk' N\n have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n constructor\n intro x\n exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n have : Nonempty Ax := hnAx.to_subtype\n have : Nonempty By := hnBy.to_subtype\n have Axf : Finite Ax := by rw [hAx]; infer_instance\n have Byf : Finite By := by rw [hBy]; infer_instance\n\n have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n rw [rdist_symm] at h2\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n -- using explicit .toFun casts as this saves a lot of heartbeats\n change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n replace hH1 : log (Nat.card H') \u2264 40 * d\u1d64[A # B] := by\n apply hH1.trans\n linarith\n replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n rw [mul_comm, \u2190 log_rpow (by norm_num)]\n norm_cast\n classical\n rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n use N, x, y, Ax, By\n refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] := rdist_of_inj hX hY e.toAddMonoidHom he_inj\n rwa [this]\n\n set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n swap\n . rw [\u2190 h] at hH2\n have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n set Af := A.toFinite.toFinset\n have hUAf : IsUniform Af UA := by\n convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UA \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n replace this := single ((\u2119).map UA) hx this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hxx : Ax = A := by\n have h : hnAx.some \u2208 Ax := hnAx.some_mem\n simp [hAx] at h \u22a2\n have := hAAx h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hAAx hz\n\n have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n set Bf := B.toFinite.toFinset\n have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UB \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n replace this := single ((\u2119).map UB) hy this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hyy : By = B := by\n have h : hnBy.some \u2208 By := hnBy.some_mem\n simp [hBy] at h \u22a2\n have := hBBy h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hBBy hz\n\n simp [hxx, hyy]\n\n have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n congr\n convert (four_logs ?_ ?_ ?_ ?_).symm\n all_goals norm_cast; exact Nat.card_pos\n _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n apply mul_le_mul_of_nonneg_right hH2\n have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n rw [\u2190 mul_le_mul_left this]\n apply le_trans _ hcard_ineq\n rw [mul_zero]\n change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n apply Real.log_nonneg\n rw [one_le_div]\n gcongr\n . apply Nat.card_mono\n . exact Set.toFinite A\n rw [hAx]; exact Set.inter_subset_left _ _\n apply Nat.card_mono\n . exact Set.toFinite B\n rw [hBy]; exact Set.inter_subset_left _ _\n norm_cast\n exact mul_pos Nat.card_pos Nat.card_pos\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n exact (mul_le_mul_left h).mp this", "proofType": "tactic", "proofLengthLines": 164, "proofLengthTokens": 7954}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n z \u2208 A := by\n contrapose! hz\n have : Disjoint {z} A := by simp [hz]\n replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n simp [hA] at this\n have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n simp [this] at h\n assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n 40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n have : Nonempty A := hnA.to_subtype\n have : Nonempty B := hnB.to_subtype\n obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n let _mH : MeasurableSpace H := \u22a4\n have h_fintype : Fintype H := Fintype.ofFinite H\n have h_torsionfree := torsion_free (G := G)\n\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n exists_isUniform_measureSpace' B\n\n rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n let N := AddSubgroup.comap \u03c6 H'\n set \u03c6' := QuotientAddGroup.mk' N\n have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n constructor\n intro x\n exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n have : Nonempty Ax := hnAx.to_subtype\n have : Nonempty By := hnBy.to_subtype\n have Axf : Finite Ax := by rw [hAx]; infer_instance\n have Byf : Finite By := by rw [hBy]; infer_instance\n\n have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n rw [rdist_symm] at h2\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n -- using explicit .toFun casts as this saves a lot of heartbeats\n change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n replace hH1 : log (Nat.card H') \u2264 40 * d\u1d64[A # B] := by\n apply hH1.trans\n linarith\n replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n rw [mul_comm, \u2190 log_rpow (by norm_num)]\n norm_cast\n classical\n rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n use N, x, y, Ax, By\n refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] := rdist_of_inj hX hY e.toAddMonoidHom he_inj\n rwa [this]\n\n set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n swap\n . rw [\u2190 h] at hH2\n have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n set Af := A.toFinite.toFinset\n have hUAf : IsUniform Af UA := by\n convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UA \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n replace this := single ((\u2119).map UA) hx this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hxx : Ax = A := by\n have h : hnAx.some \u2208 Ax := hnAx.some_mem\n simp [hAx] at h \u22a2\n have := hAAx h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hAAx hz\n\n have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n set Bf := B.toFinite.toFinset\n have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UB \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n replace this := single ((\u2119).map UB) hy this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hyy : By = B := by\n have h : hnBy.some \u2208 By := hnBy.some_mem\n simp [hBy] at h \u22a2\n have := hBBy h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hBBy hz\n\n simp [hxx, hyy]\n\n have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n congr\n convert (four_logs ?_ ?_ ?_ ?_).symm\n all_goals norm_cast; exact Nat.card_pos\n _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n apply mul_le_mul_of_nonneg_right hH2\n have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n rw [\u2190 mul_le_mul_left this]\n apply le_trans _ hcard_ineq\n rw [mul_zero]\n change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n apply Real.log_nonneg\n rw [one_le_div]\n gcongr\n . apply Nat.card_mono\n . exact Set.toFinite A\n rw [hAx]; exact Set.inter_subset_left _ _\n apply Nat.card_mono\n . exact Set.toFinite B\n rw [hBy]; exact Set.inter_subset_left _ _\n norm_cast\n exact mul_pos Nat.card_pos Nat.card_pos\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n", "theoremStatement": "/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n {H: AddSubgroup G} (A : Set H) (x : G) :\n dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A ", "theoremName": "dimension_of_shift", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c8331ef", "date": "2023-12-29"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 967, "tokenPositionInFile": 51118, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 131, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n classical\n rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n rw [\u2190 hrank]\n convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n . apply LinearEquiv.finrank_eq\n exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n intro a ha\n rw [Set.mem_image] at ha\n rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n rw [Submodule.mem_map]\n use b - v, hshift b hb\n simp [\u2190 hb']\n abel", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 524}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n z \u2208 A := by\n contrapose! hz\n have : Disjoint {z} A := by simp [hz]\n replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n simp [hA] at this\n have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n simp [this] at h\n assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n 40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n have : Nonempty A := hnA.to_subtype\n have : Nonempty B := hnB.to_subtype\n obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n let _mH : MeasurableSpace H := \u22a4\n have h_fintype : Fintype H := Fintype.ofFinite H\n have h_torsionfree := torsion_free (G := G)\n\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n exists_isUniform_measureSpace' B\n\n rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n let N := AddSubgroup.comap \u03c6 H'\n set \u03c6' := QuotientAddGroup.mk' N\n have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n constructor\n intro x\n exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n have : Nonempty Ax := hnAx.to_subtype\n have : Nonempty By := hnBy.to_subtype\n have Axf : Finite Ax := by rw [hAx]; infer_instance\n have Byf : Finite By := by rw [hBy]; infer_instance\n\n have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n rw [rdist_symm] at h2\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n -- using explicit .toFun casts as this saves a lot of heartbeats\n change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n replace hH1 : log (Nat.card H') \u2264 40 * d\u1d64[A # B] := by\n apply hH1.trans\n linarith\n replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n rw [mul_comm, \u2190 log_rpow (by norm_num)]\n norm_cast\n classical\n rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n use N, x, y, Ax, By\n refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] := rdist_of_inj hX hY e.toAddMonoidHom he_inj\n rwa [this]\n\n set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n swap\n . rw [\u2190 h] at hH2\n have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n set Af := A.toFinite.toFinset\n have hUAf : IsUniform Af UA := by\n convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UA \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n replace this := single ((\u2119).map UA) hx this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hxx : Ax = A := by\n have h : hnAx.some \u2208 Ax := hnAx.some_mem\n simp [hAx] at h \u22a2\n have := hAAx h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hAAx hz\n\n have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n set Bf := B.toFinite.toFinset\n have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UB \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n replace this := single ((\u2119).map UB) hy this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hyy : By = B := by\n have h : hnBy.some \u2208 By := hnBy.some_mem\n simp [hBy] at h \u22a2\n have := hBBy h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hBBy hz\n\n simp [hxx, hyy]\n\n have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n congr\n convert (four_logs ?_ ?_ ?_ ?_).symm\n all_goals norm_cast; exact Nat.card_pos\n _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n apply mul_le_mul_of_nonneg_right hH2\n have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n rw [\u2190 mul_le_mul_left this]\n apply le_trans _ hcard_ineq\n rw [mul_zero]\n change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n apply Real.log_nonneg\n rw [one_le_div]\n gcongr\n . apply Nat.card_mono\n . exact Set.toFinite A\n rw [hAx]; exact Set.inter_subset_left _ _\n apply Nat.card_mono\n . exact Set.toFinite B\n rw [hBy]; exact Set.inter_subset_left _ _\n norm_cast\n exact mul_pos Nat.card_pos Nat.card_pos\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n {H: AddSubgroup G} (A : Set H) (x : G) :\n dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n classical\n rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n rw [\u2190 hrank]\n convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n . apply LinearEquiv.finrank_eq\n exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n intro a ha\n rw [Set.mem_image] at ha\n rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n rw [Submodule.mem_map]\n use b - v, hshift b hb\n simp [\u2190 hb']\n abel\n\n", "theoremStatement": "lemma conclusion_transfers {A B : Set G}\n (G': AddSubgroup G) (A' B' : Set G')\n (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B ", "theoremName": "conclusion_transfers", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 986, "tokenPositionInFile": 51896, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 213, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : Nonempty A' := hA'.to_subtype\n have : Nonempty B' := hB'.to_subtype\n rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n rcases hA with \u27e8 x, hA \u27e9\n set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n have hf : Function.Injective f := by\n intro y z hyz\n simp [f] at hyz\n exact hyz\n have hA' : A = f '' A' := by\n simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n rcases hB with \u27e8 y, hB \u27e9\n set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n have hg : Function.Injective g := by\n intro y z hyz\n simp [g] at hyz\n exact hyz\n have hB' : B = g '' B' := by\n simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n use f '' A'', g '' B''\n have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n congr\n . rw [hA]\n ext y\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : x + z + -x \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use y + -x\n constructor\n . use h\n abel\n rw [hB]\n ext x\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : y + z + -y \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use x + -y\n constructor\n . use h\n abel\n\n\n refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n . simp [hA', hf, hA'']\n . simp [hB', hg, hB'']\n . simp [hA''_non]\n . simp [hB''_non]\n . convert hcard_ineq using 2\n . congr 3\n . rw [hA', Nat.card_image_of_injective hf]\n . rw [hB', Nat.card_image_of_injective hg]\n . rw [Nat.card_image_of_injective hf]\n rw [Nat.card_image_of_injective hg]\n convert LE.le.trans _ hdim_ineq using 2\n norm_cast\n apply max_le_max\n . exact dimension_of_shift A'' x\n \u00b7 exact dimension_of_shift B'' y", "proofType": "tactic", "proofLengthLines": 69, "proofLengthTokens": 1978}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n z \u2208 A := by\n contrapose! hz\n have : Disjoint {z} A := by simp [hz]\n replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n simp [hA] at this\n have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n simp [this] at h\n assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n 40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n have : Nonempty A := hnA.to_subtype\n have : Nonempty B := hnB.to_subtype\n obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n let _mH : MeasurableSpace H := \u22a4\n have h_fintype : Fintype H := Fintype.ofFinite H\n have h_torsionfree := torsion_free (G := G)\n\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n exists_isUniform_measureSpace' B\n\n rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n let N := AddSubgroup.comap \u03c6 H'\n set \u03c6' := QuotientAddGroup.mk' N\n have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n constructor\n intro x\n exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n have : Nonempty Ax := hnAx.to_subtype\n have : Nonempty By := hnBy.to_subtype\n have Axf : Finite Ax := by rw [hAx]; infer_instance\n have Byf : Finite By := by rw [hBy]; infer_instance\n\n have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n rw [rdist_symm] at h2\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n -- using explicit .toFun casts as this saves a lot of heartbeats\n change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n replace hH1 : log (Nat.card H') \u2264 40 * d\u1d64[A # B] := by\n apply hH1.trans\n linarith\n replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n rw [mul_comm, \u2190 log_rpow (by norm_num)]\n norm_cast\n classical\n rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n use N, x, y, Ax, By\n refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] := rdist_of_inj hX hY e.toAddMonoidHom he_inj\n rwa [this]\n\n set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n swap\n . rw [\u2190 h] at hH2\n have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n set Af := A.toFinite.toFinset\n have hUAf : IsUniform Af UA := by\n convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UA \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n replace this := single ((\u2119).map UA) hx this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hxx : Ax = A := by\n have h : hnAx.some \u2208 Ax := hnAx.some_mem\n simp [hAx] at h \u22a2\n have := hAAx h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hAAx hz\n\n have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n set Bf := B.toFinite.toFinset\n have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UB \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n replace this := single ((\u2119).map UB) hy this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hyy : By = B := by\n have h : hnBy.some \u2208 By := hnBy.some_mem\n simp [hBy] at h \u22a2\n have := hBBy h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hBBy hz\n\n simp [hxx, hyy]\n\n have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n congr\n convert (four_logs ?_ ?_ ?_ ?_).symm\n all_goals norm_cast; exact Nat.card_pos\n _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n apply mul_le_mul_of_nonneg_right hH2\n have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n rw [\u2190 mul_le_mul_left this]\n apply le_trans _ hcard_ineq\n rw [mul_zero]\n change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n apply Real.log_nonneg\n rw [one_le_div]\n gcongr\n . apply Nat.card_mono\n . exact Set.toFinite A\n rw [hAx]; exact Set.inter_subset_left _ _\n apply Nat.card_mono\n . exact Set.toFinite B\n rw [hBy]; exact Set.inter_subset_left _ _\n norm_cast\n exact mul_pos Nat.card_pos Nat.card_pos\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n {H: AddSubgroup G} (A : Set H) (x : G) :\n dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n classical\n rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n rw [\u2190 hrank]\n convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n . apply LinearEquiv.finrank_eq\n exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n intro a ha\n rw [Set.mem_image] at ha\n rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n rw [Submodule.mem_map]\n use b - v, hshift b hb\n simp [\u2190 hb']\n abel\n\nlemma conclusion_transfers {A B : Set G}\n (G': AddSubgroup G) (A' B' : Set G')\n (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B := by\n have : Nonempty A' := hA'.to_subtype\n have : Nonempty B' := hB'.to_subtype\n rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n rcases hA with \u27e8 x, hA \u27e9\n set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n have hf : Function.Injective f := by\n intro y z hyz\n simp [f] at hyz\n exact hyz\n have hA' : A = f '' A' := by\n simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n rcases hB with \u27e8 y, hB \u27e9\n set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n have hg : Function.Injective g := by\n intro y z hyz\n simp [g] at hyz\n exact hyz\n have hB' : B = g '' B' := by\n simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n use f '' A'', g '' B''\n have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n congr\n . rw [hA]\n ext y\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : x + z + -x \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use y + -x\n constructor\n . use h\n abel\n rw [hB]\n ext x\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : y + z + -y \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use x + -y\n constructor\n . use h\n abel\n\n\n refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n . simp [hA', hf, hA'']\n . simp [hB', hg, hB'']\n . simp [hA''_non]\n . simp [hB''_non]\n . convert hcard_ineq using 2\n . congr 3\n . rw [hA', Nat.card_image_of_injective hf]\n . rw [hB', Nat.card_image_of_injective hg]\n . rw [Nat.card_image_of_injective hf]\n rw [Nat.card_image_of_injective hg]\n convert LE.le.trans _ hdim_ineq using 2\n norm_cast\n apply max_le_max\n . exact dimension_of_shift A'' x\n \u00b7 exact dimension_of_shift B'' y\n\n", "theoremStatement": "/-- If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that\n\\[\\log\\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A'\\rvert\\lvert B'\\rvert}\\leq 34 d[U_A;U_B]\\]\nsuch that $\\max(\\dim A',\\dim B')\\leq \\frac{40}{\\log 2} d[U_A;U_B]$. -/\nlemma weak_PFR_asymm (A B : Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) : WeakPFRAsymmConclusion A B ", "theoremName": "weak_PFR_asymm", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 1060, "tokenPositionInFile": 54134, "theoremPositionInFile": 28}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 10, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 442, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let P : \u2115 \u2192 Prop := fun M \u21a6 (\u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G) (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G) (_hG_sing: MeasurableSingletonClass G) (A B: Set G) (_hA_fin: Finite A) (_hB_fin: Finite B) (_hA_non: A.Nonempty) (_hB_non: B.Nonempty) (_hM : (Nat.card A) + (Nat.card B) \u2264 M), WeakPFRAsymmConclusion A B)\n suffices \u2200 M, (\u2200 M', M' < M \u2192 P M') \u2192 P M by\n set M := (Nat.card A) + (Nat.card B)\n have hM : (Nat.card A) + (Nat.card B) \u2264 M := Nat.le_refl _\n convert (Nat.strong_induction_on (p := P) M this) G \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a _ \u2039_\u203a A B \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a hM\n intro M h_induct\n -- wlog we can assume A, B are not in cosets of a smaller subgroup\n suffices \u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G)\n (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)\n (_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B) (_hA_non : A.Nonempty) (_hB_non : B.Nonempty) (_hM : Nat.card A + Nat.card B \u2264 M)\n (_hnot : NotInCoset A B), WeakPFRAsymmConclusion A B by\n intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM\n obtain \u27e8 G', A', B', hAA', hBB', hnot' \u27e9 := wlog_notInCoset hA_non hB_non\n have hG'_fin : Module.Finite \u2124 G' :=\n Module.Finite.iff_fg (N := AddSubgroup.toIntSubmodule G').2 (IsNoetherian.noetherian _)\n\n have hG'_free : Module.Free \u2124 G' := by\n rcases Submodule.nonempty_basis_of_pid (Module.Free.chooseBasis \u2124 G) (AddSubgroup.toIntSubmodule G') with \u27e8 n, \u27e8 b \u27e9 \u27e9\n exact Module.Free.of_basis b\n have hAA'_card: Nat.card A = Nat.card A' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hAA'.card_congr\n have hBB'_card: Nat.card B = Nat.card B' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hBB'.card_congr\n have hA_non' : Nonempty A := Set.nonempty_coe_sort.mpr hA_non\n have hB_non' : Nonempty B := Set.nonempty_coe_sort.mpr hB_non\n\n rw [hAA'_card, hBB'_card] at hM\n\n have hA'_nonfin : A'.Nonempty \u2227 Finite A' := by\n have := Nat.card_pos (\u03b1 := A)\n rw [hAA'_card, Nat.card_pos_iff] at this\n exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n have hB'_nonfin : B'.Nonempty \u2227 Finite B' := by\n have := Nat.card_pos (\u03b1 := B)\n rw [hBB'_card, Nat.card_pos_iff] at this\n exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n obtain \u27e8 hA'_non, hA'_fin \u27e9 := hA'_nonfin\n obtain \u27e8 hB'_non, hB'_fin \u27e9 := hB'_nonfin\n\n replace this := this G' _ hG'_free hG'_fin (by infer_instance) (by infer_instance) (by infer_instance) A' B' hA'_fin hB'_fin hA'_non hB'_non hM hnot'\n exact conclusion_transfers G' A' B' hAA' hBB' hA'_non hB'_non this\n intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM hnot\n rcases weak_PFR_asymm_prelim A B hA_non hB_non with \u27e8 N, x, y, Ax, By, hAx_non, hBy_non, hAx_fin, hBy_fin, hAx, hBy, hdim, hcard\u27e9\n have hAxA : Ax \u2286 A := by rw [hAx]; simp\n have hByB : By \u2286 B := by rw [hBy]; simp\n have hA_pos : (0 : \u211d) < Nat.card A := Nat.cast_pos.mpr (@Nat.card_pos _ hA_non.to_subtype _)\n have hB_pos : (0 : \u211d) < Nat.card B := Nat.cast_pos.mpr (@Nat.card_pos _ hB_non.to_subtype _)\n\n rcases lt_or_ge (Nat.card Ax + Nat.card By) (Nat.card A + Nat.card B) with h | h\n . replace h := h_induct (Nat.card Ax + Nat.card By) (h.trans_le hM) G hG_comm hG_free hG_fin hG_count hG_mes hG_sing Ax By (Set.finite_coe_iff.mpr hAx_fin) (Set.finite_coe_iff.mpr hBy_fin) hAx_non hBy_non (Eq.le rfl)\n rcases h with \u27e8 A', B', hA', hB', hA'_non, hB'_non, hcard_ineq, hdim_ineq \u27e9\n use A', B'\n have hAx_fin' := Set.finite_coe_iff.mpr hAx_fin\n have hBy_fin' := Set.finite_coe_iff.mpr hBy_fin\n have hA'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hAx_fin hA')\n have hB'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hBy_fin hB')\n have hAx_non' := Set.nonempty_coe_sort.mpr hAx_non\n have hBy_non' := Set.nonempty_coe_sort.mpr hBy_non\n have hA'_non' := Set.nonempty_coe_sort.mpr hA'_non\n have hB'_non' := Set.nonempty_coe_sort.mpr hB'_non\n have hAx_pos : (0 : \u211d) < Nat.card Ax := Nat.cast_pos.mpr Nat.card_pos\n have hBy_pos : (0 : \u211d) < Nat.card By := Nat.cast_pos.mpr Nat.card_pos\n have hA'_pos : (0 : \u211d) < Nat.card A' := Nat.cast_pos.mpr Nat.card_pos\n have hB'_pos : (0 : \u211d) < Nat.card B' := Nat.cast_pos.mpr Nat.card_pos\n have hAxA_le : (Nat.card Ax : \u211d) \u2264 (Nat.card A : \u211d) := Nat.cast_le.mpr (Nat.card_mono A.toFinite hAxA)\n have hByB_le : (Nat.card By : \u211d) \u2264 (Nat.card B : \u211d) := Nat.cast_le.mpr (Nat.card_mono B.toFinite hByB)\n\n refine \u27e8 hA'.trans hAxA, hB'.trans hByB, hA'_non, hB'_non, ?_, ?_ \u27e9\n . rw [four_logs hA_pos hB_pos hA'_pos hB'_pos]\n rw [four_logs hAx_pos hBy_pos hA'_pos hB'_pos] at hcard_ineq\n linarith only [hcard, hcard_ineq]\n apply hdim_ineq.trans\n gcongr\n linarith only [Real.log_le_log hAx_pos hAxA_le, Real.log_le_log hBy_pos hByB_le, hcard]\n use A, B\n refine \u27e8 Eq.subset rfl, Eq.subset rfl, hA_non, hB_non, ?_, ?_ \u27e9\n . have := hA_non.to_subtype\n have := hB_non.to_subtype\n apply LE.le.trans _ <| mul_nonneg (by norm_num) <| rdist_set_nonneg A B\n rw [div_self (by positivity)]\n simp\n have hAx_eq : Ax = A := by\n apply Set.Finite.eq_of_subset_of_card_le A.toFinite hAxA\n linarith only [h, Nat.card_mono B.toFinite hByB]\n have hBy_eq : By = B := by\n apply Set.Finite.eq_of_subset_of_card_le B.toFinite hByB\n linarith only [h, Nat.card_mono A.toFinite hAxA]\n have hN : N = \u22a4 := by\n have : (A-A) \u222a (B-B) \u2286 N := by\n rw [\u2190 hAx_eq, \u2190 hBy_eq, hAx, hBy]\n intro z hz\n simp [Set.mem_sub] at hz\n convert (QuotientAddGroup.eq_zero_iff z).mp ?_\n . infer_instance\n rcases hz with \u27e8 a, \u27e8 -, ha\u27e9, a', \u27e8-, ha'\u27e9, haa' \u27e9 | \u27e8 b, \u27e8 -, hb\u27e9, b', \u27e8 -,hb'\u27e9, hbb' \u27e9\n . rw [\u2190 haa']; simp [ha, ha']\n rw [\u2190 hbb']; simp [hb, hb']\n rw [\u2190 AddSubgroup.closure_le, hnot] at this\n exact top_le_iff.mp this\n have : Nat.card (G \u29f8 N) = 1 := by\n rw [Nat.card_eq_one_iff_unique]\n constructor\n . rw [hN]\n exact QuotientAddGroup.subsingleton_quotient_top\n infer_instance\n simp [this] at hdim\n rw [\u2190 le_div_iff' (by positivity)] at hdim\n convert LE.le.trans ?_ hdim using 1\n . field_simp\n simp\n exact \u27e8 dimension_le_rank A, dimension_le_rank B \u27e9", "proofType": "tactic", "proofLengthLines": 109, "proofLengthTokens": 6396}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n z \u2208 A := by\n contrapose! hz\n have : Disjoint {z} A := by simp [hz]\n replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n simp [hA] at this\n have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n simp [this] at h\n assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n 40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n have : Nonempty A := hnA.to_subtype\n have : Nonempty B := hnB.to_subtype\n obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n let _mH : MeasurableSpace H := \u22a4\n have h_fintype : Fintype H := Fintype.ofFinite H\n have h_torsionfree := torsion_free (G := G)\n\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n exists_isUniform_measureSpace' B\n\n rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n let N := AddSubgroup.comap \u03c6 H'\n set \u03c6' := QuotientAddGroup.mk' N\n have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n constructor\n intro x\n exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n have : Nonempty Ax := hnAx.to_subtype\n have : Nonempty By := hnBy.to_subtype\n have Axf : Finite Ax := by rw [hAx]; infer_instance\n have Byf : Finite By := by rw [hBy]; infer_instance\n\n have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n rw [rdist_symm] at h2\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n -- using explicit .toFun casts as this saves a lot of heartbeats\n change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n replace hH1 : log (Nat.card H') \u2264 40 * d\u1d64[A # B] := by\n apply hH1.trans\n linarith\n replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n rw [mul_comm, \u2190 log_rpow (by norm_num)]\n norm_cast\n classical\n rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n use N, x, y, Ax, By\n refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] := rdist_of_inj hX hY e.toAddMonoidHom he_inj\n rwa [this]\n\n set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n swap\n . rw [\u2190 h] at hH2\n have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n set Af := A.toFinite.toFinset\n have hUAf : IsUniform Af UA := by\n convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UA \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n replace this := single ((\u2119).map UA) hx this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hxx : Ax = A := by\n have h : hnAx.some \u2208 Ax := hnAx.some_mem\n simp [hAx] at h \u22a2\n have := hAAx h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hAAx hz\n\n have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n set Bf := B.toFinite.toFinset\n have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UB \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n replace this := single ((\u2119).map UB) hy this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hyy : By = B := by\n have h : hnBy.some \u2208 By := hnBy.some_mem\n simp [hBy] at h \u22a2\n have := hBBy h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hBBy hz\n\n simp [hxx, hyy]\n\n have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n congr\n convert (four_logs ?_ ?_ ?_ ?_).symm\n all_goals norm_cast; exact Nat.card_pos\n _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n apply mul_le_mul_of_nonneg_right hH2\n have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n rw [\u2190 mul_le_mul_left this]\n apply le_trans _ hcard_ineq\n rw [mul_zero]\n change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n apply Real.log_nonneg\n rw [one_le_div]\n gcongr\n . apply Nat.card_mono\n . exact Set.toFinite A\n rw [hAx]; exact Set.inter_subset_left _ _\n apply Nat.card_mono\n . exact Set.toFinite B\n rw [hBy]; exact Set.inter_subset_left _ _\n norm_cast\n exact mul_pos Nat.card_pos Nat.card_pos\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n {H: AddSubgroup G} (A : Set H) (x : G) :\n dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n classical\n rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n rw [\u2190 hrank]\n convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n . apply LinearEquiv.finrank_eq\n exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n intro a ha\n rw [Set.mem_image] at ha\n rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n rw [Submodule.mem_map]\n use b - v, hshift b hb\n simp [\u2190 hb']\n abel\n\nlemma conclusion_transfers {A B : Set G}\n (G': AddSubgroup G) (A' B' : Set G')\n (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B := by\n have : Nonempty A' := hA'.to_subtype\n have : Nonempty B' := hB'.to_subtype\n rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n rcases hA with \u27e8 x, hA \u27e9\n set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n have hf : Function.Injective f := by\n intro y z hyz\n simp [f] at hyz\n exact hyz\n have hA' : A = f '' A' := by\n simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n rcases hB with \u27e8 y, hB \u27e9\n set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n have hg : Function.Injective g := by\n intro y z hyz\n simp [g] at hyz\n exact hyz\n have hB' : B = g '' B' := by\n simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n use f '' A'', g '' B''\n have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n congr\n . rw [hA]\n ext y\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : x + z + -x \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use y + -x\n constructor\n . use h\n abel\n rw [hB]\n ext x\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : y + z + -y \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use x + -y\n constructor\n . use h\n abel\n\n\n refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n . simp [hA', hf, hA'']\n . simp [hB', hg, hB'']\n . simp [hA''_non]\n . simp [hB''_non]\n . convert hcard_ineq using 2\n . congr 3\n . rw [hA', Nat.card_image_of_injective hf]\n . rw [hB', Nat.card_image_of_injective hg]\n . rw [Nat.card_image_of_injective hf]\n rw [Nat.card_image_of_injective hg]\n convert LE.le.trans _ hdim_ineq using 2\n norm_cast\n apply max_le_max\n . exact dimension_of_shift A'' x\n \u00b7 exact dimension_of_shift B'' y\n\n/-- If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that\n\\[\\log\\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A'\\rvert\\lvert B'\\rvert}\\leq 34 d[U_A;U_B]\\]\nsuch that $\\max(\\dim A',\\dim B')\\leq \\frac{40}{\\log 2} d[U_A;U_B]$. -/\nlemma weak_PFR_asymm (A B : Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) : WeakPFRAsymmConclusion A B := by\n let P : \u2115 \u2192 Prop := fun M \u21a6 (\u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G) (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G) (_hG_sing: MeasurableSingletonClass G) (A B: Set G) (_hA_fin: Finite A) (_hB_fin: Finite B) (_hA_non: A.Nonempty) (_hB_non: B.Nonempty) (_hM : (Nat.card A) + (Nat.card B) \u2264 M), WeakPFRAsymmConclusion A B)\n suffices \u2200 M, (\u2200 M', M' < M \u2192 P M') \u2192 P M by\n set M := (Nat.card A) + (Nat.card B)\n have hM : (Nat.card A) + (Nat.card B) \u2264 M := Nat.le_refl _\n convert (Nat.strong_induction_on (p := P) M this) G \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a _ \u2039_\u203a A B \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a hM\n intro M h_induct\n -- wlog we can assume A, B are not in cosets of a smaller subgroup\n suffices \u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G)\n (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)\n (_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B) (_hA_non : A.Nonempty) (_hB_non : B.Nonempty) (_hM : Nat.card A + Nat.card B \u2264 M)\n (_hnot : NotInCoset A B), WeakPFRAsymmConclusion A B by\n intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM\n obtain \u27e8 G', A', B', hAA', hBB', hnot' \u27e9 := wlog_notInCoset hA_non hB_non\n have hG'_fin : Module.Finite \u2124 G' :=\n Module.Finite.iff_fg (N := AddSubgroup.toIntSubmodule G').2 (IsNoetherian.noetherian _)\n\n have hG'_free : Module.Free \u2124 G' := by\n rcases Submodule.nonempty_basis_of_pid (Module.Free.chooseBasis \u2124 G) (AddSubgroup.toIntSubmodule G') with \u27e8 n, \u27e8 b \u27e9 \u27e9\n exact Module.Free.of_basis b\n have hAA'_card: Nat.card A = Nat.card A' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hAA'.card_congr\n have hBB'_card: Nat.card B = Nat.card B' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hBB'.card_congr\n have hA_non' : Nonempty A := Set.nonempty_coe_sort.mpr hA_non\n have hB_non' : Nonempty B := Set.nonempty_coe_sort.mpr hB_non\n\n rw [hAA'_card, hBB'_card] at hM\n\n have hA'_nonfin : A'.Nonempty \u2227 Finite A' := by\n have := Nat.card_pos (\u03b1 := A)\n rw [hAA'_card, Nat.card_pos_iff] at this\n exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n have hB'_nonfin : B'.Nonempty \u2227 Finite B' := by\n have := Nat.card_pos (\u03b1 := B)\n rw [hBB'_card, Nat.card_pos_iff] at this\n exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n obtain \u27e8 hA'_non, hA'_fin \u27e9 := hA'_nonfin\n obtain \u27e8 hB'_non, hB'_fin \u27e9 := hB'_nonfin\n\n replace this := this G' _ hG'_free hG'_fin (by infer_instance) (by infer_instance) (by infer_instance) A' B' hA'_fin hB'_fin hA'_non hB'_non hM hnot'\n exact conclusion_transfers G' A' B' hAA' hBB' hA'_non hB'_non this\n intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM hnot\n rcases weak_PFR_asymm_prelim A B hA_non hB_non with \u27e8 N, x, y, Ax, By, hAx_non, hBy_non, hAx_fin, hBy_fin, hAx, hBy, hdim, hcard\u27e9\n have hAxA : Ax \u2286 A := by rw [hAx]; simp\n have hByB : By \u2286 B := by rw [hBy]; simp\n have hA_pos : (0 : \u211d) < Nat.card A := Nat.cast_pos.mpr (@Nat.card_pos _ hA_non.to_subtype _)\n have hB_pos : (0 : \u211d) < Nat.card B := Nat.cast_pos.mpr (@Nat.card_pos _ hB_non.to_subtype _)\n\n rcases lt_or_ge (Nat.card Ax + Nat.card By) (Nat.card A + Nat.card B) with h | h\n . replace h := h_induct (Nat.card Ax + Nat.card By) (h.trans_le hM) G hG_comm hG_free hG_fin hG_count hG_mes hG_sing Ax By (Set.finite_coe_iff.mpr hAx_fin) (Set.finite_coe_iff.mpr hBy_fin) hAx_non hBy_non (Eq.le rfl)\n rcases h with \u27e8 A', B', hA', hB', hA'_non, hB'_non, hcard_ineq, hdim_ineq \u27e9\n use A', B'\n have hAx_fin' := Set.finite_coe_iff.mpr hAx_fin\n have hBy_fin' := Set.finite_coe_iff.mpr hBy_fin\n have hA'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hAx_fin hA')\n have hB'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hBy_fin hB')\n have hAx_non' := Set.nonempty_coe_sort.mpr hAx_non\n have hBy_non' := Set.nonempty_coe_sort.mpr hBy_non\n have hA'_non' := Set.nonempty_coe_sort.mpr hA'_non\n have hB'_non' := Set.nonempty_coe_sort.mpr hB'_non\n have hAx_pos : (0 : \u211d) < Nat.card Ax := Nat.cast_pos.mpr Nat.card_pos\n have hBy_pos : (0 : \u211d) < Nat.card By := Nat.cast_pos.mpr Nat.card_pos\n have hA'_pos : (0 : \u211d) < Nat.card A' := Nat.cast_pos.mpr Nat.card_pos\n have hB'_pos : (0 : \u211d) < Nat.card B' := Nat.cast_pos.mpr Nat.card_pos\n have hAxA_le : (Nat.card Ax : \u211d) \u2264 (Nat.card A : \u211d) := Nat.cast_le.mpr (Nat.card_mono A.toFinite hAxA)\n have hByB_le : (Nat.card By : \u211d) \u2264 (Nat.card B : \u211d) := Nat.cast_le.mpr (Nat.card_mono B.toFinite hByB)\n\n refine \u27e8 hA'.trans hAxA, hB'.trans hByB, hA'_non, hB'_non, ?_, ?_ \u27e9\n . rw [four_logs hA_pos hB_pos hA'_pos hB'_pos]\n rw [four_logs hAx_pos hBy_pos hA'_pos hB'_pos] at hcard_ineq\n linarith only [hcard, hcard_ineq]\n apply hdim_ineq.trans\n gcongr\n linarith only [Real.log_le_log hAx_pos hAxA_le, Real.log_le_log hBy_pos hByB_le, hcard]\n use A, B\n refine \u27e8 Eq.subset rfl, Eq.subset rfl, hA_non, hB_non, ?_, ?_ \u27e9\n . have := hA_non.to_subtype\n have := hB_non.to_subtype\n apply LE.le.trans _ <| mul_nonneg (by norm_num) <| rdist_set_nonneg A B\n rw [div_self (by positivity)]\n simp\n have hAx_eq : Ax = A := by\n apply Set.Finite.eq_of_subset_of_card_le A.toFinite hAxA\n linarith only [h, Nat.card_mono B.toFinite hByB]\n have hBy_eq : By = B := by\n apply Set.Finite.eq_of_subset_of_card_le B.toFinite hByB\n linarith only [h, Nat.card_mono A.toFinite hAxA]\n have hN : N = \u22a4 := by\n have : (A-A) \u222a (B-B) \u2286 N := by\n rw [\u2190 hAx_eq, \u2190 hBy_eq, hAx, hBy]\n intro z hz\n simp [Set.mem_sub] at hz\n convert (QuotientAddGroup.eq_zero_iff z).mp ?_\n . infer_instance\n rcases hz with \u27e8 a, \u27e8 -, ha\u27e9, a', \u27e8-, ha'\u27e9, haa' \u27e9 | \u27e8 b, \u27e8 -, hb\u27e9, b', \u27e8 -,hb'\u27e9, hbb' \u27e9\n . rw [\u2190 haa']; simp [ha, ha']\n rw [\u2190 hbb']; simp [hb, hb']\n rw [\u2190 AddSubgroup.closure_le, hnot] at this\n exact top_le_iff.mp this\n have : Nat.card (G \u29f8 N) = 1 := by\n rw [Nat.card_eq_one_iff_unique]\n constructor\n . rw [hN]\n exact QuotientAddGroup.subsingleton_quotient_top\n infer_instance\n simp [this] at hdim\n rw [\u2190 le_div_iff' (by positivity)] at hdim\n convert LE.le.trans ?_ hdim using 1\n . field_simp\n simp\n exact \u27e8 dimension_le_rank A, dimension_le_rank B \u27e9\n\n", "theoremStatement": "/-- If $A\\subseteq \\mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\\leq \\log K$ then there exists a non-empty $A'\\subseteq A$ such that\n$\\lvert A'\\rvert\\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A'\\leq \\frac{40}{\\log 2} \\log K$. -/\nlemma weak_PFR {A : Set G} [Finite A] {K : \u211d} (hA : A.Nonempty) (hK: 0 < K) (hdist: d\u1d64[A # A] \u2264 log K):\n \u2203 A' : Set G, A' \u2286 A \u2227 (Nat.card A') \u2265 K^(-17 : \u211d) * (Nat.card A)\n \u2227 (dimension A') \u2264 (40 / log 2) * log K ", "theoremName": "weak_PFR", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 1174, "tokenPositionInFile": 60954, "theoremPositionInFile": 29}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 352, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rcases weak_PFR_asymm A A hA hA with \u27e8A', A'', hA', hA'', hA'nonempty, hA''nonempty, hcard, hdim\u27e9\n\n have : \u2203 B : Set G, B \u2286 A \u2227 (Nat.card B) \u2265 (Nat.card A') \u2227 (Nat.card B) \u2265 (Nat.card A'') \u2227 (dimension B) \u2264\nmax (dimension A') (dimension A'') := by\n rcases lt_or_ge (Nat.card A') (Nat.card A'') with h | h\n . use A''\n exact \u27e8hA'', by linarith, by linarith, le_max_right _ _\u27e9\n use A'\n exact \u27e8hA', by linarith, by linarith, le_max_left _ _\u27e9\n\n rcases this with \u27e8B, hB, hBcard, hBcard', hBdim\u27e9\n use B\n have hApos : Nat.card A > 0 := by\n rw [gt_iff_lt, Nat.card_pos_iff]\n exact \u27e8hA.to_subtype, inferInstance\u27e9\n have hA'pos : Nat.card A' > 0 := by\n rw [gt_iff_lt, Nat.card_pos_iff]\n refine \u27e8 hA'nonempty.to_subtype, Finite.Set.subset _ hA' \u27e9\n have hA''pos : Nat.card A'' > 0 := by\n rw [gt_iff_lt, Nat.card_pos_iff]\n refine \u27e8 hA''nonempty.to_subtype, Finite.Set.subset _ hA'' \u27e9\n have hBpos : Nat.card B > 0 := by linarith\n\n refine \u27e8hB, ?_, ?_\u27e9\n . have := calc 2 * log ((Nat.card A) / (Nat.card B))\n _ = log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card B) * (Nat.card B)) ) := by\n convert (log_pow (((Nat.card A) : \u211d)/(Nat.card B)) 2).symm\n field_simp\n rw [\u2190 pow_two, \u2190 pow_two]\n _ \u2264 log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card A') * (Nat.card A'')) ) := by\n apply log_le_log\n . positivity\n gcongr\n _ \u2264 34 * d\u1d64[A # A] := hcard\n _ \u2264 34 * log K := mul_le_mul_of_nonneg_left hdist (by linarith)\n _ = 2 * (17 * log K) := by ring\n _ = 2 * log (K^17) := by\n congr\n convert (log_pow K 17).symm\n rw [mul_le_mul_left (by norm_num), log_le_log_iff (by positivity) (by positivity), div_le_iff (by positivity), <- mul_inv_le_iff (by positivity), <-ge_iff_le, mul_comm] at this\n convert this using 2\n convert zpow_neg K 17 using 1\n norm_cast\n calc (dimension B : \u211d)\n _ \u2264 (((max (dimension A') (dimension A'')) : \u2115) : \u211d) := by norm_cast\n _ \u2264 (40 / log 2) * d\u1d64[A # A] := hdim\n _ \u2264 (40 / log 2) * log K := mul_le_mul_of_nonneg_left hdist (by positivity)", "proofType": "tactic", "proofLengthLines": 47, "proofLengthTokens": 2092}} +{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n obtain \u27e8x, hx\u27e9 := hA\n obtain \u27e8y, hy\u27e9 := hB\n set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n AddSubgroup.subset_closure\n have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n Subtype.range_coe_subtype, SetLike.mem_coe]\n exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n AddSubgroup.subset_closure\n have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n refine \u27e8G', A', B', hA, hB, ?_\u27e9\n unfold NotInCoset\n convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n rw [preimage_sub, preimage_sub]\n simp [A', B', preimage_image_preimage]\n all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$. -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n refine entropy_comp_of_injective \u03bcA ?_ f ?_\n \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n group\n have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n exact hY'\u2081_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n exact hY'\u2082_ident.entropy_eq.symm\n have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n group\n have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n (measurable_const_smul 2)\n rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n let f (g : G) := 2 \u2022 g\n exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n group\n have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n rw [two_mul]\n convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n rw [\u2190 two_nsmul]\n linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n have :=\n calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n map_add, ElementaryAddCommGroup.add_self]\n rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n : MeasurableSingletonClass (G \u29f8 H) :=\n \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n let p : refPackage \u03a9 \u03a9' G := {\n X\u2080\u2081 := X\n X\u2080\u2082 := Y\n hmeas1 := hX\n hmeas2 := hY\n \u03b7 := 1/8\n h\u03b7 := by norm_num\n h\u03b7' := by norm_num }\n obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n let \u03c8 := (QuotientAddGroup.mk' H)\n use H\n haveI : Finite H := Subtype.finite\n -- Note that H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n (H[U] - H[X]) + (H[U] - H[Y]) := by\n rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n gcongr\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n _ = 2 * (d[X # U] + d[Y # U]) := by ring\n _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n _ = 20 * d[X # Y] := by ring\n -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n 20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n \u00b7 classical\n obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n set G' := G \u29f8 H\n have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n (by simp [AddSubgroup.zero_mem])\n\n obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n ((measurable_discrete _).comp hY)\n have H_ne_bot: H' \u2260 \u22a5 := by\n by_contra!\n rcases this with rfl\n have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n let H'' := H'.comap \u03c8\n use H''\n\n rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n set H' := H''.map \u03c8\n\n have Hlt : H < H'' := by\n have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n exact Ne.bot_lt H_ne_bot\n\n let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n Function.comp.assoc] at hup'\n\n have cond : log (Nat.card H'') \u2264\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n have hcard\u2081 := Nat.card_congr <|\n (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n calc\n log (Nat.card H'')\n _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n _ = log (Nat.card H') + log (Nat.card H) := by\n rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n field_simp; ring\n\n have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n \u00b7 calc\n log (Nat.card H'')\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n \u00b7 calc\n H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n \u00b7 use \u22a5\n constructor\n \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n Nat.cast_one, log_one]\n positivity\n \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n exact hE \u27e81, by norm_num, by\n norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n 34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n refine \u27e8H, ?_, ?_\u27e9\n \u00b7 convert h\n norm_num\n \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n rw [\u2190 Finset.sum_div]\n apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n classical\n haveI := Fintype.ofFinite A\n rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n norm_cast\n congr; ext\n rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n (\u03c6 : G \u2192+ H)\n {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n \u2203 (x y : H) (Ax By: Set G),\n Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n have : Nonempty A := hA.to_subtype\n have : Nonempty B := hB.to_subtype\n haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n let p (x : H) (y : H) : \u211d :=\n (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n have :\n \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n calc\n _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n hUB' ((measurable_discrete _).comp hUB')]\n refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n unfold_let p; beta_reduce\n rewrite [Finset.mem_product, not_and_or] at h_notin\n suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n refine h_notin.imp ?_ ?_\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_AX \u27e8a, ha\u27e9)\n \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n rintro h \u27e8a, ha, rfl\u27e9\n exact h (h_BY \u27e8a, ha\u27e9)\n unfold_let M\n unfold entropy\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n apply tsum_congr; intro; congr\n rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n Set.preimage_comp, hUA_coe.measure_preimage hUA',\n Set.preimage_comp, hUB_coe.measure_preimage hUB']\n simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n have h_sum : \u2211 x in X, \u2211 y in Y,\n (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n M * d[UA # UB] :=\n calc\n _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n exact congrArg (HAdd.hAdd _) (by group)\n _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n simp_rw [Finset.mul_sum]\n congr; ext; congr; ext; group\n _ \u2264 M * d[UA # UB] := by\n rewrite [\u2190 left_distrib]\n apply mul_le_mul_of_nonneg_left\n \u00b7 linarith\n \u00b7 unfold_let M\n linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n have : \u2203 x : X, \u2203 y : Y,\n M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n M * d[UA # UB] := by\n let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n by_contra hc; push_neg at hc\n replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n exact not_le_of_gt hc h_sum\n obtain \u27e8x, y, hxy\u27e9 := this\n refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n rewrite [\u2190 inv_div, Real.log_inv]\n show _ * -log (p x.val y.val) \u2264 M * _\n linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n apply Nat.find_le\n existsi S , v\n exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n rw [dimension, \u2190hs]\n apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n classical\n let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n have : Function.RightInverse g f := by\n intro n\n by_cases h : n = 0\n . simp [g, h, hf]\n \u00b7 simp [g, h, Function.surjInv_eq hs n]\n have hg : g 0 = 0 := by simp [g]\n have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n intro F\n use Finsupp.mapRange g hg F\n rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n convert Finsupp.mapRange_id F\n convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n rintro x hx hn\n rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n rcases hn with \u27e8 n, hn, hn' \u27e9\n apply_fun Module.Free.repr \u2124 G at hn'\n simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n. Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n have hH_elem : ElementaryAddCommGroup H 2 := by\n apply ElementaryAddCommGroup.quotient_group (by decide)\n intro x; rw [AddMonoidHom.mem_range]\n use x\n rw [zsmulAddGroupHom_apply]\n norm_cast\n let B := Module.Free.ChooseBasisIndex \u2124 G\n let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n have hB_fin : Fintype B := by infer_instance\n have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n have hH_module : Module (ZMod 2) H := by infer_instance\n let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n intro x hx\n simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n rcases hx with \u27e8y, rfl\u27e9\n ext b\n simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n left\n exact ZMod.nat_cast_self 2\n let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n have hsur : Function.Surjective g := by\n have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n have h2 := h1.comp bG.repr.surjective\n have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n ext x b\n simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n rw [QuotientAddGroup.lift_mk]\n simp [mod, f]\n rw [h3] at h2\n apply Function.Surjective.of_comp h2\n have hinj : Function.Injective g := by\n rw [injective_iff_map_eq_zero]\n intro x hx\n rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n simp only [QuotientAddGroup.mk'_apply, g] at hx\n rw [QuotientAddGroup.lift_mk] at hx\n simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n let z' := (Finsupp.equivFunOnFinite).symm z\n change \u2200 b : B, 2 * z' b = (f y) b at hx\n let x' := bG.repr.symm z'\n rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n simp_rw [\u03c8, zsmulAddGroupHom_apply]\n use x'\n change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n congr\n ext b\n rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n have bH : Basis B (ZMod 2) H := by\n constructor\n exact {\n toFun := g\n invFun := g'\n left_inv := hg'.1\n right_inv := hg'.2\n map_add' := AddMonoidHom.map_add _\n map_smul' := by\n intro r x\n rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n rw [intCast_smul, intCast_smul]\n exact AddMonoidHom.map_zsmul g x n\n }\n have hH_fin : Fintype H := Module.fintypeOfFintype bH\n have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n congr\n exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n let H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := mk' G\u2082\n let N := AddSubgroup.comap \u03c6 H'\n have h1 : G\u2082 \u2264 N := by\n intro x hx\n rw [\u2190 eq_zero_iff] at hx\n have : \u03c6 x = 0 := hx\n simp [N, this, AddSubgroup.zero_mem H']\n set H'' := AddSubgroup.map (mk' G\u2082) N\n have h2 : H' = H'' := by\n change H' = AddSubgroup.map (mk' G\u2082) N\n rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n simp\n let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n let e2 := quotientAddEquivOfEq h2\n set e := e2.trans e1\n use e\n intro x\n convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n z \u2208 A := by\n contrapose! hz\n have : Disjoint {z} A := by simp [hz]\n replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n simp [hA] at this\n have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n simp [this] at h\n assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n 40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n have : Nonempty A := hnA.to_subtype\n have : Nonempty B := hnB.to_subtype\n obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n set G\u2082 := AddMonoidHom.range \u03c8\n set H := G \u29f8 G\u2082\n let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n let _mH : MeasurableSpace H := \u22a4\n have h_fintype : Fintype H := Fintype.ofFinite H\n have h_torsionfree := torsion_free (G := G)\n\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n exists_isUniform_measureSpace' B\n\n rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n let N := AddSubgroup.comap \u03c6 H'\n set \u03c6' := QuotientAddGroup.mk' N\n have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n constructor\n intro x\n exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n have : Nonempty Ax := hnAx.to_subtype\n have : Nonempty By := hnBy.to_subtype\n have Axf : Finite Ax := by rw [hAx]; infer_instance\n have Byf : Finite By := by rw [hBy]; infer_instance\n\n have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n rw [rdist_symm] at h2\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n -- using explicit .toFun casts as this saves a lot of heartbeats\n change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n replace hH1 : log (Nat.card H') \u2264 40 * d\u1d64[A # B] := by\n apply hH1.trans\n linarith\n replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n rw [mul_comm, \u2190 log_rpow (by norm_num)]\n norm_cast\n classical\n rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n use N, x, y, Ax, By\n refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] := rdist_of_inj hX hY e.toAddMonoidHom he_inj\n rwa [this]\n\n set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n swap\n . rw [\u2190 h] at hH2\n have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n set Af := A.toFinite.toFinset\n have hUAf : IsUniform Af UA := by\n convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UA \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n replace this := single ((\u2119).map UA) hx this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hxx : Ax = A := by\n have h : hnAx.some \u2208 Ax := hnAx.some_mem\n simp [hAx] at h \u22a2\n have := hAAx h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hAAx hz\n\n have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n set Bf := B.toFinite.toFinset\n have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n have : (Measure.map UB \u2119).real {z} > 0 := by\n rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n positivity\n have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n replace this := single ((\u2119).map UB) hy this\n rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n have hyy : By = B := by\n have h : hnBy.some \u2208 By := hnBy.some_mem\n simp [hBy] at h \u22a2\n have := hBBy h.1\n simp [h.2] at this\n intro z hz\n simp [this]\n convert hBBy hz\n\n simp [hxx, hyy]\n\n have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n congr\n convert (four_logs ?_ ?_ ?_ ?_).symm\n all_goals norm_cast; exact Nat.card_pos\n _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n apply mul_le_mul_of_nonneg_right hH2\n have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n rw [\u2190 mul_le_mul_left this]\n apply le_trans _ hcard_ineq\n rw [mul_zero]\n change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n apply Real.log_nonneg\n rw [one_le_div]\n gcongr\n . apply Nat.card_mono\n . exact Set.toFinite A\n rw [hAx]; exact Set.inter_subset_left _ _\n apply Nat.card_mono\n . exact Set.toFinite B\n rw [hBy]; exact Set.inter_subset_left _ _\n norm_cast\n exact mul_pos Nat.card_pos Nat.card_pos\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n {H: AddSubgroup G} (A : Set H) (x : G) :\n dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n classical\n rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n rw [\u2190 hrank]\n convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n . apply LinearEquiv.finrank_eq\n exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n intro a ha\n rw [Set.mem_image] at ha\n rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n rw [Submodule.mem_map]\n use b - v, hshift b hb\n simp [\u2190 hb']\n abel\n\nlemma conclusion_transfers {A B : Set G}\n (G': AddSubgroup G) (A' B' : Set G')\n (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B := by\n have : Nonempty A' := hA'.to_subtype\n have : Nonempty B' := hB'.to_subtype\n rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n rcases hA with \u27e8 x, hA \u27e9\n set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n have hf : Function.Injective f := by\n intro y z hyz\n simp [f] at hyz\n exact hyz\n have hA' : A = f '' A' := by\n simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n rcases hB with \u27e8 y, hB \u27e9\n set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n have hg : Function.Injective g := by\n intro y z hyz\n simp [g] at hyz\n exact hyz\n have hB' : B = g '' B' := by\n simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n use f '' A'', g '' B''\n have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n congr\n . rw [hA]\n ext y\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : x + z + -x \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use y + -x\n constructor\n . use h\n abel\n rw [hB]\n ext x\n simp [Set.mem_vadd_set]\n constructor\n . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n have : y + z + -y \u2208 G' := by simp [w]\n use this\n simp\n convert hw\n rintro \u27e8 h, ha \u27e9\n use x + -y\n constructor\n . use h\n abel\n\n\n refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n . simp [hA', hf, hA'']\n . simp [hB', hg, hB'']\n . simp [hA''_non]\n . simp [hB''_non]\n . convert hcard_ineq using 2\n . congr 3\n . rw [hA', Nat.card_image_of_injective hf]\n . rw [hB', Nat.card_image_of_injective hg]\n . rw [Nat.card_image_of_injective hf]\n rw [Nat.card_image_of_injective hg]\n convert LE.le.trans _ hdim_ineq using 2\n norm_cast\n apply max_le_max\n . exact dimension_of_shift A'' x\n \u00b7 exact dimension_of_shift B'' y\n\n/-- If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that\n\\[\\log\\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A'\\rvert\\lvert B'\\rvert}\\leq 34 d[U_A;U_B]\\]\nsuch that $\\max(\\dim A',\\dim B')\\leq \\frac{40}{\\log 2} d[U_A;U_B]$. -/\nlemma weak_PFR_asymm (A B : Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) : WeakPFRAsymmConclusion A B := by\n let P : \u2115 \u2192 Prop := fun M \u21a6 (\u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G) (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G) (_hG_sing: MeasurableSingletonClass G) (A B: Set G) (_hA_fin: Finite A) (_hB_fin: Finite B) (_hA_non: A.Nonempty) (_hB_non: B.Nonempty) (_hM : (Nat.card A) + (Nat.card B) \u2264 M), WeakPFRAsymmConclusion A B)\n suffices \u2200 M, (\u2200 M', M' < M \u2192 P M') \u2192 P M by\n set M := (Nat.card A) + (Nat.card B)\n have hM : (Nat.card A) + (Nat.card B) \u2264 M := Nat.le_refl _\n convert (Nat.strong_induction_on (p := P) M this) G \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a _ \u2039_\u203a A B \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a hM\n intro M h_induct\n -- wlog we can assume A, B are not in cosets of a smaller subgroup\n suffices \u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G)\n (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)\n (_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B) (_hA_non : A.Nonempty) (_hB_non : B.Nonempty) (_hM : Nat.card A + Nat.card B \u2264 M)\n (_hnot : NotInCoset A B), WeakPFRAsymmConclusion A B by\n intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM\n obtain \u27e8 G', A', B', hAA', hBB', hnot' \u27e9 := wlog_notInCoset hA_non hB_non\n have hG'_fin : Module.Finite \u2124 G' :=\n Module.Finite.iff_fg (N := AddSubgroup.toIntSubmodule G').2 (IsNoetherian.noetherian _)\n\n have hG'_free : Module.Free \u2124 G' := by\n rcases Submodule.nonempty_basis_of_pid (Module.Free.chooseBasis \u2124 G) (AddSubgroup.toIntSubmodule G') with \u27e8 n, \u27e8 b \u27e9 \u27e9\n exact Module.Free.of_basis b\n have hAA'_card: Nat.card A = Nat.card A' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hAA'.card_congr\n have hBB'_card: Nat.card B = Nat.card B' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hBB'.card_congr\n have hA_non' : Nonempty A := Set.nonempty_coe_sort.mpr hA_non\n have hB_non' : Nonempty B := Set.nonempty_coe_sort.mpr hB_non\n\n rw [hAA'_card, hBB'_card] at hM\n\n have hA'_nonfin : A'.Nonempty \u2227 Finite A' := by\n have := Nat.card_pos (\u03b1 := A)\n rw [hAA'_card, Nat.card_pos_iff] at this\n exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n have hB'_nonfin : B'.Nonempty \u2227 Finite B' := by\n have := Nat.card_pos (\u03b1 := B)\n rw [hBB'_card, Nat.card_pos_iff] at this\n exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n obtain \u27e8 hA'_non, hA'_fin \u27e9 := hA'_nonfin\n obtain \u27e8 hB'_non, hB'_fin \u27e9 := hB'_nonfin\n\n replace this := this G' _ hG'_free hG'_fin (by infer_instance) (by infer_instance) (by infer_instance) A' B' hA'_fin hB'_fin hA'_non hB'_non hM hnot'\n exact conclusion_transfers G' A' B' hAA' hBB' hA'_non hB'_non this\n intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM hnot\n rcases weak_PFR_asymm_prelim A B hA_non hB_non with \u27e8 N, x, y, Ax, By, hAx_non, hBy_non, hAx_fin, hBy_fin, hAx, hBy, hdim, hcard\u27e9\n have hAxA : Ax \u2286 A := by rw [hAx]; simp\n have hByB : By \u2286 B := by rw [hBy]; simp\n have hA_pos : (0 : \u211d) < Nat.card A := Nat.cast_pos.mpr (@Nat.card_pos _ hA_non.to_subtype _)\n have hB_pos : (0 : \u211d) < Nat.card B := Nat.cast_pos.mpr (@Nat.card_pos _ hB_non.to_subtype _)\n\n rcases lt_or_ge (Nat.card Ax + Nat.card By) (Nat.card A + Nat.card B) with h | h\n . replace h := h_induct (Nat.card Ax + Nat.card By) (h.trans_le hM) G hG_comm hG_free hG_fin hG_count hG_mes hG_sing Ax By (Set.finite_coe_iff.mpr hAx_fin) (Set.finite_coe_iff.mpr hBy_fin) hAx_non hBy_non (Eq.le rfl)\n rcases h with \u27e8 A', B', hA', hB', hA'_non, hB'_non, hcard_ineq, hdim_ineq \u27e9\n use A', B'\n have hAx_fin' := Set.finite_coe_iff.mpr hAx_fin\n have hBy_fin' := Set.finite_coe_iff.mpr hBy_fin\n have hA'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hAx_fin hA')\n have hB'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hBy_fin hB')\n have hAx_non' := Set.nonempty_coe_sort.mpr hAx_non\n have hBy_non' := Set.nonempty_coe_sort.mpr hBy_non\n have hA'_non' := Set.nonempty_coe_sort.mpr hA'_non\n have hB'_non' := Set.nonempty_coe_sort.mpr hB'_non\n have hAx_pos : (0 : \u211d) < Nat.card Ax := Nat.cast_pos.mpr Nat.card_pos\n have hBy_pos : (0 : \u211d) < Nat.card By := Nat.cast_pos.mpr Nat.card_pos\n have hA'_pos : (0 : \u211d) < Nat.card A' := Nat.cast_pos.mpr Nat.card_pos\n have hB'_pos : (0 : \u211d) < Nat.card B' := Nat.cast_pos.mpr Nat.card_pos\n have hAxA_le : (Nat.card Ax : \u211d) \u2264 (Nat.card A : \u211d) := Nat.cast_le.mpr (Nat.card_mono A.toFinite hAxA)\n have hByB_le : (Nat.card By : \u211d) \u2264 (Nat.card B : \u211d) := Nat.cast_le.mpr (Nat.card_mono B.toFinite hByB)\n\n refine \u27e8 hA'.trans hAxA, hB'.trans hByB, hA'_non, hB'_non, ?_, ?_ \u27e9\n . rw [four_logs hA_pos hB_pos hA'_pos hB'_pos]\n rw [four_logs hAx_pos hBy_pos hA'_pos hB'_pos] at hcard_ineq\n linarith only [hcard, hcard_ineq]\n apply hdim_ineq.trans\n gcongr\n linarith only [Real.log_le_log hAx_pos hAxA_le, Real.log_le_log hBy_pos hByB_le, hcard]\n use A, B\n refine \u27e8 Eq.subset rfl, Eq.subset rfl, hA_non, hB_non, ?_, ?_ \u27e9\n . have := hA_non.to_subtype\n have := hB_non.to_subtype\n apply LE.le.trans _ <| mul_nonneg (by norm_num) <| rdist_set_nonneg A B\n rw [div_self (by positivity)]\n simp\n have hAx_eq : Ax = A := by\n apply Set.Finite.eq_of_subset_of_card_le A.toFinite hAxA\n linarith only [h, Nat.card_mono B.toFinite hByB]\n have hBy_eq : By = B := by\n apply Set.Finite.eq_of_subset_of_card_le B.toFinite hByB\n linarith only [h, Nat.card_mono A.toFinite hAxA]\n have hN : N = \u22a4 := by\n have : (A-A) \u222a (B-B) \u2286 N := by\n rw [\u2190 hAx_eq, \u2190 hBy_eq, hAx, hBy]\n intro z hz\n simp [Set.mem_sub] at hz\n convert (QuotientAddGroup.eq_zero_iff z).mp ?_\n . infer_instance\n rcases hz with \u27e8 a, \u27e8 -, ha\u27e9, a', \u27e8-, ha'\u27e9, haa' \u27e9 | \u27e8 b, \u27e8 -, hb\u27e9, b', \u27e8 -,hb'\u27e9, hbb' \u27e9\n . rw [\u2190 haa']; simp [ha, ha']\n rw [\u2190 hbb']; simp [hb, hb']\n rw [\u2190 AddSubgroup.closure_le, hnot] at this\n exact top_le_iff.mp this\n have : Nat.card (G \u29f8 N) = 1 := by\n rw [Nat.card_eq_one_iff_unique]\n constructor\n . rw [hN]\n exact QuotientAddGroup.subsingleton_quotient_top\n infer_instance\n simp [this] at hdim\n rw [\u2190 le_div_iff' (by positivity)] at hdim\n convert LE.le.trans ?_ hdim using 1\n . field_simp\n simp\n exact \u27e8 dimension_le_rank A, dimension_le_rank B \u27e9\n\n/-- If $A\\subseteq \\mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\\leq \\log K$ then there exists a non-empty $A'\\subseteq A$ such that\n$\\lvert A'\\rvert\\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A'\\leq \\frac{40}{\\log 2} \\log K$. -/\nlemma weak_PFR {A : Set G} [Finite A] {K : \u211d} (hA : A.Nonempty) (hK: 0 < K) (hdist: d\u1d64[A # A] \u2264 log K):\n \u2203 A' : Set G, A' \u2286 A \u2227 (Nat.card A') \u2265 K^(-17 : \u211d) * (Nat.card A)\n \u2227 (dimension A') \u2264 (40 / log 2) * log K := by\n rcases weak_PFR_asymm A A hA hA with \u27e8A', A'', hA', hA'', hA'nonempty, hA''nonempty, hcard, hdim\u27e9\n\n have : \u2203 B : Set G, B \u2286 A \u2227 (Nat.card B) \u2265 (Nat.card A') \u2227 (Nat.card B) \u2265 (Nat.card A'') \u2227 (dimension B) \u2264\nmax (dimension A') (dimension A'') := by\n rcases lt_or_ge (Nat.card A') (Nat.card A'') with h | h\n . use A''\n exact \u27e8hA'', by linarith, by linarith, le_max_right _ _\u27e9\n use A'\n exact \u27e8hA', by linarith, by linarith, le_max_left _ _\u27e9\n\n rcases this with \u27e8B, hB, hBcard, hBcard', hBdim\u27e9\n use B\n have hApos : Nat.card A > 0 := by\n rw [gt_iff_lt, Nat.card_pos_iff]\n exact \u27e8hA.to_subtype, inferInstance\u27e9\n have hA'pos : Nat.card A' > 0 := by\n rw [gt_iff_lt, Nat.card_pos_iff]\n refine \u27e8 hA'nonempty.to_subtype, Finite.Set.subset _ hA' \u27e9\n have hA''pos : Nat.card A'' > 0 := by\n rw [gt_iff_lt, Nat.card_pos_iff]\n refine \u27e8 hA''nonempty.to_subtype, Finite.Set.subset _ hA'' \u27e9\n have hBpos : Nat.card B > 0 := by linarith\n\n refine \u27e8hB, ?_, ?_\u27e9\n . have := calc 2 * log ((Nat.card A) / (Nat.card B))\n _ = log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card B) * (Nat.card B)) ) := by\n convert (log_pow (((Nat.card A) : \u211d)/(Nat.card B)) 2).symm\n field_simp\n rw [\u2190 pow_two, \u2190 pow_two]\n _ \u2264 log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card A') * (Nat.card A'')) ) := by\n apply log_le_log\n . positivity\n gcongr\n _ \u2264 34 * d\u1d64[A # A] := hcard\n _ \u2264 34 * log K := mul_le_mul_of_nonneg_left hdist (by linarith)\n _ = 2 * (17 * log K) := by ring\n _ = 2 * log (K^17) := by\n congr\n convert (log_pow K 17).symm\n rw [mul_le_mul_left (by norm_num), log_le_log_iff (by positivity) (by positivity), div_le_iff (by positivity), <- mul_inv_le_iff (by positivity), <-ge_iff_le, mul_comm] at this\n convert this using 2\n convert zpow_neg K 17 using 1\n norm_cast\n calc (dimension B : \u211d)\n _ \u2264 (((max (dimension A') (dimension A'')) : \u2115) : \u211d) := by norm_cast\n _ \u2264 (40 / log 2) * d\u1d64[A # A] := hdim\n _ \u2264 (40 / log 2) * log K := mul_le_mul_of_nonneg_left hdist (by positivity)\n\n", "theoremStatement": "/-- Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A-A\\rvert\\leq K\\lvert A\\rvert$.\nThere exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A' \\leq \\frac{40}{\\log 2} \\log K$.-/\ntheorem weak_PFR_int {A : Set G} [Finite A] (hnA : A.Nonempty) {K : \u211d} (hK : 0 < K)\n (hA: Nat.card (A-A) \u2264 K * Nat.card A) :\n \u2203 A' : Set G, A' \u2286 A \u2227 Nat.card A' \u2265 K ^ (-17 : \u211d) * (Nat.card A) \u2227\n dimension A' \u2264 (40 / log 2) * log K ", "theoremName": "weak_PFR_int", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 1228, "tokenPositionInFile": 63501, "theoremPositionInFile": 30}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 243, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply weak_PFR hnA hK ((rdist_set_le A A hnA hnA).trans _)\n suffices log (Nat.card (A-A)) \u2264 log K + log (Nat.card A) by linarith\n rw [\u2190 log_mul (by positivity) _]\n . apply log_le_log _ hA\n norm_cast\n have : Nonempty (A-A) := by\n exact Set.Nonempty.coe_sort (Set.Nonempty.sub hnA hnA)\n apply Nat.card_pos\n norm_cast\n apply ne_of_gt (@Nat.card_pos _ hnA.to_subtype _)", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 392}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\n", "theoremStatement": "lemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) ", "theoremName": "continuous_tau_restrict_probabilityMeasure", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "6854e2a", "date": "2023-11-20"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 72, "tokenPositionInFile": 2703, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 10, "numPremises": 63, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 618}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n", "theoremStatement": "/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] ", "theoremName": "ProbabilityTheory.IdentDistrib.tau_eq", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 87, "tokenPositionInFile": 3541, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 30, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 167}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n", "theoremStatement": "/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n [MeasureSpace \u03a9']\n {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' ", "theoremName": "ProbabilityTheory.IdentDistrib.tau_minimizes", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 106, "tokenPositionInFile": 4692, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 25, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 54}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n [MeasureSpace \u03a9']\n {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n", "theoremStatement": "/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\n \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] ", "theoremName": "tau_min_exists_measure", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "96a97b5", "date": "2023-11-27"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 114, "tokenPositionInFile": 5090, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 88, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n have : DiscreteTopology G := \u27e8rfl\u27e9\n have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n rw [isMinOn_univ_iff] at h\u03bc\n let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n exact h\u03bc \u03bd", "proofType": "tactic", "proofLengthLines": 15, "proofLengthTokens": 964}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n [MeasureSpace \u03a9']\n {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\n \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n have : DiscreteTopology G := \u27e8rfl\u27e9\n have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n rw [isMinOn_univ_iff] at h\u03bc\n let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n exact h\u03bc \u03bd\n\n", "theoremStatement": "/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227\n tau_minimizes p X\u2081 X\u2082 ", "theoremName": "tau_minimizer_exists", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "13e55a8", "date": "2023-11-28"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 136, "tokenPositionInFile": 6414, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 50, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let \u03bc := (tau_min_exists_measure p).choose\n have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 653}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n [MeasureSpace \u03a9']\n {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\n \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n have : DiscreteTopology G := \u27e8rfl\u27e9\n have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n rw [isMinOn_univ_iff] at h\u03bc\n let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227\n tau_minimizes p X\u2081 X\u2082 := by\n let \u03bc := (tau_min_exists_measure p).choose\n have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\n", "theoremStatement": "lemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] ", "theoremName": "is_tau_min", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "828bcd7", "date": "2023-11-20"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 158, "tokenPositionInFile": 7617, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 31, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n (isProbabilityMeasure_map h2.aemeasurable)", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 340}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n [MeasureSpace \u03a9']\n {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\n \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n have : DiscreteTopology G := \u27e8rfl\u27e9\n have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n rw [isMinOn_univ_iff] at h\u03bc\n let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227\n tau_minimizes p X\u2081 X\u2082 := by\n let \u03bc := (tau_min_exists_measure p).choose\n have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\nlemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] := by\n let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n (isProbabilityMeasure_map h2.aemeasurable)\n\n", "theoremStatement": "/-- Let `X\u2081` and `X\u2082` be tau-minimizers associated to `p`, with $d[X_1,X_2]=k$, then\n$$ d[X'_1;X'_2] \\geq\n k - \\eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2] - d[X^0_2;X_2] )$$\nfor any $G$-valued random variables $X'_1,X'_2$.\n-/\nlemma distance_ge_of_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081] - d[p.X\u2080\u2081 # X\u2081]) - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082] - d[p.X\u2080\u2082 # X\u2082])\n \u2264 d[X'\u2081 # X'\u2082] ", "theoremName": "distance_ge_of_min", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 167, "tokenPositionInFile": 8088, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 135, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have Z := is_tau_min p h h1 h2\n simp [tau] at Z\n linarith", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 67}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n [MeasureSpace \u03a9']\n {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\n \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n have : DiscreteTopology G := \u27e8rfl\u27e9\n have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n rw [isMinOn_univ_iff] at h\u03bc\n let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227\n tau_minimizes p X\u2081 X\u2082 := by\n let \u03bc := (tau_min_exists_measure p).choose\n have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\nlemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] := by\n let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n (isProbabilityMeasure_map h2.aemeasurable)\n\n/-- Let `X\u2081` and `X\u2082` be tau-minimizers associated to `p`, with $d[X_1,X_2]=k$, then\n$$ d[X'_1;X'_2] \\geq\n k - \\eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2] - d[X^0_2;X_2] )$$\nfor any $G$-valued random variables $X'_1,X'_2$.\n-/\nlemma distance_ge_of_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081] - d[p.X\u2080\u2081 # X\u2081]) - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082] - d[p.X\u2080\u2082 # X\u2082])\n \u2264 d[X'\u2081 # X'\u2082] := by\n have Z := is_tau_min p h h1 h2\n simp [tau] at Z\n linarith\n\n", "theoremStatement": "/-- Version of `distance_ge_of_min` with the measures made explicit. -/\nlemma distance_ge_of_min' {\u03a9'\u2081 \u03a9'\u2082 : Type*} (h : tau_minimizes p X\u2081 X\u2082)\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082] {\u03bc : Measure \u03a9'\u2081} {\u03bc' : Measure \u03a9'\u2082}\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] {X'\u2081: \u03a9'\u2081 \u2192 G} {X'\u2082: \u03a9'\u2082 \u2192 G}\n (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081; \u2119 # X'\u2081; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082; \u2119 # X'\u2082; \u03bc'] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081; \u03bc # X'\u2082; \u03bc'] ", "theoremName": "distance_ge_of_min'", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 179, "tokenPositionInFile": 8618, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 25, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n set M1 : MeasureSpace \u03a9'\u2081 := { volume := \u03bc }\n set M2 : MeasureSpace \u03a9'\u2082 := { volume := \u03bc' }\n exact distance_ge_of_min p h h1 h2", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 137}} +{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable (\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*) [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n[IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n /-- The first variable in a package. -/\n X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n /-- The second variable in a package. -/\n X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n hmeas1 : Measurable X\u2080\u2081\n hmeas2 : Measurable X\u2080\u2082\n \u03b7 : \u211d\n h\u03b7 : 0 < \u03b7\n h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n continuous_rdist_restrict_probabilityMeasure\n have obs\u2083 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n simp only [tau]\n rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n [MeasureSpace \u03a9']\n {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\n \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n have : DiscreteTopology G := \u27e8rfl\u27e9\n have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n rw [isMinOn_univ_iff] at h\u03bc\n let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227\n tau_minimizes p X\u2081 X\u2082 := by\n let \u03bc := (tau_min_exists_measure p).choose\n have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\nlemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] := by\n let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n (isProbabilityMeasure_map h2.aemeasurable)\n\n/-- Let `X\u2081` and `X\u2082` be tau-minimizers associated to `p`, with $d[X_1,X_2]=k$, then\n$$ d[X'_1;X'_2] \\geq\n k - \\eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2] - d[X^0_2;X_2] )$$\nfor any $G$-valued random variables $X'_1,X'_2$.\n-/\nlemma distance_ge_of_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081] - d[p.X\u2080\u2081 # X\u2081]) - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082] - d[p.X\u2080\u2082 # X\u2082])\n \u2264 d[X'\u2081 # X'\u2082] := by\n have Z := is_tau_min p h h1 h2\n simp [tau] at Z\n linarith\n\n/-- Version of `distance_ge_of_min` with the measures made explicit. -/\nlemma distance_ge_of_min' {\u03a9'\u2081 \u03a9'\u2082 : Type*} (h : tau_minimizes p X\u2081 X\u2082)\n [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082] {\u03bc : Measure \u03a9'\u2081} {\u03bc' : Measure \u03a9'\u2082}\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] {X'\u2081: \u03a9'\u2081 \u2192 G} {X'\u2082: \u03a9'\u2082 \u2192 G}\n (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081; \u2119 # X'\u2081; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082; \u2119 # X'\u2082; \u03bc'] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081; \u03bc # X'\u2082; \u03bc'] := by\n set M1 : MeasureSpace \u03a9'\u2081 := { volume := \u03bc }\n set M2 : MeasureSpace \u03a9'\u2082 := { volume := \u03bc' }\n exact distance_ge_of_min p h h1 h2\n\n\nopen BigOperators\n\n", "theoremStatement": "/-- For any $G$-valued random variables $X'_1,X'_2$ and random variables $Z,W$, one can lower\nbound $d[X'_1|Z;X'_2|W]$ by\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] ).$$\n-/\nlemma condRuzsaDistance_ge_of_min [MeasurableSingletonClass G]\n [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T]\n (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082)\n (Z : \u03a9'\u2081 \u2192 S) (W : \u03a9'\u2082 \u2192 T) (hZ : Measurable Z) (hW : Measurable W) :\n d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081 | Z] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082 | W] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081 | Z # X'\u2082 | W] ", "theoremName": "condRuzsaDistance_ge_of_min", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "12546ab", "date": "2023-12-20"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 193, "tokenPositionInFile": 9279, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 15, "numPremises": 196, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hz (a : \u211d) : a = \u2211 z in FiniteRange.toFinset Z, (\u2119 (Z \u207b\u00b9' {z})).toReal * a := by\n simp_rw [\u2190 Finset.sum_mul,\u2190 Measure.map_apply hZ (MeasurableSet.singleton _), Finset.sum_toReal_measure_singleton]\n rw [FiniteRange.full hZ]\n simp\n have hw (a : \u211d) : a = \u2211 w in FiniteRange.toFinset W, (\u2119 (W \u207b\u00b9' {w})).toReal * a := by\n simp_rw [\u2190 Finset.sum_mul,\u2190 Measure.map_apply hW (MeasurableSet.singleton _), Finset.sum_toReal_measure_singleton]\n rw [FiniteRange.full hW]\n simp\n rw [condRuzsaDist_eq_sum h1 hZ h2 hW, condRuzsaDist'_eq_sum h1 hZ, hz d[X\u2081 # X\u2082],\n hz d[p.X\u2080\u2081 # X\u2081], hz (p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082 | W] - d[p.X\u2080\u2082 # X\u2082])),\n \u2190 Finset.sum_sub_distrib, Finset.mul_sum, \u2190 Finset.sum_sub_distrib, \u2190 Finset.sum_sub_distrib]\n apply Finset.sum_le_sum\n intro z _\n rw [condRuzsaDist'_eq_sum h2 hW, hw d[p.X\u2080\u2082 # X\u2082],\n hw ((\u2119 (Z \u207b\u00b9' {z})).toReal * d[X\u2081 # X\u2082] - p.\u03b7 * ((\u2119 (Z \u207b\u00b9' {z})).toReal *\n d[p.X\u2080\u2081 ; \u2119 # X'\u2081 ; \u2119[|Z \u2190 z]] - (\u2119 (Z \u207b\u00b9' {z})).toReal * d[p.X\u2080\u2081 # X\u2081])),\n \u2190 Finset.sum_sub_distrib, Finset.mul_sum, Finset.mul_sum, \u2190 Finset.sum_sub_distrib]\n apply Finset.sum_le_sum\n intro w _\n rcases eq_or_ne (\u2119 (Z \u207b\u00b9' {z})) 0 with hpz | hpz\n . simp [hpz]\n rcases eq_or_ne (\u2119 (W \u207b\u00b9' {w})) 0 with hpw | hpw\n . simp [hpw]\n set \u03bc := (h\u03a9\u2081.volume)[|Z \u2190 z]\n have h\u03bc : IsProbabilityMeasure \u03bc := cond_isProbabilityMeasure \u2119 hpz\n set \u03bc' := \u2119[|W \u2190 w]\n have h\u03bc' : IsProbabilityMeasure \u03bc' := cond_isProbabilityMeasure \u2119 hpw\n suffices d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081; volume # X'\u2081; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) -\n p.\u03b7 * (d[p.X\u2080\u2082; volume # X'\u2082; \u03bc'] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081 ; \u03bc # X'\u2082; \u03bc'] by\n replace this := mul_le_mul_of_nonneg_left this (show 0 \u2264 (\u2119 (Z \u207b\u00b9' {z})).toReal * (\u2119 (W \u207b\u00b9' {w})).toReal by positivity)\n convert this using 1\n ring\n exact distance_ge_of_min' p h h1 h2", "proofType": "tactic", "proofLengthLines": 33, "proofLengthTokens": 1805}} +{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n ext \u03c9; simp [sub_eq_zero, eq_comm]\n rw [this, measureReal_iUnion_fintype _\n (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n \u00b7 intro g g' hgg'\n apply Set.disjoint_iff_inter_eq_empty.2\n ext a\n simp (config := {contextual := True}) [hgg']\n classical\n let W : Finset G := A \u2229 B\n calc\n \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply sum_congr _ _ (fun g \u21a6 ?_)\n rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n measurableSet_discrete _]\n _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply (Finset.sum_subset W.subset_univ _).symm\n intro i _ hi\n replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n rcases hi with h'i|h'i\n \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n congr\n rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n classical\n let W := fun \u03c9 \u21a6 V \u03c9 + x\n have Wunif : IsUniform (B + {x} : Set G) W := by\n convert Vunif.comp (add_left_injective x)\n simp\n have Wmeas : Measurable W := Vmeas.add_const _\n have UWindep : IndepFun U W := by\n have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n exact hindep.comp measurable_id this\n have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n ext \u03c9\n simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n abel_nf\n have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n rw [h] at Wunif\n rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n congr 3\n . rw [add_singleton]; simp\n convert Finset.card_vadd_finset (AddOpposite.op x) B\n . simp\n simp\n\nend ProbabilityTheory\n\n\n", "theoremStatement": "/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K ", "theoremName": "PFR_conjecture_pos_aux", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2ac9116", "date": "2023-12-26"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 100, "tokenPositionInFile": 4248, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 98, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 522}} +{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n ext \u03c9; simp [sub_eq_zero, eq_comm]\n rw [this, measureReal_iUnion_fintype _\n (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n \u00b7 intro g g' hgg'\n apply Set.disjoint_iff_inter_eq_empty.2\n ext a\n simp (config := {contextual := True}) [hgg']\n classical\n let W : Finset G := A \u2229 B\n calc\n \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply sum_congr _ _ (fun g \u21a6 ?_)\n rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n measurableSet_discrete _]\n _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply (Finset.sum_subset W.subset_univ _).symm\n intro i _ hi\n replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n rcases hi with h'i|h'i\n \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n congr\n rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n classical\n let W := fun \u03c9 \u21a6 V \u03c9 + x\n have Wunif : IsUniform (B + {x} : Set G) W := by\n convert Vunif.comp (add_left_injective x)\n simp\n have Wmeas : Measurable W := Vmeas.add_const _\n have UWindep : IndepFun U W := by\n have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n exact hindep.comp measurable_id this\n have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n ext \u03c9\n simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n abel_nf\n have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n rw [h] at Wunif\n rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n congr 3\n . rw [add_singleton]; simp\n convert Finset.card_vadd_finset (AddOpposite.op x) B\n . simp\n simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\n", "theoremStatement": "lemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K ", "theoremName": "PFR_conjecture_pos_aux'", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2ac9116", "date": "2023-12-26"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 112, "tokenPositionInFile": 5055, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 99, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 522}} +{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n ext \u03c9; simp [sub_eq_zero, eq_comm]\n rw [this, measureReal_iUnion_fintype _\n (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n \u00b7 intro g g' hgg'\n apply Set.disjoint_iff_inter_eq_empty.2\n ext a\n simp (config := {contextual := True}) [hgg']\n classical\n let W : Finset G := A \u2229 B\n calc\n \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply sum_congr _ _ (fun g \u21a6 ?_)\n rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n measurableSet_discrete _]\n _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply (Finset.sum_subset W.subset_univ _).symm\n intro i _ hi\n replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n rcases hi with h'i|h'i\n \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n congr\n rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n classical\n let W := fun \u03c9 \u21a6 V \u03c9 + x\n have Wunif : IsUniform (B + {x} : Set G) W := by\n convert Vunif.comp (add_left_injective x)\n simp\n have Wmeas : Measurable W := Vmeas.add_const _\n have UWindep : IndepFun U W := by\n have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n exact hindep.comp measurable_id this\n have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n ext \u03c9\n simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n abel_nf\n have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n rw [h] at Wunif\n rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n congr 3\n . rw [add_singleton]; simp\n convert Finset.card_vadd_finset (AddOpposite.op x) B\n . simp\n simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n", "theoremStatement": "/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K ", "theoremName": "rdist_le_of_isUniform_of_card_add_le", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2ac9116", "date": "2023-12-26"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 126, "tokenPositionInFile": 5928, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 263, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8A_pos, AA_pos, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n have IU : d[U # U'] \u2264 log K := by\n have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n . simp\n exact Iff.rfl\n . measurability\n filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n simp\n exact Set.sub_mem_sub h1 h2\n have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n apply (log_le_log AA_pos hA).trans (le_of_eq _)\n rw [log_mul K_pos.ne' A_pos.ne']\n-- have : H[U + U'] = H[U - U'] := by congr; simp\n rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n linarith\n rwa [idU.rdist_eq idU'] at IU", "proofType": "tactic", "proofLengthLines": 21, "proofLengthTokens": 1122}} +{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n ext \u03c9; simp [sub_eq_zero, eq_comm]\n rw [this, measureReal_iUnion_fintype _\n (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n \u00b7 intro g g' hgg'\n apply Set.disjoint_iff_inter_eq_empty.2\n ext a\n simp (config := {contextual := True}) [hgg']\n classical\n let W : Finset G := A \u2229 B\n calc\n \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply sum_congr _ _ (fun g \u21a6 ?_)\n rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n measurableSet_discrete _]\n _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply (Finset.sum_subset W.subset_univ _).symm\n intro i _ hi\n replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n rcases hi with h'i|h'i\n \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n congr\n rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n classical\n let W := fun \u03c9 \u21a6 V \u03c9 + x\n have Wunif : IsUniform (B + {x} : Set G) W := by\n convert Vunif.comp (add_left_injective x)\n simp\n have Wmeas : Measurable W := Vmeas.add_const _\n have UWindep : IndepFun U W := by\n have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n exact hindep.comp measurable_id this\n have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n ext \u03c9\n simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n abel_nf\n have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n rw [h] at Wunif\n rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n congr 3\n . rw [add_singleton]; simp\n convert Finset.card_vadd_finset (AddOpposite.op x) B\n . simp\n simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n obtain \u27e8A_pos, AA_pos, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n have IU : d[U # U'] \u2264 log K := by\n have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n . simp\n exact Iff.rfl\n . measurability\n filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n simp\n exact Set.sub_mem_sub h1 h2\n have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n apply (log_le_log AA_pos hA).trans (le_of_eq _)\n rw [log_mul K_pos.ne' A_pos.ne']\n-- have : H[U + U'] = H[U - U'] := by congr; simp\n rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n linarith\n rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\n", "theoremStatement": "lemma sumset_eq_sub : A + A = A - A ", "theoremName": "sumset_eq_sub", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "93e5ee0", "date": "2024-01-12"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 155, "tokenPositionInFile": 7476, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 37, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n congr! 1 with a _ b _\n show a + b = a - b\n simp", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 99}} +{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n ext \u03c9; simp [sub_eq_zero, eq_comm]\n rw [this, measureReal_iUnion_fintype _\n (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n \u00b7 intro g g' hgg'\n apply Set.disjoint_iff_inter_eq_empty.2\n ext a\n simp (config := {contextual := True}) [hgg']\n classical\n let W : Finset G := A \u2229 B\n calc\n \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply sum_congr _ _ (fun g \u21a6 ?_)\n rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n measurableSet_discrete _]\n _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply (Finset.sum_subset W.subset_univ _).symm\n intro i _ hi\n replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n rcases hi with h'i|h'i\n \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n congr\n rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n classical\n let W := fun \u03c9 \u21a6 V \u03c9 + x\n have Wunif : IsUniform (B + {x} : Set G) W := by\n convert Vunif.comp (add_left_injective x)\n simp\n have Wmeas : Measurable W := Vmeas.add_const _\n have UWindep : IndepFun U W := by\n have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n exact hindep.comp measurable_id this\n have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n ext \u03c9\n simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n abel_nf\n have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n rw [h] at Wunif\n rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n congr 3\n . rw [add_singleton]; simp\n convert Finset.card_vadd_finset (AddOpposite.op x) B\n . simp\n simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n obtain \u27e8A_pos, AA_pos, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n have IU : d[U # U'] \u2264 log K := by\n have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n . simp\n exact Iff.rfl\n . measurability\n filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n simp\n exact Set.sub_mem_sub h1 h2\n have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n apply (log_le_log AA_pos hA).trans (le_of_eq _)\n rw [log_mul K_pos.ne' A_pos.ne']\n-- have : H[U + U'] = H[U - U'] := by congr; simp\n rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n linarith\n rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\nlemma sumset_eq_sub : A + A = A - A := by\n rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n congr! 1 with a _ b _\n show a + b = a - b\n simp\n\n", "theoremStatement": "/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\nlemma PFR_conjecture_aux (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card H \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card H \u2227 A \u2286 c + H ", "theoremName": "PFR_conjecture_aux", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "b94162e", "date": "2023-11-28"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 161, "tokenPositionInFile": 7613, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 33, "numPremises": 483, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n classical\n let _mG : MeasurableSpace G := \u22a4\n rw [sumset_eq_sub] at hA\n have : MeasurableSingletonClass G := \u27e8\u03bb _ \u21a6 trivial\u27e9\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n let A' := A.toFinite.toFinset\n have h\u2080A' : Finset.Nonempty A' := by\n simp [A', Finset.Nonempty]\n exact h\u2080A\n have hAA' : A' = A := Finite.coe_toFinset (toFinite A)\n rcases exists_isUniform_measureSpace A' h\u2080A' with \u27e8\u03a9\u2080, m\u03a9\u2080, UA, hP\u2080, UAmeas, UAunif, -, -\u27e9\n rw [hAA'] at UAunif\n have : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n rw [\u2190 sumset_eq_sub] at hA\n let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/9, (by norm_num), (by norm_num)\u27e9\n -- entropic PFR gives a subgroup `H` which is close to `A` for the Rusza distance\n rcases entropic_PFR_conjecture p (by norm_num) with \u27e8H, \u03a9\u2081, m\u03a9\u2081, UH, hP\u2081, UHmeas, UHunif, hUH\u27e9\n rcases independent_copies_two UAmeas UHmeas\n with \u27e8\u03a9, m\u03a9, VA, VH, hP, VAmeas, VHmeas, Vindep, idVA, idVH\u27e9\n have VAunif : IsUniform A VA := UAunif.of_identDistrib idVA.symm $ measurableSet_discrete _\n have VA'unif := VAunif\n rw [\u2190 hAA'] at VA'unif\n have VHunif : IsUniform H VH := UHunif.of_identDistrib idVH.symm $ measurableSet_discrete _\n let H' := (H:Set G).toFinite.toFinset\n have hHH' : H' = (H:Set G) := Finite.coe_toFinset (toFinite (H:Set G))\n have VH'unif := VHunif\n rw [\u2190 hHH'] at VH'unif\n\n have : d[VA # VH] \u2264 11/2 * log K := by rw [idVA.rdist_eq idVH]; linarith\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos\n positivity\n have VA_ent : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have VH_ent : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n have Icard : |log (Nat.card A) - log (Nat.card (H : Set G))| \u2264 11 * log K := by\n rw [\u2190 VA_ent, \u2190 VH_ent]\n apply (diff_ent_le_rdist VAmeas VHmeas).trans\n linarith\n have IAH : Nat.card A \u2264 K ^ 11 * Nat.card (H : Set G) := by\n have : log (Nat.card A) \u2264 log K * 11 + log (Nat.card (H : Set G)) := by\n linarith [(le_abs_self _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log A_pos).symm\n \u00b7 rw [exp_add, exp_log H_pos, \u2190 rpow_def_of_pos K_pos]\n have IHA : Nat.card (H : Set G) \u2264 K ^ 11 * Nat.card A := by\n have : log (Nat.card (H : Set G)) \u2264 log K * 11 + log (Nat.card A) := by\n linarith [(neg_le_abs _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log H_pos).symm\n \u00b7 rw [exp_add, exp_log A_pos, \u2190 rpow_def_of_pos K_pos]\n -- entropic PFR shows that the entropy of `VA - VH` is small\n have I : log K * (-11/2) + log (Nat.card A) * (-1/2) + log (Nat.card (H : Set G)) * (-1/2)\n \u2264 - H[VA - VH] := by\n rw [Vindep.rdist_eq VAmeas VHmeas] at this\n have : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n linarith\n -- therefore, there exists a point `x\u2080` which is attained by `VA - VH` with a large probability\n obtain \u27e8x\u2080, h\u2080\u27e9 : \u2203 x\u2080 : G, rexp (- H[VA - VH]) \u2264 (\u2119 : Measure \u03a9).real ((VA - VH) \u207b\u00b9' {x\u2080}) :=\n prob_ge_exp_neg_entropy' _ ((VAmeas.sub VHmeas).comp measurable_id')\n -- massage the previous inequality to get that `A \u2229 (H + {x\u2080})` is large\n have J : K ^ (-11/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (1/2) \u2264\n Nat.card (A \u2229 (H + {x\u2080}) : Set G) := by\n rw [VA'unif.measureReal_preimage_sub VAmeas VH'unif VHmeas Vindep] at h\u2080\n have := (Real.exp_monotone I).trans h\u2080\n have hAA'_card : Nat.card A' = Nat.card A := congrArg Nat.card (congrArg Subtype hAA')\n have hHH'_card : Nat.card H' = Nat.card (H : Set G) := congrArg Nat.card (congrArg Subtype hHH')\n rw [hAA'_card, hHH'_card, le_div_iff] at this\n convert this using 1\n . rw [exp_add, exp_add, \u2190 rpow_def_of_pos K_pos, \u2190 rpow_def_of_pos A_pos, \u2190 rpow_def_of_pos H_pos]\n rpow_ring\n norm_num\n . rw [hAA', hHH']\n positivity\n\n have Hne : Set.Nonempty (A \u2229 (H + {x\u2080} : Set G)) := by\n by_contra h'\n have : (0 : \u211d) < Nat.card (A \u2229 (H + {x\u2080}) : Set G) := lt_of_lt_of_le (by positivity) J\n simp only [Nat.card_eq_fintype_card, card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,\n not_nonempty_iff_eq_empty.1 h'] at this\n /- use Rusza covering lemma to cover `A` by few translates of `A \u2229 (H + {x\u2080}) - A \u2229 (H + {x\u2080})`\n (which is contained in `H`). The number of translates is at most\n `#(A + (A \u2229 (H + {x\u2080}))) / #(A \u2229 (H + {x\u2080}))`, where the numerator is controlled as this is\n a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/\n rcases Set.exists_subset_add_sub (toFinite A) (toFinite (A \u2229 ((H + {x\u2080} : Set G)))) Hne with\n \u27e8u, hu, Au, -\u27e9\n have Iu : Nat.card u \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n have : (0 : \u211d) \u2264 Nat.card u := by simp\n have Z1 := mul_le_mul_of_nonneg_left J this\n have Z2 : (Nat.card u * Nat.card (A \u2229 (H + {x\u2080}) : Set G) : \u211d)\n \u2264 Nat.card (A + A \u2229 (\u2191H + {x\u2080})) := by norm_cast\n have Z3 : (Nat.card (A + A \u2229 (\u2191H + {x\u2080})) : \u211d) \u2264 K * Nat.card A := by\n apply le_trans _ hA\n simp only [Nat.cast_le]\n apply Nat.card_mono (toFinite _)\n apply add_subset_add_left (inter_subset_left _ _)\n have : 0 \u2264 K ^ (11/2) * Nat.card A ^ (-1/2) * Nat.card (H : Set G) ^ (-1/2) := by positivity\n have T := mul_le_mul_of_nonneg_left ((Z1.trans Z2).trans Z3) this\n convert T using 1 <;> rpow_ring <;> norm_num\n have A_subset_uH : A \u2286 u + H := by\n rw [add_sub_assoc] at Au\n refine Au.trans $ add_subset_add_left $\n (sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_\n rw [add_sub_add_comm, singleton_sub_singleton, sub_self]\n simp\n exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9", "proofType": "tactic", "proofLengthLines": 107, "proofLengthTokens": 5872}} +{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n ext \u03c9; simp [sub_eq_zero, eq_comm]\n rw [this, measureReal_iUnion_fintype _\n (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n \u00b7 intro g g' hgg'\n apply Set.disjoint_iff_inter_eq_empty.2\n ext a\n simp (config := {contextual := True}) [hgg']\n classical\n let W : Finset G := A \u2229 B\n calc\n \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply sum_congr _ _ (fun g \u21a6 ?_)\n rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n measurableSet_discrete _]\n _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply (Finset.sum_subset W.subset_univ _).symm\n intro i _ hi\n replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n rcases hi with h'i|h'i\n \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n congr\n rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n classical\n let W := fun \u03c9 \u21a6 V \u03c9 + x\n have Wunif : IsUniform (B + {x} : Set G) W := by\n convert Vunif.comp (add_left_injective x)\n simp\n have Wmeas : Measurable W := Vmeas.add_const _\n have UWindep : IndepFun U W := by\n have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n exact hindep.comp measurable_id this\n have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n ext \u03c9\n simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n abel_nf\n have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n rw [h] at Wunif\n rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n congr 3\n . rw [add_singleton]; simp\n convert Finset.card_vadd_finset (AddOpposite.op x) B\n . simp\n simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n obtain \u27e8A_pos, AA_pos, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n have IU : d[U # U'] \u2264 log K := by\n have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n . simp\n exact Iff.rfl\n . measurability\n filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n simp\n exact Set.sub_mem_sub h1 h2\n have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n apply (log_le_log AA_pos hA).trans (le_of_eq _)\n rw [log_mul K_pos.ne' A_pos.ne']\n-- have : H[U + U'] = H[U - U'] := by congr; simp\n rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n linarith\n rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\nlemma sumset_eq_sub : A + A = A - A := by\n rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n congr! 1 with a _ b _\n show a + b = a - b\n simp\n\n/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\nlemma PFR_conjecture_aux (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card H \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card H \u2227 A \u2286 c + H := by\n classical\n let _mG : MeasurableSpace G := \u22a4\n rw [sumset_eq_sub] at hA\n have : MeasurableSingletonClass G := \u27e8\u03bb _ \u21a6 trivial\u27e9\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n let A' := A.toFinite.toFinset\n have h\u2080A' : Finset.Nonempty A' := by\n simp [A', Finset.Nonempty]\n exact h\u2080A\n have hAA' : A' = A := Finite.coe_toFinset (toFinite A)\n rcases exists_isUniform_measureSpace A' h\u2080A' with \u27e8\u03a9\u2080, m\u03a9\u2080, UA, hP\u2080, UAmeas, UAunif, -, -\u27e9\n rw [hAA'] at UAunif\n have : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n rw [\u2190 sumset_eq_sub] at hA\n let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/9, (by norm_num), (by norm_num)\u27e9\n -- entropic PFR gives a subgroup `H` which is close to `A` for the Rusza distance\n rcases entropic_PFR_conjecture p (by norm_num) with \u27e8H, \u03a9\u2081, m\u03a9\u2081, UH, hP\u2081, UHmeas, UHunif, hUH\u27e9\n rcases independent_copies_two UAmeas UHmeas\n with \u27e8\u03a9, m\u03a9, VA, VH, hP, VAmeas, VHmeas, Vindep, idVA, idVH\u27e9\n have VAunif : IsUniform A VA := UAunif.of_identDistrib idVA.symm $ measurableSet_discrete _\n have VA'unif := VAunif\n rw [\u2190 hAA'] at VA'unif\n have VHunif : IsUniform H VH := UHunif.of_identDistrib idVH.symm $ measurableSet_discrete _\n let H' := (H:Set G).toFinite.toFinset\n have hHH' : H' = (H:Set G) := Finite.coe_toFinset (toFinite (H:Set G))\n have VH'unif := VHunif\n rw [\u2190 hHH'] at VH'unif\n\n have : d[VA # VH] \u2264 11/2 * log K := by rw [idVA.rdist_eq idVH]; linarith\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos\n positivity\n have VA_ent : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have VH_ent : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n have Icard : |log (Nat.card A) - log (Nat.card (H : Set G))| \u2264 11 * log K := by\n rw [\u2190 VA_ent, \u2190 VH_ent]\n apply (diff_ent_le_rdist VAmeas VHmeas).trans\n linarith\n have IAH : Nat.card A \u2264 K ^ 11 * Nat.card (H : Set G) := by\n have : log (Nat.card A) \u2264 log K * 11 + log (Nat.card (H : Set G)) := by\n linarith [(le_abs_self _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log A_pos).symm\n \u00b7 rw [exp_add, exp_log H_pos, \u2190 rpow_def_of_pos K_pos]\n have IHA : Nat.card (H : Set G) \u2264 K ^ 11 * Nat.card A := by\n have : log (Nat.card (H : Set G)) \u2264 log K * 11 + log (Nat.card A) := by\n linarith [(neg_le_abs _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log H_pos).symm\n \u00b7 rw [exp_add, exp_log A_pos, \u2190 rpow_def_of_pos K_pos]\n -- entropic PFR shows that the entropy of `VA - VH` is small\n have I : log K * (-11/2) + log (Nat.card A) * (-1/2) + log (Nat.card (H : Set G)) * (-1/2)\n \u2264 - H[VA - VH] := by\n rw [Vindep.rdist_eq VAmeas VHmeas] at this\n have : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n linarith\n -- therefore, there exists a point `x\u2080` which is attained by `VA - VH` with a large probability\n obtain \u27e8x\u2080, h\u2080\u27e9 : \u2203 x\u2080 : G, rexp (- H[VA - VH]) \u2264 (\u2119 : Measure \u03a9).real ((VA - VH) \u207b\u00b9' {x\u2080}) :=\n prob_ge_exp_neg_entropy' _ ((VAmeas.sub VHmeas).comp measurable_id')\n -- massage the previous inequality to get that `A \u2229 (H + {x\u2080})` is large\n have J : K ^ (-11/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (1/2) \u2264\n Nat.card (A \u2229 (H + {x\u2080}) : Set G) := by\n rw [VA'unif.measureReal_preimage_sub VAmeas VH'unif VHmeas Vindep] at h\u2080\n have := (Real.exp_monotone I).trans h\u2080\n have hAA'_card : Nat.card A' = Nat.card A := congrArg Nat.card (congrArg Subtype hAA')\n have hHH'_card : Nat.card H' = Nat.card (H : Set G) := congrArg Nat.card (congrArg Subtype hHH')\n rw [hAA'_card, hHH'_card, le_div_iff] at this\n convert this using 1\n . rw [exp_add, exp_add, \u2190 rpow_def_of_pos K_pos, \u2190 rpow_def_of_pos A_pos, \u2190 rpow_def_of_pos H_pos]\n rpow_ring\n norm_num\n . rw [hAA', hHH']\n positivity\n\n have Hne : Set.Nonempty (A \u2229 (H + {x\u2080} : Set G)) := by\n by_contra h'\n have : (0 : \u211d) < Nat.card (A \u2229 (H + {x\u2080}) : Set G) := lt_of_lt_of_le (by positivity) J\n simp only [Nat.card_eq_fintype_card, card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,\n not_nonempty_iff_eq_empty.1 h'] at this\n /- use Rusza covering lemma to cover `A` by few translates of `A \u2229 (H + {x\u2080}) - A \u2229 (H + {x\u2080})`\n (which is contained in `H`). The number of translates is at most\n `#(A + (A \u2229 (H + {x\u2080}))) / #(A \u2229 (H + {x\u2080}))`, where the numerator is controlled as this is\n a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/\n rcases Set.exists_subset_add_sub (toFinite A) (toFinite (A \u2229 ((H + {x\u2080} : Set G)))) Hne with\n \u27e8u, hu, Au, -\u27e9\n have Iu : Nat.card u \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n have : (0 : \u211d) \u2264 Nat.card u := by simp\n have Z1 := mul_le_mul_of_nonneg_left J this\n have Z2 : (Nat.card u * Nat.card (A \u2229 (H + {x\u2080}) : Set G) : \u211d)\n \u2264 Nat.card (A + A \u2229 (\u2191H + {x\u2080})) := by norm_cast\n have Z3 : (Nat.card (A + A \u2229 (\u2191H + {x\u2080})) : \u211d) \u2264 K * Nat.card A := by\n apply le_trans _ hA\n simp only [Nat.cast_le]\n apply Nat.card_mono (toFinite _)\n apply add_subset_add_left (inter_subset_left _ _)\n have : 0 \u2264 K ^ (11/2) * Nat.card A ^ (-1/2) * Nat.card (H : Set G) ^ (-1/2) := by positivity\n have T := mul_le_mul_of_nonneg_left ((Z1.trans Z2).trans Z3) this\n convert T using 1 <;> rpow_ring <;> norm_num\n have A_subset_uH : A \u2286 u + H := by\n rw [add_sub_assoc] at Au\n refine Au.trans $ add_subset_add_left $\n (sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_\n rw [add_sub_add_comm, singleton_sub_singleton, sub_self]\n simp\n exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9\n\n\n", "theoremStatement": "/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{12}$ cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c < 2 * K ^ 12 \u2227 Nat.card H \u2264 Nat.card A \u2227 A \u2286 c + H ", "theoremName": "PFR_conjecture", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "b94162e", "date": "2023-11-28"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 278, "tokenPositionInFile": 14154, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 342, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K :=\n PFR_conjecture_pos_aux' h\u2080A hA\n -- consider the subgroup `H` given by Lemma `PFR_conjecture_aux`.\n obtain \u27e8H, c, hc, IHA, IAH, A_subs_cH\u27e9 : \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card (H : Set G) \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card (H : Set G)\n \u2227 A \u2286 c + H :=\n PFR_conjecture_aux h\u2080A hA\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos; positivity\n rcases le_or_lt (Nat.card (H : Set G)) (Nat.card A) with h|h\n -- If `#H \u2264 #A`, then `H` satisfies the conclusion of the theorem\n \u00b7 refine \u27e8H, c, ?_, h, A_subs_cH\u27e9\n calc\n Nat.card c \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := hc\n _ \u2264 K ^ (13/2) * (K ^ 11 * Nat.card (H : Set G)) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n gcongr\n _ = K ^ 12 := by rpow_ring; norm_num\n _ < 2 * K ^ 12 := by linarith [show 0 < K ^ 12 by positivity]\n -- otherwise, we decompose `H` into cosets of one of its subgroups `H'`, chosen so that\n -- `#A / 2 < #H' \u2264 #A`. This `H'` satisfies the desired conclusion.\n \u00b7 obtain \u27e8H', IH'A, IAH', H'H\u27e9 : \u2203 H' : AddSubgroup G, Nat.card (H' : Set G) \u2264 Nat.card A\n \u2227 Nat.card A < 2 * Nat.card (H' : Set G) \u2227 H' \u2264 H := by\n have A_pos' : 0 < Nat.card A := mod_cast A_pos\n exact ElementaryAddCommGroup.exists_subgroup_subset_card_le Nat.prime_two H h.le A_pos'.ne'\n have : (Nat.card A / 2 : \u211d) < Nat.card (H' : Set G) := by\n rw [div_lt_iff zero_lt_two, mul_comm]; norm_cast\n have H'_pos : (0 : \u211d) < Nat.card (H' : Set G) := by\n have : 0 < Nat.card (H' : Set G) := Nat.card_pos; positivity\n obtain \u27e8u, HH'u, hu\u27e9 := AddSubgroup.exists_left_transversal_of_le H'H\n refine \u27e8H', c + u, ?_, IH'A, by rwa [add_assoc, HH'u]\u27e9\n calc\n (Nat.card (c + u) : \u211d)\n \u2264 Nat.card c * Nat.card u := mod_cast card_add_le\n _ \u2264 (K ^ (13/2) * (Nat.card A) ^ (1 / 2) * (Nat.card (H : Set G) ^ (-1 / 2)))\n * (Nat.card (H : Set G) / Nat.card (H' : Set G)) := by\n gcongr\n apply le_of_eq\n rw [eq_div_iff H'_pos.ne']\n norm_cast\n _ < (K ^ (13/2) * (Nat.card A) ^ (1 / 2) * (Nat.card (H : Set G) ^ (-1 / 2)))\n * (Nat.card (H : Set G) / (Nat.card A / 2)) := by\n gcongr\n _ = 2 * K ^ (13/2) * (Nat.card A) ^ (-1/2) * (Nat.card (H : Set G)) ^ (1/2) := by\n have : (0 : \u211d) < Nat.card H := H_pos\n field_simp\n rpow_ring\n norm_num\n _ \u2264 2 * K ^ (13/2) * (Nat.card A) ^ (-1/2) * (K ^ 11 * Nat.card A) ^ (1/2) := by\n gcongr\n _ = 2 * K ^ 12 := by\n rpow_ring\n norm_num", "proofType": "tactic", "proofLengthLines": 53, "proofLengthTokens": 2786}} +{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n ext \u03c9; simp [sub_eq_zero, eq_comm]\n rw [this, measureReal_iUnion_fintype _\n (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n \u00b7 intro g g' hgg'\n apply Set.disjoint_iff_inter_eq_empty.2\n ext a\n simp (config := {contextual := True}) [hgg']\n classical\n let W : Finset G := A \u2229 B\n calc\n \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply sum_congr _ _ (fun g \u21a6 ?_)\n rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n measurableSet_discrete _]\n _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n apply (Finset.sum_subset W.subset_univ _).symm\n intro i _ hi\n replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n rcases hi with h'i|h'i\n \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n congr\n rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n classical\n let W := fun \u03c9 \u21a6 V \u03c9 + x\n have Wunif : IsUniform (B + {x} : Set G) W := by\n convert Vunif.comp (add_left_injective x)\n simp\n have Wmeas : Measurable W := Vmeas.add_const _\n have UWindep : IndepFun U W := by\n have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n exact hindep.comp measurable_id this\n have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n ext \u03c9\n simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n abel_nf\n have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n rw [h] at Wunif\n rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n congr 3\n . rw [add_singleton]; simp\n convert Finset.card_vadd_finset (AddOpposite.op x) B\n . simp\n simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n simp [Nat.cast_pos, Nat.card_pos_iff]\n have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n obtain \u27e8A_pos, AA_pos, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n have IU : d[U # U'] \u2264 log K := by\n have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n . simp\n exact Iff.rfl\n . measurability\n filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n simp\n exact Set.sub_mem_sub h1 h2\n have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n apply (log_le_log AA_pos hA).trans (le_of_eq _)\n rw [log_mul K_pos.ne' A_pos.ne']\n-- have : H[U + U'] = H[U - U'] := by congr; simp\n rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n linarith\n rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\nlemma sumset_eq_sub : A + A = A - A := by\n rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n congr! 1 with a _ b _\n show a + b = a - b\n simp\n\n/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\nlemma PFR_conjecture_aux (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card H \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card H \u2227 A \u2286 c + H := by\n classical\n let _mG : MeasurableSpace G := \u22a4\n rw [sumset_eq_sub] at hA\n have : MeasurableSingletonClass G := \u27e8\u03bb _ \u21a6 trivial\u27e9\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K :=\n PFR_conjecture_pos_aux h\u2080A hA\n let A' := A.toFinite.toFinset\n have h\u2080A' : Finset.Nonempty A' := by\n simp [A', Finset.Nonempty]\n exact h\u2080A\n have hAA' : A' = A := Finite.coe_toFinset (toFinite A)\n rcases exists_isUniform_measureSpace A' h\u2080A' with \u27e8\u03a9\u2080, m\u03a9\u2080, UA, hP\u2080, UAmeas, UAunif, -, -\u27e9\n rw [hAA'] at UAunif\n have : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n rw [\u2190 sumset_eq_sub] at hA\n let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/9, (by norm_num), (by norm_num)\u27e9\n -- entropic PFR gives a subgroup `H` which is close to `A` for the Rusza distance\n rcases entropic_PFR_conjecture p (by norm_num) with \u27e8H, \u03a9\u2081, m\u03a9\u2081, UH, hP\u2081, UHmeas, UHunif, hUH\u27e9\n rcases independent_copies_two UAmeas UHmeas\n with \u27e8\u03a9, m\u03a9, VA, VH, hP, VAmeas, VHmeas, Vindep, idVA, idVH\u27e9\n have VAunif : IsUniform A VA := UAunif.of_identDistrib idVA.symm $ measurableSet_discrete _\n have VA'unif := VAunif\n rw [\u2190 hAA'] at VA'unif\n have VHunif : IsUniform H VH := UHunif.of_identDistrib idVH.symm $ measurableSet_discrete _\n let H' := (H:Set G).toFinite.toFinset\n have hHH' : H' = (H:Set G) := Finite.coe_toFinset (toFinite (H:Set G))\n have VH'unif := VHunif\n rw [\u2190 hHH'] at VH'unif\n\n have : d[VA # VH] \u2264 11/2 * log K := by rw [idVA.rdist_eq idVH]; linarith\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos\n positivity\n have VA_ent : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have VH_ent : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n have Icard : |log (Nat.card A) - log (Nat.card (H : Set G))| \u2264 11 * log K := by\n rw [\u2190 VA_ent, \u2190 VH_ent]\n apply (diff_ent_le_rdist VAmeas VHmeas).trans\n linarith\n have IAH : Nat.card A \u2264 K ^ 11 * Nat.card (H : Set G) := by\n have : log (Nat.card A) \u2264 log K * 11 + log (Nat.card (H : Set G)) := by\n linarith [(le_abs_self _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log A_pos).symm\n \u00b7 rw [exp_add, exp_log H_pos, \u2190 rpow_def_of_pos K_pos]\n have IHA : Nat.card (H : Set G) \u2264 K ^ 11 * Nat.card A := by\n have : log (Nat.card (H : Set G)) \u2264 log K * 11 + log (Nat.card A) := by\n linarith [(neg_le_abs _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log H_pos).symm\n \u00b7 rw [exp_add, exp_log A_pos, \u2190 rpow_def_of_pos K_pos]\n -- entropic PFR shows that the entropy of `VA - VH` is small\n have I : log K * (-11/2) + log (Nat.card A) * (-1/2) + log (Nat.card (H : Set G)) * (-1/2)\n \u2264 - H[VA - VH] := by\n rw [Vindep.rdist_eq VAmeas VHmeas] at this\n have : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n linarith\n -- therefore, there exists a point `x\u2080` which is attained by `VA - VH` with a large probability\n obtain \u27e8x\u2080, h\u2080\u27e9 : \u2203 x\u2080 : G, rexp (- H[VA - VH]) \u2264 (\u2119 : Measure \u03a9).real ((VA - VH) \u207b\u00b9' {x\u2080}) :=\n prob_ge_exp_neg_entropy' _ ((VAmeas.sub VHmeas).comp measurable_id')\n -- massage the previous inequality to get that `A \u2229 (H + {x\u2080})` is large\n have J : K ^ (-11/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (1/2) \u2264\n Nat.card (A \u2229 (H + {x\u2080}) : Set G) := by\n rw [VA'unif.measureReal_preimage_sub VAmeas VH'unif VHmeas Vindep] at h\u2080\n have := (Real.exp_monotone I).trans h\u2080\n have hAA'_card : Nat.card A' = Nat.card A := congrArg Nat.card (congrArg Subtype hAA')\n have hHH'_card : Nat.card H' = Nat.card (H : Set G) := congrArg Nat.card (congrArg Subtype hHH')\n rw [hAA'_card, hHH'_card, le_div_iff] at this\n convert this using 1\n . rw [exp_add, exp_add, \u2190 rpow_def_of_pos K_pos, \u2190 rpow_def_of_pos A_pos, \u2190 rpow_def_of_pos H_pos]\n rpow_ring\n norm_num\n . rw [hAA', hHH']\n positivity\n\n have Hne : Set.Nonempty (A \u2229 (H + {x\u2080} : Set G)) := by\n by_contra h'\n have : (0 : \u211d) < Nat.card (A \u2229 (H + {x\u2080}) : Set G) := lt_of_lt_of_le (by positivity) J\n simp only [Nat.card_eq_fintype_card, card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,\n not_nonempty_iff_eq_empty.1 h'] at this\n /- use Rusza covering lemma to cover `A` by few translates of `A \u2229 (H + {x\u2080}) - A \u2229 (H + {x\u2080})`\n (which is contained in `H`). The number of translates is at most\n `#(A + (A \u2229 (H + {x\u2080}))) / #(A \u2229 (H + {x\u2080}))`, where the numerator is controlled as this is\n a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/\n rcases Set.exists_subset_add_sub (toFinite A) (toFinite (A \u2229 ((H + {x\u2080} : Set G)))) Hne with\n \u27e8u, hu, Au, -\u27e9\n have Iu : Nat.card u \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n have : (0 : \u211d) \u2264 Nat.card u := by simp\n have Z1 := mul_le_mul_of_nonneg_left J this\n have Z2 : (Nat.card u * Nat.card (A \u2229 (H + {x\u2080}) : Set G) : \u211d)\n \u2264 Nat.card (A + A \u2229 (\u2191H + {x\u2080})) := by norm_cast\n have Z3 : (Nat.card (A + A \u2229 (\u2191H + {x\u2080})) : \u211d) \u2264 K * Nat.card A := by\n apply le_trans _ hA\n simp only [Nat.cast_le]\n apply Nat.card_mono (toFinite _)\n apply add_subset_add_left (inter_subset_left _ _)\n have : 0 \u2264 K ^ (11/2) * Nat.card A ^ (-1/2) * Nat.card (H : Set G) ^ (-1/2) := by positivity\n have T := mul_le_mul_of_nonneg_left ((Z1.trans Z2).trans Z3) this\n convert T using 1 <;> rpow_ring <;> norm_num\n have A_subset_uH : A \u2286 u + H := by\n rw [add_sub_assoc] at Au\n refine Au.trans $ add_subset_add_left $\n (sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_\n rw [add_sub_add_comm, singleton_sub_singleton, sub_self]\n simp\n exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9\n\n\n/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{12}$ cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c < 2 * K ^ 12 \u2227 Nat.card H \u2264 Nat.card A \u2227 A \u2286 c + H := by\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K :=\n PFR_conjecture_pos_aux' h\u2080A hA\n -- consider the subgroup `H` given by Lemma `PFR_conjecture_aux`.\n obtain \u27e8H, c, hc, IHA, IAH, A_subs_cH\u27e9 : \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card (H : Set G) \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card (H : Set G)\n \u2227 A \u2286 c + H :=\n PFR_conjecture_aux h\u2080A hA\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos; positivity\n rcases le_or_lt (Nat.card (H : Set G)) (Nat.card A) with h|h\n -- If `#H \u2264 #A`, then `H` satisfies the conclusion of the theorem\n \u00b7 refine \u27e8H, c, ?_, h, A_subs_cH\u27e9\n calc\n Nat.card c \u2264 K ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := hc\n _ \u2264 K ^ (13/2) * (K ^ 11 * Nat.card (H : Set G)) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n gcongr\n _ = K ^ 12 := by rpow_ring; norm_num\n _ < 2 * K ^ 12 := by linarith [show 0 < K ^ 12 by positivity]\n -- otherwise, we decompose `H` into cosets of one of its subgroups `H'`, chosen so that\n -- `#A / 2 < #H' \u2264 #A`. This `H'` satisfies the desired conclusion.\n \u00b7 obtain \u27e8H', IH'A, IAH', H'H\u27e9 : \u2203 H' : AddSubgroup G, Nat.card (H' : Set G) \u2264 Nat.card A\n \u2227 Nat.card A < 2 * Nat.card (H' : Set G) \u2227 H' \u2264 H := by\n have A_pos' : 0 < Nat.card A := mod_cast A_pos\n exact ElementaryAddCommGroup.exists_subgroup_subset_card_le Nat.prime_two H h.le A_pos'.ne'\n have : (Nat.card A / 2 : \u211d) < Nat.card (H' : Set G) := by\n rw [div_lt_iff zero_lt_two, mul_comm]; norm_cast\n have H'_pos : (0 : \u211d) < Nat.card (H' : Set G) := by\n have : 0 < Nat.card (H' : Set G) := Nat.card_pos; positivity\n obtain \u27e8u, HH'u, hu\u27e9 := AddSubgroup.exists_left_transversal_of_le H'H\n refine \u27e8H', c + u, ?_, IH'A, by rwa [add_assoc, HH'u]\u27e9\n calc\n (Nat.card (c + u) : \u211d)\n \u2264 Nat.card c * Nat.card u := mod_cast card_add_le\n _ \u2264 (K ^ (13/2) * (Nat.card A) ^ (1 / 2) * (Nat.card (H : Set G) ^ (-1 / 2)))\n * (Nat.card (H : Set G) / Nat.card (H' : Set G)) := by\n gcongr\n apply le_of_eq\n rw [eq_div_iff H'_pos.ne']\n norm_cast\n _ < (K ^ (13/2) * (Nat.card A) ^ (1 / 2) * (Nat.card (H : Set G) ^ (-1 / 2)))\n * (Nat.card (H : Set G) / (Nat.card A / 2)) := by\n gcongr\n _ = 2 * K ^ (13/2) * (Nat.card A) ^ (-1/2) * (Nat.card (H : Set G)) ^ (1/2) := by\n have : (0 : \u211d) < Nat.card H := H_pos\n field_simp\n rpow_ring\n norm_num\n _ \u2264 2 * K ^ (13/2) * (Nat.card A) ^ (-1/2) * (K ^ 11 * Nat.card A) ^ (1/2) := by\n gcongr\n _ = 2 * K ^ 12 := by\n rpow_ring\n norm_num\n\n", "theoremStatement": "/-- Corollary of `PFR_conjecture` in which the ambient group is not required to be finite (but) then\n$H$ and $c$ are finite. -/\ntheorem PFR_conjecture' {G : Type*} [AddCommGroup G] [ElementaryAddCommGroup G 2]\n {A : Set G} {K : \u211d} (h\u2080A : A.Nonempty) (Afin : A.Finite)\n (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G), c.Finite \u2227 (H : Set G).Finite \u2227\n Nat.card c < 2 * K ^ 12 \u2227 Nat.card H \u2264 Nat.card A \u2227 A \u2286 c + H ", "theoremName": "PFR_conjecture'", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "6ffc2bd", "date": "2023-11-28"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 338, "tokenPositionInFile": 17371, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 104, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let G' := AddSubgroup.closure A\n let G'fin : Fintype G' := by\n exact Finite.fintype (ElementaryAddCommGroup.finite_closure Afin)\n have G'Elem : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.subgroup _\n let \u03b9 : G'\u2192+ G := G'.subtype\n have \u03b9_inj : Injective \u03b9 := AddSubgroup.subtype_injective G'\n let A' : Set G' := \u03b9 \u207b\u00b9' A\n have A_rg : A \u2286 range \u03b9 := by\n simp only [AddSubgroup.coeSubtype, Subtype.range_coe_subtype, SetLike.mem_coe, G', \u03b9]\n exact AddSubgroup.subset_closure\n have cardA' : Nat.card A' = Nat.card A := Nat.card_preimage_of_injective \u03b9_inj A_rg\n have hA' : Nat.card (A' + A') \u2264 K * Nat.card A' := by\n rwa [cardA', \u2190 preimage_add _ \u03b9_inj A_rg A_rg,\n Nat.card_preimage_of_injective \u03b9_inj (add_subset_range _ A_rg A_rg)]\n rcases PFR_conjecture (h\u2080A.preimage' A_rg) hA' with \u27e8H', c', hc', hH', hH'\u2082\u27e9\n refine \u27e8AddSubgroup.map \u03b9 H', \u03b9 '' c', toFinite _, toFinite (\u03b9 '' H'), ?_, ?_, fun x hx \u21a6 ?_\u27e9\n \u00b7 rwa [Nat.card_image_of_injective \u03b9_inj]\n \u00b7 erw [Nat.card_image_of_injective \u03b9_inj, \u2190 cardA']\n exact hH'\n \u00b7 erw [\u2190 image_add]\n exact \u27e8\u27e8x, AddSubgroup.subset_closure hx\u27e9, hH'\u2082 hx, rfl\u27e9", "proofType": "tactic", "proofLengthLines": 21, "proofLengthTokens": 1144}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n", "theoremStatement": "/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k ", "theoremName": "rdist_add_rdist_add_condMutual_eq", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 50, "tokenPositionInFile": 1690, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 159, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 519}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n", "theoremStatement": "/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) ", "theoremName": "rdist_of_sums_ge", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 72, "tokenPositionInFile": 2571, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 32, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 61}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n", "theoremStatement": "/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) ", "theoremName": "condRuzsaDist_of_sums_ge", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 80, "tokenPositionInFile": 2972, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 41, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 90}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n", "theoremStatement": "/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 ", "theoremName": "diff_rdist_le_1", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "0ec25b1", "date": "2023-11-22"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 89, "tokenPositionInFile": 3453, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 10, "numPremises": 94, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n \u00b7 exact h\u2082.entropy_eq", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 275}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n \u00b7 exact h\u2082.entropy_eq\n\n", "theoremStatement": "/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 ", "theoremName": "diff_rdist_le_2", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "96a97b5", "date": "2023-11-27"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 96, "tokenPositionInFile": 3939, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 97, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n \u00b7 rw [rdist_symm]\n exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n \u00b7 exact h\u2081.entropy_eq", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 264}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n \u00b7 rw [rdist_symm]\n exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n \u00b7 exact h\u2081.entropy_eq\n\n", "theoremStatement": "/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 ", "theoremName": "diff_rdist_le_3", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "ba82447", "date": "2023-11-28"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 104, "tokenPositionInFile": 4426, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 96, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n \u00b7 apply h\u2082.entropy_eq", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 245}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n \u00b7 rw [rdist_symm]\n exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n \u00b7 exact h\u2081.entropy_eq\n\n/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n \u00b7 apply h\u2082.entropy_eq\n\n", "theoremStatement": "/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n \\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\nlemma diff_rdist_le_4 : d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 ", "theoremName": "diff_rdist_le_4", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "ba82447", "date": "2023-11-28"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 112, "tokenPositionInFile": 4906, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 98, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le''' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 3\n \u00b7 rw [rdist_symm, (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081]\n \u00b7 apply h\u2081.entropy_eq", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 257}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n \u00b7 rw [rdist_symm]\n exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n \u00b7 exact h\u2081.entropy_eq\n\n/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n \u00b7 apply h\u2082.entropy_eq\n\n/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n \\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\nlemma diff_rdist_le_4 : d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le''' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 3\n \u00b7 rw [rdist_symm, (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081]\n \u00b7 apply h\u2081.entropy_eq\n\n", "theoremStatement": "/-- We have $I_1 \\leq 2 \\eta k$ -/\nlemma first_estimate : I\u2081 \u2264 2 * p.\u03b7 * k ", "theoremName": "first_estimate", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 120, "tokenPositionInFile": 5399, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 19, "numPremises": 239, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have v1 := rdist_add_rdist_add_condMutual_eq X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n have v2 := rdist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n have v3 := condRuzsaDist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a (by measurability) (by measurability)\n have v4 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_1 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n have v5 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_2 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n have v6 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_3 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n have v7 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_4 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n simp only [inv_eq_one_div] at *\n linarith [v1, v2, v3, v4, v5, v6, v7]", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 690}} +{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n (fun i => by fin_cases i <;> assumption)\n rw [h0, h1, h2, h3] at h\n have heq : d[X\u2082' # X\u2081'] = k := by\n rw [rdist_symm]\n apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n rw [heq] at h\n convert h.symm using 1\n \u00b7 congr 2 <;> abel\n \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n \u00b7 rw [rdist_symm]\n exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n \u00b7 exact h\u2081.entropy_eq\n\n/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n \u00b7 apply h\u2082.entropy_eq\n\n/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n \\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\nlemma diff_rdist_le_4 : d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n convert condRuzsaDist_diff_le''' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 3\n \u00b7 rw [rdist_symm, (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081]\n \u00b7 apply h\u2081.entropy_eq\n\n/-- We have $I_1 \\leq 2 \\eta k$ -/\nlemma first_estimate : I\u2081 \u2264 2 * p.\u03b7 * k := by\n have v1 := rdist_add_rdist_add_condMutual_eq X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n have v2 := rdist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n have v3 := condRuzsaDist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a (by measurability) (by measurability)\n have v4 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_1 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n have v5 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_2 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n have v6 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_3 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n have v7 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_4 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n simp only [inv_eq_one_div] at *\n linarith [v1, v2, v3, v4, v5, v6, v7]\n\n", "theoremStatement": "/--\n$$\\mathbb{H}[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} \\mathbb{H}[X_1]+\\tfrac{1}{2} \\mathbb{H}[X_2] + (2 + \\eta) k - I_1.$$\n-/\nlemma ent_ofsum_le : H[X\u2081 + X\u2082 + X\u2081' + X\u2082'] \u2264 H[X\u2081]/2 + H[X\u2082]/2 + (2+p.\u03b7)*k - I\u2081 ", "theoremName": "ent_ofsum_le", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 132, "tokenPositionInFile": 6166, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 21, "numPremises": 340, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let D := d[X\u2081 + X\u2082' # X\u2082 + X\u2081']\n let Dcc := d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n let D1 := d[p.X\u2080\u2081 # X\u2081]\n let Dc1 := d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082']\n let D2 := d[p.X\u2080\u2082 # X\u2082]\n let Dc2 := d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081']\n have lem68 : D + Dcc + I\u2081 = 2 * k :=\n rdist_add_rdist_add_condMutual_eq _ _ _ _ hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep\n have lem610 : Dcc \u2265 k - p.\u03b7 * (Dc1 - D1) - p.\u03b7 * (Dc2 - D2) :=\n condRuzsaDist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 (by measurability) (by measurability) h_min\n have lem611c : Dc1 - D1 \u2264 k / 2 + H[X\u2081] / 4 - H[X\u2082] / 4 :=\n diff_rdist_le_3 p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082' h\u2082 h_indep\n have lem611d : Dc2 - D2 \u2264 k / 2 + H[X\u2082] / 4 - H[X\u2081] / 4 :=\n diff_rdist_le_4 p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2082 hX\u2081' h\u2081 h_indep\n have aux : D + I\u2081 \u2264 (1 + p.\u03b7) * k := by\n calc D + I\u2081\n \u2264 k + p.\u03b7 * (Dc1 - D1) + p.\u03b7 * (Dc2 - D2) := ?_\n _ \u2264 k + p.\u03b7 * (k / 2 + H[X\u2081] / 4 - H[X\u2082] / 4) + p.\u03b7 * (k / 2 + H[X\u2082] / 4 - H[X\u2081] / 4) := ?_\n _ = (1 + p.\u03b7) * k := by ring\n \u00b7 convert add_le_add lem68.le (neg_le_neg lem610) using 1 <;> ring\n \u00b7 refine add_le_add (add_le_add (le_refl _) ?_) ?_\n \u00b7 apply (mul_le_mul_left p.h\u03b7).mpr lem611c\n \u00b7 apply (mul_le_mul_left p.h\u03b7).mpr lem611d\n have ent_sub_eq_ent_add : H[X\u2081 + X\u2082' - (X\u2082 + X\u2081')] = H[X\u2081 + X\u2082' + (X\u2082 + X\u2081')] := by simp\n have rw\u2081 : X\u2081 + X\u2082' + (X\u2082 + X\u2081') = X\u2081 + X\u2082 + X\u2081' + X\u2082' := by abel\n have ind_aux : IndepFun (X\u2081 + X\u2082') (X\u2082 + X\u2081') := by\n exact iIndepFun.indepFun_add_add h_indep (fun i \u21a6 by fin_cases i <;> assumption) 0 2 1 3\n (by decide) (by decide) (by decide) (by decide)\n have ind : D = H[X\u2081 + X\u2082' - (X\u2082 + X\u2081')] - H[X\u2081 + X\u2082'] / 2 - H[X\u2082 + X\u2081'] / 2 :=\n @IndepFun.rdist_eq \u03a9 G _ \u2119 _ _ _ (X\u2081 + X\u2082') _ (X\u2082 + X\u2081') ind_aux (by measurability) (by measurability)\n rw [ind, ent_sub_eq_ent_add, rw\u2081] at aux\n have obs : H[X\u2081 + X\u2082 + X\u2081' + X\u2082'] \u2264 H[X\u2081 + X\u2082'] / 2 + H[X\u2082 + X\u2081'] / 2 + (1 + p.\u03b7) * k - I\u2081 := by\n linarith\n have rw\u2082 : H[X\u2081 + X\u2082'] = k + H[X\u2081]/2 + H[X\u2082]/2 := by\n have HX\u2082_eq : H[X\u2082] = H[X\u2082'] :=\n congr_arg (fun (\u03bc : Measure G) \u21a6 measureEntropy (\u03bc := \u03bc)) h\u2082.map_eq\n have k_eq : k = H[X\u2081 - X\u2082'] - H[X\u2081] / 2 - H[X\u2082'] / 2 := by\n have k_eq_aux : k = d[X\u2081 # X\u2082'] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082\n rw [k_eq_aux]\n exact (h_indep.indepFun (show (0 : Fin 4) \u2260 2 by decide)).rdist_eq hX\u2081 hX\u2082'\n rw [k_eq, \u2190 ElementaryAddCommGroup.sub_eq_add, \u2190 HX\u2082_eq]\n ring\n have rw\u2083 : H[X\u2082 + X\u2081'] = k + H[X\u2081]/2 + H[X\u2082]/2 := by\n have HX\u2081_eq : H[X\u2081] = H[X\u2081'] :=\n congr_arg (fun (\u03bc : Measure G) \u21a6 measureEntropy (\u03bc := \u03bc)) h\u2081.map_eq\n have k_eq' : k = H[X\u2081' - X\u2082] - H[X\u2081'] / 2 - H[X\u2082] / 2 := by\n have k_eq_aux : k = d[X\u2081' # X\u2082] :=\n IdentDistrib.rdist_eq h\u2081 (IdentDistrib.refl hX\u2082.aemeasurable)\n rw [k_eq_aux]\n exact IndepFun.rdist_eq (h_indep.indepFun (show (3 : Fin 4) \u2260 1 by decide)) hX\u2081' hX\u2082\n rw [add_comm X\u2082 X\u2081', k_eq', \u2190 ElementaryAddCommGroup.sub_eq_add, \u2190 HX\u2081_eq]\n ring\n calc H[X\u2081 + X\u2082 + X\u2081' + X\u2082']\n \u2264 H[X\u2081 + X\u2082'] / 2 + H[X\u2082 + X\u2081'] / 2 + (1 + p.\u03b7) * k - I\u2081 := obs\n _ = (k + H[X\u2081] / 2 + H[X\u2082] / 2) / 2\n + (k + H[X\u2081] / 2 + H[X\u2082] / 2) / 2 + (1 + p.\u03b7) * k - I\u2081 := by rw [rw\u2082, rw\u2083]\n _ = H[X\u2081] / 2 + H[X\u2082] / 2 + (2 + p.\u03b7) * k - I\u2081 := by ring", "proofType": "tactic", "proofLengthLines": 57, "proofLengthTokens": 3340}} diff --git a/minictx/pfr_cross.jsonl b/minictx/pfr_cross.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..68c2c395be45a348369dc880481c40cfccb273af --- /dev/null +++ b/minictx/pfr_cross.jsonl @@ -0,0 +1,43 @@ +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n", "theoremStatement": "@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] ", "theoremName": "measureEntropy_inv", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 19, "tokenPositionInFile": 507, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 20, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 69}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n", "theoremStatement": "@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] ", "theoremName": "measureEntropy_div_comm", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 23, "tokenPositionInFile": 675, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 46, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 102}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_inv", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 32, "tokenPositionInFile": 946, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 31, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 62}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_div_comm", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 36, "tokenPositionInFile": 1123, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 60, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 64}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 44, "tokenPositionInFile": 1397, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 52, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 88}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 51, "tokenPositionInFile": 1776, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 54, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 87}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 58, "tokenPositionInFile": 2164, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 73, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 291}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 69, "tokenPositionInFile": 2746, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 73, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 290}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_div", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 80, "tokenPositionInFile": 3338, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 47, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ (fun _ \u21a6 div_right_injective) h\u03ba", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 98}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ (fun _ \u21a6 div_right_injective) h\u03ba\n\n", "theoremStatement": "@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_div", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 87, "tokenPositionInFile": 3728, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 70, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_div (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 / p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 / p.1) := rfl\n simp_rw [this]\n rw [\u2190 entropy_div_comm]", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 318}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ (fun _ \u21a6 div_right_injective) h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_div (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 / p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 / p.1) := rfl\n simp_rw [this]\n rw [\u2190 entropy_div_comm]\n\n", "theoremStatement": "@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 99, "tokenPositionInFile": 4338, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 73, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 165}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ (fun _ \u21a6 div_right_injective) h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_div (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 / p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 / p.1) := rfl\n simp_rw [this]\n rw [\u2190 entropy_div_comm]\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba\u27e9\n\n", "theoremStatement": "@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 109, "tokenPositionInFile": 4818, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 76, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 167}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ (fun _ \u21a6 div_right_injective) h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_div (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 / p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 / p.1) := rfl\n simp_rw [this]\n rw [\u2190 entropy_div_comm]\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba\u27e9\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba\u27e9\n\n", "theoremStatement": "@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_div", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5f2281a", "date": "2023-12-23"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 119, "tokenPositionInFile": 5310, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 71, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_div _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_div _ _ h\u03ba\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 165}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ (fun _ \u21a6 div_right_injective) h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_div (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 / p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 / p.1) := rfl\n simp_rw [this]\n rw [\u2190 entropy_div_comm]\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba\u27e9\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba\u27e9\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_div _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_div _ _ h\u03ba\u27e9\n\n", "theoremStatement": "@[to_additive]\nlemma max_entropy_le_entropy_mul_prod\n (\u03ba : kernel T G) [IsMarkovKernel \u03ba] (\u03b7 : kernel T G) [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 \u03bc) :\n max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc]) \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.max_entropy_le_entropy_mul_prod", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "737f79b", "date": "2023-12-24"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 129, "tokenPositionInFile": 5789, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 93, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n calc max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc])\n = max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc]) - Ik[\u03ba \u00d7\u2096 \u03b7, \u03bc] := by\n rw [mutualInfo_prod _ h\u03ba h\u03b7, sub_zero]\n _ \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n convert max_entropy_sub_mutualInfo_le_entropy_mul (\u03ba \u00d7\u2096 \u03b7) \u03bc (h\u03ba.prod h\u03b7)\n \u00b7 simp\n \u00b7 simp", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 318}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport Mathlib.MeasureTheory.Group.Arithmetic\n/-!\n# Kernel entropy and mutual information in a commutative group\n\n## Main definitions\n\n## Main results\n\n-/\nopen MeasureTheory ProbabilityTheory\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace G] [MeasurableSingletonClass G] [Group G]\n [MeasurableDiv\u2082 G] [MeasurableMul\u2082 G] [Countable G]\n {\u03ba : kernel T G} {\u03bc : Measure T}\n\n@[to_additive (attr := simp)]\nlemma measureEntropy_inv (\u03bc : Measure G) : Hm[\u03bc.map (\u00b7\u207b\u00b9)] = Hm[\u03bc] :=\n measureEntropy_map_of_injective \u03bc _ measurable_inv inv_injective\n\n@[to_additive]\nlemma measureEntropy_div_comm (\u03bc : Measure (G \u00d7 G)) :\n Hm[\u03bc.map fun p \u21a6 p.1 / p.2] = Hm[\u03bc.map fun p \u21a6 p.2 / p.1] := by\n rw [\u2190 measureEntropy_inv, Measure.map_map measurable_inv measurable_div]\n congr with x\n simp\n\nnamespace ProbabilityTheory.kernel\n\n@[to_additive]\nlemma entropy_inv (\u03ba : kernel T G) (\u03bc : Measure T) : Hk[map \u03ba (\u00b7\u207b\u00b9) measurable_inv, \u03bc] = Hk[\u03ba, \u03bc] :=\n entropy_map_of_injective \u03ba \u03bc inv_injective measurable_inv\n\n@[to_additive]\nlemma entropy_div_comm (\u03ba : kernel T (G \u00d7 G)) (\u03bc : Measure T) :\n Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc]\n = Hk[map \u03ba (fun p \u21a6 p.2 / p.1) (measurable_snd.div measurable_fst), \u03bc] := by\n rw [\u2190 entropy_inv, kernel.map_map]\n congr with x\n simp\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_right_injective h\u03ba\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ mul_left_injective h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul' (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.2 * p.1) \u2218 Prod.swap = (fun p \u21a6 p.1 * p.2) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_mul (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 * p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 * p.1) := rfl\n simp_rw [this]\n\n@[to_additive]\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] :=\n entropy_snd_sub_mutualInfo_le_entropy_map_of_injective \u03ba \u03bc _ (fun _ \u21a6 div_right_injective) h\u03ba\n\n@[to_additive]\nlemma entropy_fst_sub_mutualInfo_le_entropy_map_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[fst \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n have h := entropy_snd_sub_mutualInfo_le_entropy_map_div (swapRight \u03ba) \u03bc h\u03ba.swapRight\n simp only [snd_swapRight, mutualInfo_swapRight, map_swapRight] at h\n refine h.trans_eq ?_\n have : (fun p : G \u00d7 G \u21a6 p.1 / p.2) \u2218 Prod.swap = (fun p \u21a6 p.2 / p.1) := rfl\n simp_rw [this]\n rw [\u2190 entropy_div_comm]\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul _ _ h\u03ba\u27e9\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul'\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.2 * p.1) $ measurable_discrete _, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_mul' _ _ h\u03ba\u27e9\n\n@[to_additive]\nlemma max_entropy_sub_mutualInfo_le_entropy_div\n (\u03ba : kernel T (G \u00d7 G)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n max (Hk[fst \u03ba, \u03bc]) (Hk[snd \u03ba, \u03bc]) - Ik[\u03ba, \u03bc]\n \u2264 Hk[map \u03ba (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_fst_sub_mutualInfo_le_entropy_map_div _ _ h\u03ba,\n entropy_snd_sub_mutualInfo_le_entropy_map_div _ _ h\u03ba\u27e9\n\n@[to_additive]\nlemma max_entropy_le_entropy_mul_prod\n (\u03ba : kernel T G) [IsMarkovKernel \u03ba] (\u03b7 : kernel T G) [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 \u03bc) :\n max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc]) \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n calc max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc])\n = max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc]) - Ik[\u03ba \u00d7\u2096 \u03b7, \u03bc] := by\n rw [mutualInfo_prod _ h\u03ba h\u03b7, sub_zero]\n _ \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1 * p.2) measurable_mul, \u03bc] := by\n convert max_entropy_sub_mutualInfo_le_entropy_mul (\u03ba \u00d7\u2096 \u03b7) \u03bc (h\u03ba.prod h\u03b7)\n \u00b7 simp\n \u00b7 simp\n\n", "theoremStatement": "@[to_additive max_entropy_le_entropy_sub_prod]\nlemma max_entropy_le_entropy_div_prod\n (\u03ba : kernel T G) [IsMarkovKernel \u03ba] (\u03b7 : kernel T G) [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 \u03bc) :\n max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc]) \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.max_entropy_le_entropy_div_prod", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "737f79b", "date": "2023-12-24"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Group.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Group", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Group.jsonl", "positionMetadata": {"lineInFile": 143, "tokenPositionInFile": 6460, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 91, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n calc max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc])\n = max (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bc]) - Ik[\u03ba \u00d7\u2096 \u03b7, \u03bc] := by\n rw [mutualInfo_prod _ h\u03ba h\u03b7, sub_zero]\n _ \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1 / p.2) measurable_div, \u03bc] := by\n convert max_entropy_sub_mutualInfo_le_entropy_div (\u03ba \u00d7\u2096 \u03b7) \u03bc (h\u03ba.prod h\u03b7)\n \u00b7 simp\n \u00b7 simp", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 318}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n", "theoremStatement": "@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_mul_const", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "6146b3f", "date": "2023-12-15"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 19, "tokenPositionInFile": 658, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 69}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n", "theoremStatement": "/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_mul_right", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 24, "tokenPositionInFile": 848, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 159}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n", "theoremStatement": "/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_mul_right'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 31, "tokenPositionInFile": 1200, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 160}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n", "theoremStatement": "/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_mul_left", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 38, "tokenPositionInFile": 1554, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 27, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 101}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n", "theoremStatement": "/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_mul_left'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 44, "tokenPositionInFile": 1847, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 27, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 102}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n", "theoremStatement": "/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_inv_right", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 50, "tokenPositionInFile": 2142, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 159}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n", "theoremStatement": "/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_inv_left", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 57, "tokenPositionInFile": 2487, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 159}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n", "theoremStatement": "/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_div_right", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 64, "tokenPositionInFile": 2831, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 31, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 159}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n", "theoremStatement": "/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_div_right'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 71, "tokenPositionInFile": 3183, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 31, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 160}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n", "theoremStatement": "/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_div_left", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 78, "tokenPositionInFile": 3537, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 25, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 101}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n", "theoremStatement": "/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_div_left'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 84, "tokenPositionInFile": 3830, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 25, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 102}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n", "theoremStatement": "/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_inv", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 90, "tokenPositionInFile": 4125, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 55}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n", "theoremStatement": "/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_div_comm", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 95, "tokenPositionInFile": 4351, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 32, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 51}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n", "theoremStatement": "/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] ", "theoremName": "ProbabilityTheory.condEntropy_mul_left", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 105, "tokenPositionInFile": 4648, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 23, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 75}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n", "theoremStatement": "/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] ", "theoremName": "ProbabilityTheory.condEntropy_mul_right", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 111, "tokenPositionInFile": 4911, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 23, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 74}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n", "theoremStatement": "/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] ", "theoremName": "ProbabilityTheory.condEntropy_div_left", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "5d4d48f", "date": "2024-01-08"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 117, "tokenPositionInFile": 5173, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 83}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n", "theoremStatement": "/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] ", "theoremName": "ProbabilityTheory.condEntropy_div_right", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 123, "tokenPositionInFile": 5443, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 82}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n", "theoremStatement": "/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] ", "theoremName": "ProbabilityTheory.mutualInfo_mul_right", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 133, "tokenPositionInFile": 5750, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 83, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 88}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n", "theoremStatement": "/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 146, "tokenPositionInFile": 6276, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 42, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 135}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n", "theoremStatement": "/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 153, "tokenPositionInFile": 6635, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 42, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 135}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n", "theoremStatement": "/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 160, "tokenPositionInFile": 6995, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 40, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 135}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY\n\n", "theoremStatement": "/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div'", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 167, "tokenPositionInFile": 7354, "theoremPositionInFile": 21}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 41, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mutualInfo_comm hX hY, entropy_sub_mutualInfo_eq_condEntropy hY hX,\n \u2190 condEntropy_div_left hY hX]\n exact condEntropy_le_entropy _ (hX.div hY) hX", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 161}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [mutualInfo_comm hX hY, entropy_sub_mutualInfo_eq_condEntropy hY hX,\n \u2190 condEntropy_div_left hY hX]\n exact condEntropy_le_entropy _ (hX.div hY) hX\n\n", "theoremStatement": "/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] ", "theoremName": "ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_mul", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 175, "tokenPositionInFile": 7740, "theoremPositionInFile": 22}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 54, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_mul hX hY, entropy_sub_mutualInfo_le_entropy_mul' hX hY\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 144}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [mutualInfo_comm hX hY, entropy_sub_mutualInfo_eq_condEntropy hY hX,\n \u2190 condEntropy_div_left hY hX]\n exact condEntropy_le_entropy _ (hX.div hY) hX\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_mul hX hY, entropy_sub_mutualInfo_le_entropy_mul' hX hY\u27e9\n\n", "theoremStatement": "/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] ", "theoremName": "ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_div", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 182, "tokenPositionInFile": 8150, "theoremPositionInFile": 23}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 52, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_div hX hY, entropy_sub_mutualInfo_le_entropy_div' hX hY\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 144}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [mutualInfo_comm hX hY, entropy_sub_mutualInfo_eq_condEntropy hY hX,\n \u2190 condEntropy_div_left hY hX]\n exact condEntropy_le_entropy _ (hX.div hY) hX\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_mul hX hY, entropy_sub_mutualInfo_le_entropy_mul' hX hY\u27e9\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_div hX hY, entropy_sub_mutualInfo_le_entropy_div' hX hY\u27e9\n\n", "theoremStatement": "/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X * Y | Z]$$ -/\n@[to_additive \"$$\\\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\\\leq H[X + Y | Z]$$\"]\nlemma max_condEntropy_sub_condMutualInfo_le_condEntropy_mul {Z : \u03a9 \u2192 T} [FiniteRange Z]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) :\n max H[X | Z ; \u03bc] H[Y | Z ; \u03bc] - I[X : Y | Z ; \u03bc] \u2264 H[X * Y | Z ; \u03bc] ", "theoremName": "ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_mul", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "9d960b7", "date": "2023-12-14"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 189, "tokenPositionInFile": 8560, "theoremPositionInFile": 24}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 18, "numPremises": 93, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condMutualInfo_comm hX hY, condEntropy_eq_kernel_entropy hX hZ,\n condEntropy_eq_kernel_entropy hY hZ, condMutualInfo_eq_kernel_mutualInfo hY hX hZ,\n condEntropy_eq_kernel_entropy (show Measurable (X * Y) from hX.mul hY) hZ]\n rw [kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ \u03bc).symm,\n kernel.entropy_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc).symm,\n max_comm]\n refine (kernel.max_entropy_sub_mutualInfo_le_entropy_mul' _ _ ?_).trans_eq ?_\n . exact kernel.aefiniteKernelSupport_condDistrib _ _ _ (hY.prod_mk hX) hZ\n have h := condDistrib_comp (hY.prod_mk hX) hZ \u03bc (fun x \u21a6 x.2 * x.1)\n rw [kernel.entropy_congr h.symm]\n rfl", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 740}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [mutualInfo_comm hX hY, entropy_sub_mutualInfo_eq_condEntropy hY hX,\n \u2190 condEntropy_div_left hY hX]\n exact condEntropy_le_entropy _ (hX.div hY) hX\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_mul hX hY, entropy_sub_mutualInfo_le_entropy_mul' hX hY\u27e9\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_div hX hY, entropy_sub_mutualInfo_le_entropy_div' hX hY\u27e9\n\n/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X * Y | Z]$$ -/\n@[to_additive \"$$\\\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\\\leq H[X + Y | Z]$$\"]\nlemma max_condEntropy_sub_condMutualInfo_le_condEntropy_mul {Z : \u03a9 \u2192 T} [FiniteRange Z]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) :\n max H[X | Z ; \u03bc] H[Y | Z ; \u03bc] - I[X : Y | Z ; \u03bc] \u2264 H[X * Y | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condMutualInfo_comm hX hY, condEntropy_eq_kernel_entropy hX hZ,\n condEntropy_eq_kernel_entropy hY hZ, condMutualInfo_eq_kernel_mutualInfo hY hX hZ,\n condEntropy_eq_kernel_entropy (show Measurable (X * Y) from hX.mul hY) hZ]\n rw [kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ \u03bc).symm,\n kernel.entropy_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc).symm,\n max_comm]\n refine (kernel.max_entropy_sub_mutualInfo_le_entropy_mul' _ _ ?_).trans_eq ?_\n . exact kernel.aefiniteKernelSupport_condDistrib _ _ _ (hY.prod_mk hX) hZ\n have h := condDistrib_comp (hY.prod_mk hX) hZ \u03bc (fun x \u21a6 x.2 * x.1)\n rw [kernel.entropy_congr h.symm]\n rfl\n\n", "theoremStatement": "/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X / Y | Z]$$ -/\n@[to_additive \"$$\\\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\\\leq H[X - Y | Z]$$\"]\nlemma max_condEntropy_sub_condMutualInfo_le_condEntropy_div {Z : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n (max H[X | Z ; \u03bc] H[Y | Z ; \u03bc]) - I[X : Y | Z ; \u03bc] \u2264 H[X / Y | Z ; \u03bc] ", "theoremName": "ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_div", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "9d960b7", "date": "2023-12-14"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 207, "tokenPositionInFile": 9678, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 19, "numPremises": 94, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condMutualInfo_comm hX hY, condEntropy_eq_kernel_entropy hX hZ,\n condEntropy_eq_kernel_entropy hY hZ, condMutualInfo_eq_kernel_mutualInfo hY hX hZ,\n condEntropy_eq_kernel_entropy ?_ hZ]\n swap ; \u00b7 exact hX.div hY\n rw [kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ \u03bc).symm,\n kernel.entropy_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc).symm, max_comm]\n refine (kernel.max_entropy_sub_mutualInfo_le_entropy_div _ _ ?_).trans_eq ?_\n . exact kernel.aefiniteKernelSupport_condDistrib _ _ _ (hY.prod_mk hX) hZ\n rw [kernel.entropy_div_comm]\n have h := condDistrib_comp (hY.prod_mk hX) hZ \u03bc (fun x \u21a6 x.2 / x.1)\n rw [kernel.entropy_congr h.symm]\n rfl", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 755}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [mutualInfo_comm hX hY, entropy_sub_mutualInfo_eq_condEntropy hY hX,\n \u2190 condEntropy_div_left hY hX]\n exact condEntropy_le_entropy _ (hX.div hY) hX\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_mul hX hY, entropy_sub_mutualInfo_le_entropy_mul' hX hY\u27e9\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_div hX hY, entropy_sub_mutualInfo_le_entropy_div' hX hY\u27e9\n\n/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X * Y | Z]$$ -/\n@[to_additive \"$$\\\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\\\leq H[X + Y | Z]$$\"]\nlemma max_condEntropy_sub_condMutualInfo_le_condEntropy_mul {Z : \u03a9 \u2192 T} [FiniteRange Z]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) :\n max H[X | Z ; \u03bc] H[Y | Z ; \u03bc] - I[X : Y | Z ; \u03bc] \u2264 H[X * Y | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condMutualInfo_comm hX hY, condEntropy_eq_kernel_entropy hX hZ,\n condEntropy_eq_kernel_entropy hY hZ, condMutualInfo_eq_kernel_mutualInfo hY hX hZ,\n condEntropy_eq_kernel_entropy (show Measurable (X * Y) from hX.mul hY) hZ]\n rw [kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ \u03bc).symm,\n kernel.entropy_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc).symm,\n max_comm]\n refine (kernel.max_entropy_sub_mutualInfo_le_entropy_mul' _ _ ?_).trans_eq ?_\n . exact kernel.aefiniteKernelSupport_condDistrib _ _ _ (hY.prod_mk hX) hZ\n have h := condDistrib_comp (hY.prod_mk hX) hZ \u03bc (fun x \u21a6 x.2 * x.1)\n rw [kernel.entropy_congr h.symm]\n rfl\n\n/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X / Y | Z]$$ -/\n@[to_additive \"$$\\\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\\\leq H[X - Y | Z]$$\"]\nlemma max_condEntropy_sub_condMutualInfo_le_condEntropy_div {Z : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n (max H[X | Z ; \u03bc] H[Y | Z ; \u03bc]) - I[X : Y | Z ; \u03bc] \u2264 H[X / Y | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condMutualInfo_comm hX hY, condEntropy_eq_kernel_entropy hX hZ,\n condEntropy_eq_kernel_entropy hY hZ, condMutualInfo_eq_kernel_mutualInfo hY hX hZ,\n condEntropy_eq_kernel_entropy ?_ hZ]\n swap ; \u00b7 exact hX.div hY\n rw [kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ \u03bc).symm,\n kernel.entropy_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc).symm, max_comm]\n refine (kernel.max_entropy_sub_mutualInfo_le_entropy_div _ _ ?_).trans_eq ?_\n . exact kernel.aefiniteKernelSupport_condDistrib _ _ _ (hY.prod_mk hX) hZ\n rw [kernel.entropy_div_comm]\n have h := condDistrib_comp (hY.prod_mk hX) hZ \u03bc (fun x \u21a6 x.2 / x.1)\n rw [kernel.entropy_congr h.symm]\n rfl\n\n", "theoremStatement": "/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X * Y]$$. -/\n@[to_additive \"If $X, Y$ are independent, then $$\\\\max(H[X], H[Y]) \\\\leq H[X + Y]$$\"]\nlemma max_entropy_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) :\n max H[X ; \u03bc] H[Y ; \u03bc] \u2264 H[X * Y ; \u03bc] ", "theoremName": "ProbabilityTheory.max_entropy_le_entropy_mul", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 227, "tokenPositionInFile": 10842, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 56, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simpa [h.mutualInfo_eq_zero hX hY] using max_entropy_sub_mutualInfo_le_entropy_mul hX hY (\u03bc := \u03bc)", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 105}} +{"srcContext": "import PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.Kernel.Group\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {G : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable G] [Countable T] [Countable U]\n [Nonempty G] [Nonempty T] [Nonempty U]\n [MeasurableSpace G] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass G] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n [Group G] {X Y : \u03a9 \u2192 G} {\u03bc : Measure \u03a9}\n [FiniteRange X] [FiniteRange Y]\n\nnamespace ProbabilityTheory\nsection entropy\n\n@[to_additive (attr := simp)]\nlemma entropy_mul_const (hX : Measurable X) (c : G) :\n H[X * fun _ \u21a6 c; \u03bc] = H[X ; \u03bc] := by\n apply entropy_comp_of_injective \u03bc hX _ $ mul_left_injective c\n\n/-- $H[X, X * Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X + Y] = H[X, Y]$\"]\nlemma entropy_mul_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X * Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[X, Y * X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y + X] = H[X, Y]$\"]\nlemma entropy_mul_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y * X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.mulRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ $ Equiv.injective _\n\n/-- $H[Y * X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y + X, Y] = H[X, Y]$\"]\nlemma entropy_mul_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y * X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.mul hX) hY _).trans $ (entropy_mul_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X * Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X + Y, Y] = H[X, Y]$\"]\nlemma entropy_mul_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X * Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.mul hY) hY _).trans $ (entropy_mul_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X, Y\u207b\u00b9] = H[X, Y]$ -/\n@[to_additive \"$H[X, -Y] = H[X, Y]$\"]\nlemma entropy_inv_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u207b\u00b9\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodCongr (Equiv.inv _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X\u207b\u00b9, Y] = H[X, Y]$ -/\n@[to_additive \"$H[-X, Y] = H[X, Y]$\"]\nlemma entropy_inv_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X\u207b\u00b9, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.inv _).prodCongr (Equiv.refl _) \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, X / Y] = H[X, Y]$ -/\n@[to_additive \"$H[X, X - Y] = H[X, Y]$\"]\nlemma entropy_div_right (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, X / Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divLeft \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[X, Y / X] = H[X, Y]$ -/\n@[to_additive \"$H[X, Y - X] = H[X, Y]$\"]\nlemma entropy_div_right' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y / X\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := by\n change H[(Equiv.refl _).prodShear Equiv.divRight \u2218 \u27e8X, Y\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hX.prod_mk hY) _ (Equiv.injective _)\n\n/-- $H[Y / X, Y] = H[X, Y]$ -/\n@[to_additive \"$H[Y - X, Y] = H[X, Y]$\"]\nlemma entropy_div_left (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8Y / X, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hY.div hX) hY _).trans $ (entropy_div_right hY hX _).trans $ entropy_comm hY hX _\n\n/-- $H[X / Y, Y] = H[X, Y]$ -/\n@[to_additive \"$H[X - Y, Y] = H[X, Y]$\"]\nlemma entropy_div_left' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X / Y, Y\u27e9; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] :=\n (entropy_comm (hX.div hY) hY _).trans $ (entropy_div_right' hY hX _).trans $ entropy_comm hY hX _\n\n/-- If $X$ is $G$-valued, then $H[X\u207b\u00b9]=H[X]$. -/\n@[to_additive \"If $X$ is $G$-valued, then $H[-X]=H[X]$.\"]\nlemma entropy_inv (hX : Measurable X) : H[X\u207b\u00b9 ; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (\u00b7\u207b\u00b9) inv_injective\n\n/-- $$H[X / Y] = H[Y / X]$$ -/\n@[to_additive \"$$H[X - Y] = H[Y - X]$$\"]\nlemma entropy_div_comm {Y : \u03a9 \u2192 G} (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y ; \u03bc] = H[Y / X ; \u03bc] := by rw [\u2190 inv_div]; exact entropy_inv (hY.div hX)\n\nend entropy\n\nsection condEntropy\nvariable [IsFiniteMeasure \u03bc]\n\n/-- $$H[Y * X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y + X | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y * X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y * x) mul_right_injective\n\n/-- $$H[X * Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X + Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_mul_right (hX : Measurable X) (hY : Measurable Y) :\n H[X * Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x * y) mul_left_injective\n\n/-- $$H[Y / X | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[Y - X | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_left (hX : Measurable X) (hY : Measurable Y) :\n H[Y / X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 y / x) fun _ \u21a6 div_right_injective\n\n/-- $$H[X / Y | Y] = H[X | Y]$$ -/\n@[to_additive \"$$H[X - Y | Y] = H[X | Y]$$\"]\nlemma condEntropy_div_right (hX : Measurable X) (hY : Measurable Y) :\n H[X / Y | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n condEntropy_of_injective \u03bc hX hY (fun y x \u21a6 x / y) fun _ \u21a6 div_left_injective\n\nend condEntropy\n\nsection mutualInfo\n\n/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\n@[to_additive \"$I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.\"]\nlemma mutualInfo_mul_right (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (h : IndepFun X Y \u03bc) :\n I[X : X * Y ; \u03bc] = H[X * Y ; \u03bc] - H[Y ; \u03bc] := by\n rw [mutualInfo_def, entropy_mul_right hX hY, h.entropy_pair_eq_add hX hY]\n abel\n\nend mutualInfo\n\nsection IsProbabilityMeasure\nvariable [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G} [FiniteRange Y]\n\n/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_mul_right hX hY]\n exact condEntropy_le_entropy _ (hX.mul hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_mul' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy' hX hY, \u2190 condEntropy_mul_left hY hX]\n exact condEntropy_le_entropy _ (hX.mul hY) hX\n\n/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[X] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n H[X ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [entropy_sub_mutualInfo_eq_condEntropy hX hY, \u2190 condEntropy_div_right hX hY]\n exact condEntropy_le_entropy _ (hX.div hY) hY\n\n/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$H[Y] - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma entropy_sub_mutualInfo_le_entropy_div' (hX : Measurable X) (hY : Measurable Y) :\n H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [mutualInfo_comm hX hY, entropy_sub_mutualInfo_eq_condEntropy hY hX,\n \u2190 condEntropy_div_left hY hX]\n exact condEntropy_le_entropy _ (hX.div hY) hX\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X + Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_mul hX hY, entropy_sub_mutualInfo_le_entropy_mul' hX hY\u27e9\n\n/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X / Y]$$ -/\n@[to_additive \"$$\\\\max(H[X], H[Y]) - I[X : Y] \\\\leq H[X - Y]$$\"]\nlemma max_entropy_sub_mutualInfo_le_entropy_div (hX : Measurable X) (hY : Measurable Y) :\n max H[X ; \u03bc] H[Y ; \u03bc] - I[X : Y ; \u03bc] \u2264 H[X / Y ; \u03bc] := by\n rw [\u2190 max_sub_sub_right, max_le_iff]\n exact \u27e8entropy_sub_mutualInfo_le_entropy_div hX hY, entropy_sub_mutualInfo_le_entropy_div' hX hY\u27e9\n\n/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X * Y | Z]$$ -/\n@[to_additive \"$$\\\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\\\leq H[X + Y | Z]$$\"]\nlemma max_condEntropy_sub_condMutualInfo_le_condEntropy_mul {Z : \u03a9 \u2192 T} [FiniteRange Z]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) :\n max H[X | Z ; \u03bc] H[Y | Z ; \u03bc] - I[X : Y | Z ; \u03bc] \u2264 H[X * Y | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condMutualInfo_comm hX hY, condEntropy_eq_kernel_entropy hX hZ,\n condEntropy_eq_kernel_entropy hY hZ, condMutualInfo_eq_kernel_mutualInfo hY hX hZ,\n condEntropy_eq_kernel_entropy (show Measurable (X * Y) from hX.mul hY) hZ]\n rw [kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ \u03bc).symm,\n kernel.entropy_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc).symm,\n max_comm]\n refine (kernel.max_entropy_sub_mutualInfo_le_entropy_mul' _ _ ?_).trans_eq ?_\n . exact kernel.aefiniteKernelSupport_condDistrib _ _ _ (hY.prod_mk hX) hZ\n have h := condDistrib_comp (hY.prod_mk hX) hZ \u03bc (fun x \u21a6 x.2 * x.1)\n rw [kernel.entropy_congr h.symm]\n rfl\n\n/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X / Y | Z]$$ -/\n@[to_additive \"$$\\\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\\\leq H[X - Y | Z]$$\"]\nlemma max_condEntropy_sub_condMutualInfo_le_condEntropy_div {Z : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n (max H[X | Z ; \u03bc] H[Y | Z ; \u03bc]) - I[X : Y | Z ; \u03bc] \u2264 H[X / Y | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condMutualInfo_comm hX hY, condEntropy_eq_kernel_entropy hX hZ,\n condEntropy_eq_kernel_entropy hY hZ, condMutualInfo_eq_kernel_mutualInfo hY hX hZ,\n condEntropy_eq_kernel_entropy ?_ hZ]\n swap ; \u00b7 exact hX.div hY\n rw [kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ \u03bc).symm,\n kernel.entropy_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc).symm, max_comm]\n refine (kernel.max_entropy_sub_mutualInfo_le_entropy_div _ _ ?_).trans_eq ?_\n . exact kernel.aefiniteKernelSupport_condDistrib _ _ _ (hY.prod_mk hX) hZ\n rw [kernel.entropy_div_comm]\n have h := condDistrib_comp (hY.prod_mk hX) hZ \u03bc (fun x \u21a6 x.2 / x.1)\n rw [kernel.entropy_congr h.symm]\n rfl\n\n/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X * Y]$$. -/\n@[to_additive \"If $X, Y$ are independent, then $$\\\\max(H[X], H[Y]) \\\\leq H[X + Y]$$\"]\nlemma max_entropy_le_entropy_mul (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) :\n max H[X ; \u03bc] H[Y ; \u03bc] \u2264 H[X * Y ; \u03bc] := by\n simpa [h.mutualInfo_eq_zero hX hY] using max_entropy_sub_mutualInfo_le_entropy_mul hX hY (\u03bc := \u03bc)\n\n", "theoremStatement": "/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X / Y]$$. -/\n@[to_additive \"If $X, Y$ are independent, then $$\\\\max(H[X], H[Y]) \\\\leq H[X - Y]$$.\"]\nlemma max_entropy_le_entropy_div (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) :\n max H[X ; \u03bc] H[Y ; \u03bc] \u2264 H[X / Y ; \u03bc] ", "theoremName": "ProbabilityTheory.max_entropy_le_entropy_div", "fileCreated": {"commit": "81571e5", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fd", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/Group.lean", "module": "PFR.ForMathlib.Entropy.Group", "jsonFile": "PFR.ForMathlib.Entropy.Group.jsonl", "positionMetadata": {"lineInFile": 233, "tokenPositionInFile": 11247, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 54, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simpa [h.mutualInfo_eq_zero hX hY] using max_entropy_sub_mutualInfo_le_entropy_div hX hY (\u03bc := \u03bc)", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 105}} diff --git a/minictx/pnt.jsonl b/minictx/pnt.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f549149e746363fb4a3d4737cae15508800e0c60 --- /dev/null +++ b/minictx/pnt.jsonl @@ -0,0 +1,85 @@ +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\n", "theoremStatement": "lemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y ", "theoremName": "HIntegral_symm", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 24, "tokenPositionInFile": 961, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 27, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= integral_symm _ _", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 20}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\n", "theoremStatement": "lemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 ", "theoremName": "VIntegral_symm", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 26, "tokenPositionInFile": 1050, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 67, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 67}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\n", "theoremStatement": "lemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) ", "theoremName": "verticalIntegral_split_three", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 76, "tokenPositionInFile": 3759, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 85, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 249}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\n", "theoremStatement": "lemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) ", "theoremName": "DiffVertRect_eq_UpperLowerUs", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 89, "tokenPositionInFile": 4509, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 134, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 355}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\n", "theoremStatement": "theorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) ", "theoremName": "existsDifferentiableOn_of_bddAbove", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 113, "tokenPositionInFile": 5854, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 62, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 175}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\n", "theoremStatement": "theorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 ", "theoremName": "HolomorphicOn.vanishesOnRectangle", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 130, "tokenPositionInFile": 6770, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 37, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 78}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\n", "theoremStatement": "theorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w ", "theoremName": "RectangleIntegral_congr", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 140, "tokenPositionInFile": 7082, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 115, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 346}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\n", "theoremStatement": "theorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w ", "theoremName": "RectangleIntegral'_congr", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 150, "tokenPositionInFile": 7559, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 49, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [RectangleIntegral', RectangleIntegral_congr h]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 58}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\n", "theoremStatement": "theorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z ", "theoremName": "rectangleIntegral_symm", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 154, "tokenPositionInFile": 7751, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 83, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 235}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\n", "theoremStatement": "theorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w ", "theoremName": "rectangleIntegral_symm_re", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 160, "tokenPositionInFile": 8097, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 113, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 116}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\n", "theoremStatement": "theorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w ", "theoremName": "RectangleBorderIntegrable.add", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 172, "tokenPositionInFile": 8702, "theoremPositionInFile": 22}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 106, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 299}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\n", "theoremStatement": "theorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w ", "theoremName": "ContinuousOn.rectangleBorder_integrable", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 181, "tokenPositionInFile": 9225, "theoremPositionInFile": 23}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 59, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9", "proofType": "term", "proofLengthLines": 4, "proofLengthTokens": 343}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\n", "theoremStatement": "theorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w ", "theoremName": "ContinuousOn.rectangleBorderIntegrable", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 188, "tokenPositionInFile": 9700, "theoremPositionInFile": 24}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 80}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\n", "theoremStatement": "theorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w ", "theoremName": "ContinuousOn.rectangleBorderNoPIntegrable", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 192, "tokenPositionInFile": 9905, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 43, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 166}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\n", "theoremStatement": "theorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w ", "theoremName": "HolomorphicOn.rectangleBorderIntegrable'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 197, "tokenPositionInFile": 10246, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 32, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 100}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\n", "theoremStatement": "theorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w ", "theoremName": "HolomorphicOn.rectangleBorderIntegrable", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 201, "tokenPositionInFile": 10510, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 16, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n hf.continuousOn.rectangleBorderIntegrable", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 46}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\n", "theoremStatement": "lemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) ", "theoremName": "RectangleIntegralHSplit'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 227, "tokenPositionInFile": 11905, "theoremPositionInFile": 29}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 86, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)", "proofType": "term", "proofLengthLines": 5, "proofLengthTokens": 414}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\n", "theoremStatement": "lemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) ", "theoremName": "RectangleIntegralVSplit", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 238, "tokenPositionInFile": 12629, "theoremPositionInFile": 30}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 131, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 341}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\n", "theoremStatement": "lemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) ", "theoremName": "RectangleIntegralVSplit'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 252, "tokenPositionInFile": 13512, "theoremPositionInFile": 31}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 86, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)", "proofType": "term", "proofLengthLines": 5, "proofLengthTokens": 426}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\n", "theoremStatement": "lemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 ", "theoremName": "RectanglePullToNhdOfPole'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 263, "tokenPositionInFile": 14248, "theoremPositionInFile": 32}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 15, "numPremises": 246, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]", "proofType": "tactic", "proofLengthLines": 38, "proofLengthTokens": 2376}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\n", "theoremStatement": "lemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) ", "theoremName": "RectanglePullToNhdOfPole''", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "9730de7", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 338, "tokenPositionInFile": 18605, "theoremPositionInFile": 34}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 91, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 136}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\n", "theoremStatement": "theorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b ", "theoremName": "ResidueTheoremAtOrigin_aux1c", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "facd4bc", "date": "2024-02-09"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 346, "tokenPositionInFile": 19068, "theoremPositionInFile": 35}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 87, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 85}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\n", "theoremStatement": "theorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b ", "theoremName": "ResidueTheoremAtOrigin_aux1c'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "8a7b39a", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 351, "tokenPositionInFile": 19280, "theoremPositionInFile": 36}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 98, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 85}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\n", "theoremStatement": "theorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b ", "theoremName": "ResidueTheoremAtOrigin_aux2c", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "8a7b39a", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 356, "tokenPositionInFile": 19493, "theoremPositionInFile": 37}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 113, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 85}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\n", "theoremStatement": "theorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b ", "theoremName": "ResidueTheoremAtOrigin_aux2c'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "8a7b39a", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 361, "tokenPositionInFile": 19709, "theoremPositionInFile": 38}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 123, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 85}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\n", "theoremStatement": "theorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w ", "theoremName": "RectangleIntegral.const_smul", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "e5d82fd", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 366, "tokenPositionInFile": 19927, "theoremPositionInFile": 39}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 96, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 95}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\n", "theoremStatement": "theorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w ", "theoremName": "RectangleIntegral.const_mul'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "e5d82fd", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 370, "tokenPositionInFile": 20162, "theoremPositionInFile": 40}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 171, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 84}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\n", "theoremStatement": "theorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) ", "theoremName": "RectangleIntegral.translate", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "cd8d627", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 374, "tokenPositionInFile": 20388, "theoremPositionInFile": 41}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 109, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 172}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\n", "theoremStatement": "theorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) ", "theoremName": "RectangleIntegral.translate'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "e5d82fd", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 379, "tokenPositionInFile": 20709, "theoremPositionInFile": 42}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 51, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 65}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\n", "theoremStatement": "lemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) ", "theoremName": "Complex.inv_re_add_im", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 383, "tokenPositionInFile": 20926, "theoremPositionInFile": 43}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 179, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 92}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\n", "theoremStatement": "lemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 ", "theoremName": "sq_add_sq_ne_zero", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 386, "tokenPositionInFile": 21096, "theoremPositionInFile": 44}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 131, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 51}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\n", "theoremStatement": "lemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) ", "theoremName": "continuous_self_div_sq_add_sq", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 388, "tokenPositionInFile": 21206, "theoremPositionInFile": 45}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 48, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 98}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\n", "theoremStatement": "lemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 ", "theoremName": "integral_self_div_sq_add_sq", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 391, "tokenPositionInFile": 21397, "theoremPositionInFile": 46}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 222, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 566}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\nlemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 := by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _\n\n", "theoremStatement": "lemma integral_const_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, y / (x ^ 2 + y ^ 2) =\n arctan (x\u2082 / y) - arctan (x\u2081 / y) ", "theoremName": "integral_const_div_sq_add_sq", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 402, "tokenPositionInFile": 22117, "theoremPositionInFile": 47}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 184, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n nth_rewrite 1 [\u2190 div_mul_cancel\u2080 x\u2081 hy, \u2190 div_mul_cancel\u2080 x\u2082 hy]\n simp_rw [\u2190 mul_integral_comp_mul_right, \u2190 integral_const_mul, \u2190 integral_one_div_one_add_sq]\n exact integral_congr <| \u03bb x _ => by field_simp; ring", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 222}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\nlemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 := by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _\n\nlemma integral_const_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, y / (x ^ 2 + y ^ 2) =\n arctan (x\u2082 / y) - arctan (x\u2081 / y) := by\n nth_rewrite 1 [\u2190 div_mul_cancel\u2080 x\u2081 hy, \u2190 div_mul_cancel\u2080 x\u2082 hy]\n simp_rw [\u2190 mul_integral_comp_mul_right, \u2190 integral_const_mul, \u2190 integral_one_div_one_add_sq]\n exact integral_congr <| \u03bb x _ => by field_simp; ring\n\n", "theoremStatement": "lemma integral_const_div_self_add_im (hy : y \u2260 0) : \u222b x : \u211d in x\u2081..x\u2082, A / (x + y * I) =\n A * (Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2) -\n A * I * (arctan (x\u2082 / y) - arctan (x\u2081 / y)) ", "theoremName": "integral_const_div_self_add_im", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 408, "tokenPositionInFile": 22466, "theoremPositionInFile": 48}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 236, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have e1 {x : \u211d} : A / (x + y * I) = A * x / (x ^ 2 + y ^ 2) - A * I * y / (x ^ 2 + y ^ 2) := by\n ring_nf ; simp_rw [inv_re_add_im] ; ring\n have e2 : IntervalIntegrable (fun x \u21a6 A * x / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n simp_rw [mul_div_assoc] ; norm_cast\n exact continuous_const.mul <| continuous_ofReal.comp <| continuous_self_div_sq_add_sq hy\n have e3 : IntervalIntegrable (fun x \u21a6 A * I * y / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n refine continuous_const.div (by continuity) (\u03bb x => ?_)\n norm_cast ; exact sq_add_sq_ne_zero hy\n simp_rw [integral_congr (\u03bb _ _ => e1), integral_sub e2 e3, mul_div_assoc]\n norm_cast\n simp_rw [integral_const_mul, intervalIntegral.integral_ofReal, integral_self_div_sq_add_sq hy,\n integral_const_div_sq_add_sq hy]", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 858}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\nlemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 := by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _\n\nlemma integral_const_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, y / (x ^ 2 + y ^ 2) =\n arctan (x\u2082 / y) - arctan (x\u2081 / y) := by\n nth_rewrite 1 [\u2190 div_mul_cancel\u2080 x\u2081 hy, \u2190 div_mul_cancel\u2080 x\u2082 hy]\n simp_rw [\u2190 mul_integral_comp_mul_right, \u2190 integral_const_mul, \u2190 integral_one_div_one_add_sq]\n exact integral_congr <| \u03bb x _ => by field_simp; ring\n\nlemma integral_const_div_self_add_im (hy : y \u2260 0) : \u222b x : \u211d in x\u2081..x\u2082, A / (x + y * I) =\n A * (Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2) -\n A * I * (arctan (x\u2082 / y) - arctan (x\u2081 / y)) := by\n have e1 {x : \u211d} : A / (x + y * I) = A * x / (x ^ 2 + y ^ 2) - A * I * y / (x ^ 2 + y ^ 2) := by\n ring_nf ; simp_rw [inv_re_add_im] ; ring\n have e2 : IntervalIntegrable (fun x \u21a6 A * x / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n simp_rw [mul_div_assoc] ; norm_cast\n exact continuous_const.mul <| continuous_ofReal.comp <| continuous_self_div_sq_add_sq hy\n have e3 : IntervalIntegrable (fun x \u21a6 A * I * y / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n refine continuous_const.div (by continuity) (\u03bb x => ?_)\n norm_cast ; exact sq_add_sq_ne_zero hy\n simp_rw [integral_congr (\u03bb _ _ => e1), integral_sub e2 e3, mul_div_assoc]\n norm_cast\n simp_rw [integral_const_mul, intervalIntegral.integral_ofReal, integral_self_div_sq_add_sq hy,\n integral_const_div_sq_add_sq hy]\n\n", "theoremStatement": "lemma integral_const_div_re_add_self (hx : x \u2260 0) : \u222b y : \u211d in y\u2081..y\u2082, A / (x + y * I) =\n A / I * (Real.log (y\u2082 ^ 2 + (-x) ^ 2) / 2 - Real.log (y\u2081 ^ 2 + (-x) ^ 2) / 2) -\n A / I * I * (arctan (y\u2082 / -x) - arctan (y\u2081 / -x)) ", "theoremName": "integral_const_div_re_add_self", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 426, "tokenPositionInFile": 23537, "theoremPositionInFile": 49}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 203, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have l1 {y : \u211d} : A / (x + y * I) = A / I / (y + \u2191(-x) * I) := by\n have e1 : x + y * I \u2260 0 := by contrapose! hx ; simpa using congr_arg re hx\n have e2 : y + -(x * I) \u2260 0 := by contrapose! hx ; simpa using congr_arg im hx\n field_simp ; ring_nf ; simp\n have l2 : -x \u2260 0 := by rwa [neg_ne_zero]\n simp_rw [l1, integral_const_div_self_add_im l2]", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 359}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\nlemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 := by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _\n\nlemma integral_const_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, y / (x ^ 2 + y ^ 2) =\n arctan (x\u2082 / y) - arctan (x\u2081 / y) := by\n nth_rewrite 1 [\u2190 div_mul_cancel\u2080 x\u2081 hy, \u2190 div_mul_cancel\u2080 x\u2082 hy]\n simp_rw [\u2190 mul_integral_comp_mul_right, \u2190 integral_const_mul, \u2190 integral_one_div_one_add_sq]\n exact integral_congr <| \u03bb x _ => by field_simp; ring\n\nlemma integral_const_div_self_add_im (hy : y \u2260 0) : \u222b x : \u211d in x\u2081..x\u2082, A / (x + y * I) =\n A * (Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2) -\n A * I * (arctan (x\u2082 / y) - arctan (x\u2081 / y)) := by\n have e1 {x : \u211d} : A / (x + y * I) = A * x / (x ^ 2 + y ^ 2) - A * I * y / (x ^ 2 + y ^ 2) := by\n ring_nf ; simp_rw [inv_re_add_im] ; ring\n have e2 : IntervalIntegrable (fun x \u21a6 A * x / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n simp_rw [mul_div_assoc] ; norm_cast\n exact continuous_const.mul <| continuous_ofReal.comp <| continuous_self_div_sq_add_sq hy\n have e3 : IntervalIntegrable (fun x \u21a6 A * I * y / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n refine continuous_const.div (by continuity) (\u03bb x => ?_)\n norm_cast ; exact sq_add_sq_ne_zero hy\n simp_rw [integral_congr (\u03bb _ _ => e1), integral_sub e2 e3, mul_div_assoc]\n norm_cast\n simp_rw [integral_const_mul, intervalIntegral.integral_ofReal, integral_self_div_sq_add_sq hy,\n integral_const_div_sq_add_sq hy]\n\nlemma integral_const_div_re_add_self (hx : x \u2260 0) : \u222b y : \u211d in y\u2081..y\u2082, A / (x + y * I) =\n A / I * (Real.log (y\u2082 ^ 2 + (-x) ^ 2) / 2 - Real.log (y\u2081 ^ 2 + (-x) ^ 2) / 2) -\n A / I * I * (arctan (y\u2082 / -x) - arctan (y\u2081 / -x)) := by\n have l1 {y : \u211d} : A / (x + y * I) = A / I / (y + \u2191(-x) * I) := by\n have e1 : x + y * I \u2260 0 := by contrapose! hx ; simpa using congr_arg re hx\n have e2 : y + -(x * I) \u2260 0 := by contrapose! hx ; simpa using congr_arg im hx\n field_simp ; ring_nf ; simp\n have l2 : -x \u2260 0 := by rwa [neg_ne_zero]\n simp_rw [l1, integral_const_div_self_add_im l2]\n\n", "theoremStatement": "lemma ResidueTheoremAtOrigin' {z w c : \u2102} (h1 : z.re < 0) (h2 : z.im < 0) (h3 : 0 < w.re) (h4 : 0 < w.im) :\n RectangleIntegral (\u03bb s => c / s) z w = 2 * I * \u03c0 * c ", "theoremName": "ResidueTheoremAtOrigin'", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "69828a9", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 436, "tokenPositionInFile": 24125, "theoremPositionInFile": 50}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 312, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp only [RectangleIntegral, HIntegral, VIntegral, smul_eq_mul]\n rw [integral_const_div_re_add_self h1.ne, integral_const_div_re_add_self h3.ne.symm]\n rw [integral_const_div_self_add_im h2.ne, integral_const_div_self_add_im h4.ne.symm]\n have l1 : z.im * w.re\u207b\u00b9 = (w.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have l3 := arctan_inv_of_neg <| mul_neg_of_pos_of_neg h3 <| inv_lt_zero.mpr h2\n have l4 : w.im * z.re\u207b\u00b9 = (z.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have l6 := arctan_inv_of_neg <| mul_neg_of_neg_of_pos h1 <| inv_pos.mpr h4\n have r1 : z.im * z.re\u207b\u00b9 = (z.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have r3 := arctan_inv_of_pos <| mul_pos_of_neg_of_neg h1 <| inv_lt_zero.mpr h2\n have r4 : w.im * w.re\u207b\u00b9 = (w.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have r6 := arctan_inv_of_pos <| mul_pos h3 <| inv_pos.mpr h4\n ring_nf\n simp only [one_div, inv_I, mul_neg, neg_mul, I_sq, one_mul, neg_neg, arctan_neg, ofReal_neg, sub_neg_eq_add]\n rw [l1, l3, l4, l6, r1, r3, r4, r6]\n ring_nf\n simp only [I_sq, ofReal_sub, ofReal_mul, ofReal_ofNat, ofReal_div, ofReal_neg, ofReal_one]\n ring_nf", "proofType": "tactic", "proofLengthLines": 17, "proofLengthTokens": 1052}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\nlemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 := by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _\n\nlemma integral_const_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, y / (x ^ 2 + y ^ 2) =\n arctan (x\u2082 / y) - arctan (x\u2081 / y) := by\n nth_rewrite 1 [\u2190 div_mul_cancel\u2080 x\u2081 hy, \u2190 div_mul_cancel\u2080 x\u2082 hy]\n simp_rw [\u2190 mul_integral_comp_mul_right, \u2190 integral_const_mul, \u2190 integral_one_div_one_add_sq]\n exact integral_congr <| \u03bb x _ => by field_simp; ring\n\nlemma integral_const_div_self_add_im (hy : y \u2260 0) : \u222b x : \u211d in x\u2081..x\u2082, A / (x + y * I) =\n A * (Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2) -\n A * I * (arctan (x\u2082 / y) - arctan (x\u2081 / y)) := by\n have e1 {x : \u211d} : A / (x + y * I) = A * x / (x ^ 2 + y ^ 2) - A * I * y / (x ^ 2 + y ^ 2) := by\n ring_nf ; simp_rw [inv_re_add_im] ; ring\n have e2 : IntervalIntegrable (fun x \u21a6 A * x / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n simp_rw [mul_div_assoc] ; norm_cast\n exact continuous_const.mul <| continuous_ofReal.comp <| continuous_self_div_sq_add_sq hy\n have e3 : IntervalIntegrable (fun x \u21a6 A * I * y / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n refine continuous_const.div (by continuity) (\u03bb x => ?_)\n norm_cast ; exact sq_add_sq_ne_zero hy\n simp_rw [integral_congr (\u03bb _ _ => e1), integral_sub e2 e3, mul_div_assoc]\n norm_cast\n simp_rw [integral_const_mul, intervalIntegral.integral_ofReal, integral_self_div_sq_add_sq hy,\n integral_const_div_sq_add_sq hy]\n\nlemma integral_const_div_re_add_self (hx : x \u2260 0) : \u222b y : \u211d in y\u2081..y\u2082, A / (x + y * I) =\n A / I * (Real.log (y\u2082 ^ 2 + (-x) ^ 2) / 2 - Real.log (y\u2081 ^ 2 + (-x) ^ 2) / 2) -\n A / I * I * (arctan (y\u2082 / -x) - arctan (y\u2081 / -x)) := by\n have l1 {y : \u211d} : A / (x + y * I) = A / I / (y + \u2191(-x) * I) := by\n have e1 : x + y * I \u2260 0 := by contrapose! hx ; simpa using congr_arg re hx\n have e2 : y + -(x * I) \u2260 0 := by contrapose! hx ; simpa using congr_arg im hx\n field_simp ; ring_nf ; simp\n have l2 : -x \u2260 0 := by rwa [neg_ne_zero]\n simp_rw [l1, integral_const_div_self_add_im l2]\n\nlemma ResidueTheoremAtOrigin' {z w c : \u2102} (h1 : z.re < 0) (h2 : z.im < 0) (h3 : 0 < w.re) (h4 : 0 < w.im) :\n RectangleIntegral (\u03bb s => c / s) z w = 2 * I * \u03c0 * c := by\n simp only [RectangleIntegral, HIntegral, VIntegral, smul_eq_mul]\n rw [integral_const_div_re_add_self h1.ne, integral_const_div_re_add_self h3.ne.symm]\n rw [integral_const_div_self_add_im h2.ne, integral_const_div_self_add_im h4.ne.symm]\n have l1 : z.im * w.re\u207b\u00b9 = (w.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have l3 := arctan_inv_of_neg <| mul_neg_of_pos_of_neg h3 <| inv_lt_zero.mpr h2\n have l4 : w.im * z.re\u207b\u00b9 = (z.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have l6 := arctan_inv_of_neg <| mul_neg_of_neg_of_pos h1 <| inv_pos.mpr h4\n have r1 : z.im * z.re\u207b\u00b9 = (z.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have r3 := arctan_inv_of_pos <| mul_pos_of_neg_of_neg h1 <| inv_lt_zero.mpr h2\n have r4 : w.im * w.re\u207b\u00b9 = (w.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have r6 := arctan_inv_of_pos <| mul_pos h3 <| inv_pos.mpr h4\n ring_nf\n simp only [one_div, inv_I, mul_neg, neg_mul, I_sq, one_mul, neg_neg, arctan_neg, ofReal_neg, sub_neg_eq_add]\n rw [l1, l3, l4, l6, r1, r3, r4, r6]\n ring_nf\n simp only [I_sq, ofReal_sub, ofReal_mul, ofReal_ofNat, ofReal_div, ofReal_neg, ofReal_one]\n ring_nf\n\n", "theoremStatement": "theorem ResidueTheoremInRectangle (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\n (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p) : RectangleIntegral' (\u03bb s => c / (s - p)) z w = c ", "theoremName": "ResidueTheoremInRectangle", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "19f1f08", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 456, "tokenPositionInFile": 25344, "theoremPositionInFile": 51}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 185, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [rectangle_mem_nhds_iff, mem_reProdIm, uIoo_of_le zRe_le_wRe, uIoo_of_le zIm_le_wIm]\n at pInRectInterior\n rw [RectangleIntegral.translate', RectangleIntegral']\n have : 1 / (2 * \u2191\u03c0 * I) * (2 * I * \u2191\u03c0 * c) = c := by field_simp [two_pi_I_ne_zero] ; ring\n rwa [ResidueTheoremAtOrigin'] ; all_goals { simp [*] }", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 325}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\nlemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 := by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _\n\nlemma integral_const_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, y / (x ^ 2 + y ^ 2) =\n arctan (x\u2082 / y) - arctan (x\u2081 / y) := by\n nth_rewrite 1 [\u2190 div_mul_cancel\u2080 x\u2081 hy, \u2190 div_mul_cancel\u2080 x\u2082 hy]\n simp_rw [\u2190 mul_integral_comp_mul_right, \u2190 integral_const_mul, \u2190 integral_one_div_one_add_sq]\n exact integral_congr <| \u03bb x _ => by field_simp; ring\n\nlemma integral_const_div_self_add_im (hy : y \u2260 0) : \u222b x : \u211d in x\u2081..x\u2082, A / (x + y * I) =\n A * (Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2) -\n A * I * (arctan (x\u2082 / y) - arctan (x\u2081 / y)) := by\n have e1 {x : \u211d} : A / (x + y * I) = A * x / (x ^ 2 + y ^ 2) - A * I * y / (x ^ 2 + y ^ 2) := by\n ring_nf ; simp_rw [inv_re_add_im] ; ring\n have e2 : IntervalIntegrable (fun x \u21a6 A * x / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n simp_rw [mul_div_assoc] ; norm_cast\n exact continuous_const.mul <| continuous_ofReal.comp <| continuous_self_div_sq_add_sq hy\n have e3 : IntervalIntegrable (fun x \u21a6 A * I * y / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n refine continuous_const.div (by continuity) (\u03bb x => ?_)\n norm_cast ; exact sq_add_sq_ne_zero hy\n simp_rw [integral_congr (\u03bb _ _ => e1), integral_sub e2 e3, mul_div_assoc]\n norm_cast\n simp_rw [integral_const_mul, intervalIntegral.integral_ofReal, integral_self_div_sq_add_sq hy,\n integral_const_div_sq_add_sq hy]\n\nlemma integral_const_div_re_add_self (hx : x \u2260 0) : \u222b y : \u211d in y\u2081..y\u2082, A / (x + y * I) =\n A / I * (Real.log (y\u2082 ^ 2 + (-x) ^ 2) / 2 - Real.log (y\u2081 ^ 2 + (-x) ^ 2) / 2) -\n A / I * I * (arctan (y\u2082 / -x) - arctan (y\u2081 / -x)) := by\n have l1 {y : \u211d} : A / (x + y * I) = A / I / (y + \u2191(-x) * I) := by\n have e1 : x + y * I \u2260 0 := by contrapose! hx ; simpa using congr_arg re hx\n have e2 : y + -(x * I) \u2260 0 := by contrapose! hx ; simpa using congr_arg im hx\n field_simp ; ring_nf ; simp\n have l2 : -x \u2260 0 := by rwa [neg_ne_zero]\n simp_rw [l1, integral_const_div_self_add_im l2]\n\nlemma ResidueTheoremAtOrigin' {z w c : \u2102} (h1 : z.re < 0) (h2 : z.im < 0) (h3 : 0 < w.re) (h4 : 0 < w.im) :\n RectangleIntegral (\u03bb s => c / s) z w = 2 * I * \u03c0 * c := by\n simp only [RectangleIntegral, HIntegral, VIntegral, smul_eq_mul]\n rw [integral_const_div_re_add_self h1.ne, integral_const_div_re_add_self h3.ne.symm]\n rw [integral_const_div_self_add_im h2.ne, integral_const_div_self_add_im h4.ne.symm]\n have l1 : z.im * w.re\u207b\u00b9 = (w.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have l3 := arctan_inv_of_neg <| mul_neg_of_pos_of_neg h3 <| inv_lt_zero.mpr h2\n have l4 : w.im * z.re\u207b\u00b9 = (z.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have l6 := arctan_inv_of_neg <| mul_neg_of_neg_of_pos h1 <| inv_pos.mpr h4\n have r1 : z.im * z.re\u207b\u00b9 = (z.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have r3 := arctan_inv_of_pos <| mul_pos_of_neg_of_neg h1 <| inv_lt_zero.mpr h2\n have r4 : w.im * w.re\u207b\u00b9 = (w.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have r6 := arctan_inv_of_pos <| mul_pos h3 <| inv_pos.mpr h4\n ring_nf\n simp only [one_div, inv_I, mul_neg, neg_mul, I_sq, one_mul, neg_neg, arctan_neg, ofReal_neg, sub_neg_eq_add]\n rw [l1, l3, l4, l6, r1, r3, r4, r6]\n ring_nf\n simp only [I_sq, ofReal_sub, ofReal_mul, ofReal_ofNat, ofReal_div, ofReal_neg, ofReal_one]\n ring_nf\n\ntheorem ResidueTheoremInRectangle (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\n (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p) : RectangleIntegral' (\u03bb s => c / (s - p)) z w = c := by\n simp [rectangle_mem_nhds_iff, mem_reProdIm, uIoo_of_le zRe_le_wRe, uIoo_of_le zIm_le_wIm]\n at pInRectInterior\n rw [RectangleIntegral.translate', RectangleIntegral']\n have : 1 / (2 * \u2191\u03c0 * I) * (2 * I * \u2191\u03c0 * c) = c := by field_simp [two_pi_I_ne_zero] ; ring\n rwa [ResidueTheoremAtOrigin'] ; all_goals { simp [*] }\n\n/-%%\n\\begin{lemma}[ResidueTheoremAtOrigin]\\label{ResidueTheoremAtOrigin}\n\\lean{ResidueTheoremAtOrigin}\\leanok\nThe rectangle (square) integral of $f(s) = 1/s$ with corners $-1-i$ and $1+i$ is equal to $2\\pi i$.\n\\end{lemma}\n%%-/\n", "theoremStatement": "lemma ResidueTheoremAtOrigin : RectangleIntegral' (fun s \u21a6 1 / s) (-1 - I) (1 + I) = 1 ", "theoremName": "ResidueTheoremAtOrigin", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "897cabd", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 470, "tokenPositionInFile": 26080, "theoremPositionInFile": 52}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 191, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [RectangleIntegral', ResidueTheoremAtOrigin']\n all_goals { field_simp [pi_ne_zero] <;> ring }", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 105}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\nimport Mathlib.Analysis.Complex.RemovableSingularity\nimport Mathlib.Analysis.Analytic.Meromorphic\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.MeasureTheory.Measure.Lebesgue.Integral\nimport PrimeNumberTheoremAnd.Rectangle\n\nopen Complex BigOperators Nat Classical Real Topology Filter Set MeasureTheory intervalIntegral\n\nopen scoped Interval\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E] {f g : \u2102 \u2192 E}\n {z w p c A : \u2102} {x x\u2081 x\u2082 y y\u2081 y\u2082 \u03c3 : \u211d}\n\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := \u222b x in x\u2081..x\u2082, f (x + y * I)\n\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := I \u2022 \u222b y in y\u2081..y\u2082, f (x + y * I)\n\nnoncomputable def HIntegral' (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 HIntegral f x\u2081 x\u2082 y\n\nnoncomputable def VIntegral' (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E := (1 / (2 * \u03c0 * I)) \u2022 VIntegral f x y\u2081 y\u2082\n\nlemma HIntegral_symm : HIntegral f x\u2081 x\u2082 y = - HIntegral f x\u2082 x\u2081 y := integral_symm _ _\n\nlemma VIntegral_symm : VIntegral f x y\u2081 y\u2082 = - VIntegral f x y\u2082 y\u2081 := by\n simp_rw [VIntegral, integral_symm y\u2081 y\u2082, smul_neg, neg_neg]\n\n/-%%\n\\begin{definition}[RectangleIntegral]\\label{RectangleIntegral}\\lean{RectangleIntegral}\\leanok\nA RectangleIntegral of a function $f$ is one over a rectangle determined by $z$ and $w$ in $\\C$.\nWe will sometimes denote it by $\\int_{z}^{w} f$. (There is also a primed version, which is $1/(2\\pi i)$ times the original.)\n\\end{definition}\n%%-/\n/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`. -/\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E := HIntegral f z.re w.re z.im -\n HIntegral f z.re w.re w.im + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\n\n/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n `z` and `w` in `\u2102`, divided by `2 * \u03c0 * I`. -/\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\n\n/-% ** Wrong delimiter on purpose **\nAn UpperUIntegral is the integral of a function over a |\\_| shape.\n\\begin{definition}\\label{UpperUIntegral}\\lean{UpperUIntegral}\\leanok\nAn UpperUIntegral of a function $f$ comes from $\\sigma+i\\infty$ down to $\\sigma+iT$, over to $\\sigma'+iT$, and back up to $\\sigma'+i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def UpperUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' T +\n I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3' + y * I)) - I \u2022 (\u222b y : \u211d in Ici T, f (\u03c3 + y * I))\n\n/-% ** Wrong delimiter on purpose **\nA LowerUIntegral is the integral of a function over a |-| shape.\n\\begin{definition}[LowerUIntegral]\\label{LowerUIntegral}\\lean{LowerUIntegral}\\leanok\nA LowerUIntegral of a function $f$ comes from $\\sigma-i\\infty$ up to $\\sigma-iT$, over to $\\sigma'-iT$, and back down to $\\sigma'-i\\infty$.\n\\end{definition}\n%-/\nnoncomputable def LowerUIntegral (f : \u2102 \u2192 E) (\u03c3 \u03c3' T : \u211d) : E := HIntegral f \u03c3 \u03c3' (-T) -\n I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3' + y * I)) + I \u2022 (\u222b y : \u211d in Iic (-T), f (\u03c3 + y * I))\n\n/-%%\nIt is very convenient to define integrals along vertical lines in the complex plane, as follows.\n\\begin{definition}[VerticalIntegral]\\label{VerticalIntegral}\\lean{VerticalIntegral}\\leanok\nLet $f$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$, and let $\\sigma$ be a real number. Then we define\n$$\\int_{(\\sigma)}f(s)ds = \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}f(s)ds.$$\n\\end{definition}\n%%-/\nnoncomputable def VerticalIntegral (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E := I \u2022 \u222b t : \u211d, f (\u03c3 + t * I)\n\n--%% We also have a version with a factor of $1/(2\\pi i)$.\nnoncomputable abbrev VerticalIntegral' (f : \u2102 \u2192 E) (\u03c3 : \u211d) : E :=\n (1 / (2 * \u03c0 * I)) \u2022 VerticalIntegral f \u03c3\n\nlemma verticalIntegral_split_three (a b : \u211d) (hf : Integrable (fun t : \u211d \u21a6 f (\u03c3 + t * I))) :\n VerticalIntegral f \u03c3 = I \u2022 (\u222b t in Iic a, f (\u03c3 + t * I)) + VIntegral f \u03c3 a b\n + I \u2022 \u222b t in Ici b, f (\u03c3 + t * I) := by\n simp_rw [VerticalIntegral, VIntegral, \u2190 smul_add]\n congr\n rw [\u2190 intervalIntegral.integral_Iic_sub_Iic hf.restrict hf.restrict, add_sub_cancel,\n integral_Iic_eq_integral_Iio, intervalIntegral.integral_Iio_add_Ici hf.restrict hf.restrict]\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[DiffVertRect_eq_UpperLowerUs]\\label{DiffVertRect_eq_UpperLowerUs}\\lean{DiffVertRect_eq_UpperLowerUs}\\leanok\nThe difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.\n\\end{lemma}\n%-/\nlemma DiffVertRect_eq_UpperLowerUs {\u03c3 \u03c3' T : \u211d}\n (f_int_\u03c3 : Integrable (fun (t : \u211d) \u21a6 f (\u03c3 + t * I)))\n (f_int_\u03c3' : Integrable (fun (t : \u211d) \u21a6 f (\u03c3' + t * I))) :\n (VerticalIntegral f \u03c3') - (VerticalIntegral f \u03c3) - (RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) =\n (UpperUIntegral f \u03c3 \u03c3' T) - (LowerUIntegral f \u03c3 \u03c3' T) := by\n rw [verticalIntegral_split_three (-T) T f_int_\u03c3, verticalIntegral_split_three (-T) T f_int_\u03c3']\n simp only [smul_eq_mul, RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im,\n ofReal_im, mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add,\n zero_sub, add_im, UpperUIntegral, LowerUIntegral]\n abel\n/-%\n\\begin{proof}\\uses{UpperUIntegral, LowerUIntegral}\\leanok\nFollows directly from the definitions.\n\\end{proof}\n%-/\n\n/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\nabbrev HolomorphicOn (f : \u2102 \u2192 E) (s : Set \u2102) : Prop := DifferentiableOn \u2102 f s\n\n/-%%\n\\begin{theorem}[existsDifferentiableOn_of_bddAbove]\\label{existsDifferentiableOn_of_bddAbove}\\lean{existsDifferentiableOn_of_bddAbove}\\leanok\nIf $f$ is differentiable on a set $s$ except at $c\\in s$, and $f$ is bounded above on $s\\setminus\\{c\\}$, then there exists a differentiable function $g$ on $s$ such that $f$ and $g$ agree on $s\\setminus\\{c\\}$.\n\\end{theorem}\n%%-/\ntheorem existsDifferentiableOn_of_bddAbove [CompleteSpace E] {s : Set \u2102} {c : \u2102} (hc : s \u2208 nhds c)\n (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm \u2218 f '' (s \\ {c}))) :\n \u2203 (g : \u2102 \u2192 E), HolomorphicOn g s \u2227 (Set.EqOn f g (s \\ {c})) :=\n \u27e8Function.update f c (limUnder (\ud835\udcdd[{c}\u1d9c] c) f),\n differentiableOn_update_limUnder_of_bddAbove hc hd hb,\n fun z hz \u21a6 if h : z = c then (hz.2 h).elim else by simp [h]\u27e9\n/-%%\n\\begin{proof}\\leanok\nThis is the Riemann Removable Singularity Theorem, slightly rephrased from what's in Mathlib. (We don't care what the function $g$ is, just that it's holomorphic.)\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{theorem}[HolomorphicOn.vanishesOnRectangle]\\label{HolomorphicOn.vanishesOnRectangle}\\lean{HolomorphicOn.vanishesOnRectangle}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$, then the integral of $f$ over the rectangle with corners $z$ and $w$ is $0$.\n\\end{theorem}\n%%-/\ntheorem HolomorphicOn.vanishesOnRectangle [CompleteSpace E] {U : Set \u2102}\n (f_holo : HolomorphicOn f U) (hU : Rectangle z w \u2286 U) :\n RectangleIntegral f z w = 0 :=\n integral_boundary_rect_eq_zero_of_differentiableOn f z w (f_holo.mono hU)\n/-%%\n\\begin{proof}\\leanok\nThis is in a Mathlib PR.\n\\end{proof}\n%%-/\n\ntheorem RectangleIntegral_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral f z w = RectangleIntegral g z w := by\n unfold RectangleIntegral VIntegral\n congr 2; swap; congr 1; swap; congr 1\n all_goals refine intervalIntegral.integral_congr fun _ _ \u21a6 h ?_\n \u00b7 exact Or.inl <| Or.inl <| Or.inl \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inl <| Or.inr \u27e8by simpa, by simp\u27e9\n \u00b7 exact Or.inr \u27e8by simp, by simpa\u27e9\n \u00b7 exact Or.inl <| Or.inl <| Or.inr \u27e8by simp, by simpa\u27e9\n\ntheorem RectangleIntegral'_congr (h : Set.EqOn f g (RectangleBorder z w)) :\n RectangleIntegral' f z w = RectangleIntegral' g z w := by\n rw [RectangleIntegral', RectangleIntegral_congr h]\n\ntheorem rectangleIntegral_symm (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f z w = RectangleIntegral f w z := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, intervalIntegral.integral_symm w.re,\n intervalIntegral.integral_symm w.im, sub_neg_eq_add, smul_neg, sub_neg_eq_add, \u2190 sub_eq_add_neg,\n neg_add_eq_sub, sub_add_eq_add_sub]\n\ntheorem rectangleIntegral_symm_re (f : \u2102 \u2192 E) (z w : \u2102) :\n RectangleIntegral f (w.re + z.im * I) (z.re + w.im * I) = - RectangleIntegral f z w := by\n simp [RectangleIntegral, \u2190 sub_eq_zero]\n rw [HIntegral_symm (y := z.im), HIntegral_symm (y := w.im)]\n abel\n\ndef RectangleBorderIntegrable (f : \u2102 \u2192 E) (z w : \u2102) : Prop :=\n IntervalIntegrable (fun x => f (x + z.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun x => f (x + w.im * I)) volume z.re w.re \u2227\n IntervalIntegrable (fun y => f (w.re + y * I)) volume z.im w.im \u2227\n IntervalIntegrable (fun y => f (z.re + y * I)) volume z.im w.im\n\ntheorem RectangleBorderIntegrable.add {f g : \u2102 \u2192 E} (hf : RectangleBorderIntegrable f z w)\n (hg : RectangleBorderIntegrable g z w) :\n RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n rw [intervalIntegral.integral_add hf.1 hg.1, intervalIntegral.integral_add hf.2.1 hg.2.1,\n intervalIntegral.integral_add hf.2.2.1 hg.2.2.1, intervalIntegral.integral_add hf.2.2.2 hg.2.2.2]\n rw [\u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\ntheorem ContinuousOn.rectangleBorder_integrable (hf : ContinuousOn f (RectangleBorder z w)) :\n RectangleBorderIntegrable f z w :=\n \u27e8(hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_im z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_right_re z w)).intervalIntegrable,\n (hf.comp (by fun_prop) (mapsTo_rectangleBorder_left_re z w)).intervalIntegrable\u27e9\n\ntheorem ContinuousOn.rectangleBorderIntegrable (hf : ContinuousOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n (hf.mono (rectangleBorder_subset_rectangle z w)).rectangleBorder_integrable\n\ntheorem ContinuousOn.rectangleBorderNoPIntegrable (hf : ContinuousOn f (Rectangle z w \\ {p}))\n (pNotOnBorder : p \u2209 RectangleBorder z w) : RectangleBorderIntegrable f z w := by\n refine (hf.mono (Set.subset_diff.mpr ?_)).rectangleBorder_integrable\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem HolomorphicOn.rectangleBorderIntegrable' (hf : HolomorphicOn f (Rectangle z w \\ {p}))\n (hp : Rectangle z w \u2208 nhds p) : RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderNoPIntegrable (not_mem_rectangleBorder_of_rectangle_mem_nhds hp)\n\ntheorem HolomorphicOn.rectangleBorderIntegrable (hf : HolomorphicOn f (Rectangle z w)) :\n RectangleBorderIntegrable f z w :=\n hf.continuousOn.rectangleBorderIntegrable\n\n/--\nGiven `x\u2080 a x\u2081 : \u211d`, and `y\u2080 y\u2081 : \u211d` and a function `f : \u2102 \u2192 \u2102` so that\nboth `(t : \u211d) \u21a6 f(t + y\u2080 * I)` and `(t : \u211d) \u21a6 f(t + y\u2081 * I)` are integrable over both\n`t \u2208 Icc x\u2080 a` and `t \u2208 Icc a x\u2081`, we have that\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I)` is the sum of\n`RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I)` and\n`RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I)`.\n-/\nlemma RectangleIntegralHSplit {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_x\u2080_a_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_bot : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2080 * I)) volume a x\u2081)\n (f_int_x\u2080_a_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume x\u2080 a)\n (f_int_a_x\u2081_top : IntervalIntegrable (fun x => f (\u2191x + \u2191y\u2081 * I)) volume a x\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_bot f_int_a_x\u2081_bot,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_x\u2080_a_top f_int_a_x\u2081_top]\n abel\n\nlemma RectangleIntegralHSplit' {a x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (ha : a \u2208 [[x\u2080, x\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (a + y\u2081 * I) +\n RectangleIntegral f (a + y\u2080 * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralHSplit\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc left_mem_uIcc ha) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.1) (uIcc_subset_uIcc ha right_mem_uIcc) le_rfl)\n\nlemma RectangleIntegralVSplit {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d}\n (f_int_y\u2080_b_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_left : IntervalIntegrable (fun y => f (x\u2080 + y * I)) volume b y\u2081)\n (f_int_y\u2080_b_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume y\u2080 b)\n (f_int_b_y\u2081_right : IntervalIntegrable (fun y => f (x\u2081 + y * I)) volume b y\u2081) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) := by\n dsimp [RectangleIntegral, HIntegral, VIntegral]\n simp only [mul_one, mul_zero, add_zero, zero_add, sub_self]\n rw [\u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_left f_int_b_y\u2081_left,\n \u2190 intervalIntegral.integral_add_adjacent_intervals f_int_y\u2080_b_right f_int_b_y\u2081_right, \u2190 sub_eq_zero]\n simp only [smul_add]; abel\n\nlemma RectangleIntegralVSplit' {b x\u2080 x\u2081 y\u2080 y\u2081 : \u211d} (hb : b \u2208 [[y\u2080, y\u2081]])\n (hf : RectangleBorderIntegrable f (\u2191x\u2080 + \u2191y\u2080 * I) (\u2191x\u2081 + \u2191y\u2081 * I)) :\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + y\u2081 * I) =\n RectangleIntegral f (x\u2080 + y\u2080 * I) (x\u2081 + b * I) +\n RectangleIntegral f (x\u2080 + b * I) (x\u2081 + y\u2081 * I) :=\n RectangleIntegralVSplit\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.2) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc left_mem_uIcc hb) le_rfl)\n (IntervalIntegrable.mono (by simpa using hf.2.2.1) (uIcc_subset_uIcc hb right_mem_uIcc) le_rfl)\n\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\n (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p) (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\n (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\ {p})) :\n RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082 := by\n obtain \u27e8hz\u2080_re, hz\u2080_im, hz\u2081_re, hz\u2081_im\u27e9 := h_orientation\n have := rect_subset_iff.mp hz\n rw [Rectangle, uIcc_of_le hz\u2080_re, uIcc_of_le hz\u2080_im] at this\n obtain \u27e8\u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9, \u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := this\n obtain \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9 := (uIoo_of_le hz\u2081_re) \u25b8 (uIoo_of_le hz\u2081_im) \u25b8 rectangle_mem_nhds_iff.mp hp\n obtain \u27e8_, _, _, _\u27e9 := show p.re < z\u2082.re \u2227 p.re < z\u2083.re \u2227 p.im < z\u2082.im \u2227 p.im < z\u2083.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n obtain \u27e8_, _, _, _\u27e9 := show z\u2080.re < p.re \u2227 z\u2081.re < p.re \u2227 z\u2080.im < p.im \u2227 z\u2081.im < p.im from\n \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n\n have fCont := fHolo.continuousOn\n\n have hbot : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have htop : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) := ?_\n have hleft : RectangleBorderIntegrable f (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n have hright : RectangleBorderIntegrable f (\u2191z\u2081.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) := ?_\n all_goals try {\n refine (fCont.mono (rectangleBorder_subset_punctured_rect ?_ ?_)).rectangleBorder_integrable\n \u00b7 simp_all\n \u00b7 simpa using \u27e8by linarith, by linarith, by linarith, by linarith\u27e9\n }\n have hbot' : z\u2081.im \u2208 [[z\u2080.im, z\u2083.im]] := ?_\n have htop' : z\u2082.im \u2208 [[z\u2081.im, z\u2083.im]] := ?_\n have hleft' : z\u2081.re \u2208 [[z\u2080.re, z\u2083.re]] := ?_\n have hright' : z\u2082.re \u2208 [[z\u2081.re, z\u2083.re]] := ?_\n all_goals try { rw [Set.uIcc_of_le]; constructor; all_goals assumption }\n have hbot'' : Rectangle (\u2191z\u2080.re + \u2191z\u2080.im * I) (\u2191z\u2083.re + \u2191z\u2081.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have htop'' : Rectangle (\u2191z\u2080.re + \u2191z\u2082.im * I) (\u2191z\u2083.re + \u2191z\u2083.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hleft'' : Rectangle (\u2191z\u2080.re + \u2191z\u2081.im * I) (\u2191z\u2081.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n have hright'' : Rectangle (\u2191z\u2082.re + \u2191z\u2081.im * I) (\u2191z\u2083.re + \u2191z\u2082.im * I) \u2286 Rectangle z\u2080 z\u2083 \\ {p} := ?_\n all_goals try { apply rectangle_subset_punctured_rect <;> simp_all }\n\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2083,\n RectangleIntegralVSplit' hbot' hbot, fHolo.vanishesOnRectangle hbot'', zero_add,\n RectangleIntegralVSplit' htop' htop, fHolo.vanishesOnRectangle htop'', add_zero,\n RectangleIntegralHSplit' hleft' hleft, fHolo.vanishesOnRectangle hleft'', zero_add,\n RectangleIntegralHSplit' hright' hright, fHolo.vanishesOnRectangle hright'', add_zero,\n re_add_im, re_add_im]\n\n/-%%\nThe next lemma allows to zoom a big rectangle down to a small square, centered at a pole.\n\n\\begin{lemma}[RectanglePullToNhdOfPole]\\label{RectanglePullToNhdOfPole}\\lean{RectanglePullToNhdOfPole}\\leanok\nIf $f$ is holomorphic on a rectangle $z$ and $w$ except at a point $p$, then the integral of $f$\nover the rectangle with corners $z$ and $w$ is the same as the integral of $f$ over a small square\ncentered at $p$.\n\\end{lemma}\n%%-/\n/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102} (zRe_lt_wRe : z.re \u2264 w.re)\n (zIm_lt_wIm : z.im \u2264 w.im) (hp : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral f z w = RectangleIntegral f (-c - I * c + p) (c + I * c + p) := by\n/-%%\n\\begin{proof}\\uses{HolomorphicOn.vanishesOnRectangle}\\leanok\nChop the big rectangle with two vertical cuts and two horizontal cuts into smaller rectangles,\nthe middle one being the desired square. The integral over each of the outer rectangles\nvanishes, since $f$ is holomorphic there. (The constant $c$ being ``small enough'' here just means\nthat the inner square is strictly contained in the big rectangle.)\n%%-/\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' zero_lt_one, SmallSquareInRectangle hp]\n intro c \u27e8cpos, _\u27e9 hc\n simp_rw [mul_comm I]\n exact RectanglePullToNhdOfPole' (by simp_all [cpos.le])\n (square_mem_nhds p (ne_of_gt cpos)) hc fHolo\n--%%\\end{proof}\n\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102} (zRe_le_wRe : z.re \u2264 w.re)\n (zIm_le_wIm : z.im \u2264 w.im) (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\n RectangleIntegral' f z w = RectangleIntegral' f (-c - I * c + p) (c + I * c + p) := by\n filter_upwards [RectanglePullToNhdOfPole zRe_le_wRe zIm_le_wIm pInRectInterior fHolo] with c h\n simp_rw [RectangleIntegral', h]\n\ntheorem ResidueTheoremAtOrigin_aux1c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y + I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux1c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (y - I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem ResidueTheoremAtOrigin_aux2c' (a b : \u211d) :\n let f : \u211d \u2192 \u2102 := fun y => (-1 + y * I)\u207b\u00b9\n IntervalIntegrable f volume a b :=\n (ContinuousOn.inv\u2080 (by fun_prop) (by simp [Complex.ext_iff])).intervalIntegrable\n\ntheorem RectangleIntegral.const_smul (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral f z w := by\n simp [RectangleIntegral, HIntegral, VIntegral, smul_add, smul_sub, smul_smul, mul_comm]\n\ntheorem RectangleIntegral.const_mul' (f : \u2102 \u2192 E) (z w c : \u2102) :\n RectangleIntegral' (fun s => c \u2022 f s) z w = c \u2022 RectangleIntegral' f z w := by\n simp [RectangleIntegral', RectangleIntegral.const_smul, smul_smul] ; ring_nf\n\ntheorem RectangleIntegral.translate (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p) := by\n simp_rw [RectangleIntegral, HIntegral, VIntegral, sub_re, sub_im, \u2190 intervalIntegral.integral_comp_sub_right]\n congr <;> ext <;> congr 1 <;> simp [Complex.ext_iff]\n\ntheorem RectangleIntegral.translate' (f : \u2102 \u2192 E) (z w p : \u2102) :\n RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p) := by\n simp_rw [RectangleIntegral', RectangleIntegral.translate]\n\nlemma Complex.inv_re_add_im : (x + y * I)\u207b\u00b9 = (x - I * y) / (x ^ 2 + y ^ 2) := by\n rw [Complex.inv_def, div_eq_mul_inv] ; congr <;> simp [conj_ofReal, normSq] <;> ring\n\nlemma sq_add_sq_ne_zero (hy : y \u2260 0) : x ^ 2 + y ^ 2 \u2260 0 := by linarith [sq_nonneg x, (sq_pos_iff y).mpr hy]\n\nlemma continuous_self_div_sq_add_sq (hy : y \u2260 0) : Continuous fun x => x / (x ^ 2 + y ^ 2) :=\n continuous_id.div (continuous_id.pow 2 |>.add continuous_const) (\u03bb _ => sq_add_sq_ne_zero hy)\n\nlemma integral_self_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, x / (x ^ 2 + y ^ 2) =\n Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2 := by\n let f (x : \u211d) : \u211d := Real.log (x ^ 2 + y ^ 2) / 2\n have e1 {x} := HasDerivAt.add_const (by simpa using hasDerivAt_pow 2 x) (y ^ 2)\n have e2 {x} : HasDerivAt f (x / (x ^ 2 + y ^ 2)) x := by\n convert (e1.log (sq_add_sq_ne_zero hy)).div_const 2 using 1 ; field_simp ; ring\n have e3 : deriv f = \u03bb x => x / (x ^ 2 + y ^ 2) := funext (\u03bb _ => e2.deriv)\n have e4 : Continuous (deriv f) := by simpa only [e3] using continuous_self_div_sq_add_sq hy\n simp_rw [\u2190 e2.deriv]\n exact integral_deriv_eq_sub (\u03bb _ _ => e2.differentiableAt) <| e4.intervalIntegrable _ _\n\nlemma integral_const_div_sq_add_sq (hy : y \u2260 0) : \u222b x in x\u2081..x\u2082, y / (x ^ 2 + y ^ 2) =\n arctan (x\u2082 / y) - arctan (x\u2081 / y) := by\n nth_rewrite 1 [\u2190 div_mul_cancel\u2080 x\u2081 hy, \u2190 div_mul_cancel\u2080 x\u2082 hy]\n simp_rw [\u2190 mul_integral_comp_mul_right, \u2190 integral_const_mul, \u2190 integral_one_div_one_add_sq]\n exact integral_congr <| \u03bb x _ => by field_simp; ring\n\nlemma integral_const_div_self_add_im (hy : y \u2260 0) : \u222b x : \u211d in x\u2081..x\u2082, A / (x + y * I) =\n A * (Real.log (x\u2082 ^ 2 + y ^ 2) / 2 - Real.log (x\u2081 ^ 2 + y ^ 2) / 2) -\n A * I * (arctan (x\u2082 / y) - arctan (x\u2081 / y)) := by\n have e1 {x : \u211d} : A / (x + y * I) = A * x / (x ^ 2 + y ^ 2) - A * I * y / (x ^ 2 + y ^ 2) := by\n ring_nf ; simp_rw [inv_re_add_im] ; ring\n have e2 : IntervalIntegrable (fun x \u21a6 A * x / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n simp_rw [mul_div_assoc] ; norm_cast\n exact continuous_const.mul <| continuous_ofReal.comp <| continuous_self_div_sq_add_sq hy\n have e3 : IntervalIntegrable (fun x \u21a6 A * I * y / (x ^ 2 + y ^ 2)) volume x\u2081 x\u2082 := by\n apply Continuous.intervalIntegrable\n refine continuous_const.div (by continuity) (\u03bb x => ?_)\n norm_cast ; exact sq_add_sq_ne_zero hy\n simp_rw [integral_congr (\u03bb _ _ => e1), integral_sub e2 e3, mul_div_assoc]\n norm_cast\n simp_rw [integral_const_mul, intervalIntegral.integral_ofReal, integral_self_div_sq_add_sq hy,\n integral_const_div_sq_add_sq hy]\n\nlemma integral_const_div_re_add_self (hx : x \u2260 0) : \u222b y : \u211d in y\u2081..y\u2082, A / (x + y * I) =\n A / I * (Real.log (y\u2082 ^ 2 + (-x) ^ 2) / 2 - Real.log (y\u2081 ^ 2 + (-x) ^ 2) / 2) -\n A / I * I * (arctan (y\u2082 / -x) - arctan (y\u2081 / -x)) := by\n have l1 {y : \u211d} : A / (x + y * I) = A / I / (y + \u2191(-x) * I) := by\n have e1 : x + y * I \u2260 0 := by contrapose! hx ; simpa using congr_arg re hx\n have e2 : y + -(x * I) \u2260 0 := by contrapose! hx ; simpa using congr_arg im hx\n field_simp ; ring_nf ; simp\n have l2 : -x \u2260 0 := by rwa [neg_ne_zero]\n simp_rw [l1, integral_const_div_self_add_im l2]\n\nlemma ResidueTheoremAtOrigin' {z w c : \u2102} (h1 : z.re < 0) (h2 : z.im < 0) (h3 : 0 < w.re) (h4 : 0 < w.im) :\n RectangleIntegral (\u03bb s => c / s) z w = 2 * I * \u03c0 * c := by\n simp only [RectangleIntegral, HIntegral, VIntegral, smul_eq_mul]\n rw [integral_const_div_re_add_self h1.ne, integral_const_div_re_add_self h3.ne.symm]\n rw [integral_const_div_self_add_im h2.ne, integral_const_div_self_add_im h4.ne.symm]\n have l1 : z.im * w.re\u207b\u00b9 = (w.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have l3 := arctan_inv_of_neg <| mul_neg_of_pos_of_neg h3 <| inv_lt_zero.mpr h2\n have l4 : w.im * z.re\u207b\u00b9 = (z.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have l6 := arctan_inv_of_neg <| mul_neg_of_neg_of_pos h1 <| inv_pos.mpr h4\n have r1 : z.im * z.re\u207b\u00b9 = (z.re * z.im\u207b\u00b9)\u207b\u00b9 := by group\n have r3 := arctan_inv_of_pos <| mul_pos_of_neg_of_neg h1 <| inv_lt_zero.mpr h2\n have r4 : w.im * w.re\u207b\u00b9 = (w.re * w.im\u207b\u00b9)\u207b\u00b9 := by group\n have r6 := arctan_inv_of_pos <| mul_pos h3 <| inv_pos.mpr h4\n ring_nf\n simp only [one_div, inv_I, mul_neg, neg_mul, I_sq, one_mul, neg_neg, arctan_neg, ofReal_neg, sub_neg_eq_add]\n rw [l1, l3, l4, l6, r1, r3, r4, r6]\n ring_nf\n simp only [I_sq, ofReal_sub, ofReal_mul, ofReal_ofNat, ofReal_div, ofReal_neg, ofReal_one]\n ring_nf\n\ntheorem ResidueTheoremInRectangle (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\n (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p) : RectangleIntegral' (\u03bb s => c / (s - p)) z w = c := by\n simp [rectangle_mem_nhds_iff, mem_reProdIm, uIoo_of_le zRe_le_wRe, uIoo_of_le zIm_le_wIm]\n at pInRectInterior\n rw [RectangleIntegral.translate', RectangleIntegral']\n have : 1 / (2 * \u2191\u03c0 * I) * (2 * I * \u2191\u03c0 * c) = c := by field_simp [two_pi_I_ne_zero] ; ring\n rwa [ResidueTheoremAtOrigin'] ; all_goals { simp [*] }\n\n/-%%\n\\begin{lemma}[ResidueTheoremAtOrigin]\\label{ResidueTheoremAtOrigin}\n\\lean{ResidueTheoremAtOrigin}\\leanok\nThe rectangle (square) integral of $f(s) = 1/s$ with corners $-1-i$ and $1+i$ is equal to $2\\pi i$.\n\\end{lemma}\n%%-/\nlemma ResidueTheoremAtOrigin : RectangleIntegral' (fun s \u21a6 1 / s) (-1 - I) (1 + I) = 1 := by\n rw [RectangleIntegral', ResidueTheoremAtOrigin']\n all_goals { field_simp [pi_ne_zero] <;> ring }\n/-%%\n\\begin{proof}\\leanok\nThis is a special case of the more general result above.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[ResidueTheoremOnRectangleWithSimplePole]\\label{ResidueTheoremOnRectangleWithSimplePole}\n\\lean{ResidueTheoremOnRectangleWithSimplePole}\\leanok\nSuppose that $f$ is a holomorphic function on a rectangle, except for a simple pole\nat $p$. By the latter, we mean that there is a function $g$ holomorphic on the rectangle such that, $f = g + A/(s-p)$ for some $A\\in\\C$. Then the integral of $f$ over the\nrectangle is $A$.\n\\end{lemma}\n%%-/\n-- TODO: generalize to `f g : \u2102 \u2192 E`\n", "theoremStatement": "lemma ResidueTheoremOnRectangleWithSimplePole {f g : \u2102 \u2192 \u2102} {z w p A : \u2102}\n (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\n (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\n (gHolo : HolomorphicOn g (Rectangle z w))\n (principalPart : Set.EqOn (f - fun s \u21a6 A / (s - p)) (g) (Rectangle z w \\ {p})) :\n RectangleIntegral' f z w = A ", "theoremName": "ResidueTheoremOnRectangleWithSimplePole", "fileCreated": {"commit": "f36a520", "date": "2024-01-22"}, "theoremCreated": {"commit": "2282e1a", "date": "2024-02-18"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ResidueCalcOnRectangles.lean", "module": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "jsonFile": "PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl", "positionMetadata": {"lineInFile": 488, "tokenPositionInFile": 26862, "theoremPositionInFile": 53}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 10, "repositoryPremises": true, "numRepositoryPremises": 14, "numPremises": 171, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n have principalPart' : Set.EqOn f (g + (fun s \u21a6 A / (s - p))) (Rectangle z w \\ {p}) :=\n fun s hs => by rw [Pi.add_apply, \u2190 principalPart hs, Pi.sub_apply, sub_add_cancel]\n\n have : Set.EqOn f (g + (fun s \u21a6 A / (s - p))) (RectangleBorder z w) :=\n principalPart'.mono <| Set.subset_diff.mpr \u27e8rectangleBorder_subset_rectangle z w,\n disjoint_singleton_right.mpr (not_mem_rectangleBorder_of_rectangle_mem_nhds pInRectInterior)\u27e9\n rw [RectangleIntegral'_congr this]\n\n have t1 : RectangleBorderIntegrable g z w := gHolo.rectangleBorderIntegrable\n have t2 : HolomorphicOn (fun s \u21a6 A / (s - p)) (Rectangle z w \\ {p}) := by\n apply DifferentiableOn.mono (t := {p}\u1d9c)\n \u00b7 apply DifferentiableOn.div\n \u00b7 exact differentiableOn_const _\n \u00b7 exact DifferentiableOn.sub differentiableOn_id (differentiableOn_const _)\n \u00b7 exact fun x hx => by rw [sub_ne_zero]; exact hx\n \u00b7 rintro s \u27e8_, hs\u27e9 ; exact hs\n have t3 : RectangleBorderIntegrable (fun s \u21a6 A / (s - p)) z w :=\n HolomorphicOn.rectangleBorderIntegrable' t2 pInRectInterior\n\n rw [RectangleIntegral', RectangleBorderIntegrable.add t1 t3, smul_add]\n rw [gHolo.vanishesOnRectangle (by rfl), smul_zero, zero_add]\n\n exact ResidueTheoremInRectangle zRe_le_wRe zIm_le_wIm pInRectInterior", "proofType": "tactic", "proofLengthLines": 24, "proofLengthTokens": 1263}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\n", "theoremStatement": "lemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) ", "theoremName": "Square_apply", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 43, "tokenPositionInFile": 1256, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 178, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 101}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n", "theoremStatement": "/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t ", "theoremName": "preimage_equivRealProd_prod", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "0b22b41", "date": "2024-02-06"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 49, "tokenPositionInFile": 1494, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 14, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n", "theoremStatement": "@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t ", "theoremName": "preimage_equivRealProdCLM_reProdIm", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "f1c959f", "date": "2024-02-19"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 52, "tokenPositionInFile": 1655, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 45, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n rfl", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 8}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n", "theoremStatement": "@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm ", "theoremName": "ContinuousLinearEquiv.coe_toLinearEquiv_symm", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "1d47e66", "date": "2024-02-19"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 57, "tokenPositionInFile": 1780, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n rfl", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 8}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n", "theoremStatement": "/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 ", "theoremName": "reProdIm_subset_iff", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "0b22b41", "date": "2024-02-06"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 66, "tokenPositionInFile": 2190, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 23, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 78}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n", "theoremStatement": "/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 ", "theoremName": "reProdIm_subset_iff'", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "0b22b41", "date": "2024-02-06"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 71, "tokenPositionInFile": 2451, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n reProdIm_subset_iff.trans prod_subset_prod_iff", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 51}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n", "theoremStatement": "/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} ", "theoremName": "segment_reProdIm_segment_eq_convexHull", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 76, "tokenPositionInFile": 2681, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 159, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 162}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n", "theoremStatement": "/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U ", "theoremName": "rectangle_in_convex", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 83, "tokenPositionInFile": 3189, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 81, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 128}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\n", "theoremStatement": "lemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im ", "theoremName": "mem_Rect", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "91651fb", "date": "2024-02-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 91, "tokenPositionInFile": 3640, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 28, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 86}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\n", "theoremStatement": "lemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c ", "theoremName": "square_neg", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "a926b33", "date": "2024-02-17"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 96, "tokenPositionInFile": 3895, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 48, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simpa [Square] using Rectangle.symm", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 43}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n", "theoremStatement": "@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b ", "theoremName": "uIoo_of_le", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "a926b33", "date": "2024-02-17"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 101, "tokenPositionInFile": 4081, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 25, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 52}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\n", "theoremStatement": "theorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b ", "theoremName": "Set.left_not_mem_uIoo", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "2282e1a", "date": "2024-02-18"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 105, "tokenPositionInFile": 4233, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 30, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 74}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\n", "theoremStatement": "theorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b ", "theoremName": "Set.right_not_mem_uIoo", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "2282e1a", "date": "2024-02-18"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 108, "tokenPositionInFile": 4368, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 30, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 76}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\n", "theoremStatement": "theorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a ", "theoremName": "Set.ne_left_of_mem_uIoo", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "2282e1a", "date": "2024-02-18"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 111, "tokenPositionInFile": 4506, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 10, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 43}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\n", "theoremStatement": "theorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b ", "theoremName": "Set.ne_right_of_mem_uIoo", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "2282e1a", "date": "2024-02-18"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 114, "tokenPositionInFile": 4627, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 10, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 44}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\n", "theoremStatement": "lemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w ", "theoremName": "left_mem_rect", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "4e8cdf2", "date": "2024-02-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 117, "tokenPositionInFile": 4750, "theoremPositionInFile": 21}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= \u27e8left_mem_uIcc, left_mem_uIcc\u27e9", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 33}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\n", "theoremStatement": "lemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w ", "theoremName": "right_mem_rect", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "4e8cdf2", "date": "2024-02-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 119, "tokenPositionInFile": 4835, "theoremPositionInFile": 22}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= \u27e8right_mem_uIcc, right_mem_uIcc\u27e9", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 35}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\n", "theoremStatement": "lemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w ", "theoremName": "rect_subset_iff", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "4e8cdf2", "date": "2024-02-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 121, "tokenPositionInFile": 4923, "theoremPositionInFile": 23}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 42, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 543}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\n", "theoremStatement": "lemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) ", "theoremName": "RectSubRect", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "f887328", "date": "2024-02-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 132, "tokenPositionInFile": 5586, "theoremPositionInFile": 24}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 154, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 128}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\n", "theoremStatement": "lemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 ", "theoremName": "RectSubRect'", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "f887328", "date": "2024-02-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 139, "tokenPositionInFile": 5978, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 25, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 149}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\n", "theoremStatement": "lemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w ", "theoremName": "rectangleBorder_subset_rectangle", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 146, "tokenPositionInFile": 6376, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 27, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 201}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n", "theoremStatement": "/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} ", "theoremName": "rectangle_disjoint_singleton", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "a39f250", "date": "2024-02-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 154, "tokenPositionInFile": 6666, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 44, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 275}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\n", "theoremStatement": "lemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} ", "theoremName": "rectangleBorder_disjoint_singleton", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 166, "tokenPositionInFile": 7201, "theoremPositionInFile": 28}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 48, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 196}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\n", "theoremStatement": "lemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} ", "theoremName": "rectangle_subset_punctured_rect", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 173, "tokenPositionInFile": 7557, "theoremPositionInFile": 29}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 38, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 91}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\n", "theoremStatement": "lemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} ", "theoremName": "rectangleBorder_subset_punctured_rect", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 181, "tokenPositionInFile": 8008, "theoremPositionInFile": 30}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 39, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 153}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\n", "theoremStatement": "lemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) ", "theoremName": "rectangle_mem_nhds_iff", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "3727e0f", "date": "2024-02-19"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 190, "tokenPositionInFile": 8459, "theoremPositionInFile": 31}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 62, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 109}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n", "theoremStatement": "/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} ", "theoremName": "horizontalSegment_eq", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 194, "tokenPositionInFile": 8688, "theoremPositionInFile": 32}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 89, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9", "proofType": "term", "proofLengthLines": 2, "proofLengthTokens": 142}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n", "theoremStatement": "/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] ", "theoremName": "verticalSegment_eq", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 201, "tokenPositionInFile": 9049, "theoremPositionInFile": 33}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 89, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9", "proofType": "term", "proofLengthLines": 2, "proofLengthTokens": 142}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) ", "theoremName": "mapsTo_rectangle_left_re", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 208, "tokenPositionInFile": 9408, "theoremPositionInFile": 34}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 68, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 39}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) ", "theoremName": "mapsTo_rectangle_right_re", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 212, "tokenPositionInFile": 9566, "theoremPositionInFile": 35}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 68, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 39}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) ", "theoremName": "mapsTo_rectangle_left_im", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 216, "tokenPositionInFile": 9725, "theoremPositionInFile": 36}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 68, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 39}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) ", "theoremName": "mapsTo_rectangle_right_im", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 220, "tokenPositionInFile": 9882, "theoremPositionInFile": 37}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 68, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 39}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) ", "theoremName": "mapsTo_rectangleBorder_left_re", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 224, "tokenPositionInFile": 10040, "theoremPositionInFile": 38}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 44, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 101}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\n", "theoremStatement": "lemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) ", "theoremName": "mapsTo_rectangleBorder_right_re", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 228, "tokenPositionInFile": 10272, "theoremPositionInFile": 39}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 44, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 101}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\n", "theoremStatement": "lemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) ", "theoremName": "mapsTo_rectangleBorder_left_im", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 232, "tokenPositionInFile": 10505, "theoremPositionInFile": 40}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 43, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 103}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\n", "theoremStatement": "lemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) ", "theoremName": "mapsTo_rectangleBorder_right_im", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 236, "tokenPositionInFile": 10738, "theoremPositionInFile": 41}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 45, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 103}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\n", "theoremStatement": "lemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) ", "theoremName": "mapsTo_rectangle_left_re_NoP", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 240, "tokenPositionInFile": 10972, "theoremPositionInFile": 42}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 45, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 177}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) ", "theoremName": "mapsTo_rectangle_right_re_NoP", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 245, "tokenPositionInFile": 11327, "theoremPositionInFile": 43}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 45, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 178}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangle_left_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) ", "theoremName": "mapsTo_rectangle_left_im_NoP", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 250, "tokenPositionInFile": 11684, "theoremPositionInFile": 44}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 45, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine (mapsTo_rectangleBorder_left_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 177}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_left_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\n", "theoremStatement": "lemma mapsTo_rectangle_right_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) ", "theoremName": "mapsTo_rectangle_right_im_NoP", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "6c9e2b8", "date": "2024-02-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 255, "tokenPositionInFile": 12038, "theoremPositionInFile": 45}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 45, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine (mapsTo_rectangleBorder_right_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 178}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_left_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\n", "theoremStatement": "theorem not_mem_rectangleBorder_of_rectangle_mem_nhds {z w p : \u2102} (hp : Rectangle z w \u2208 \ud835\udcdd p) :\n p \u2209 RectangleBorder z w ", "theoremName": "not_mem_rectangleBorder_of_rectangle_mem_nhds", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "2282e1a", "date": "2024-02-18"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 260, "tokenPositionInFile": 12394, "theoremPositionInFile": 46}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 49, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine Set.disjoint_right.mp (rectangleBorder_disjoint_singleton ?_) rfl\n have h1 := rectangle_mem_nhds_iff.mp hp\n exact \u27e8Set.ne_left_of_mem_uIoo h1.1, Set.ne_right_of_mem_uIoo h1.1,\n Set.ne_left_of_mem_uIoo h1.2, Set.ne_right_of_mem_uIoo h1.2\u27e9", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 257}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_left_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem not_mem_rectangleBorder_of_rectangle_mem_nhds {z w p : \u2102} (hp : Rectangle z w \u2208 \ud835\udcdd p) :\n p \u2209 RectangleBorder z w := by\n refine Set.disjoint_right.mp (rectangleBorder_disjoint_singleton ?_) rfl\n have h1 := rectangle_mem_nhds_iff.mp hp\n exact \u27e8Set.ne_left_of_mem_uIoo h1.1, Set.ne_right_of_mem_uIoo h1.1,\n Set.ne_left_of_mem_uIoo h1.2, Set.ne_right_of_mem_uIoo h1.2\u27e9\n\n", "theoremStatement": "theorem Complex.nhds_hasBasis_square (p : \u2102) : (\ud835\udcdd p).HasBasis (0 < \u00b7) (Square p \u00b7) ", "theoremName": "Complex.nhds_hasBasis_square", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "f1c959f", "date": "2024-02-19"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 267, "tokenPositionInFile": 12776, "theoremPositionInFile": 47}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 281, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n suffices (\ud835\udcdd p.re \u00d7\u02e2 \ud835\udcdd p.im).HasBasis (0 < .) (equivRealProdCLM.symm.toHomeomorph \u207b\u00b9' Square p .)\n by simpa only [\u2190 nhds_prod_eq, Homeomorph.map_nhds_eq, Homeomorph.image_preimage]\n using this.map equivRealProdCLM.symm.toHomeomorph\n apply ((nhds_basis_Icc_pos p.re).prod_same_index_mono (nhds_basis_Icc_pos p.im) ?_ ?_).congr\n \u00b7 intro; rfl\n \u00b7 intros\n rw [\u2190 uIcc_of_lt (by linarith), \u2190 uIcc_of_lt (by linarith)]\n simpa [Square, Rectangle] using by ring_nf\n all_goals exact (antitone_const_tsub.Icc (monotone_id.const_add _)).monotoneOn _", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 561}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_left_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem not_mem_rectangleBorder_of_rectangle_mem_nhds {z w p : \u2102} (hp : Rectangle z w \u2208 \ud835\udcdd p) :\n p \u2209 RectangleBorder z w := by\n refine Set.disjoint_right.mp (rectangleBorder_disjoint_singleton ?_) rfl\n have h1 := rectangle_mem_nhds_iff.mp hp\n exact \u27e8Set.ne_left_of_mem_uIoo h1.1, Set.ne_right_of_mem_uIoo h1.1,\n Set.ne_left_of_mem_uIoo h1.2, Set.ne_right_of_mem_uIoo h1.2\u27e9\n\ntheorem Complex.nhds_hasBasis_square (p : \u2102) : (\ud835\udcdd p).HasBasis (0 < \u00b7) (Square p \u00b7) := by\n suffices (\ud835\udcdd p.re \u00d7\u02e2 \ud835\udcdd p.im).HasBasis (0 < .) (equivRealProdCLM.symm.toHomeomorph \u207b\u00b9' Square p .)\n by simpa only [\u2190 nhds_prod_eq, Homeomorph.map_nhds_eq, Homeomorph.image_preimage]\n using this.map equivRealProdCLM.symm.toHomeomorph\n apply ((nhds_basis_Icc_pos p.re).prod_same_index_mono (nhds_basis_Icc_pos p.im) ?_ ?_).congr\n \u00b7 intro; rfl\n \u00b7 intros\n rw [\u2190 uIcc_of_lt (by linarith), \u2190 uIcc_of_lt (by linarith)]\n simpa [Square, Rectangle] using by ring_nf\n all_goals exact (antitone_const_tsub.Icc (monotone_id.const_add _)).monotoneOn _\n\n", "theoremStatement": "lemma square_mem_nhds (p : \u2102) {c : \u211d} (hc : c \u2260 0) :\n Square p c \u2208 \ud835\udcdd p ", "theoremName": "square_mem_nhds", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "a926b33", "date": "2024-02-17"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 278, "tokenPositionInFile": 13422, "theoremPositionInFile": 48}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 63, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n wlog hc_pos : 0 < c generalizing c with h\n \u00b7 rw [\u2190 square_neg]\n exact h (neg_ne_zero.mpr hc) <| neg_pos.mpr <| hc.lt_of_le <| not_lt.mp hc_pos\n exact (nhds_hasBasis_square p).mem_of_mem hc_pos", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 205}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_left_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem not_mem_rectangleBorder_of_rectangle_mem_nhds {z w p : \u2102} (hp : Rectangle z w \u2208 \ud835\udcdd p) :\n p \u2209 RectangleBorder z w := by\n refine Set.disjoint_right.mp (rectangleBorder_disjoint_singleton ?_) rfl\n have h1 := rectangle_mem_nhds_iff.mp hp\n exact \u27e8Set.ne_left_of_mem_uIoo h1.1, Set.ne_right_of_mem_uIoo h1.1,\n Set.ne_left_of_mem_uIoo h1.2, Set.ne_right_of_mem_uIoo h1.2\u27e9\n\ntheorem Complex.nhds_hasBasis_square (p : \u2102) : (\ud835\udcdd p).HasBasis (0 < \u00b7) (Square p \u00b7) := by\n suffices (\ud835\udcdd p.re \u00d7\u02e2 \ud835\udcdd p.im).HasBasis (0 < .) (equivRealProdCLM.symm.toHomeomorph \u207b\u00b9' Square p .)\n by simpa only [\u2190 nhds_prod_eq, Homeomorph.map_nhds_eq, Homeomorph.image_preimage]\n using this.map equivRealProdCLM.symm.toHomeomorph\n apply ((nhds_basis_Icc_pos p.re).prod_same_index_mono (nhds_basis_Icc_pos p.im) ?_ ?_).congr\n \u00b7 intro; rfl\n \u00b7 intros\n rw [\u2190 uIcc_of_lt (by linarith), \u2190 uIcc_of_lt (by linarith)]\n simpa [Square, Rectangle] using by ring_nf\n all_goals exact (antitone_const_tsub.Icc (monotone_id.const_add _)).monotoneOn _\n\nlemma square_mem_nhds (p : \u2102) {c : \u211d} (hc : c \u2260 0) :\n Square p c \u2208 \ud835\udcdd p := by\n wlog hc_pos : 0 < c generalizing c with h\n \u00b7 rw [\u2190 square_neg]\n exact h (neg_ne_zero.mpr hc) <| neg_pos.mpr <| hc.lt_of_le <| not_lt.mp hc_pos\n exact (nhds_hasBasis_square p).mem_of_mem hc_pos\n\n", "theoremStatement": "lemma square_subset_square {p : \u2102} {c\u2081 c\u2082 : \u211d} (hc\u2081 : 0 < c\u2081) (hc : c\u2081 \u2264 c\u2082) :\n Square p c\u2081 \u2286 Square p c\u2082 ", "theoremName": "square_subset_square", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "88ff156", "date": "2024-02-19"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 285, "tokenPositionInFile": 13703, "theoremPositionInFile": 49}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 184, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply RectSubRect' <;> simpa using by linarith", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 54}} +{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n simp [Rectangle, uIcc_comm]\n\nlemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w := by\n simp [Rectangle, uIcc_comm]\n\nend Rectangle\n\n/-%%\nThe border of a rectangle is the union of its four sides.\n\\begin{definition}[RectangleBorder]\\label{RectangleBorder}\\lean{RectangleBorder}\\leanok\nA Rectangle's border, given corners $z$ and $w$ is the union of the four sides.\n\\end{definition}\n%%-/\n/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 {z.im} \u222a {z.re} \u00d7\u2102 [[z.im, w.im]] \u222a [[z.re, w.re]] \u00d7\u2102 {w.im} \u222a {w.re} \u00d7\u2102 [[z.im, w.im]]\n\ndef Square (p : \u2102) (c : \u211d) : Set \u2102 := Rectangle (-c - c * I + p) (c + c * I + p)\n\nlemma Square_apply (p : \u2102) (cpos : c > 0) :\n Square p c = Icc (-c + p.re) (c + p.re) \u00d7\u2102 Icc (-c + p.im) (c + p.im) := by\n rw [Square, Rectangle, uIcc_of_le (by simp; linarith), uIcc_of_le (by simp; linarith)]\n simp\n\n-- From PR #9598\n/-- The preimage under `equivRealProd` of `s \u00d7\u02e2 t` is `s \u00d7\u2102 t`. -/\nlemma preimage_equivRealProd_prod (s t : Set \u211d) : equivRealProd \u207b\u00b9' (s \u00d7\u02e2 t) = s \u00d7\u2102 t := rfl\n\n@[simp]\ntheorem preimage_equivRealProdCLM_reProdIm (s t : Set \u211d) :\n equivRealProdCLM.symm \u207b\u00b9' (s \u00d7\u2102 t) = s \u00d7\u02e2 t :=\n rfl\n\n@[simp]\ntheorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type*} {S : Type*} [Semiring R] [Semiring S] {\u03c3 : R \u2192+* S}\n {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type*) [TopologicalSpace M]\n [AddCommMonoid M] {M\u2082 : Type*} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n [Module S M\u2082] (e : M \u2243SL[\u03c3] M\u2082) :\n \u21d1e.toLinearEquiv.symm = e.symm :=\n rfl\n\n-- From PR #9598\n/-- The inequality `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` holds in `\u2102` iff it holds in `\u211d \u00d7 \u211d`. -/\nlemma reProdIm_subset_iff {s s\u2081 t t\u2081 : Set \u211d} : s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u00d7\u02e2 t \u2286 s\u2081 \u00d7\u02e2 t\u2081 := by\n simp_rw [\u2190 preimage_equivRealProd_prod, equivRealProd.preimage_subset]\n\n-- From PR #9598\n/-- If `s \u2286 s\u2081 \u2286 \u211d` and `t \u2286 t\u2081 \u2286 \u211d`, then `s \u00d7 t \u2286 s\u2081 \u00d7 t\u2081` in `\u2102`. -/\nlemma reProdIm_subset_iff' {s s\u2081 t t\u2081 : Set \u211d} :\n s \u00d7\u2102 t \u2286 s\u2081 \u00d7\u2102 t\u2081 \u2194 s \u2286 s\u2081 \u2227 t \u2286 t\u2081 \u2228 s = \u2205 \u2228 t = \u2205 :=\n reProdIm_subset_iff.trans prod_subset_prod_iff\n\n/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\n of two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\nlemma segment_reProdIm_segment_eq_convexHull (z w : \u2102) :\n [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] = convexHull \u211d {z, z.re + w.im * I, w.re + z.im * I, w} := by\n simp_rw [\u2190 segment_eq_uIcc, \u2190 convexHull_pair, \u2190 convexHull_reProdIm, reProdIm]\n exact congrArg _ <| Set.ext <| by simpa [Complex.ext_iff] using by tauto\n\n/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\n rectangle is. Golfed from mathlib4\\#9598.-/\nlemma rectangle_in_convex {U : Set \u2102} (U_convex : Convex \u211d U) {z w : \u2102} (hz : z \u2208 U)\n (hw : w \u2208 U) (hzw : (z.re + w.im * I) \u2208 U) (hwz : (w.re + z.im * I) \u2208 U) :\n Rectangle z w \u2286 U := by\n rw [Rectangle, segment_reProdIm_segment_eq_convexHull]\n exact convexHull_min (by simp_all [insert_subset_iff]) U_convex\n\nlemma mem_Rect {z w : \u2102} (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im) (p : \u2102) :\n p \u2208 Rectangle z w \u2194 z.re \u2264 p.re \u2227 p.re \u2264 w.re \u2227 z.im \u2264 p.im \u2227 p.im \u2264 w.im := by\n rw [Rectangle, uIcc_of_le zRe_lt_wRe, uIcc_of_le zIm_lt_wIm]\n exact and_assoc\n\nlemma square_neg (p : \u2102) (c : \u211d) : Square p (-c) = Square p c := by\n simpa [Square] using Rectangle.symm\n\ndef Set.uIoo {\u03b1 : Type*} [Lattice \u03b1] (a b : \u03b1) : Set \u03b1 := Ioo (a \u2293 b) (a \u2294 b)\n\n@[simp]\ntheorem uIoo_of_le {\u03b1 : Type*} [Lattice \u03b1] {a b : \u03b1} (h : a \u2264 b) : Set.uIoo a b = Ioo a b := by\n rw [uIoo, inf_eq_left.2 h, sup_eq_right.2 h]\n\ntheorem Set.left_not_mem_uIoo {a b : \u211d} : a \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (left_lt_sup.mp h2) (le_of_not_le (inf_lt_left.mp h1))\n\ntheorem Set.right_not_mem_uIoo {a b : \u211d} : b \u2209 Set.uIoo a b :=\n fun \u27e8h1, h2\u27e9 \u21a6 (right_lt_sup.mp h2) (le_of_not_le (inf_lt_right.mp h1))\n\ntheorem Set.ne_left_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 a :=\n fun h \u21a6 Set.left_not_mem_uIoo (h \u25b8 hc)\n\ntheorem Set.ne_right_of_mem_uIoo {a b c : \u211d} (hc : c \u2208 Set.uIoo a b) : c \u2260 b :=\n fun h \u21a6 Set.right_not_mem_uIoo (h \u25b8 hc)\n\nlemma left_mem_rect (z w : \u2102) : z \u2208 Rectangle z w := \u27e8left_mem_uIcc, left_mem_uIcc\u27e9\n\nlemma right_mem_rect (z w : \u2102) : w \u2208 Rectangle z w := \u27e8right_mem_uIcc, right_mem_uIcc\u27e9\n\nlemma rect_subset_iff {z w z' w' : \u2102} :\n Rectangle z' w' \u2286 Rectangle z w \u2194 z' \u2208 Rectangle z w \u2227 w' \u2208 Rectangle z w := by\n use fun h \u21a6 \u27e8h (left_mem_rect z' w'), h (right_mem_rect z' w')\u27e9\n intro \u27e8\u27e8\u27e8hz're_ge, hz're_le\u27e9, \u27e8hz'im_ge, hz'im_le\u27e9\u27e9,\n \u27e8\u27e8hw're_ge, hw're_le\u27e9, \u27e8hw'im_ge, hw'im_le\u27e9\u27e9\u27e9 x \u27e8\u27e8hxre_ge, hxre_le\u27e9, \u27e8hxim_ge, hxim_le\u27e9\u27e9\n refine \u27e8\u27e8?_, ?_\u27e9, \u27e8?_, ?_\u27e9\u27e9\n \u00b7 exact (le_inf hz're_ge hw're_ge).trans hxre_ge\n \u00b7 exact (le_sup_iff.mp hxre_le).casesOn (fun h \u21a6 h.trans hz're_le) (fun h \u21a6 h.trans hw're_le)\n \u00b7 exact (le_inf hz'im_ge hw'im_ge).trans hxim_ge\n \u00b7 exact (le_sup_iff.mp hxim_le).casesOn (fun h \u21a6 h.trans hz'im_le) (fun h \u21a6 h.trans hw'im_le)\n\nlemma RectSubRect {x\u2080 x\u2081 x\u2082 x\u2083 y\u2080 y\u2081 y\u2082 y\u2083 : \u211d} (x\u2080_le_x\u2081 : x\u2080 \u2264 x\u2081) (x\u2081_le_x\u2082 : x\u2081 \u2264 x\u2082)\n (x\u2082_le_x\u2083 : x\u2082 \u2264 x\u2083) (y\u2080_le_y\u2081 : y\u2080 \u2264 y\u2081) (y\u2081_le_y\u2082 : y\u2081 \u2264 y\u2082) (y\u2082_le_y\u2083 : y\u2082 \u2264 y\u2083) :\n Rectangle (x\u2081 + y\u2081 * I) (x\u2082 + y\u2082 * I) \u2286 Rectangle (x\u2080 + y\u2080 * I) (x\u2083 + y\u2083 * I) := by\n rw [rect_subset_iff, mem_Rect, mem_Rect]\n refine \u27e8\u27e8?_, ?_, ?_, ?_\u27e9, ?_, ?_, ?_, ?_\u27e9\n all_goals simpa using by linarith\n\nlemma RectSubRect' {z\u2080 z\u2081 z\u2082 z\u2083 : \u2102} (x\u2080_le_x\u2081 : z\u2080.re \u2264 z\u2081.re) (x\u2081_le_x\u2082 : z\u2081.re \u2264 z\u2082.re)\n (x\u2082_le_x\u2083 : z\u2082.re \u2264 z\u2083.re) (y\u2080_le_y\u2081 : z\u2080.im \u2264 z\u2081.im) (y\u2081_le_y\u2082 : z\u2081.im \u2264 z\u2082.im)\n (y\u2082_le_y\u2083 : z\u2082.im \u2264 z\u2083.im) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 := by\n rw [\u2190 re_add_im z\u2080, \u2190 re_add_im z\u2081, \u2190 re_add_im z\u2082, \u2190 re_add_im z\u2083]\n exact RectSubRect x\u2080_le_x\u2081 x\u2081_le_x\u2082 x\u2082_le_x\u2083 y\u2080_le_y\u2081 y\u2081_le_y\u2082 y\u2082_le_y\u2083\n\nlemma rectangleBorder_subset_rectangle (z w : \u2102) : RectangleBorder z w \u2286 Rectangle z w := by\n intro x hx\n obtain \u27e8\u27e8h | h\u27e9 | h\u27e9 | h := hx\n \u00b7 exact \u27e8h.1, h.2 \u25b8 left_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 left_mem_uIcc, h.2\u27e9\n \u00b7 exact \u27e8h.1, h.2 \u25b8 right_mem_uIcc\u27e9\n \u00b7 exact \u27e8h.1 \u25b8 right_mem_uIcc, h.2\u27e9\n\n/-- Note: try using `by simp` for `h`. -/\nlemma rectangle_disjoint_singleton {z w p : \u2102}\n (h : (p.re < z.re \u2227 p.re < w.re) \u2228 (p.im < z.im \u2227 p.im < w.im) \u2228\n (z.re < p.re \u2227 w.re < p.re) \u2228 (z.im < p.im \u2227 w.im < p.im)) :\n Disjoint (Rectangle z w) {p} := by\n refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)\n obtain h | h | h | h := h\n \u00b7 exact Or.inl (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_lt h.1 h.2)\n \u00b7 exact Or.inl (not_mem_uIcc_of_gt h.1 h.2)\n \u00b7 exact Or.inr (not_mem_uIcc_of_gt h.1 h.2)\n\nlemma rectangleBorder_disjoint_singleton {z w p : \u2102}\n (h : p.re \u2260 z.re \u2227 p.re \u2260 w.re \u2227 p.im \u2260 z.im \u2227 p.im \u2260 w.im) :\n Disjoint (RectangleBorder z w) {p} := by\n refine disjoint_singleton_right.mpr ?_\n simp_rw [RectangleBorder, Set.mem_union, not_or]\n exact \u27e8\u27e8\u27e8fun hc \u21a6 h.2.2.1 hc.2, fun hc \u21a6 h.1 hc.1\u27e9, fun hc \u21a6 h.2.2.2 hc.2\u27e9, fun hc \u21a6 h.2.1 hc.1\u27e9\n\nlemma rectangle_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : (p.re < z\u2081.re \u2227 p.re < z\u2082.re) \u2228 (p.im < z\u2081.im \u2227 p.im < z\u2082.im) \u2228\n (z\u2081.re < p.re \u2227 z\u2082.re < p.re) \u2228 (z\u2081.im < p.im \u2227 z\u2082.im < p.im)) :\n Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8by apply RectSubRect' <;> tauto, rectangle_disjoint_singleton hp\u27e9\n\nlemma rectangleBorder_subset_punctured_rect {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\n (hz : z\u2080.re \u2264 z\u2081.re \u2227 z\u2081.re \u2264 z\u2082.re \u2227 z\u2082.re \u2264 z\u2083.re \u2227\n z\u2080.im \u2264 z\u2081.im \u2227 z\u2081.im \u2264 z\u2082.im \u2227 z\u2082.im \u2264 z\u2083.im)\n (hp : p.re \u2260 z\u2081.re \u2227 p.re \u2260 z\u2082.re \u2227 p.im \u2260 z\u2081.im \u2227 p.im \u2260 z\u2082.im) :\n RectangleBorder z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083 \\ {p} :=\n Set.subset_diff.mpr \u27e8\n (rectangleBorder_subset_rectangle _ _).trans (by apply RectSubRect' <;> tauto),\n rectangleBorder_disjoint_singleton hp\u27e9\n\nlemma rectangle_mem_nhds_iff {z w p : \u2102} : Rectangle z w \u2208 \ud835\udcdd p \u2194\n p \u2208 (Set.uIoo z.re w.re) \u00d7\u2102 (Set.uIoo z.im w.im) := by\n simp_rw [\u2190 mem_interior_iff_mem_nhds, Rectangle, Complex.interior_reProdIm, uIoo, uIcc, interior_Icc]\n\n/-- A real segment `[a\u2081, a\u2082]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma horizontalSegment_eq (a\u2081 a\u2082 b : \u211d) :\n (fun (x : \u211d) \u21a6 x + b * I) '' [[a\u2081, a\u2082]] = [[a\u2081, a\u2082]] \u00d7\u2102 {b} :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\n/-- A vertical segment `[b\u2081, b\u2082]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\nlemma verticalSegment_eq (a b\u2081 b\u2082 : \u211d) :\n (fun (y : \u211d) \u21a6 a + y * I) '' [[b\u2081, b\u2082]] = {a} \u00d7\u2102 [[b\u2081, b\u2082]] :=\n Set.ext fun _ => \u27e8fun hx \u21a6 hx.casesOn fun _ \u27e8_, hx\u27e9 \u21a6 by simpa [\u2190 hx, reProdIm],\n fun hx \u21a6 hx.casesOn (by simp_all [Complex.ext_iff])\u27e9\n\nlemma mapsTo_rectangle_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp, by simp [hx]\u27e9\n\nlemma mapsTo_rectangle_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangle_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w) :=\n fun _ hx \u21a6 \u27e8by simp [hx], by simp\u27e9\n\nlemma mapsTo_rectangleBorder_left_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_re (z w : \u2102) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [verticalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_left_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangleBorder_right_im (z w : \u2102) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (RectangleBorder z w) :=\n (Set.mapsTo_image _ _).mono subset_rfl fun _ \u21a6 by simp_all [horizontalSegment_eq, RectangleBorder]\n\nlemma mapsTo_rectangle_left_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191z.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_re_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (y : \u211d) => \u2191w.re + \u2191y * I) [[z.im, w.im]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_re z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_left_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + z.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_left_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\nlemma mapsTo_rectangle_right_im_NoP (z w : \u2102) {p : \u2102} (pNotOnBorder : p \u2209 RectangleBorder z w) :\n MapsTo (fun (x : \u211d) => \u2191x + w.im * I) [[z.re, w.re]] (Rectangle z w \\ {p}) := by\n refine (mapsTo_rectangleBorder_right_im z w).mono_right (Set.subset_diff.mpr ?_)\n exact \u27e8rectangleBorder_subset_rectangle z w, disjoint_singleton_right.mpr pNotOnBorder\u27e9\n\ntheorem not_mem_rectangleBorder_of_rectangle_mem_nhds {z w p : \u2102} (hp : Rectangle z w \u2208 \ud835\udcdd p) :\n p \u2209 RectangleBorder z w := by\n refine Set.disjoint_right.mp (rectangleBorder_disjoint_singleton ?_) rfl\n have h1 := rectangle_mem_nhds_iff.mp hp\n exact \u27e8Set.ne_left_of_mem_uIoo h1.1, Set.ne_right_of_mem_uIoo h1.1,\n Set.ne_left_of_mem_uIoo h1.2, Set.ne_right_of_mem_uIoo h1.2\u27e9\n\ntheorem Complex.nhds_hasBasis_square (p : \u2102) : (\ud835\udcdd p).HasBasis (0 < \u00b7) (Square p \u00b7) := by\n suffices (\ud835\udcdd p.re \u00d7\u02e2 \ud835\udcdd p.im).HasBasis (0 < .) (equivRealProdCLM.symm.toHomeomorph \u207b\u00b9' Square p .)\n by simpa only [\u2190 nhds_prod_eq, Homeomorph.map_nhds_eq, Homeomorph.image_preimage]\n using this.map equivRealProdCLM.symm.toHomeomorph\n apply ((nhds_basis_Icc_pos p.re).prod_same_index_mono (nhds_basis_Icc_pos p.im) ?_ ?_).congr\n \u00b7 intro; rfl\n \u00b7 intros\n rw [\u2190 uIcc_of_lt (by linarith), \u2190 uIcc_of_lt (by linarith)]\n simpa [Square, Rectangle] using by ring_nf\n all_goals exact (antitone_const_tsub.Icc (monotone_id.const_add _)).monotoneOn _\n\nlemma square_mem_nhds (p : \u2102) {c : \u211d} (hc : c \u2260 0) :\n Square p c \u2208 \ud835\udcdd p := by\n wlog hc_pos : 0 < c generalizing c with h\n \u00b7 rw [\u2190 square_neg]\n exact h (neg_ne_zero.mpr hc) <| neg_pos.mpr <| hc.lt_of_le <| not_lt.mp hc_pos\n exact (nhds_hasBasis_square p).mem_of_mem hc_pos\n\nlemma square_subset_square {p : \u2102} {c\u2081 c\u2082 : \u211d} (hc\u2081 : 0 < c\u2081) (hc : c\u2081 \u2264 c\u2082) :\n Square p c\u2081 \u2286 Square p c\u2082 := by\n apply RectSubRect' <;> simpa using by linarith\n\n", "theoremStatement": "lemma SmallSquareInRectangle {z w p : \u2102} (pInRectInterior : Rectangle z w \u2208 nhds p) :\n \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0, Square p c \u2286 Rectangle z w ", "theoremName": "SmallSquareInRectangle", "fileCreated": {"commit": "2b6bc53", "date": "2024-02-20"}, "theoremCreated": {"commit": "9b8e7fa", "date": "2024-02-17"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Rectangle.lean", "module": "PrimeNumberTheoremAnd.Rectangle", "jsonFile": "PrimeNumberTheoremAnd.Rectangle.jsonl", "positionMetadata": {"lineInFile": 289, "tokenPositionInFile": 13868, "theoremPositionInFile": 50}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 57, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8\u03b5, h\u03b50, h\u03b5\u27e9 := ((Complex.nhds_hasBasis_square p).1 _).mp pInRectInterior\n filter_upwards [Ioo_mem_nhdsWithin_Ioi' (h\u03b50)] with _ \u27e8h\u03b5'0, h\u03b5'\u27e9\n exact subset_trans (square_subset_square h\u03b5'0 h\u03b5'.le) h\u03b5", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 215}} diff --git a/minictx/scilean.jsonl b/minictx/scilean.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..63791099631c901780ebe9b4883abefe99028005 --- /dev/null +++ b/minictx/scilean.jsonl @@ -0,0 +1,46 @@ +{"srcContext": "import Mathlib.Algebra.Field.Defs\nimport Mathlib.GroupTheory.GroupAction.Defs\nimport Mathlib.Tactic.FunProp\n\nimport SciLean.Util.SorryProof\n\nset_option linter.unusedVariables false\n\n-- Some missing theorems -------------------------------------------------------\n--------------------------------------------------------------------------------\n\n", "theoremStatement": "theorem Function.invFun_comp' [Nonempty \u03b1] {f : \u03b1 \u2192 \u03b2} (hf : f.Injective) {x : \u03b1} :\n f.invFun (f x) = x ", "theoremName": "Function.invFun_comp'", "fileCreated": {"commit": "a95431a4", "date": "2023-08-19"}, "theoremCreated": {"commit": "c0cf2eb5", "date": "2024-04-12"}, "file": "scilean/SciLean/Core/FunctionPropositions/Bijective.lean", "module": "SciLean.Core.FunctionPropositions.Bijective", "jsonFile": "SciLean.Core.FunctionPropositions.Bijective.jsonl", "positionMetadata": {"lineInFile": 12, "tokenPositionInFile": 345, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Opposites", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Tactic.FunProp.Decl", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "SciLean.Util.SorryProof"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n suffices (f.invFun \u2218 f) x = x by assumption\n rw[Function.invFun_comp hf]\n rfl", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 87}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.Distribution.ParametricDistribFwdDeriv\n\nnamespace SciLean\n\nopen MeasureTheory\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W]\n {X} [MeasurableSpace X] [Vec R X]\n {Y} [Vec R Y] [Module \u211d Y]\n\nset_default_scalar R\n\n@[fun_trans]\ntheorem Rand.\ud835\udd3c.arg_r.cderiv_rule (r : W \u2192 Rand X) (f : X \u2192 Y) :\n cderiv R (fun w => (r w).\ud835\udd3c f)\n =\n fun w dw =>\n let d := parDistribDeriv (fun w => (r w).\u2119.toDistribution (R:=R)) w dw\n d.extAction f (fun r \u22b8 fun y \u22b8 ((r \u2022 y) : Y)) := sorry_proof\n\n@[fun_trans]\ntheorem Rand.\ud835\udd3c.arg_rf.cderiv_rule' (r : W \u2192 Rand X) (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n cderiv R (fun w => (r w).\ud835\udd3c (f w))\n =\n fun w dw =>\n let dr := parDistribFwdDeriv (fun w => (r w).\u2119.toDistribution (R:=R)) w dw\n let df := fun x => fwdDeriv R (f \u00b7 x) w dw\n dr.extAction df (fun rdr \u22b8 fun ydy \u22b8 rdr.1\u2022ydy.2 + rdr.2\u2022ydy.1) := sorry_proof\n\n\n\n", "theoremStatement": "theorem Rand.\ud835\udd3c_deriv_as_distribDeriv {X} [Vec R X] [MeasureSpace X]\n (r : W \u2192 Rand X) (f : W \u2192 X \u2192 Y) :\n cderiv R (fun w => (r w).\ud835\udd3c (f w))\n =\n fun w dw =>\n parDistribDeriv (fun w => (fun x => ((r w).pdf R volume x) \u2022 f w x).toDistribution (R:=R)) w dw |>.integrate ", "theoremName": "SciLean.Rand.\ud835\udd3c_deriv_as_distribDeriv", "fileCreated": {"commit": "b3d5680d", "date": "2024-04-12"}, "theoremCreated": {"commit": "c0cf2eb5", "date": "2024-04-12"}, "file": "scilean/SciLean/Core/Rand/ExpectedValue.lean", "module": "SciLean.Core.Rand.ExpectedValue", "jsonFile": "SciLean.Core.Rand.ExpectedValue.jsonl", "positionMetadata": {"lineInFile": 36, "tokenPositionInFile": 920, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 44, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation", "SciLean.Core.Distribution.ParametricDistribDeriv", "SciLean.Core.Distribution.ParametricDistribFwdDeriv"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 8}} +{"srcContext": "import Mathlib.Data.Erased\nimport Mathlib.Control.Random\nimport Mathlib.MeasureTheory.Integral.Bochner\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\n\nimport SciLean.Core.FunctionPropositions.Bijective\nimport SciLean.Core.Objects.Scalar\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Rand.SimpAttr\n\nimport Mathlib.MeasureTheory.Measure.GiryMonad\n\nopen MeasureTheory ENNReal BigOperators Finset\n\nnamespace SciLean\n\nabbrev erase (a : \u03b1) : Erased \u03b1 := .mk a\n\n@[simp,ftrans_simp]\ntheorem erase_out {\u03b1} (a : \u03b1) : (erase a).out = a := by simp[erase]\n\n\n/-- `x : Rand X` is a random variable of type `X`\n\nYou can:\n - generate sample with `x.get : IO X`\n - get probability measure with `x.\u2119 : Measure X`\n\nThe internal fields `spec` and `rand` are just an internal implementation of `Rand` and should not\nbe accessed by normal users.\n\nTODO: Hide implementation using quotients or something like that\n-/\nstructure Rand (X : Type _) where\n /-- `spec` defines a probability measure by computing an expectation. This means if `x : Rand X`\n corresponds to a probability measure `\u03bc` then for `\u03c6 : X \u2192 \u211d`\n ```\n x.spec.out \u03c6 = \u222b x, \u03c6 x \u2202\u03bc\n ```\n\n Using `(X\u2192\u211d)\u2192\u211d` instead of `Measure X` for the specification of random variables has the\n advantage that we can reuse Lean's `do` notation.\n -/\n spec : Erased ((X\u2192\u211d)\u2192\u211d)\n /-- `rand` is a pseudo randon number generator implemented using the \"Standard\" number generator\n -/\n rand : StateM StdGen X\n\n\nnamespace Rand\n\ndef _root_.Function.IsMeasure {X} [MeasurableSpace X] (F : (X \u2192 \u211d) \u2192 \u211d) : Prop :=\n \u2203 \u03bc : Measure X, \u2200 (f : X \u2192 \u211d), F f = \u222b x, f x \u2202\u03bc\n\nopen Classical in\n/-- Probability measure of a random variable -/\n@[pp_dot]\nnoncomputable\ndef \u2119 {X} [MeasurableSpace X] (r : Rand X) : Measure X :=\n if h : r.spec.out.IsMeasure then\n choose h\n else\n 0\n\n/-- Specification of `x : Rand X` is really saying that it is a probability measure. -/\nclass LawfulRand (x : Rand X) [MeasurableSpace X] where\n is_measure : x.spec.out.IsMeasure\n is_prob : IsProbabilityMeasure x.\u2119\n\nvariable {X Y Z : Type _}\n [MeasurableSpace X]\n [MeasurableSpace Y]\n\ninstance instIsProbabilityMeasure\u2119 (x : Rand X) [inst : LawfulRand x] : IsProbabilityMeasure (x.\u2119) := inst.is_prob\n\n\n/-- Extensionality of random variable.\n\nWARNING: This theorem is inconsistent!!! The random generators `x.rand` and `y.rand` might differ.\n We are not trying to model pseudo-random numbers. We assume that every random number\n generator is a true random number generator. Thus the result of any probabilistic program\n should be independent on the exact generator up to some randomness.\n\nTODO: We might quotient all the random number generators corresponding to the measure `x.\u2119` under\n the assumption that they are all true random generators. I believe that such type would be\n a singleton i.e. all the random number generators are all the same.\n-/\n@[ext]\naxiom ext (x y : Rand X) : x.spec.out = y.spec.out \u2192 x = y\n\n\n/-- Generate rundom number using IO randomness -/\ndef get (x : Rand X) : IO X := do\n let stdGen \u2190 IO.stdGenRef.get\n let (res, new) := x.rand stdGen\n let _ \u2190 IO.stdGenRef.set new\n pure res\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\ninstance : Monad Rand where\n pure x := {\n spec := erase (fun \u03c6 => \u03c6 x),\n rand := pure x\n }\n bind x f := {\n spec := erase (fun \u03c6 => x.spec.out (fun x => (f x).spec.out \u03c6)),\n rand := bind x.rand (fun x => (f x).rand)\n }\n\n\ninstance : LawfulMonad Rand where\n bind_pure_comp := by intros; rfl\n bind_map := by intros; rfl\n pure_bind := by intros; ext; simp[Bind.bind,Pure.pure]\n bind_assoc := by intros; ext; simp[Bind.bind,Pure.pure]\n map_const := by intros; ext; rfl\n id_map := by intros; ext; simp[Bind.bind,Pure.pure,id,Functor.map]\n seqLeft_eq := by intros; ext; simp[Bind.bind,Pure.pure,Seq.seq,Function.const,Functor.map,SeqLeft.seqLeft]\n seqRight_eq := by intros; ext; simp[Bind.bind,Pure.pure,Seq.seq,Function.const,Functor.map,SeqRight.seqRight]\n pure_seq := by intros; ext; simp[Bind.bind,Pure.pure,Seq.seq,Functor.map]\n\n\n-- this needs some integrability and lawfulness of Rand\ntheorem swap_bind (f : X \u2192 Y \u2192 Z) (x : Rand X) (y : Rand Y) :\n (do let x' \u2190 x; let y' \u2190 y; pure (f x' y'))\n =\n (do let y' \u2190 y; let x' \u2190 x; pure (f x' y')) := by\n sorry_proof\n\n\n@[simp, ftrans_simp]\ntheorem pure_\u2119 (x : X) : (pure x : Rand X).\u2119 = Measure.dirac x := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\ninstance [Add X] : HAdd X (Rand X) (Rand X) := \u27e8fun x' x => do\n let x'' \u2190 x\n pure (x' + x'')\u27e9\n\ninstance [Add X] : HAdd (Rand X) X (Rand X) := \u27e8fun x x' => do\n let x'' \u2190 x\n pure (x'' + x')\u27e9\n\n-- instance [Add X] : HAdd (Rand X) (Rand X) (Rand X) := \u27e8fun x y =>\n-- let x' ~ x\n-- let y' ~ y\n-- pure (x' + y')\u27e9\n\n-- todo: add simp theorems that inline these operations\n\n----------------------------------------------------------------------------------------------------\n-- Simple Random Variable functions ----------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nabbrev map (r : Rand X) (f : X \u2192 Y) : Rand Y := do\n let x' \u2190 r\n return f x'\n\n/-- Marginal distribution for the first component of a pair. -/\n@[pp_dot]\nabbrev fst (r : Rand (X\u00d7Y)) : Rand X := do\n let (x,_) \u2190 r\n return x\n\n/-- Marginal distribution for the second component of a pair. -/\n@[pp_dot]\nabbrev snd (r : Rand (X\u00d7Y)) : Rand Y := do\n let (_,y) \u2190 r\n return y\n\n\n@[simp, ftrans_simp]\ntheorem map_\u2119 (r : Rand X) (f : X \u2192 Y) :\n (r.map f).\u2119 = r.\u2119.map f := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Expected Value ----------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection ExpectedValue\n\nvariable\n {R} [RealScalar R]\n [AddCommGroup Y] [Module \u211d Y]\n [AddCommGroup Z] [Module \u211d Z]\n -- [AddCommGroup U] [TopologicalSpace U] [TopologicalAddGroup U] [Module \u211d U] [LocallyConveUSpace \u211d U]\n {U} [AddCommGroup U] [Module \u211d U]\n\n@[pp_dot]\nnoncomputable\ndef \ud835\udd3c (r : Rand X) (\u03c6 : X \u2192 Y) : Y := \u222b' x, \u03c6 x \u2202r.\u2119\n\n@[simp, ftrans_simp, rand_push_E]\ntheorem pure_\ud835\udd3c (x : X) (\u03c6 : X \u2192 Y) :\n (pure (f:=Rand) x).\ud835\udd3c \u03c6 = \u03c6 x := by simp [\ud835\udd3c]\n\n-- What are the right assumptions here? Lambda lawfulness of `x` and `f x'` and integrability of `\u03c6`\n@[rand_push_E]\ntheorem bind_E (r : Rand X) (f : X \u2192 Rand Y) (\u03c6 : Y \u2192 Z) :\n (r >>= f).\ud835\udd3c \u03c6 = r.\ud835\udd3c (fun x' => (f x').\ud835\udd3c \u03c6) := by simp[\ud835\udd3c]; sorry_proof\n\n-- todo: We might want this to hold without lawfulness\n-- consider adding as a property inside of `Distribution` or `Rand`\n@[simp, ftrans_simp, rand_push_E]\ntheorem E_zero (r : Rand X) :\n r.\ud835\udd3c (fun _ => (0 : U)) = 0 := by simp[\ud835\udd3c]\n\n@[simp, ftrans_simp, add_pull, rand_push_E]\ntheorem E_add (r : Rand X) (\u03c6 \u03c8 : X \u2192 U)\n (h\u03c6 : CIntegrable \u03c6 r.\u2119) (h\u03c8 : CIntegrable \u03c8 r.\u2119) :\n r.\ud835\udd3c (fun x => \u03c6 x + \u03c8 x) = r.\ud835\udd3c \u03c6 + r.\ud835\udd3c \u03c8 := by\n simp[\ud835\udd3c]; rw[cintegral_add] <;> assumption\n\n@[simp, ftrans_simp, smul_pull, rand_push_E]\ntheorem E_smul (r : Rand X) (\u03c6 : X \u2192 \u211d) (y : Y) :\n r.\ud835\udd3c (fun x' => \u03c6 x' \u2022 y) = r.\ud835\udd3c \u03c6 \u2022 y := by sorry_proof\n\n", "theoremStatement": "theorem reparameterize [Nonempty X] (f : X \u2192 Y) (hf : f.Injective) {r : Rand X} {\u03c6 : X \u2192 Z} :\n r.\ud835\udd3c \u03c6\n =\n let invf := f.invFun\n (r.map f).\ud835\udd3c (fun y => \u03c6 (invf y)) ", "theoremName": "SciLean.Rand.reparameterize", "fileCreated": {"commit": "68228219", "date": "2024-02-23"}, "theoremCreated": {"commit": "c0cf2eb5", "date": "2024-04-12"}, "file": "scilean/SciLean/Core/Rand/Rand.lean", "module": "SciLean.Core.Rand.Rand", "jsonFile": "SciLean.Core.Rand.Rand.jsonl", "positionMetadata": {"lineInFile": 228, "tokenPositionInFile": 7815, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 26, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [\ud835\udd3c,Function.invFun_comp' hf]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 41}} +{"srcContext": "import Mathlib.Data.Set.Defs\nimport Mathlib.Data.Set.Image\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nimport SciLean.Core.Objects.Scalar\nimport SciLean.Util.SorryProof\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _}\n\nattribute [fun_trans] Set.preimage\nattribute [fun_trans] Set.image\n\n\nattribute [fun_trans] Set.preimage_id Set.preimage_id'\n\nnamespace Set\n\nopen Classical in\n@[fun_trans]\ntheorem preimage_const' (b : \u03b2) (s : Set \u03b2) :\n (fun _ : \u03b1 => b) \u207b\u00b9' s = if b \u2208 s then univ else \u2205 := by apply preimage_const\n\n@[fun_trans]\ntheorem preimage_comp' (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) :\n preimage (fun x => f (g x))\n =\n fun s => g \u207b\u00b9' (f \u207b\u00b9' s) := rfl\n\n\n----------------------------------------------------------------------------------------------------\n\n", "theoremStatement": "@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (B : Set \u03b2) (C : Set \u03b3) :\n preimage (fun x => (f x, g x)) (B.prod C)\n =\n f \u207b\u00b9' B \u2229 g \u207b\u00b9' C ", "theoremName": "Set.Prod.mk.arg_fstsnd.preimage_rule_prod", "fileCreated": {"commit": "67090616", "date": "2024-03-27"}, "theoremCreated": {"commit": "67090616", "date": "2024-03-27"}, "file": "scilean/SciLean/Core/FunctionTransformations/Preimage.lean", "module": "SciLean.Core.FunctionTransformations.Preimage", "jsonFile": "SciLean.Core.FunctionTransformations.Preimage.jsonl", "positionMetadata": {"lineInFile": 34, "tokenPositionInFile": 767, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Std.Tactic.OpenPrivate", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Util.CompileInductive", "Mathlib.Data.Set.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Module", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Util.SorryProof", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.Data.Set.Defs\nimport Mathlib.Data.Set.Image\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nimport SciLean.Core.Objects.Scalar\nimport SciLean.Util.SorryProof\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _}\n\nattribute [fun_trans] Set.preimage\nattribute [fun_trans] Set.image\n\n\nattribute [fun_trans] Set.preimage_id Set.preimage_id'\n\nnamespace Set\n\nopen Classical in\n@[fun_trans]\ntheorem preimage_const' (b : \u03b2) (s : Set \u03b2) :\n (fun _ : \u03b1 => b) \u207b\u00b9' s = if b \u2208 s then univ else \u2205 := by apply preimage_const\n\n@[fun_trans]\ntheorem preimage_comp' (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) :\n preimage (fun x => f (g x))\n =\n fun s => g \u207b\u00b9' (f \u207b\u00b9' s) := rfl\n\n\n----------------------------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (B : Set \u03b2) (C : Set \u03b3) :\n preimage (fun x => (f x, g x)) (B.prod C)\n =\n f \u207b\u00b9' B \u2229 g \u207b\u00b9' C := sorry_proof\n\n@[fun_trans]\ntheorem Prod.mk.arg_fst.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (c : \u03b3) :\n preimage (fun x => (f x, c))\n =\n fun s => f \u207b\u00b9' {y | (y,c) \u2208 s} := sorry_proof\n\n", "theoremStatement": "@[fun_trans]\ntheorem Prod.mk.arg_snd.preimage_rule_prod (b : \u03b2) (g : \u03b1 \u2192 \u03b3) :\n preimage (fun x => (b, g x))\n =\n fun s => g \u207b\u00b9' {z | (b,z) \u2208 s} ", "theoremName": "Set.Prod.mk.arg_snd.preimage_rule_prod", "fileCreated": {"commit": "67090616", "date": "2024-03-27"}, "theoremCreated": {"commit": "f6fb5a0e", "date": "2024-03-27"}, "file": "scilean/SciLean/Core/FunctionTransformations/Preimage.lean", "module": "SciLean.Core.FunctionTransformations.Preimage", "jsonFile": "SciLean.Core.FunctionTransformations.Preimage.jsonl", "positionMetadata": {"lineInFile": 46, "tokenPositionInFile": 1134, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Std.Tactic.OpenPrivate", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Util.CompileInductive", "Mathlib.Data.Set.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Module", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Util.SorryProof", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.Data.Set.Defs\nimport Mathlib.Data.Set.Image\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nimport SciLean.Core.Objects.Scalar\nimport SciLean.Util.SorryProof\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _}\n\nattribute [fun_trans] Set.preimage\nattribute [fun_trans] Set.image\n\n\nattribute [fun_trans] Set.preimage_id Set.preimage_id'\n\nnamespace Set\n\nopen Classical in\n@[fun_trans]\ntheorem preimage_const' (b : \u03b2) (s : Set \u03b2) :\n (fun _ : \u03b1 => b) \u207b\u00b9' s = if b \u2208 s then univ else \u2205 := by apply preimage_const\n\n@[fun_trans]\ntheorem preimage_comp' (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) :\n preimage (fun x => f (g x))\n =\n fun s => g \u207b\u00b9' (f \u207b\u00b9' s) := rfl\n\n\n----------------------------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (B : Set \u03b2) (C : Set \u03b3) :\n preimage (fun x => (f x, g x)) (B.prod C)\n =\n f \u207b\u00b9' B \u2229 g \u207b\u00b9' C := sorry_proof\n\n@[fun_trans]\ntheorem Prod.mk.arg_fst.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (c : \u03b3) :\n preimage (fun x => (f x, c))\n =\n fun s => f \u207b\u00b9' {y | (y,c) \u2208 s} := sorry_proof\n\n@[fun_trans]\ntheorem Prod.mk.arg_snd.preimage_rule_prod (b : \u03b2) (g : \u03b1 \u2192 \u03b3) :\n preimage (fun x => (b, g x))\n =\n fun s => g \u207b\u00b9' {z | (b,z) \u2208 s} := sorry_proof\n\n\nopen SciLean\nvariable {R} [RealScalar R] -- probably generalize following to LinearlyOrderedAddCommGroup\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x + x') (Ioo a b)\n =\n Ioo (a - x') (b - x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a1.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x' + x) (Ioo a b)\n =\n Ioo (a - x') (b - x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x - x') (Ioo a b)\n =\n Ioo (a + x') (b + x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a1.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x' - x) (Ioo a b)\n =\n Ioo (x' - b) (x' - a) := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem Neg.neg.arg_a1.preimage_rule_Ioo (a b : R) :\n preimage (fun x : R => - x) (Ioo a b)\n =\n Ioo (-b) (-a) := by ext; simp; sorry_proof\n\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Preimage1 ---------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- todo: turn into function transformation once we have `fun_trans` supporting two argument functions\ndef preimage1 {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (C : Set \u03b3) : Set \u03b1 := \u22c3 b, (f \u00b7 b) \u207b\u00b9' C\n\n\n-- @[simp, ftrans_simp]\n-- theorem preimage1_id {\u03b1 \u03b2} (s : Set (\u03b1\u00d7\u03b2)) :\n-- s.preimage1 (fun (a : \u03b1) (b : \u03b2) => (a,b)) = \u22c3 b, {a | (a, b) \u2208 s} := sorry_proof\n\n-- this probably needs non-empty `\u03b2`\n", "theoremStatement": "@[simp, ftrans_simp]\ntheorem preimage1_id1 {\u03b1 \u03b2} (A : Set \u03b1) :\n A.preimage1 (fun (a : \u03b1) (_ : \u03b2) => a) = A ", "theoremName": "Set.preimage1_id1", "fileCreated": {"commit": "67090616", "date": "2024-03-27"}, "theoremCreated": {"commit": "f6fb5a0e", "date": "2024-03-27"}, "file": "scilean/SciLean/Core/FunctionTransformations/Preimage.lean", "module": "SciLean.Core.FunctionTransformations.Preimage", "jsonFile": "SciLean.Core.FunctionTransformations.Preimage.jsonl", "positionMetadata": {"lineInFile": 102, "tokenPositionInFile": 2979, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 4, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Std.Tactic.OpenPrivate", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Util.CompileInductive", "Mathlib.Data.Set.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Module", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Util.SorryProof", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.Data.Set.Defs\nimport Mathlib.Data.Set.Image\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nimport SciLean.Core.Objects.Scalar\nimport SciLean.Util.SorryProof\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _}\n\nattribute [fun_trans] Set.preimage\nattribute [fun_trans] Set.image\n\n\nattribute [fun_trans] Set.preimage_id Set.preimage_id'\n\nnamespace Set\n\nopen Classical in\n@[fun_trans]\ntheorem preimage_const' (b : \u03b2) (s : Set \u03b2) :\n (fun _ : \u03b1 => b) \u207b\u00b9' s = if b \u2208 s then univ else \u2205 := by apply preimage_const\n\n@[fun_trans]\ntheorem preimage_comp' (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) :\n preimage (fun x => f (g x))\n =\n fun s => g \u207b\u00b9' (f \u207b\u00b9' s) := rfl\n\n\n----------------------------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (B : Set \u03b2) (C : Set \u03b3) :\n preimage (fun x => (f x, g x)) (B.prod C)\n =\n f \u207b\u00b9' B \u2229 g \u207b\u00b9' C := sorry_proof\n\n@[fun_trans]\ntheorem Prod.mk.arg_fst.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (c : \u03b3) :\n preimage (fun x => (f x, c))\n =\n fun s => f \u207b\u00b9' {y | (y,c) \u2208 s} := sorry_proof\n\n@[fun_trans]\ntheorem Prod.mk.arg_snd.preimage_rule_prod (b : \u03b2) (g : \u03b1 \u2192 \u03b3) :\n preimage (fun x => (b, g x))\n =\n fun s => g \u207b\u00b9' {z | (b,z) \u2208 s} := sorry_proof\n\n\nopen SciLean\nvariable {R} [RealScalar R] -- probably generalize following to LinearlyOrderedAddCommGroup\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x + x') (Ioo a b)\n =\n Ioo (a - x') (b - x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a1.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x' + x) (Ioo a b)\n =\n Ioo (a - x') (b - x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x - x') (Ioo a b)\n =\n Ioo (a + x') (b + x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a1.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x' - x) (Ioo a b)\n =\n Ioo (x' - b) (x' - a) := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem Neg.neg.arg_a1.preimage_rule_Ioo (a b : R) :\n preimage (fun x : R => - x) (Ioo a b)\n =\n Ioo (-b) (-a) := by ext; simp; sorry_proof\n\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Preimage1 ---------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- todo: turn into function transformation once we have `fun_trans` supporting two argument functions\ndef preimage1 {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (C : Set \u03b3) : Set \u03b1 := \u22c3 b, (f \u00b7 b) \u207b\u00b9' C\n\n\n-- @[simp, ftrans_simp]\n-- theorem preimage1_id {\u03b1 \u03b2} (s : Set (\u03b1\u00d7\u03b2)) :\n-- s.preimage1 (fun (a : \u03b1) (b : \u03b2) => (a,b)) = \u22c3 b, {a | (a, b) \u2208 s} := sorry_proof\n\n-- this probably needs non-empty `\u03b2`\n@[simp, ftrans_simp]\ntheorem preimage1_id1 {\u03b1 \u03b2} (A : Set \u03b1) :\n A.preimage1 (fun (a : \u03b1) (_ : \u03b2) => a) = A := sorry_proof\n\n-- this probably needs non-empty `B`\n", "theoremStatement": "@[simp, ftrans_simp]\ntheorem preimage1_id2 {\u03b1 \u03b2} (B : Set \u03b2) :\n B.preimage1 (fun (_ : \u03b1) (b : \u03b2) => b) = Set.univ ", "theoremName": "Set.preimage1_id2", "fileCreated": {"commit": "67090616", "date": "2024-03-27"}, "theoremCreated": {"commit": "f6fb5a0e", "date": "2024-03-27"}, "file": "scilean/SciLean/Core/FunctionTransformations/Preimage.lean", "module": "SciLean.Core.FunctionTransformations.Preimage", "jsonFile": "SciLean.Core.FunctionTransformations.Preimage.jsonl", "positionMetadata": {"lineInFile": 107, "tokenPositionInFile": 3143, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 5, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Std.Tactic.OpenPrivate", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Util.CompileInductive", "Mathlib.Data.Set.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Module", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Util.SorryProof", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.Data.Set.Defs\nimport Mathlib.Data.Set.Image\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nimport SciLean.Core.Objects.Scalar\nimport SciLean.Util.SorryProof\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _}\n\nattribute [fun_trans] Set.preimage\nattribute [fun_trans] Set.image\n\n\nattribute [fun_trans] Set.preimage_id Set.preimage_id'\n\nnamespace Set\n\nopen Classical in\n@[fun_trans]\ntheorem preimage_const' (b : \u03b2) (s : Set \u03b2) :\n (fun _ : \u03b1 => b) \u207b\u00b9' s = if b \u2208 s then univ else \u2205 := by apply preimage_const\n\n@[fun_trans]\ntheorem preimage_comp' (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) :\n preimage (fun x => f (g x))\n =\n fun s => g \u207b\u00b9' (f \u207b\u00b9' s) := rfl\n\n\n----------------------------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (B : Set \u03b2) (C : Set \u03b3) :\n preimage (fun x => (f x, g x)) (B.prod C)\n =\n f \u207b\u00b9' B \u2229 g \u207b\u00b9' C := sorry_proof\n\n@[fun_trans]\ntheorem Prod.mk.arg_fst.preimage_rule_prod (f : \u03b1 \u2192 \u03b2) (c : \u03b3) :\n preimage (fun x => (f x, c))\n =\n fun s => f \u207b\u00b9' {y | (y,c) \u2208 s} := sorry_proof\n\n@[fun_trans]\ntheorem Prod.mk.arg_snd.preimage_rule_prod (b : \u03b2) (g : \u03b1 \u2192 \u03b3) :\n preimage (fun x => (b, g x))\n =\n fun s => g \u207b\u00b9' {z | (b,z) \u2208 s} := sorry_proof\n\n\nopen SciLean\nvariable {R} [RealScalar R] -- probably generalize following to LinearlyOrderedAddCommGroup\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x + x') (Ioo a b)\n =\n Ioo (a - x') (b - x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a1.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x' + x) (Ioo a b)\n =\n Ioo (a - x') (b - x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x - x') (Ioo a b)\n =\n Ioo (a + x') (b + x') := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a1.preimage_rule_Ioo (x' a b : R) :\n preimage (fun x : R => x' - x) (Ioo a b)\n =\n Ioo (x' - b) (x' - a) := by ext; simp; sorry_proof\n\n@[fun_trans]\ntheorem Neg.neg.arg_a1.preimage_rule_Ioo (a b : R) :\n preimage (fun x : R => - x) (Ioo a b)\n =\n Ioo (-b) (-a) := by ext; simp; sorry_proof\n\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Preimage1 ---------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- todo: turn into function transformation once we have `fun_trans` supporting two argument functions\ndef preimage1 {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (C : Set \u03b3) : Set \u03b1 := \u22c3 b, (f \u00b7 b) \u207b\u00b9' C\n\n\n-- @[simp, ftrans_simp]\n-- theorem preimage1_id {\u03b1 \u03b2} (s : Set (\u03b1\u00d7\u03b2)) :\n-- s.preimage1 (fun (a : \u03b1) (b : \u03b2) => (a,b)) = \u22c3 b, {a | (a, b) \u2208 s} := sorry_proof\n\n-- this probably needs non-empty `\u03b2`\n@[simp, ftrans_simp]\ntheorem preimage1_id1 {\u03b1 \u03b2} (A : Set \u03b1) :\n A.preimage1 (fun (a : \u03b1) (_ : \u03b2) => a) = A := sorry_proof\n\n-- this probably needs non-empty `B`\n@[simp, ftrans_simp]\ntheorem preimage1_id2 {\u03b1 \u03b2} (B : Set \u03b2) :\n B.preimage1 (fun (_ : \u03b1) (b : \u03b2) => b) = Set.univ := sorry_proof\n\nopen Classical in\n", "theoremStatement": "@[simp, ftrans_simp]\ntheorem preimage1_const {\u03b1 \u03b2 \u03b3} (c : \u03b3) (C : Set \u03b3) :\n C.preimage1 (fun (_ : \u03b1) (_ : \u03b2) => c) = if c \u2208 C then Set.univ else \u2205 ", "theoremName": "Set.preimage1_const", "fileCreated": {"commit": "67090616", "date": "2024-03-27"}, "theoremCreated": {"commit": "f6fb5a0e", "date": "2024-03-27"}, "file": "scilean/SciLean/Core/FunctionTransformations/Preimage.lean", "module": "SciLean.Core.FunctionTransformations.Preimage", "jsonFile": "SciLean.Core.FunctionTransformations.Preimage.jsonl", "positionMetadata": {"lineInFile": 112, "tokenPositionInFile": 3295, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Std.Tactic.OpenPrivate", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Util.CompileInductive", "Mathlib.Data.Set.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Module", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Util.SorryProof", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FloatAsReal\n-- import SciLean.Modules.Prob.DRand\n-- import SciLean.Modules.Prob.FDRand\n\nnamespace SciLean.Rand\n\nvariable\n {R} [RealScalar R]\n -- {X} [NormedAddCommGroup X] [NormedSpace \u211d X] [NormedSpace R X] [CompleteSpace X] [MeasurableSpace X]\n -- {Y} [NormedAddCommGroup Y] [NormedSpace \u211d Y] [NormedSpace R Y] [CompleteSpace Y] [MeasurableSpace Y]\n -- {Z} [NormedAddCommGroup Z] [NormedSpace \u211d Z] [NormedSpace R Z] [CompleteSpace Z] [MeasurableSpace Z]\n {X : Type _} [MeasurableSpace X] [AddCommGroup X] [Module \u211d X]\n {Y : Type _} [AddCommGroup Y] [Module \u211d Y] [MeasurableSpace Y]\n {Z : Type _} [AddCommGroup Z] [Module \u211d Z]\n\nopen Rand\n\n@[rand_pull_E]\ntheorem bind_pull_mean (x : Rand X) (f : X \u2192 Rand Y) :\n (x >>= (fun x' => pure (f x').mean)).mean\n =\n (x >>= f).mean := by simp[rand_push_E]\n\n@[rand_push_E]\ntheorem ite_push_E {c} [Decidable c] (t e : Rand X) (\u03c6 : X \u2192 Y):\n (if c then t else e).\ud835\udd3c \u03c6 = (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c6) := by\n if h : c then simp[h] else simp[h]\n\n-- I don't think this is a desirable `rand_pull_E` theorem as it duplicates the if statement\n-- @[rand_pull_E]\ntheorem ite_pull_E {c} [Decidable c] (t e : Rand X) (\u03c6 \u03c8 : X \u2192 Y):\n (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c8) = (if c then t else e).\ud835\udd3c (if c then \u03c6 else \u03c8) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_push_E]\ntheorem ite_push_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t else e).mean = (if c then t.mean else e.mean) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E]\ntheorem ite_pull_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t.mean else e.mean) = (if c then t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_t {c} [Decidable c] (t : Rand X) (e : X) :\n (if c then t.mean else e) = (if c then t else pure e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_f {c} [Decidable c] (t : X) (e : Rand X) :\n (if c then t else e.mean) = (if c then pure t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n-- this has messed up universes\n@[rand_pull_E]\ntheorem pull_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n (fun x => r.\ud835\udd3c (fun y => f x y))\n =\n r.\ud835\udd3c (fun y x => f x y) := sorry_proof\n\n-- this has messed up universes\n@[rand_push_E]\ntheorem push_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n r.\ud835\udd3c (fun y x => f x y)\n =\n (fun x => r.\ud835\udd3c (fun y => f x y)) := sorry_proof\n\n-- can't be simp as it has variable head\ntheorem pull_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n (f (r.\ud835\udd3c \u03c6)) = r.\ud835\udd3c (fun x => f (\u03c6 x)) := by sorry_proof -- have := hf; sorry_proof\n\n@[rand_push_E]\ntheorem push_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n r.\ud835\udd3c (fun x => f (\u03c6 x)) = (f (r.\ud835\udd3c \u03c6)) := by rw[pull_E_affine (hf:=hf)]\n\n", "theoremStatement": "@[rand_pull_E]\ntheorem pull_mean_add (x y : Rand X) :\n x.mean + y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' + y' ", "theoremName": "SciLean.Rand.pull_mean_add", "fileCreated": {"commit": "c09d1780", "date": "2024-02-24"}, "theoremCreated": {"commit": "f6fb5a0e", "date": "2024-03-27"}, "file": "scilean/SciLean/Core/Rand/PushPullExpectation.lean", "module": "SciLean.Core.Rand.PushPullExpectation", "jsonFile": "SciLean.Core.Rand.PushPullExpectation.jsonl", "positionMetadata": {"lineInFile": 81, "tokenPositionInFile": 2927, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.Objects.IsomorphicType.RealToFloat", "SciLean.Core.FunctionTransformations.Isomorph.RealToFloat", "SciLean.Core.Objects.IsReal", "SciLean.Core.FloatAsReal"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FloatAsReal\n-- import SciLean.Modules.Prob.DRand\n-- import SciLean.Modules.Prob.FDRand\n\nnamespace SciLean.Rand\n\nvariable\n {R} [RealScalar R]\n -- {X} [NormedAddCommGroup X] [NormedSpace \u211d X] [NormedSpace R X] [CompleteSpace X] [MeasurableSpace X]\n -- {Y} [NormedAddCommGroup Y] [NormedSpace \u211d Y] [NormedSpace R Y] [CompleteSpace Y] [MeasurableSpace Y]\n -- {Z} [NormedAddCommGroup Z] [NormedSpace \u211d Z] [NormedSpace R Z] [CompleteSpace Z] [MeasurableSpace Z]\n {X : Type _} [MeasurableSpace X] [AddCommGroup X] [Module \u211d X]\n {Y : Type _} [AddCommGroup Y] [Module \u211d Y] [MeasurableSpace Y]\n {Z : Type _} [AddCommGroup Z] [Module \u211d Z]\n\nopen Rand\n\n@[rand_pull_E]\ntheorem bind_pull_mean (x : Rand X) (f : X \u2192 Rand Y) :\n (x >>= (fun x' => pure (f x').mean)).mean\n =\n (x >>= f).mean := by simp[rand_push_E]\n\n@[rand_push_E]\ntheorem ite_push_E {c} [Decidable c] (t e : Rand X) (\u03c6 : X \u2192 Y):\n (if c then t else e).\ud835\udd3c \u03c6 = (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c6) := by\n if h : c then simp[h] else simp[h]\n\n-- I don't think this is a desirable `rand_pull_E` theorem as it duplicates the if statement\n-- @[rand_pull_E]\ntheorem ite_pull_E {c} [Decidable c] (t e : Rand X) (\u03c6 \u03c8 : X \u2192 Y):\n (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c8) = (if c then t else e).\ud835\udd3c (if c then \u03c6 else \u03c8) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_push_E]\ntheorem ite_push_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t else e).mean = (if c then t.mean else e.mean) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E]\ntheorem ite_pull_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t.mean else e.mean) = (if c then t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_t {c} [Decidable c] (t : Rand X) (e : X) :\n (if c then t.mean else e) = (if c then t else pure e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_f {c} [Decidable c] (t : X) (e : Rand X) :\n (if c then t else e.mean) = (if c then pure t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n-- this has messed up universes\n@[rand_pull_E]\ntheorem pull_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n (fun x => r.\ud835\udd3c (fun y => f x y))\n =\n r.\ud835\udd3c (fun y x => f x y) := sorry_proof\n\n-- this has messed up universes\n@[rand_push_E]\ntheorem push_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n r.\ud835\udd3c (fun y x => f x y)\n =\n (fun x => r.\ud835\udd3c (fun y => f x y)) := sorry_proof\n\n-- can't be simp as it has variable head\ntheorem pull_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n (f (r.\ud835\udd3c \u03c6)) = r.\ud835\udd3c (fun x => f (\u03c6 x)) := by sorry_proof -- have := hf; sorry_proof\n\n@[rand_push_E]\ntheorem push_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n r.\ud835\udd3c (fun x => f (\u03c6 x)) = (f (r.\ud835\udd3c \u03c6)) := by rw[pull_E_affine (hf:=hf)]\n\n@[rand_pull_E]\ntheorem pull_mean_add (x y : Rand X) :\n x.mean + y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' + y' := sorry_proof\n\n", "theoremStatement": "@[rand_pull_E]\ntheorem pull_mean_add_1 (x : Rand X) (y : X) :\n x.mean + y\n =\n Rand.mean do\n let x' \u2190 x\n return x' + y ", "theoremName": "SciLean.Rand.pull_mean_add_1", "fileCreated": {"commit": "c09d1780", "date": "2024-02-24"}, "theoremCreated": {"commit": "23206ae3", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Rand/PushPullExpectation.lean", "module": "SciLean.Core.Rand.PushPullExpectation", "jsonFile": "SciLean.Core.Rand.PushPullExpectation.jsonl", "positionMetadata": {"lineInFile": 90, "tokenPositionInFile": 3095, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.Objects.IsomorphicType.RealToFloat", "SciLean.Core.FunctionTransformations.Isomorph.RealToFloat", "SciLean.Core.Objects.IsReal", "SciLean.Core.FloatAsReal"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FloatAsReal\n-- import SciLean.Modules.Prob.DRand\n-- import SciLean.Modules.Prob.FDRand\n\nnamespace SciLean.Rand\n\nvariable\n {R} [RealScalar R]\n -- {X} [NormedAddCommGroup X] [NormedSpace \u211d X] [NormedSpace R X] [CompleteSpace X] [MeasurableSpace X]\n -- {Y} [NormedAddCommGroup Y] [NormedSpace \u211d Y] [NormedSpace R Y] [CompleteSpace Y] [MeasurableSpace Y]\n -- {Z} [NormedAddCommGroup Z] [NormedSpace \u211d Z] [NormedSpace R Z] [CompleteSpace Z] [MeasurableSpace Z]\n {X : Type _} [MeasurableSpace X] [AddCommGroup X] [Module \u211d X]\n {Y : Type _} [AddCommGroup Y] [Module \u211d Y] [MeasurableSpace Y]\n {Z : Type _} [AddCommGroup Z] [Module \u211d Z]\n\nopen Rand\n\n@[rand_pull_E]\ntheorem bind_pull_mean (x : Rand X) (f : X \u2192 Rand Y) :\n (x >>= (fun x' => pure (f x').mean)).mean\n =\n (x >>= f).mean := by simp[rand_push_E]\n\n@[rand_push_E]\ntheorem ite_push_E {c} [Decidable c] (t e : Rand X) (\u03c6 : X \u2192 Y):\n (if c then t else e).\ud835\udd3c \u03c6 = (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c6) := by\n if h : c then simp[h] else simp[h]\n\n-- I don't think this is a desirable `rand_pull_E` theorem as it duplicates the if statement\n-- @[rand_pull_E]\ntheorem ite_pull_E {c} [Decidable c] (t e : Rand X) (\u03c6 \u03c8 : X \u2192 Y):\n (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c8) = (if c then t else e).\ud835\udd3c (if c then \u03c6 else \u03c8) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_push_E]\ntheorem ite_push_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t else e).mean = (if c then t.mean else e.mean) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E]\ntheorem ite_pull_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t.mean else e.mean) = (if c then t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_t {c} [Decidable c] (t : Rand X) (e : X) :\n (if c then t.mean else e) = (if c then t else pure e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_f {c} [Decidable c] (t : X) (e : Rand X) :\n (if c then t else e.mean) = (if c then pure t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n-- this has messed up universes\n@[rand_pull_E]\ntheorem pull_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n (fun x => r.\ud835\udd3c (fun y => f x y))\n =\n r.\ud835\udd3c (fun y x => f x y) := sorry_proof\n\n-- this has messed up universes\n@[rand_push_E]\ntheorem push_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n r.\ud835\udd3c (fun y x => f x y)\n =\n (fun x => r.\ud835\udd3c (fun y => f x y)) := sorry_proof\n\n-- can't be simp as it has variable head\ntheorem pull_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n (f (r.\ud835\udd3c \u03c6)) = r.\ud835\udd3c (fun x => f (\u03c6 x)) := by sorry_proof -- have := hf; sorry_proof\n\n@[rand_push_E]\ntheorem push_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n r.\ud835\udd3c (fun x => f (\u03c6 x)) = (f (r.\ud835\udd3c \u03c6)) := by rw[pull_E_affine (hf:=hf)]\n\n@[rand_pull_E]\ntheorem pull_mean_add (x y : Rand X) :\n x.mean + y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' + y' := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_add_1 (x : Rand X) (y : X) :\n x.mean + y\n =\n Rand.mean do\n let x' \u2190 x\n return x' + y := sorry_proof\n\n", "theoremStatement": "@[rand_pull_E]\ntheorem pull_mean_add_2 (x : X) (y : Rand X) :\n x + y.mean\n =\n Rand.mean do\n let y' \u2190 y\n return x + y' ", "theoremName": "SciLean.Rand.pull_mean_add_2", "fileCreated": {"commit": "c09d1780", "date": "2024-02-24"}, "theoremCreated": {"commit": "23206ae3", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Rand/PushPullExpectation.lean", "module": "SciLean.Core.Rand.PushPullExpectation", "jsonFile": "SciLean.Core.Rand.PushPullExpectation.jsonl", "positionMetadata": {"lineInFile": 98, "tokenPositionInFile": 3248, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.Objects.IsomorphicType.RealToFloat", "SciLean.Core.FunctionTransformations.Isomorph.RealToFloat", "SciLean.Core.Objects.IsReal", "SciLean.Core.FloatAsReal"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FloatAsReal\n-- import SciLean.Modules.Prob.DRand\n-- import SciLean.Modules.Prob.FDRand\n\nnamespace SciLean.Rand\n\nvariable\n {R} [RealScalar R]\n -- {X} [NormedAddCommGroup X] [NormedSpace \u211d X] [NormedSpace R X] [CompleteSpace X] [MeasurableSpace X]\n -- {Y} [NormedAddCommGroup Y] [NormedSpace \u211d Y] [NormedSpace R Y] [CompleteSpace Y] [MeasurableSpace Y]\n -- {Z} [NormedAddCommGroup Z] [NormedSpace \u211d Z] [NormedSpace R Z] [CompleteSpace Z] [MeasurableSpace Z]\n {X : Type _} [MeasurableSpace X] [AddCommGroup X] [Module \u211d X]\n {Y : Type _} [AddCommGroup Y] [Module \u211d Y] [MeasurableSpace Y]\n {Z : Type _} [AddCommGroup Z] [Module \u211d Z]\n\nopen Rand\n\n@[rand_pull_E]\ntheorem bind_pull_mean (x : Rand X) (f : X \u2192 Rand Y) :\n (x >>= (fun x' => pure (f x').mean)).mean\n =\n (x >>= f).mean := by simp[rand_push_E]\n\n@[rand_push_E]\ntheorem ite_push_E {c} [Decidable c] (t e : Rand X) (\u03c6 : X \u2192 Y):\n (if c then t else e).\ud835\udd3c \u03c6 = (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c6) := by\n if h : c then simp[h] else simp[h]\n\n-- I don't think this is a desirable `rand_pull_E` theorem as it duplicates the if statement\n-- @[rand_pull_E]\ntheorem ite_pull_E {c} [Decidable c] (t e : Rand X) (\u03c6 \u03c8 : X \u2192 Y):\n (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c8) = (if c then t else e).\ud835\udd3c (if c then \u03c6 else \u03c8) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_push_E]\ntheorem ite_push_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t else e).mean = (if c then t.mean else e.mean) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E]\ntheorem ite_pull_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t.mean else e.mean) = (if c then t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_t {c} [Decidable c] (t : Rand X) (e : X) :\n (if c then t.mean else e) = (if c then t else pure e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_f {c} [Decidable c] (t : X) (e : Rand X) :\n (if c then t else e.mean) = (if c then pure t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n-- this has messed up universes\n@[rand_pull_E]\ntheorem pull_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n (fun x => r.\ud835\udd3c (fun y => f x y))\n =\n r.\ud835\udd3c (fun y x => f x y) := sorry_proof\n\n-- this has messed up universes\n@[rand_push_E]\ntheorem push_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n r.\ud835\udd3c (fun y x => f x y)\n =\n (fun x => r.\ud835\udd3c (fun y => f x y)) := sorry_proof\n\n-- can't be simp as it has variable head\ntheorem pull_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n (f (r.\ud835\udd3c \u03c6)) = r.\ud835\udd3c (fun x => f (\u03c6 x)) := by sorry_proof -- have := hf; sorry_proof\n\n@[rand_push_E]\ntheorem push_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n r.\ud835\udd3c (fun x => f (\u03c6 x)) = (f (r.\ud835\udd3c \u03c6)) := by rw[pull_E_affine (hf:=hf)]\n\n@[rand_pull_E]\ntheorem pull_mean_add (x y : Rand X) :\n x.mean + y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' + y' := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_add_1 (x : Rand X) (y : X) :\n x.mean + y\n =\n Rand.mean do\n let x' \u2190 x\n return x' + y := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_add_2 (x : X) (y : Rand X) :\n x + y.mean\n =\n Rand.mean do\n let y' \u2190 y\n return x + y' := sorry_proof\n\n", "theoremStatement": "@[rand_pull_E]\ntheorem pull_mean_sub (x y : Rand X) :\n x.mean - y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' - y' ", "theoremName": "SciLean.Rand.pull_mean_sub", "fileCreated": {"commit": "c09d1780", "date": "2024-02-24"}, "theoremCreated": {"commit": "f6fb5a0e", "date": "2024-03-27"}, "file": "scilean/SciLean/Core/Rand/PushPullExpectation.lean", "module": "SciLean.Core.Rand.PushPullExpectation", "jsonFile": "SciLean.Core.Rand.PushPullExpectation.jsonl", "positionMetadata": {"lineInFile": 106, "tokenPositionInFile": 3401, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.Objects.IsomorphicType.RealToFloat", "SciLean.Core.FunctionTransformations.Isomorph.RealToFloat", "SciLean.Core.Objects.IsReal", "SciLean.Core.FloatAsReal"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.Algebra.Field.Defs\nimport Mathlib.GroupTheory.GroupAction.Defs\nimport Mathlib.Tactic.FunProp\n\nimport SciLean.Util.SorryProof\n\nset_option linter.unusedVariables false\n\n-- Some missing theorems -------------------------------------------------------\n--------------------------------------------------------------------------------\n\n", "theoremStatement": "theorem Function.invFun_comp' [Nonempty \u03b1] {f : \u03b1 \u2192 \u03b2} (hf : f.Injective) {x : \u03b1} :\n f.invFun (f x) = x ", "theoremName": "Function.invFun_comp'", "fileCreated": {"commit": "a95431a4", "date": "2023-08-19"}, "theoremCreated": {"commit": "c0cf2eb5", "date": "2024-04-12"}, "file": "scilean/SciLean/Core/FunctionPropositions/Bijective.lean", "module": "SciLean.Core.FunctionPropositions.Bijective", "jsonFile": "SciLean.Core.FunctionPropositions.Bijective.jsonl", "positionMetadata": {"lineInFile": 12, "tokenPositionInFile": 345, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Opposites", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Tactic.FunProp.Decl", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "SciLean.Util.SorryProof"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n suffices (f.invFun \u2218 f) x = x by assumption\n rw[Function.invFun_comp hf]\n rfl", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 87}} +{"srcContext": "import SciLean.Core.FunctionPropositions.CDifferentiable\nimport SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nimport SciLean.Core.Meta.GenerateLinearMapSimp\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nset_option linter.unusedVariables false\n\nopen LeanColls\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n@[fun_trans]\nnoncomputable\ndef cderiv (f : X \u2192 Y) (x dx : X) : Y := Curve.deriv (fun t : K => f (x + t\u2022dx)) 0\n\n@[ftrans_simp]\nnoncomputable\ndef scalarCDeriv (f : K \u2192 X) (t : K) : X := cderiv K f t 1\n\n\n-- Basic identities ------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n@[fun_trans]\ntheorem cderiv_of_linear (f : X \u2192 Y) (hf : IsSmoothLinearMap K f)\n : cderiv K f = fun x dx => f dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply\n (f : X \u2192 Y \u2192 Z) (x dx : X) (y : Y)\n : cderiv K f x dx y\n =\n cderiv K (fun x' => f x' y) x dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply_zero\n (f : X \u2192 Y) (x : X)\n : cderiv K f x 0\n =\n 0 := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule_simple\n (f : X \u2192 Y) (x : X) (hf : CDifferentiableAt K f x)\n : IsLinearMap K (fun dx => cderiv K f x dx) := sorry_proof\n\n#generate_linear_map_simps SciLean.cderiv.arg_dx.IsLinearMap_rule_simple\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule\n (f : X \u2192 Y) (x : X) (dx : W \u2192 X) (hf : CDifferentiableAt K f x) (hdx : IsLinearMap K dx)\n : IsLinearMap K (fun w => cderiv K f x (dx w)) := by fun_prop\n\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : \u2200 x, CDifferentiable K (fun y => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsLinearMap K (fun x => cderiv K (f x \u00b7)) := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsSmoothLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : CDifferentiable K (fun (x,y) => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsSmoothLinearMap K (fun x => cderiv K (f x \u00b7)) := by constructor; fun_prop; sorry_proof /- differentiable and linear implies smooth -/\n\n\nvariable (K)\n\n-- Basic lambda calculus rules -------------------------------------------------\n--------------------------------------------------------------------------------\n@[fun_trans]\ntheorem cderiv.id_rule :\n (cderiv K fun x : X => x) = fun _ => fun dx => dx := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.const_rule (x : X) :\n (cderiv K fun _ : Y => x) = fun _ => fun dx => 0 := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule_at\n (f : Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K f (g x)) (hg : CDifferentiableAt K g x)\n : (cderiv K fun x : X => f (g x)) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule\n (f : Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K f) (hg : CDifferentiable K g)\n : (cderiv K fun x : X => f (g x))\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule_at\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K \u21bff (x, g x))\n (hg : CDifferentiableAt K g x)\n : (cderiv K\n fun x : X =>\n let y := g x\n f x y) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K fun xy : X\u00d7Y => f xy.1 xy.2) (hg : CDifferentiable K g)\n : (cderiv K fun x : X =>\n let y := g x\n f x y)\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.apply_rule (i : \u03b9) :\n (cderiv K fun (x : (i : \u03b9) \u2192 E i) => x i)\n =\n fun _ => fun dx => dx i := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule_at\n (f : X \u2192 (i : \u03b9) \u2192 E i) (x : X) (hf : \u2200 i, CDifferentiableAt K (f \u00b7 i) x)\n : (cderiv K fun (x : X) (i : \u03b9) => f x i) x\n =\n fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule\n (f : X \u2192 (i : \u03b9) \u2192 E i) (hf : \u2200 i, CDifferentiable K (f \u00b7 i))\n : (cderiv K fun (x : X) (i : \u03b9) => f x i)\n =\n fun x => fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n\n\n--------------------------------------------------------------------------------\n-- Function Rules --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nopen SciLean\nopen LeanColls\n\nvariable\n {K : Type _} [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Prod.mk -----------------------------------v---------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.cderiv_rule_at (x : X)\n (g : X \u2192 Y) (hg : CDifferentiableAt K g x)\n (f : X \u2192 Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (g x, f x)) x\n =\n fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by\n sorry_proof\n\n\n@[fun_trans]\n theorem Prod.mk.arg_fstsnd.cderiv_rule\n (g : X \u2192 Y) (hg : CDifferentiable K g)\n (f : X \u2192 Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (g x, f x))\n =\n fun x => fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by funext x; fun_trans\n\n\n-- Prod.fst --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).1) x\n =\n fun dx => (cderiv K f x dx).1 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).1)\n =\n fun x dx => (cderiv K f x dx).1 := by funext x; fun_trans\n\n\n-- Prod.snd --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).2) x\n =\n fun dx => (cderiv K f x dx).2 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).2)\n =\n fun x => fun dx => (cderiv K f x dx).2 := by funext x; fun_trans\n\n\n-- HAdd.hAdd -------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule_at (x : X)\n (f g : X \u2192 Y) (hf : CDifferentiableAt K f x) (hg : CDifferentiableAt K g x) :\n (cderiv K fun x => f x + g x) x\n =\n fun dx =>\n cderiv K f x dx + cderiv K g x dx := by fun_trans\n\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule\n (f g : X \u2192 Y) (hf : CDifferentiable K f) (hg : CDifferentiable K g) :\n (cderiv K fun x => f x + g x)\n =\n fun x => fun dx =>\n cderiv K f x dx + cderiv K g x dx := by funext x; fun_trans\n\n", "theoremStatement": "@[fun_trans]\ntheorem HAdd.hAdd.arg_a0.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => f x + y)\n =\n fun x dx =>\n cderiv K f x dx ", "theoremName": "SciLean.HAdd.hAdd.arg_a0.cderiv_rule", "fileCreated": {"commit": "fc69a949", "date": "2023-08-07"}, "theoremCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "file": "scilean/SciLean/Core/FunctionTransformations/CDeriv.lean", "module": "SciLean.Core.FunctionTransformations.CDeriv", "jsonFile": "SciLean.Core.FunctionTransformations.CDeriv.jsonl", "positionMetadata": {"lineInFile": 274, "tokenPositionInFile": 7812, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab"]}, "proofMetadata": {"hasProof": false, "proof": ":= by funext x; sorry_proof", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 27}} +{"srcContext": "import SciLean.Core.FunctionPropositions.CDifferentiable\nimport SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nimport SciLean.Core.Meta.GenerateLinearMapSimp\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nset_option linter.unusedVariables false\n\nopen LeanColls\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n@[fun_trans]\nnoncomputable\ndef cderiv (f : X \u2192 Y) (x dx : X) : Y := Curve.deriv (fun t : K => f (x + t\u2022dx)) 0\n\n@[ftrans_simp]\nnoncomputable\ndef scalarCDeriv (f : K \u2192 X) (t : K) : X := cderiv K f t 1\n\n\n-- Basic identities ------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n@[fun_trans]\ntheorem cderiv_of_linear (f : X \u2192 Y) (hf : IsSmoothLinearMap K f)\n : cderiv K f = fun x dx => f dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply\n (f : X \u2192 Y \u2192 Z) (x dx : X) (y : Y)\n : cderiv K f x dx y\n =\n cderiv K (fun x' => f x' y) x dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply_zero\n (f : X \u2192 Y) (x : X)\n : cderiv K f x 0\n =\n 0 := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule_simple\n (f : X \u2192 Y) (x : X) (hf : CDifferentiableAt K f x)\n : IsLinearMap K (fun dx => cderiv K f x dx) := sorry_proof\n\n#generate_linear_map_simps SciLean.cderiv.arg_dx.IsLinearMap_rule_simple\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule\n (f : X \u2192 Y) (x : X) (dx : W \u2192 X) (hf : CDifferentiableAt K f x) (hdx : IsLinearMap K dx)\n : IsLinearMap K (fun w => cderiv K f x (dx w)) := by fun_prop\n\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : \u2200 x, CDifferentiable K (fun y => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsLinearMap K (fun x => cderiv K (f x \u00b7)) := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsSmoothLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : CDifferentiable K (fun (x,y) => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsSmoothLinearMap K (fun x => cderiv K (f x \u00b7)) := by constructor; fun_prop; sorry_proof /- differentiable and linear implies smooth -/\n\n\nvariable (K)\n\n-- Basic lambda calculus rules -------------------------------------------------\n--------------------------------------------------------------------------------\n@[fun_trans]\ntheorem cderiv.id_rule :\n (cderiv K fun x : X => x) = fun _ => fun dx => dx := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.const_rule (x : X) :\n (cderiv K fun _ : Y => x) = fun _ => fun dx => 0 := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule_at\n (f : Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K f (g x)) (hg : CDifferentiableAt K g x)\n : (cderiv K fun x : X => f (g x)) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule\n (f : Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K f) (hg : CDifferentiable K g)\n : (cderiv K fun x : X => f (g x))\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule_at\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K \u21bff (x, g x))\n (hg : CDifferentiableAt K g x)\n : (cderiv K\n fun x : X =>\n let y := g x\n f x y) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K fun xy : X\u00d7Y => f xy.1 xy.2) (hg : CDifferentiable K g)\n : (cderiv K fun x : X =>\n let y := g x\n f x y)\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.apply_rule (i : \u03b9) :\n (cderiv K fun (x : (i : \u03b9) \u2192 E i) => x i)\n =\n fun _ => fun dx => dx i := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule_at\n (f : X \u2192 (i : \u03b9) \u2192 E i) (x : X) (hf : \u2200 i, CDifferentiableAt K (f \u00b7 i) x)\n : (cderiv K fun (x : X) (i : \u03b9) => f x i) x\n =\n fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule\n (f : X \u2192 (i : \u03b9) \u2192 E i) (hf : \u2200 i, CDifferentiable K (f \u00b7 i))\n : (cderiv K fun (x : X) (i : \u03b9) => f x i)\n =\n fun x => fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n\n\n--------------------------------------------------------------------------------\n-- Function Rules --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nopen SciLean\nopen LeanColls\n\nvariable\n {K : Type _} [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Prod.mk -----------------------------------v---------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.cderiv_rule_at (x : X)\n (g : X \u2192 Y) (hg : CDifferentiableAt K g x)\n (f : X \u2192 Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (g x, f x)) x\n =\n fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by\n sorry_proof\n\n\n@[fun_trans]\n theorem Prod.mk.arg_fstsnd.cderiv_rule\n (g : X \u2192 Y) (hg : CDifferentiable K g)\n (f : X \u2192 Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (g x, f x))\n =\n fun x => fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by funext x; fun_trans\n\n\n-- Prod.fst --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).1) x\n =\n fun dx => (cderiv K f x dx).1 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).1)\n =\n fun x dx => (cderiv K f x dx).1 := by funext x; fun_trans\n\n\n-- Prod.snd --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).2) x\n =\n fun dx => (cderiv K f x dx).2 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).2)\n =\n fun x => fun dx => (cderiv K f x dx).2 := by funext x; fun_trans\n\n\n-- HAdd.hAdd -------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule_at (x : X)\n (f g : X \u2192 Y) (hf : CDifferentiableAt K f x) (hg : CDifferentiableAt K g x) :\n (cderiv K fun x => f x + g x) x\n =\n fun dx =>\n cderiv K f x dx + cderiv K g x dx := by fun_trans\n\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule\n (f g : X \u2192 Y) (hf : CDifferentiable K f) (hg : CDifferentiable K g) :\n (cderiv K fun x => f x + g x)\n =\n fun x => fun dx =>\n cderiv K f x dx + cderiv K g x dx := by funext x; fun_trans\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => f x + y)\n =\n fun x dx =>\n cderiv K f x dx := by funext x; sorry_proof\n\n", "theoremStatement": "@[fun_trans]\ntheorem HAdd.hAdd.arg_a1.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => y + f x)\n =\n fun x dx =>\n cderiv K f x dx ", "theoremName": "SciLean.HAdd.hAdd.arg_a1.cderiv_rule", "fileCreated": {"commit": "fc69a949", "date": "2023-08-07"}, "theoremCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "file": "scilean/SciLean/Core/FunctionTransformations/CDeriv.lean", "module": "SciLean.Core.FunctionTransformations.CDeriv", "jsonFile": "SciLean.Core.FunctionTransformations.CDeriv.jsonl", "positionMetadata": {"lineInFile": 282, "tokenPositionInFile": 7993, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab"]}, "proofMetadata": {"hasProof": false, "proof": ":= by funext x; sorry_proof", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 27}} +{"srcContext": "import SciLean.Core.FunctionPropositions.CDifferentiable\nimport SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nimport SciLean.Core.Meta.GenerateLinearMapSimp\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nset_option linter.unusedVariables false\n\nopen LeanColls\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n@[fun_trans]\nnoncomputable\ndef cderiv (f : X \u2192 Y) (x dx : X) : Y := Curve.deriv (fun t : K => f (x + t\u2022dx)) 0\n\n@[ftrans_simp]\nnoncomputable\ndef scalarCDeriv (f : K \u2192 X) (t : K) : X := cderiv K f t 1\n\n\n-- Basic identities ------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n@[fun_trans]\ntheorem cderiv_of_linear (f : X \u2192 Y) (hf : IsSmoothLinearMap K f)\n : cderiv K f = fun x dx => f dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply\n (f : X \u2192 Y \u2192 Z) (x dx : X) (y : Y)\n : cderiv K f x dx y\n =\n cderiv K (fun x' => f x' y) x dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply_zero\n (f : X \u2192 Y) (x : X)\n : cderiv K f x 0\n =\n 0 := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule_simple\n (f : X \u2192 Y) (x : X) (hf : CDifferentiableAt K f x)\n : IsLinearMap K (fun dx => cderiv K f x dx) := sorry_proof\n\n#generate_linear_map_simps SciLean.cderiv.arg_dx.IsLinearMap_rule_simple\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule\n (f : X \u2192 Y) (x : X) (dx : W \u2192 X) (hf : CDifferentiableAt K f x) (hdx : IsLinearMap K dx)\n : IsLinearMap K (fun w => cderiv K f x (dx w)) := by fun_prop\n\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : \u2200 x, CDifferentiable K (fun y => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsLinearMap K (fun x => cderiv K (f x \u00b7)) := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsSmoothLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : CDifferentiable K (fun (x,y) => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsSmoothLinearMap K (fun x => cderiv K (f x \u00b7)) := by constructor; fun_prop; sorry_proof /- differentiable and linear implies smooth -/\n\n\nvariable (K)\n\n-- Basic lambda calculus rules -------------------------------------------------\n--------------------------------------------------------------------------------\n@[fun_trans]\ntheorem cderiv.id_rule :\n (cderiv K fun x : X => x) = fun _ => fun dx => dx := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.const_rule (x : X) :\n (cderiv K fun _ : Y => x) = fun _ => fun dx => 0 := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule_at\n (f : Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K f (g x)) (hg : CDifferentiableAt K g x)\n : (cderiv K fun x : X => f (g x)) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule\n (f : Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K f) (hg : CDifferentiable K g)\n : (cderiv K fun x : X => f (g x))\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule_at\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K \u21bff (x, g x))\n (hg : CDifferentiableAt K g x)\n : (cderiv K\n fun x : X =>\n let y := g x\n f x y) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K fun xy : X\u00d7Y => f xy.1 xy.2) (hg : CDifferentiable K g)\n : (cderiv K fun x : X =>\n let y := g x\n f x y)\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.apply_rule (i : \u03b9) :\n (cderiv K fun (x : (i : \u03b9) \u2192 E i) => x i)\n =\n fun _ => fun dx => dx i := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule_at\n (f : X \u2192 (i : \u03b9) \u2192 E i) (x : X) (hf : \u2200 i, CDifferentiableAt K (f \u00b7 i) x)\n : (cderiv K fun (x : X) (i : \u03b9) => f x i) x\n =\n fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule\n (f : X \u2192 (i : \u03b9) \u2192 E i) (hf : \u2200 i, CDifferentiable K (f \u00b7 i))\n : (cderiv K fun (x : X) (i : \u03b9) => f x i)\n =\n fun x => fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n\n\n--------------------------------------------------------------------------------\n-- Function Rules --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nopen SciLean\nopen LeanColls\n\nvariable\n {K : Type _} [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Prod.mk -----------------------------------v---------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.cderiv_rule_at (x : X)\n (g : X \u2192 Y) (hg : CDifferentiableAt K g x)\n (f : X \u2192 Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (g x, f x)) x\n =\n fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by\n sorry_proof\n\n\n@[fun_trans]\n theorem Prod.mk.arg_fstsnd.cderiv_rule\n (g : X \u2192 Y) (hg : CDifferentiable K g)\n (f : X \u2192 Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (g x, f x))\n =\n fun x => fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by funext x; fun_trans\n\n\n-- Prod.fst --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).1) x\n =\n fun dx => (cderiv K f x dx).1 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).1)\n =\n fun x dx => (cderiv K f x dx).1 := by funext x; fun_trans\n\n\n-- Prod.snd --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).2) x\n =\n fun dx => (cderiv K f x dx).2 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).2)\n =\n fun x => fun dx => (cderiv K f x dx).2 := by funext x; fun_trans\n\n\n-- HAdd.hAdd -------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule_at (x : X)\n (f g : X \u2192 Y) (hf : CDifferentiableAt K f x) (hg : CDifferentiableAt K g x) :\n (cderiv K fun x => f x + g x) x\n =\n fun dx =>\n cderiv K f x dx + cderiv K g x dx := by fun_trans\n\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule\n (f g : X \u2192 Y) (hf : CDifferentiable K f) (hg : CDifferentiable K g) :\n (cderiv K fun x => f x + g x)\n =\n fun x => fun dx =>\n cderiv K f x dx + cderiv K g x dx := by funext x; fun_trans\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => f x + y)\n =\n fun x dx =>\n cderiv K f x dx := by funext x; sorry_proof\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a1.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => y + f x)\n =\n fun x dx =>\n cderiv K f x dx := by funext x; sorry_proof\n\n\n-- HSub.hSub -------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0a1.cderiv_rule_at (x : X)\n (f g : X \u2192 Y) (hf : CDifferentiableAt K f x) (hg : CDifferentiableAt K g x) :\n (cderiv K fun x => f x - g x) x\n =\n fun dx =>\n cderiv K f x dx - cderiv K g x dx := by fun_trans\n\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0a1.cderiv_rule\n (f g : X \u2192 Y) (hf : CDifferentiable K f) (hg : CDifferentiable K g) :\n (cderiv K fun x => f x - g x)\n =\n fun x => fun dx =>\n cderiv K f x dx - cderiv K g x dx := by funext x; fun_trans\n\n", "theoremStatement": "@[fun_trans]\ntheorem HSub.hSub.arg_a0.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => f x - y)\n =\n fun x dx =>\n cderiv K f x dx ", "theoremName": "SciLean.HSub.hSub.arg_a0.cderiv_rule", "fileCreated": {"commit": "fc69a949", "date": "2023-08-07"}, "theoremCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "file": "scilean/SciLean/Core/FunctionTransformations/CDeriv.lean", "module": "SciLean.Core.FunctionTransformations.CDeriv", "jsonFile": "SciLean.Core.FunctionTransformations.CDeriv.jsonl", "positionMetadata": {"lineInFile": 311, "tokenPositionInFile": 8853, "theoremPositionInFile": 30}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 12, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab"]}, "proofMetadata": {"hasProof": false, "proof": ":= by funext x; sorry_proof", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 27}} +{"srcContext": "import SciLean.Core.FunctionPropositions.CDifferentiable\nimport SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nimport SciLean.Core.Meta.GenerateLinearMapSimp\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nset_option linter.unusedVariables false\n\nopen LeanColls\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n@[fun_trans]\nnoncomputable\ndef cderiv (f : X \u2192 Y) (x dx : X) : Y := Curve.deriv (fun t : K => f (x + t\u2022dx)) 0\n\n@[ftrans_simp]\nnoncomputable\ndef scalarCDeriv (f : K \u2192 X) (t : K) : X := cderiv K f t 1\n\n\n-- Basic identities ------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n@[fun_trans]\ntheorem cderiv_of_linear (f : X \u2192 Y) (hf : IsSmoothLinearMap K f)\n : cderiv K f = fun x dx => f dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply\n (f : X \u2192 Y \u2192 Z) (x dx : X) (y : Y)\n : cderiv K f x dx y\n =\n cderiv K (fun x' => f x' y) x dx := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cderiv_apply_zero\n (f : X \u2192 Y) (x : X)\n : cderiv K f x 0\n =\n 0 := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule_simple\n (f : X \u2192 Y) (x : X) (hf : CDifferentiableAt K f x)\n : IsLinearMap K (fun dx => cderiv K f x dx) := sorry_proof\n\n#generate_linear_map_simps SciLean.cderiv.arg_dx.IsLinearMap_rule_simple\n\n@[fun_prop]\ntheorem cderiv.arg_dx.IsLinearMap_rule\n (f : X \u2192 Y) (x : X) (dx : W \u2192 X) (hf : CDifferentiableAt K f x) (hdx : IsLinearMap K dx)\n : IsLinearMap K (fun w => cderiv K f x (dx w)) := by fun_prop\n\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : \u2200 x, CDifferentiable K (fun y => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsLinearMap K (fun x => cderiv K (f x \u00b7)) := sorry_proof\n\n@[fun_prop]\ntheorem cderiv.arg_f.IsSmoothLinearMap_rule\n (f : X \u2192 Y \u2192 Z) (hf : CDifferentiable K (fun (x,y) => f x y)) (hf' : \u2200 y, IsLinearMap K (fun x => f x y)) :\n IsSmoothLinearMap K (fun x => cderiv K (f x \u00b7)) := by constructor; fun_prop; sorry_proof /- differentiable and linear implies smooth -/\n\n\nvariable (K)\n\n-- Basic lambda calculus rules -------------------------------------------------\n--------------------------------------------------------------------------------\n@[fun_trans]\ntheorem cderiv.id_rule :\n (cderiv K fun x : X => x) = fun _ => fun dx => dx := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.const_rule (x : X) :\n (cderiv K fun _ : Y => x) = fun _ => fun dx => 0 := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule_at\n (f : Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K f (g x)) (hg : CDifferentiableAt K g x)\n : (cderiv K fun x : X => f (g x)) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.comp_rule\n (f : Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K f) (hg : CDifferentiable K g)\n : (cderiv K fun x : X => f (g x))\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K f y dy\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule_at\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y) (x : X)\n (hf : CDifferentiableAt K \u21bff (x, g x))\n (hg : CDifferentiableAt K g x)\n : (cderiv K\n fun x : X =>\n let y := g x\n f x y) x\n =\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.let_rule\n (f : X \u2192 Y \u2192 Z) (g : X \u2192 Y)\n (hf : CDifferentiable K fun xy : X\u00d7Y => f xy.1 xy.2) (hg : CDifferentiable K g)\n : (cderiv K fun x : X =>\n let y := g x\n f x y)\n =\n fun x =>\n let y := g x\n fun dx =>\n let dy := cderiv K g x dx\n let dz := cderiv K (fun xy : X\u00d7Y => f xy.1 xy.2) (x,y) (dx, dy)\n dz :=\nby sorry_proof\n\n@[fun_trans]\ntheorem cderiv.apply_rule (i : \u03b9) :\n (cderiv K fun (x : (i : \u03b9) \u2192 E i) => x i)\n =\n fun _ => fun dx => dx i := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule_at\n (f : X \u2192 (i : \u03b9) \u2192 E i) (x : X) (hf : \u2200 i, CDifferentiableAt K (f \u00b7 i) x)\n : (cderiv K fun (x : X) (i : \u03b9) => f x i) x\n =\n fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n@[fun_trans]\ntheorem cderiv.pi_rule\n (f : X \u2192 (i : \u03b9) \u2192 E i) (hf : \u2200 i, CDifferentiable K (f \u00b7 i))\n : (cderiv K fun (x : X) (i : \u03b9) => f x i)\n =\n fun x => fun dx => fun i =>\n cderiv K (f \u00b7 i) x dx\n := by sorry_proof\n\n\n\n--------------------------------------------------------------------------------\n-- Function Rules --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nopen SciLean\nopen LeanColls\n\nvariable\n {K : Type _} [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Prod.mk -----------------------------------v---------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.mk.arg_fstsnd.cderiv_rule_at (x : X)\n (g : X \u2192 Y) (hg : CDifferentiableAt K g x)\n (f : X \u2192 Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (g x, f x)) x\n =\n fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by\n sorry_proof\n\n\n@[fun_trans]\n theorem Prod.mk.arg_fstsnd.cderiv_rule\n (g : X \u2192 Y) (hg : CDifferentiable K g)\n (f : X \u2192 Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (g x, f x))\n =\n fun x => fun dx =>\n (cderiv K g x dx, cderiv K f x dx) := by funext x; fun_trans\n\n\n-- Prod.fst --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).1) x\n =\n fun dx => (cderiv K f x dx).1 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.fst.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).1)\n =\n fun x dx => (cderiv K f x dx).1 := by funext x; fun_trans\n\n\n-- Prod.snd --------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule_at (x : X)\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiableAt K f x) :\n cderiv K (fun x => (f x).2) x\n =\n fun dx => (cderiv K f x dx).2 := by fun_trans\n\n@[fun_trans]\ntheorem Prod.snd.arg_self.cderiv_rule\n (f : X \u2192 Y\u00d7Z) (hf : CDifferentiable K f) :\n cderiv K (fun x => (f x).2)\n =\n fun x => fun dx => (cderiv K f x dx).2 := by funext x; fun_trans\n\n\n-- HAdd.hAdd -------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule_at (x : X)\n (f g : X \u2192 Y) (hf : CDifferentiableAt K f x) (hg : CDifferentiableAt K g x) :\n (cderiv K fun x => f x + g x) x\n =\n fun dx =>\n cderiv K f x dx + cderiv K g x dx := by fun_trans\n\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.cderiv_rule\n (f g : X \u2192 Y) (hf : CDifferentiable K f) (hg : CDifferentiable K g) :\n (cderiv K fun x => f x + g x)\n =\n fun x => fun dx =>\n cderiv K f x dx + cderiv K g x dx := by funext x; fun_trans\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => f x + y)\n =\n fun x dx =>\n cderiv K f x dx := by funext x; sorry_proof\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a1.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => y + f x)\n =\n fun x dx =>\n cderiv K f x dx := by funext x; sorry_proof\n\n\n-- HSub.hSub -------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0a1.cderiv_rule_at (x : X)\n (f g : X \u2192 Y) (hf : CDifferentiableAt K f x) (hg : CDifferentiableAt K g x) :\n (cderiv K fun x => f x - g x) x\n =\n fun dx =>\n cderiv K f x dx - cderiv K g x dx := by fun_trans\n\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0a1.cderiv_rule\n (f g : X \u2192 Y) (hf : CDifferentiable K f) (hg : CDifferentiable K g) :\n (cderiv K fun x => f x - g x)\n =\n fun x => fun dx =>\n cderiv K f x dx - cderiv K g x dx := by funext x; fun_trans\n\n@[fun_trans]\ntheorem HSub.hSub.arg_a0.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => f x - y)\n =\n fun x dx =>\n cderiv K f x dx := by funext x; sorry_proof\n\n", "theoremStatement": "@[fun_trans]\ntheorem HSub.hSub.arg_a1.cderiv_rule\n (f : X \u2192 Y) (y : Y) :\n (cderiv K fun x => y - f x)\n =\n fun x dx =>\n - cderiv K f x dx ", "theoremName": "SciLean.HSub.hSub.arg_a1.cderiv_rule", "fileCreated": {"commit": "fc69a949", "date": "2023-08-07"}, "theoremCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "file": "scilean/SciLean/Core/FunctionTransformations/CDeriv.lean", "module": "SciLean.Core.FunctionTransformations.CDeriv", "jsonFile": "SciLean.Core.FunctionTransformations.CDeriv.jsonl", "positionMetadata": {"lineInFile": 319, "tokenPositionInFile": 9034, "theoremPositionInFile": 31}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab"]}, "proofMetadata": {"hasProof": false, "proof": ":= by funext x; sorry_proof", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 27}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.Rand.Condition\n\nimport SciLean.Core.Distribution.ParametricDistribDeriv\n\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nnamespace SciLean\n\n\nopen MeasureTheory\nvariable\n {R} [RealScalar R]\n {X Z} [MeasurableSpace X] [MeasurableSpace Z]\n\n/-- Kullback\u2013Leibler divergence of `D\u2096\u2097(P\u2016Q)` -/\nnoncomputable\ndef KLDiv (P Q : Rand X) : R := P.\ud835\udd3c (fun x => Scalar.log (P.pdf R Q.\u2119 x))\n\n/-- Evidence Lower Bound\n\n```\n\ud835\udd3c_Q [log Q(Z) - log P(Z,X)]\n```\nreference: https://en.wikipedia.org/wiki/Variational_Bayesian_methods#Evidence_lower_bound\n -/\nnoncomputable\ndef ELBO {X Z} [MeasureSpace Z] [MeasureSpace X]\n (P : Rand (Z\u00d7X)) (Q : Rand Z) (x : X) : R :=\n - Q.\ud835\udd3c (fun z => Scalar.log (Q.pdf R volume z) - Scalar.log (P.pdf R volume (z,x)))\n\n\n", "theoremStatement": "/-- Express `Kullback\u2013Leibler divergence` as log evidence + ELBO -/\ntheorem kldiv_elbo\n {X Z} [MeasureSpace Z] [MeasureSpace X]\n (P : Rand (Z\u00d7X)) (Q : Rand Z) (x : X) :\n KLDiv Q (P.conditionSnd x)\n =\n (Scalar.log (P.snd.pdf R volume x)) - ELBO P Q x ", "theoremName": "SciLean.kldiv_elbo", "fileCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "theoremCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "file": "scilean/SciLean/Core/Rand/VariationalInference.lean", "module": "SciLean.Core.Rand.VariationalInference", "jsonFile": "SciLean.Core.Rand.VariationalInference.jsonl", "positionMetadata": {"lineInFile": 33, "tokenPositionInFile": 790, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 24, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.Rand.Condition", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation", "SciLean.Core.Distribution.ParametricDistribDeriv"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.Rand.Condition\n\nimport SciLean.Core.Distribution.ParametricDistribDeriv\n\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nnamespace SciLean\n\n\nopen MeasureTheory\nvariable\n {R} [RealScalar R]\n {X Z} [MeasurableSpace X] [MeasurableSpace Z]\n\n/-- Kullback\u2013Leibler divergence of `D\u2096\u2097(P\u2016Q)` -/\nnoncomputable\ndef KLDiv (P Q : Rand X) : R := P.\ud835\udd3c (fun x => Scalar.log (P.pdf R Q.\u2119 x))\n\n/-- Evidence Lower Bound\n\n```\n\ud835\udd3c_Q [log Q(Z) - log P(Z,X)]\n```\nreference: https://en.wikipedia.org/wiki/Variational_Bayesian_methods#Evidence_lower_bound\n -/\nnoncomputable\ndef ELBO {X Z} [MeasureSpace Z] [MeasureSpace X]\n (P : Rand (Z\u00d7X)) (Q : Rand Z) (x : X) : R :=\n - Q.\ud835\udd3c (fun z => Scalar.log (Q.pdf R volume z) - Scalar.log (P.pdf R volume (z,x)))\n\n\n/-- Express `Kullback\u2013Leibler divergence` as log evidence + ELBO -/\ntheorem kldiv_elbo\n {X Z} [MeasureSpace Z] [MeasureSpace X]\n (P : Rand (Z\u00d7X)) (Q : Rand Z) (x : X) :\n KLDiv Q (P.conditionSnd x)\n =\n (Scalar.log (P.snd.pdf R volume x)) - ELBO P Q x := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- KLDiv properties --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable\n {W} [Vec R W]\n [Vec R X]\n\n", "theoremStatement": "@[fun_trans]\ntheorem KLDiv.arg_P.cderiv_rule (P : W \u2192 Rand X) (Q : Rand X) :\n cderiv R (fun w => KLDiv (R:=R) (P w) Q)\n =\n fun w dw =>\n let dP := parDistribDeriv (fun w => (P w).\u2119.toDistribution (R:=R)) w dw\n dP.extAction' (fun x => Scalar.log ((P w).pdf R Q.\u2119 x) - 1) ", "theoremName": "SciLean.KLDiv.arg_P.cderiv_rule", "fileCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "theoremCreated": {"commit": "a33d3993", "date": "2024-04-09"}, "file": "scilean/SciLean/Core/Rand/VariationalInference.lean", "module": "SciLean.Core.Rand.VariationalInference", "jsonFile": "SciLean.Core.Rand.VariationalInference.jsonl", "positionMetadata": {"lineInFile": 50, "tokenPositionInFile": 1414, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 17, "numPremises": 34, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.Rand.Condition", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation", "SciLean.Core.Distribution.ParametricDistribDeriv"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.Distribution.ParametricDistribFwdDeriv\n\nnamespace SciLean\n\nopen MeasureTheory\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W]\n {X} [MeasurableSpace X] [Vec R X]\n {Y} [Vec R Y] [Module \u211d Y]\n\nset_default_scalar R\n\n", "theoremStatement": "@[fun_trans]\ntheorem Rand.\ud835\udd3c.arg_r.cderiv_rule (r : W \u2192 Rand X) (f : X \u2192 Y) :\n cderiv R (fun w => (r w).\ud835\udd3c f)\n =\n fun w dw =>\n let d := parDistribDeriv (fun w => (r w).\u2119.toDistribution (R:=R)) w dw\n d.extAction f (fun r \u22b8 fun y \u22b8 ((r \u2022 y) : Y)) ", "theoremName": "SciLean.Rand.\ud835\udd3c.arg_r.cderiv_rule", "fileCreated": {"commit": "b3d5680d", "date": "2024-04-12"}, "theoremCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "file": "scilean/SciLean/Core/Rand/ExpectedValue.lean", "module": "SciLean.Core.Rand.ExpectedValue", "jsonFile": "SciLean.Core.Rand.ExpectedValue.jsonl", "positionMetadata": {"lineInFile": 16, "tokenPositionInFile": 263, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 29, "numPremises": 59, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation", "SciLean.Core.Distribution.ParametricDistribDeriv", "SciLean.Core.Distribution.ParametricDistribFwdDeriv"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.MeasureSpaceDef\nimport Mathlib.MeasureTheory.Measure.Restrict\nimport Mathlib.MeasureTheory.Measure.Dirac\n-- import Mathlib.Data.RCLike.Basic\n-- import Mathlib.Analysis.LocallyConvex.Basic\n-- import Mathlib.Topology.Algebra.Module.LocallyConvex\n\nimport SciLean.Core.FunctionTransformations\nimport SciLean.Util.SorryProof\n\nopen MeasureTheory\n\nnamespace SciLean\n\n/-- Convenient integral - the integral I need :)\nIt should be Bochner integral but it should integrate function valued function point wise i.e.\n```\n\u222b x, fun y => f x y = fun y => \u222b x, f x y\n```\nwhere rhs can be understoods as Bochenr integral and lhs defined thie `cintegral`. -/\nnoncomputable\nopaque cintegral {\u03b1} [MeasurableSpace \u03b1] {X} [AddCommGroup X] [Module \u211d X]\n -- dragging along all of these typeclasses is really really annoying\n -- [AddCommGroup X] [TopologicalSpace X] [TopologicalAddGroup X] [Module \u211d X] [LocallyConvexSpace \u211d X]\n (f : \u03b1 \u2192 X) (\u03bc : Measure \u03b1) : X := 0\n\nopaque CIntegrable {\u03b1} [MeasurableSpace \u03b1] {X} [AddCommGroup X] [Module \u211d X]\n (f : \u03b1 \u2192 X) (\u03bc : Measure \u03b1) : Prop\n\nopen Lean Parser Term\nsyntax \"\u222b' \" funBinder (\" in \" term)? \", \" term:60 (\" \u2202\" term:70)? : term\n\nmacro_rules\n| `(\u222b' $x:funBinder, $b) => `(cintegral (fun $x => $b) (by volume_tac))\n| `(\u222b' $x:funBinder, $b \u2202$\u03bc) => `(cintegral (fun $x => $b) $\u03bc)\n| `(\u222b' $x:funBinder in $set, $b) => `(cintegral (fun $x => $b) (Measure.restrict (by volume_tac) $set))\n| `(\u222b' $x:funBinder in $set, $b \u2202$\u03bc) => `(cintegral (fun $x => $b) (Measure.restrict $\u03bc $set))\n\n\n@[app_unexpander cintegral] def unexpandCIntegral : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $f:term volume) =>\n match f with\n | `(fun $x => $b) => `(\u222b' $x, $b)\n | `(fun $x $xs* => $b) => `(\u222b' $x, fun $xs* => $b)\n | _ => throw ()\n\n | `($(_) $f:term $\u03bc) =>\n match f, \u03bc with\n -- | `(fun $x => $b), `(Measure.restrict volume $A) => `(\u222b' $x in $A, $b)\n -- | `(fun $x $xs* => $b), `(Measure.restrict volume $A) => `(\u222b' $x in $A, (fun $xs* => $b))\n | `(fun $x => $b), `(Measure.restrict $\u03bc $A) => `(\u222b' $x in $A, $b \u2202$\u03bc)\n | `(fun $x $xs* => $b), `(Measure.restrict $\u03bc $A) => `(\u222b' $x in $A, (fun $xs* => $b) \u2202$\u03bc)\n -- | `(fun $x => $b), `(volume) => `(\u222b' $x, $b)\n -- | `(fun $x $xs* => $b), `(volume) => `(\u222b' $x, (fun $xs* => $b))\n | `(fun $x => $b), _ => `(\u222b' $x, $b \u2202$\u03bc)\n | `(fun $x $xs* => $b), _ => `(\u222b' $x, (fun $xs* => $b) \u2202$\u03bc)\n | _, _ => throw ()\n\n\n | _ => throw ()\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\nsection Algebra\n\nvariable\n {\u03b1} [MeasurableSpace \u03b1]\n {\u03b2} [MeasurableSpace \u03b2]\n {X} [AddCommGroup X] [Module \u211d X]\n {Y} [AddCommGroup Y] [Module \u211d Y]\n\n@[simp, ftrans_simp]\ntheorem cintegral_zero (\u03bc : Measure \u03b1): \u222b' _, (0 : X) \u2202\u03bc = 0 := sorry_proof\n\n@[add_pull] -- @[integral_push]\ntheorem cintegral_add {f g : \u03b1 \u2192 X} {\u03bc : Measure \u03b1}\n (hf : CIntegrable f \u03bc) (hg : CIntegrable g \u03bc) :\n \u222b' x, f x + g x \u2202\u03bc = \u222b' x, f x \u2202\u03bc + \u222b' x, g x \u2202\u03bc := sorry_proof\n\n@[add_push] -- @[integral_push]\ntheorem cintegral_add' {f g : \u03b1 \u2192 X} {\u03bc : Measure \u03b1}\n (hf : CIntegrable f \u03bc) (hg : CIntegrable g \u03bc) :\n \u222b' x, f x \u2202\u03bc + \u222b' x, g x \u2202\u03bc = \u222b' x, f x + g x \u2202\u03bc := sorry_proof\n\n@[smul_pull]-- @[integral_push]\ntheorem cintegral_smul {R} [Semiring R] [Module R X] {f : \u03b1 \u2192 X} (r : R) :\n \u222b' x, r \u2022 f x \u2202\u03bc = r \u2022 \u222b' x, f x \u2202\u03bc := sorry_proof\n\n@[smul_push] -- @[integral_pull]\ntheorem cintegral_smul' {R} [Semiring R] [Module R X] {f : \u03b1 \u2192 X} (r : R) :\n r \u2022 \u222b' x, f x \u2202\u03bc = \u222b' x, r \u2022 f x \u2202\u03bc := sorry_proof\n\n@[add_pull] -- @[integral_push]\ntheorem cintegral_add_measures {f : \u03b1 \u2192 X} {\u03bc \u03bd : Measure \u03b1}\n (h\u03bc : CIntegrable f \u03bc) (h\u03bd : CIntegrable f \u03bd) :\n \u222b' x, f x \u2202(\u03bc+\u03bd) = \u222b' x, f x \u2202\u03bc + \u222b' x, f x \u2202\u03bd := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem cintegral_dirac {f : \u03b1 \u2192 X} (p : \u03b1) :\n \u222b' x, f x \u2202(Measure.dirac p) = f p := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem cintegral_unit {f : Unit \u2192 X} :\n \u222b' x, f x = f () := sorry_proof\n\n-- @[integral_push]\ntheorem cintegral_prod_mk {f : \u03b1 \u2192 X} {g : \u03b1 \u2192 Y} {\u03bc : Measure \u03b1} :\n \u222b' x, (f x, g x) \u2202\u03bc = (\u222b' x, f x \u2202\u03bc, \u222b' x, g x \u2202\u03bc) := sorry_proof\n\n-- @[integral_pull]\ntheorem cintegral_prod_mk' {f : \u03b1 \u2192 X} {g : \u03b1 \u2192 Y} {\u03bc : Measure \u03b1} :\n (\u222b' x, f x \u2202\u03bc, \u222b' x, g x \u2202\u03bc) = \u222b' x, (f x, g x) \u2202\u03bc := sorry_proof\n\ntheorem cintegral_lambda {\u03b1} (f : \u03b1 \u2192 \u03b2 \u2192 X) (\u03bc : Measure \u03b2) :\n (fun x => \u222b' y, f x y \u2202\u03bc)\n =\n \u222b' y, (fun x => f x y) \u2202\u03bc := sorry_proof\n\ntheorem cintegral.arg_f.push_lambda {\u03b1} (f : \u03b1 \u2192 \u03b2 \u2192 X) (\u03bc : Measure \u03b2) :\n (fun x => \u222b' y, f x y \u2202\u03bc)\n =\n \u222b' y, (fun x => f x y) \u2202\u03bc := sorry_proof\n\n\n", "theoremStatement": "@[simp, ftrans_simp]\ntheorem cintegral_measure_map (g : \u03b2 \u2192 X) (f : \u03b1 \u2192 \u03b2) (\u03bc : Measure \u03b1) :\n (\u222b' y, g y \u2202(\u03bc.map f))\n =\n \u222b' x, g (f x) \u2202\u03bc ", "theoremName": "SciLean.cintegral_measure_map", "fileCreated": {"commit": "8b35bac7", "date": "2024-03-18"}, "theoremCreated": {"commit": "352ee6e8", "date": "2024-04-10"}, "file": "scilean/SciLean/Core/Integral/CIntegral.lean", "module": "SciLean.Core.Integral.CIntegral", "jsonFile": "SciLean.Core.Integral.CIntegral.jsonl", "positionMetadata": {"lineInFile": 127, "tokenPositionInFile": 4822, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Util.SorryProof", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Distribution.Basic\nimport SciLean.Core.FunctionTransformations\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.Notation\n\n\nopen MeasureTheory\n\nnamespace SciLean\n\nopen Distribution\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W]\n {X} [Vec R X] [MeasureSpace X]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z] [Module \u211d Z]\n {U} [Vec R U] -- [Module \u211d U]\n {V} [Vec R V] -- [Module \u211d U]\n\n\nset_default_scalar R\n\n\nnoncomputable\ndef diracDeriv (x dx : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 cderiv R \u03c6 x dx\n\n@[fun_prop]\ndef DistribDifferentiableAt (f : X \u2192 \ud835\udc9f'(Y,Z)) (x : X) :=\n \u2200 (\u03c6 : X \u2192 \ud835\udc9f Y), CDifferentiableAt R \u03c6 x \u2192 CDifferentiableAt R (fun x => f x (\u03c6 x)) x\n\n\ntheorem distribDifferentiableAt_const_test_fun\n {f : X \u2192 \ud835\udc9f'(Y,Z)} {x : X}\n (hf : DistribDifferentiableAt f x)\n {\u03c6 : \ud835\udc9f Y} :\n CDifferentiableAt R (fun x => f x \u03c6) x := by\n apply hf\n fun_prop\n\n\n@[fun_prop]\ndef DistribDifferentiable (f : X \u2192 \ud835\udc9f'(Y,Z)) :=\n \u2200 x, DistribDifferentiableAt f x\n\n\n-- TODO:\n-- probably change the definition of `parDistribDeriv` to:\n-- \u27e8\u27e8fun \u03c6 =>\n-- if h : DistribDifferentiableAt f x then\n-- \u2202 (x':=x;dx), \u27eaf x', \u03c6\u27eb\n-- else\n-- 0 , sorry_proof\u27e9\u27e9\n-- I believe in that case the function is indeed linear in \u03c6\n\nopen Classical in\n@[fun_trans]\nnoncomputable\ndef parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) : \ud835\udc9f'(Y,Z) :=\n \u27e8fun \u03c6 => \u2202 (x':=x;dx), f x' \u03c6, sorry_proof\u27e9\n\n\n@[simp, ftrans_simp]\ntheorem action_parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) (\u03c6 : \ud835\udc9f Y) :\n parDistribDeriv f x dx \u03c6 = \u2202 (x':=x;dx), f x' \u03c6 := rfl\n\n\n----------------------------------------------------------------------------------------------------\n-- Const rule --------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.const_rule (T : \ud835\udc9f'(X,Y)) :\n DistribDifferentiable (fun _ : W => T) := by\n intro _ \u03c6 h\u03c6; simp; fun_prop\n\n@[fun_trans]\ntheorem parDistribDeriv.const_rule (T : \ud835\udc9f'(X,Y)) :\n parDistribDeriv (fun _ : W => T)\n =\n fun w dw =>\n 0 := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Pure --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem dirac.arg_xy.DistribDiffrentiable_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n DistribDifferentiable (R:=R) (fun w => dirac (x w)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp [action_dirac, dirac]\n fun_prop\n\n\n@[fun_trans]\ntheorem dirac.arg_x.parDistribDeriv_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n parDistribDeriv (R:=R) (fun w => dirac (x w))\n =\n fun w dw =>\n let xdx := fwdDeriv R x w dw\n diracDeriv xdx.1 xdx.2 := by --= (dpure (R:=R) ydy.1 ydy.2) := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv dirac diracDeriv\n simp [pure, fwdDeriv, DistribDifferentiableAt]\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Composition -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n DistribDifferentiable (fun x => f (g x)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n apply CDifferentiable.comp_rule (K:=R) (f:=fun xy : X\u00d7Y => f xy.2 (\u03c6 xy.1)) (g:=fun x => (x, g x))\n (hg:=by fun_prop)\n intro x\n sorry_proof -- is this even true ?\n\n\n@[fun_trans]\ntheorem parDistribDeriv.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n parDistribDeriv (fun x => f (g x))\n =\n fun x dx =>\n let ydy := fwdDeriv R g x dx\n parDistribDeriv f ydy.1 ydy.2 := by\n\n funext x dx; ext \u03c6\n unfold parDistribDeriv\n simp[hg]\n sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Bind --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n-- The assumptions here are definitely not right!!!\n-- I think `f` has to be `deg`\n@[fun_prop]\ntheorem Bind.bind.arg_fx.DistribDifferentiable_rule\n (f : X \u2192 Y \u2192 \ud835\udc9f'(Z,V)) (g : X \u2192 \ud835\udc9f'(Y,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (x,y) => f x y)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun x => (g x).bind (f x) L) := by\n\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp\n sorry_proof\n\n\n@[fun_trans]\ntheorem Bind.bind.arg_fx.parDistribDiff_rule\n (f : W \u2192 X \u2192 \ud835\udc9f'(Y,V)) (g : W \u2192 \ud835\udc9f'(X,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (w,x) => f w x)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => (g w).bind (f w) L)\n =\n fun w dw =>\n ((parDistribDeriv g w dw).bind (f x \u00b7 ) L)\n +\n ((g w).bind (fun x => parDistribDeriv (f \u00b7 x) w dw) L) := sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Move these around -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).restrict A) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule_simple (A : Set X) :\n IsSmoothLinearMap R (fun (T : \ud835\udc9f'(X,Y)) => T.restrict A) := sorry_proof\n\n\n@[fun_trans]\ntheorem Distribution.restrict.arg_T.parDistribDeriv_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : DistribDifferentiable T) :\n parDistribDeriv (fun w => (T w).restrict A)\n =\n fun w dw =>\n (parDistribDeriv T w dw).restrict A := sorry_proof\n\n@[fun_prop]\ntheorem Function.toDistribution.arg_f.CDifferentiable_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n CDifferentiable R (fun w => (fun x => f w x).toDistribution (R:=R)) := sorry_proof\n\n@[fun_trans]\ntheorem Function.toDistribution.arg_f.cderiv_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n cderiv R (fun w => (fun x => f w x).toDistribution (R:=R))\n =\n fun w dw =>\n (fun x =>\n let dy := cderiv R (f \u00b7 x) w dw\n dy).toDistribution := sorry_proof\n\n@[fun_trans]\ntheorem toDistribution.linear_parDistribDeriv_rule (f : W \u2192 X \u2192 Y) (L : Y \u2192 Z)\n (hL : IsSmoothLinearMap R L) :\n parDistribDeriv (fun w => (fun x => L (f w x)).toDistribution)\n =\n fun w dw =>\n parDistribDeriv (fun w => (fun x => f w x).toDistribution) w dw |>.postComp (fun y \u22b8 L y) := by\n funext w dw\n unfold parDistribDeriv Distribution.postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Integral ----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X] [MeasureSpace Y] [MeasureSpace (X\u00d7Y)]\n\nopen Notation\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule (f : W \u2192 X \u2192 Y) :\n cderiv R (fun w => \u222b' x, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule' (f : W \u2192 X \u2192 R) (A : Set X):\n cderiv R (fun w => \u222b' x in A, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).restrict A |>.extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule (f : W \u2192 X \u2192 Y \u2192 Z) :\n parDistribDeriv (fun w => (fun x => \u222b' y, f w x y).toDistribution (R:=R))\n =\n fun w dw =>\n let Tf := (fun w => (fun x => (fun y => f w x y).toDistribution (R:=R)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := by\n funext w dw\n unfold parDistribDeriv postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule' (f : W \u2192 X \u2192 Y \u2192 Z) (B : X \u2192 Set Y) :\n parDistribDeriv (fun w => (fun x => \u222b' y in B x, f w x y).toDistribution)\n =\n fun w dw =>\n let Tf := (fun w => (fun x => ((fun y => f w x y).toDistribution (R:=R)).restrict (B x)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := sorry_proof\n\n\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Add ---------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n", "theoremStatement": "@[fun_prop]\ntheorem HAdd.hAdd.arg_a0a1.DistribDifferentiable_rule (f g : W \u2192 \ud835\udc9f'(X,Y))\n (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun w => f w + g w) ", "theoremName": "SciLean.HAdd.hAdd.arg_a0a1.DistribDifferentiable_rule", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "a33d3993", "date": "2024-04-09"}, "file": "scilean/SciLean/Core/Distribution/ParametricDistribDeriv.lean", "module": "SciLean.Core.Distribution.ParametricDistribDeriv", "jsonFile": "SciLean.Core.Distribution.ParametricDistribDeriv.jsonl", "positionMetadata": {"lineInFile": 294, "tokenPositionInFile": 9726, "theoremPositionInFile": 24}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 16, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Distribution.Basic\nimport SciLean.Core.FunctionTransformations\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.Notation\n\n\nopen MeasureTheory\n\nnamespace SciLean\n\nopen Distribution\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W]\n {X} [Vec R X] [MeasureSpace X]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z] [Module \u211d Z]\n {U} [Vec R U] -- [Module \u211d U]\n {V} [Vec R V] -- [Module \u211d U]\n\n\nset_default_scalar R\n\n\nnoncomputable\ndef diracDeriv (x dx : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 cderiv R \u03c6 x dx\n\n@[fun_prop]\ndef DistribDifferentiableAt (f : X \u2192 \ud835\udc9f'(Y,Z)) (x : X) :=\n \u2200 (\u03c6 : X \u2192 \ud835\udc9f Y), CDifferentiableAt R \u03c6 x \u2192 CDifferentiableAt R (fun x => f x (\u03c6 x)) x\n\n\ntheorem distribDifferentiableAt_const_test_fun\n {f : X \u2192 \ud835\udc9f'(Y,Z)} {x : X}\n (hf : DistribDifferentiableAt f x)\n {\u03c6 : \ud835\udc9f Y} :\n CDifferentiableAt R (fun x => f x \u03c6) x := by\n apply hf\n fun_prop\n\n\n@[fun_prop]\ndef DistribDifferentiable (f : X \u2192 \ud835\udc9f'(Y,Z)) :=\n \u2200 x, DistribDifferentiableAt f x\n\n\n-- TODO:\n-- probably change the definition of `parDistribDeriv` to:\n-- \u27e8\u27e8fun \u03c6 =>\n-- if h : DistribDifferentiableAt f x then\n-- \u2202 (x':=x;dx), \u27eaf x', \u03c6\u27eb\n-- else\n-- 0 , sorry_proof\u27e9\u27e9\n-- I believe in that case the function is indeed linear in \u03c6\n\nopen Classical in\n@[fun_trans]\nnoncomputable\ndef parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) : \ud835\udc9f'(Y,Z) :=\n \u27e8fun \u03c6 => \u2202 (x':=x;dx), f x' \u03c6, sorry_proof\u27e9\n\n\n@[simp, ftrans_simp]\ntheorem action_parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) (\u03c6 : \ud835\udc9f Y) :\n parDistribDeriv f x dx \u03c6 = \u2202 (x':=x;dx), f x' \u03c6 := rfl\n\n\n----------------------------------------------------------------------------------------------------\n-- Const rule --------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.const_rule (T : \ud835\udc9f'(X,Y)) :\n DistribDifferentiable (fun _ : W => T) := by\n intro _ \u03c6 h\u03c6; simp; fun_prop\n\n@[fun_trans]\ntheorem parDistribDeriv.const_rule (T : \ud835\udc9f'(X,Y)) :\n parDistribDeriv (fun _ : W => T)\n =\n fun w dw =>\n 0 := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Pure --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem dirac.arg_xy.DistribDiffrentiable_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n DistribDifferentiable (R:=R) (fun w => dirac (x w)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp [action_dirac, dirac]\n fun_prop\n\n\n@[fun_trans]\ntheorem dirac.arg_x.parDistribDeriv_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n parDistribDeriv (R:=R) (fun w => dirac (x w))\n =\n fun w dw =>\n let xdx := fwdDeriv R x w dw\n diracDeriv xdx.1 xdx.2 := by --= (dpure (R:=R) ydy.1 ydy.2) := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv dirac diracDeriv\n simp [pure, fwdDeriv, DistribDifferentiableAt]\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Composition -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n DistribDifferentiable (fun x => f (g x)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n apply CDifferentiable.comp_rule (K:=R) (f:=fun xy : X\u00d7Y => f xy.2 (\u03c6 xy.1)) (g:=fun x => (x, g x))\n (hg:=by fun_prop)\n intro x\n sorry_proof -- is this even true ?\n\n\n@[fun_trans]\ntheorem parDistribDeriv.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n parDistribDeriv (fun x => f (g x))\n =\n fun x dx =>\n let ydy := fwdDeriv R g x dx\n parDistribDeriv f ydy.1 ydy.2 := by\n\n funext x dx; ext \u03c6\n unfold parDistribDeriv\n simp[hg]\n sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Bind --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n-- The assumptions here are definitely not right!!!\n-- I think `f` has to be `deg`\n@[fun_prop]\ntheorem Bind.bind.arg_fx.DistribDifferentiable_rule\n (f : X \u2192 Y \u2192 \ud835\udc9f'(Z,V)) (g : X \u2192 \ud835\udc9f'(Y,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (x,y) => f x y)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun x => (g x).bind (f x) L) := by\n\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp\n sorry_proof\n\n\n@[fun_trans]\ntheorem Bind.bind.arg_fx.parDistribDiff_rule\n (f : W \u2192 X \u2192 \ud835\udc9f'(Y,V)) (g : W \u2192 \ud835\udc9f'(X,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (w,x) => f w x)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => (g w).bind (f w) L)\n =\n fun w dw =>\n ((parDistribDeriv g w dw).bind (f x \u00b7 ) L)\n +\n ((g w).bind (fun x => parDistribDeriv (f \u00b7 x) w dw) L) := sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Move these around -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).restrict A) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule_simple (A : Set X) :\n IsSmoothLinearMap R (fun (T : \ud835\udc9f'(X,Y)) => T.restrict A) := sorry_proof\n\n\n@[fun_trans]\ntheorem Distribution.restrict.arg_T.parDistribDeriv_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : DistribDifferentiable T) :\n parDistribDeriv (fun w => (T w).restrict A)\n =\n fun w dw =>\n (parDistribDeriv T w dw).restrict A := sorry_proof\n\n@[fun_prop]\ntheorem Function.toDistribution.arg_f.CDifferentiable_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n CDifferentiable R (fun w => (fun x => f w x).toDistribution (R:=R)) := sorry_proof\n\n@[fun_trans]\ntheorem Function.toDistribution.arg_f.cderiv_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n cderiv R (fun w => (fun x => f w x).toDistribution (R:=R))\n =\n fun w dw =>\n (fun x =>\n let dy := cderiv R (f \u00b7 x) w dw\n dy).toDistribution := sorry_proof\n\n@[fun_trans]\ntheorem toDistribution.linear_parDistribDeriv_rule (f : W \u2192 X \u2192 Y) (L : Y \u2192 Z)\n (hL : IsSmoothLinearMap R L) :\n parDistribDeriv (fun w => (fun x => L (f w x)).toDistribution)\n =\n fun w dw =>\n parDistribDeriv (fun w => (fun x => f w x).toDistribution) w dw |>.postComp (fun y \u22b8 L y) := by\n funext w dw\n unfold parDistribDeriv Distribution.postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Integral ----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X] [MeasureSpace Y] [MeasureSpace (X\u00d7Y)]\n\nopen Notation\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule (f : W \u2192 X \u2192 Y) :\n cderiv R (fun w => \u222b' x, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule' (f : W \u2192 X \u2192 R) (A : Set X):\n cderiv R (fun w => \u222b' x in A, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).restrict A |>.extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule (f : W \u2192 X \u2192 Y \u2192 Z) :\n parDistribDeriv (fun w => (fun x => \u222b' y, f w x y).toDistribution (R:=R))\n =\n fun w dw =>\n let Tf := (fun w => (fun x => (fun y => f w x y).toDistribution (R:=R)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := by\n funext w dw\n unfold parDistribDeriv postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule' (f : W \u2192 X \u2192 Y \u2192 Z) (B : X \u2192 Set Y) :\n parDistribDeriv (fun w => (fun x => \u222b' y in B x, f w x y).toDistribution)\n =\n fun w dw =>\n let Tf := (fun w => (fun x => ((fun y => f w x y).toDistribution (R:=R)).restrict (B x)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := sorry_proof\n\n\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Add ---------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n@[fun_prop]\ntheorem HAdd.hAdd.arg_a0a1.DistribDifferentiable_rule (f g : W \u2192 \ud835\udc9f'(X,Y))\n (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun w => f w + g w) := sorry_proof\n\n\n", "theoremStatement": "@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.parDistribDeriv_rule (f g : W \u2192 \ud835\udc9f'(X,Y))\n (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => f w + g w)\n =\n fun w dw =>\n let dy := parDistribDeriv f w dw\n let dz := parDistribDeriv g w dw\n dy + dz ", "theoremName": "SciLean.HAdd.hAdd.arg_a0a1.parDistribDeriv_rule", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "a33d3993", "date": "2024-04-09"}, "file": "scilean/SciLean/Core/Distribution/ParametricDistribDeriv.lean", "module": "SciLean.Core.Distribution.ParametricDistribDeriv", "jsonFile": "SciLean.Core.Distribution.ParametricDistribDeriv.jsonl", "positionMetadata": {"lineInFile": 300, "tokenPositionInFile": 9944, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Distribution.Basic\nimport SciLean.Core.FunctionTransformations\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.Notation\n\n\nopen MeasureTheory\n\nnamespace SciLean\n\nopen Distribution\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W]\n {X} [Vec R X] [MeasureSpace X]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z] [Module \u211d Z]\n {U} [Vec R U] -- [Module \u211d U]\n {V} [Vec R V] -- [Module \u211d U]\n\n\nset_default_scalar R\n\n\nnoncomputable\ndef diracDeriv (x dx : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 cderiv R \u03c6 x dx\n\n@[fun_prop]\ndef DistribDifferentiableAt (f : X \u2192 \ud835\udc9f'(Y,Z)) (x : X) :=\n \u2200 (\u03c6 : X \u2192 \ud835\udc9f Y), CDifferentiableAt R \u03c6 x \u2192 CDifferentiableAt R (fun x => f x (\u03c6 x)) x\n\n\ntheorem distribDifferentiableAt_const_test_fun\n {f : X \u2192 \ud835\udc9f'(Y,Z)} {x : X}\n (hf : DistribDifferentiableAt f x)\n {\u03c6 : \ud835\udc9f Y} :\n CDifferentiableAt R (fun x => f x \u03c6) x := by\n apply hf\n fun_prop\n\n\n@[fun_prop]\ndef DistribDifferentiable (f : X \u2192 \ud835\udc9f'(Y,Z)) :=\n \u2200 x, DistribDifferentiableAt f x\n\n\n-- TODO:\n-- probably change the definition of `parDistribDeriv` to:\n-- \u27e8\u27e8fun \u03c6 =>\n-- if h : DistribDifferentiableAt f x then\n-- \u2202 (x':=x;dx), \u27eaf x', \u03c6\u27eb\n-- else\n-- 0 , sorry_proof\u27e9\u27e9\n-- I believe in that case the function is indeed linear in \u03c6\n\nopen Classical in\n@[fun_trans]\nnoncomputable\ndef parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) : \ud835\udc9f'(Y,Z) :=\n \u27e8fun \u03c6 => \u2202 (x':=x;dx), f x' \u03c6, sorry_proof\u27e9\n\n\n@[simp, ftrans_simp]\ntheorem action_parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) (\u03c6 : \ud835\udc9f Y) :\n parDistribDeriv f x dx \u03c6 = \u2202 (x':=x;dx), f x' \u03c6 := rfl\n\n\n----------------------------------------------------------------------------------------------------\n-- Const rule --------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.const_rule (T : \ud835\udc9f'(X,Y)) :\n DistribDifferentiable (fun _ : W => T) := by\n intro _ \u03c6 h\u03c6; simp; fun_prop\n\n@[fun_trans]\ntheorem parDistribDeriv.const_rule (T : \ud835\udc9f'(X,Y)) :\n parDistribDeriv (fun _ : W => T)\n =\n fun w dw =>\n 0 := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Pure --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem dirac.arg_xy.DistribDiffrentiable_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n DistribDifferentiable (R:=R) (fun w => dirac (x w)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp [action_dirac, dirac]\n fun_prop\n\n\n@[fun_trans]\ntheorem dirac.arg_x.parDistribDeriv_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n parDistribDeriv (R:=R) (fun w => dirac (x w))\n =\n fun w dw =>\n let xdx := fwdDeriv R x w dw\n diracDeriv xdx.1 xdx.2 := by --= (dpure (R:=R) ydy.1 ydy.2) := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv dirac diracDeriv\n simp [pure, fwdDeriv, DistribDifferentiableAt]\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Composition -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n DistribDifferentiable (fun x => f (g x)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n apply CDifferentiable.comp_rule (K:=R) (f:=fun xy : X\u00d7Y => f xy.2 (\u03c6 xy.1)) (g:=fun x => (x, g x))\n (hg:=by fun_prop)\n intro x\n sorry_proof -- is this even true ?\n\n\n@[fun_trans]\ntheorem parDistribDeriv.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n parDistribDeriv (fun x => f (g x))\n =\n fun x dx =>\n let ydy := fwdDeriv R g x dx\n parDistribDeriv f ydy.1 ydy.2 := by\n\n funext x dx; ext \u03c6\n unfold parDistribDeriv\n simp[hg]\n sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Bind --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n-- The assumptions here are definitely not right!!!\n-- I think `f` has to be `deg`\n@[fun_prop]\ntheorem Bind.bind.arg_fx.DistribDifferentiable_rule\n (f : X \u2192 Y \u2192 \ud835\udc9f'(Z,V)) (g : X \u2192 \ud835\udc9f'(Y,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (x,y) => f x y)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun x => (g x).bind (f x) L) := by\n\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp\n sorry_proof\n\n\n@[fun_trans]\ntheorem Bind.bind.arg_fx.parDistribDiff_rule\n (f : W \u2192 X \u2192 \ud835\udc9f'(Y,V)) (g : W \u2192 \ud835\udc9f'(X,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (w,x) => f w x)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => (g w).bind (f w) L)\n =\n fun w dw =>\n ((parDistribDeriv g w dw).bind (f x \u00b7 ) L)\n +\n ((g w).bind (fun x => parDistribDeriv (f \u00b7 x) w dw) L) := sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Move these around -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).restrict A) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule_simple (A : Set X) :\n IsSmoothLinearMap R (fun (T : \ud835\udc9f'(X,Y)) => T.restrict A) := sorry_proof\n\n\n@[fun_trans]\ntheorem Distribution.restrict.arg_T.parDistribDeriv_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : DistribDifferentiable T) :\n parDistribDeriv (fun w => (T w).restrict A)\n =\n fun w dw =>\n (parDistribDeriv T w dw).restrict A := sorry_proof\n\n@[fun_prop]\ntheorem Function.toDistribution.arg_f.CDifferentiable_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n CDifferentiable R (fun w => (fun x => f w x).toDistribution (R:=R)) := sorry_proof\n\n@[fun_trans]\ntheorem Function.toDistribution.arg_f.cderiv_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n cderiv R (fun w => (fun x => f w x).toDistribution (R:=R))\n =\n fun w dw =>\n (fun x =>\n let dy := cderiv R (f \u00b7 x) w dw\n dy).toDistribution := sorry_proof\n\n@[fun_trans]\ntheorem toDistribution.linear_parDistribDeriv_rule (f : W \u2192 X \u2192 Y) (L : Y \u2192 Z)\n (hL : IsSmoothLinearMap R L) :\n parDistribDeriv (fun w => (fun x => L (f w x)).toDistribution)\n =\n fun w dw =>\n parDistribDeriv (fun w => (fun x => f w x).toDistribution) w dw |>.postComp (fun y \u22b8 L y) := by\n funext w dw\n unfold parDistribDeriv Distribution.postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Integral ----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X] [MeasureSpace Y] [MeasureSpace (X\u00d7Y)]\n\nopen Notation\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule (f : W \u2192 X \u2192 Y) :\n cderiv R (fun w => \u222b' x, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule' (f : W \u2192 X \u2192 R) (A : Set X):\n cderiv R (fun w => \u222b' x in A, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).restrict A |>.extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule (f : W \u2192 X \u2192 Y \u2192 Z) :\n parDistribDeriv (fun w => (fun x => \u222b' y, f w x y).toDistribution (R:=R))\n =\n fun w dw =>\n let Tf := (fun w => (fun x => (fun y => f w x y).toDistribution (R:=R)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := by\n funext w dw\n unfold parDistribDeriv postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule' (f : W \u2192 X \u2192 Y \u2192 Z) (B : X \u2192 Set Y) :\n parDistribDeriv (fun w => (fun x => \u222b' y in B x, f w x y).toDistribution)\n =\n fun w dw =>\n let Tf := (fun w => (fun x => ((fun y => f w x y).toDistribution (R:=R)).restrict (B x)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := sorry_proof\n\n\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Add ---------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n@[fun_prop]\ntheorem HAdd.hAdd.arg_a0a1.DistribDifferentiable_rule (f g : W \u2192 \ud835\udc9f'(X,Y))\n (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun w => f w + g w) := sorry_proof\n\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.parDistribDeriv_rule (f g : W \u2192 \ud835\udc9f'(X,Y))\n (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => f w + g w)\n =\n fun w dw =>\n let dy := parDistribDeriv f w dw\n let dz := parDistribDeriv g w dw\n dy + dz := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Sub ---------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n", "theoremStatement": "@[fun_prop]\ntheorem HSub.hSub.arg_a0a1.DistribDifferentiable_rule (f g : W \u2192 \ud835\udc9f'(X,Y)) :\n -- (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun w => f w - g w) ", "theoremName": "SciLean.HSub.hSub.arg_a0a1.DistribDifferentiable_rule", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "a33d3993", "date": "2024-04-09"}, "file": "scilean/SciLean/Core/Distribution/ParametricDistribDeriv.lean", "module": "SciLean.Core.Distribution.ParametricDistribDeriv", "jsonFile": "SciLean.Core.Distribution.ParametricDistribDeriv.jsonl", "positionMetadata": {"lineInFile": 316, "tokenPositionInFile": 10570, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 16, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Distribution.Basic\nimport SciLean.Core.FunctionTransformations\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.Notation\n\n\nopen MeasureTheory\n\nnamespace SciLean\n\nopen Distribution\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W]\n {X} [Vec R X] [MeasureSpace X]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z] [Module \u211d Z]\n {U} [Vec R U] -- [Module \u211d U]\n {V} [Vec R V] -- [Module \u211d U]\n\n\nset_default_scalar R\n\n\nnoncomputable\ndef diracDeriv (x dx : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 cderiv R \u03c6 x dx\n\n@[fun_prop]\ndef DistribDifferentiableAt (f : X \u2192 \ud835\udc9f'(Y,Z)) (x : X) :=\n \u2200 (\u03c6 : X \u2192 \ud835\udc9f Y), CDifferentiableAt R \u03c6 x \u2192 CDifferentiableAt R (fun x => f x (\u03c6 x)) x\n\n\ntheorem distribDifferentiableAt_const_test_fun\n {f : X \u2192 \ud835\udc9f'(Y,Z)} {x : X}\n (hf : DistribDifferentiableAt f x)\n {\u03c6 : \ud835\udc9f Y} :\n CDifferentiableAt R (fun x => f x \u03c6) x := by\n apply hf\n fun_prop\n\n\n@[fun_prop]\ndef DistribDifferentiable (f : X \u2192 \ud835\udc9f'(Y,Z)) :=\n \u2200 x, DistribDifferentiableAt f x\n\n\n-- TODO:\n-- probably change the definition of `parDistribDeriv` to:\n-- \u27e8\u27e8fun \u03c6 =>\n-- if h : DistribDifferentiableAt f x then\n-- \u2202 (x':=x;dx), \u27eaf x', \u03c6\u27eb\n-- else\n-- 0 , sorry_proof\u27e9\u27e9\n-- I believe in that case the function is indeed linear in \u03c6\n\nopen Classical in\n@[fun_trans]\nnoncomputable\ndef parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) : \ud835\udc9f'(Y,Z) :=\n \u27e8fun \u03c6 => \u2202 (x':=x;dx), f x' \u03c6, sorry_proof\u27e9\n\n\n@[simp, ftrans_simp]\ntheorem action_parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) (\u03c6 : \ud835\udc9f Y) :\n parDistribDeriv f x dx \u03c6 = \u2202 (x':=x;dx), f x' \u03c6 := rfl\n\n\n----------------------------------------------------------------------------------------------------\n-- Const rule --------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.const_rule (T : \ud835\udc9f'(X,Y)) :\n DistribDifferentiable (fun _ : W => T) := by\n intro _ \u03c6 h\u03c6; simp; fun_prop\n\n@[fun_trans]\ntheorem parDistribDeriv.const_rule (T : \ud835\udc9f'(X,Y)) :\n parDistribDeriv (fun _ : W => T)\n =\n fun w dw =>\n 0 := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Pure --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem dirac.arg_xy.DistribDiffrentiable_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n DistribDifferentiable (R:=R) (fun w => dirac (x w)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp [action_dirac, dirac]\n fun_prop\n\n\n@[fun_trans]\ntheorem dirac.arg_x.parDistribDeriv_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n parDistribDeriv (R:=R) (fun w => dirac (x w))\n =\n fun w dw =>\n let xdx := fwdDeriv R x w dw\n diracDeriv xdx.1 xdx.2 := by --= (dpure (R:=R) ydy.1 ydy.2) := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv dirac diracDeriv\n simp [pure, fwdDeriv, DistribDifferentiableAt]\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Composition -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n DistribDifferentiable (fun x => f (g x)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n apply CDifferentiable.comp_rule (K:=R) (f:=fun xy : X\u00d7Y => f xy.2 (\u03c6 xy.1)) (g:=fun x => (x, g x))\n (hg:=by fun_prop)\n intro x\n sorry_proof -- is this even true ?\n\n\n@[fun_trans]\ntheorem parDistribDeriv.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n parDistribDeriv (fun x => f (g x))\n =\n fun x dx =>\n let ydy := fwdDeriv R g x dx\n parDistribDeriv f ydy.1 ydy.2 := by\n\n funext x dx; ext \u03c6\n unfold parDistribDeriv\n simp[hg]\n sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Bind --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n-- The assumptions here are definitely not right!!!\n-- I think `f` has to be `deg`\n@[fun_prop]\ntheorem Bind.bind.arg_fx.DistribDifferentiable_rule\n (f : X \u2192 Y \u2192 \ud835\udc9f'(Z,V)) (g : X \u2192 \ud835\udc9f'(Y,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (x,y) => f x y)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun x => (g x).bind (f x) L) := by\n\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp\n sorry_proof\n\n\n@[fun_trans]\ntheorem Bind.bind.arg_fx.parDistribDiff_rule\n (f : W \u2192 X \u2192 \ud835\udc9f'(Y,V)) (g : W \u2192 \ud835\udc9f'(X,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (w,x) => f w x)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => (g w).bind (f w) L)\n =\n fun w dw =>\n ((parDistribDeriv g w dw).bind (f x \u00b7 ) L)\n +\n ((g w).bind (fun x => parDistribDeriv (f \u00b7 x) w dw) L) := sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Move these around -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).restrict A) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule_simple (A : Set X) :\n IsSmoothLinearMap R (fun (T : \ud835\udc9f'(X,Y)) => T.restrict A) := sorry_proof\n\n\n@[fun_trans]\ntheorem Distribution.restrict.arg_T.parDistribDeriv_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : DistribDifferentiable T) :\n parDistribDeriv (fun w => (T w).restrict A)\n =\n fun w dw =>\n (parDistribDeriv T w dw).restrict A := sorry_proof\n\n@[fun_prop]\ntheorem Function.toDistribution.arg_f.CDifferentiable_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n CDifferentiable R (fun w => (fun x => f w x).toDistribution (R:=R)) := sorry_proof\n\n@[fun_trans]\ntheorem Function.toDistribution.arg_f.cderiv_rule (f : W \u2192 X \u2192 Y)\n (hf : \u2200 x, CDifferentiable R (f \u00b7 x)) :\n cderiv R (fun w => (fun x => f w x).toDistribution (R:=R))\n =\n fun w dw =>\n (fun x =>\n let dy := cderiv R (f \u00b7 x) w dw\n dy).toDistribution := sorry_proof\n\n@[fun_trans]\ntheorem toDistribution.linear_parDistribDeriv_rule (f : W \u2192 X \u2192 Y) (L : Y \u2192 Z)\n (hL : IsSmoothLinearMap R L) :\n parDistribDeriv (fun w => (fun x => L (f w x)).toDistribution)\n =\n fun w dw =>\n parDistribDeriv (fun w => (fun x => f w x).toDistribution) w dw |>.postComp (fun y \u22b8 L y) := by\n funext w dw\n unfold parDistribDeriv Distribution.postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Integral ----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X] [MeasureSpace Y] [MeasureSpace (X\u00d7Y)]\n\nopen Notation\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule (f : W \u2192 X \u2192 Y) :\n cderiv R (fun w => \u222b' x, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.cderiv_distrib_rule' (f : W \u2192 X \u2192 R) (A : Set X):\n cderiv R (fun w => \u222b' x in A, f w x)\n =\n fun w dw =>\n (parDistribDeriv (fun w => (f w \u00b7).toDistribution) w dw).restrict A |>.extAction (fun _ => (1:R)) (fun y \u22b8 fun r \u22b8 r \u2022 y) := sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule (f : W \u2192 X \u2192 Y \u2192 Z) :\n parDistribDeriv (fun w => (fun x => \u222b' y, f w x y).toDistribution (R:=R))\n =\n fun w dw =>\n let Tf := (fun w => (fun x => (fun y => f w x y).toDistribution (R:=R)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := by\n funext w dw\n unfold parDistribDeriv postComp Function.toDistribution\n ext \u03c6\n simp [ftrans_simp] -- , Distribution.mk_extAction_simproc]\n sorry_proof\n\n\n@[fun_trans]\ntheorem cintegral.arg_f.parDistribDeriv_rule' (f : W \u2192 X \u2192 Y \u2192 Z) (B : X \u2192 Set Y) :\n parDistribDeriv (fun w => (fun x => \u222b' y in B x, f w x y).toDistribution)\n =\n fun w dw =>\n let Tf := (fun w => (fun x => ((fun y => f w x y).toDistribution (R:=R)).restrict (B x)).toDistribution (R:=R))\n parDistribDeriv Tf w dw |>.postComp (fun T \u22b8 T.extAction (fun _ => (1:R)) (fun z \u22b8 fun r \u22b8 r \u2022 z)) := sorry_proof\n\n\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Add ---------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n@[fun_prop]\ntheorem HAdd.hAdd.arg_a0a1.DistribDifferentiable_rule (f g : W \u2192 \ud835\udc9f'(X,Y))\n (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun w => f w + g w) := sorry_proof\n\n\n@[fun_trans]\ntheorem HAdd.hAdd.arg_a0a1.parDistribDeriv_rule (f g : W \u2192 \ud835\udc9f'(X,Y))\n (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => f w + g w)\n =\n fun w dw =>\n let dy := parDistribDeriv f w dw\n let dz := parDistribDeriv g w dw\n dy + dz := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Sub ---------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n@[fun_prop]\ntheorem HSub.hSub.arg_a0a1.DistribDifferentiable_rule (f g : W \u2192 \ud835\udc9f'(X,Y)) :\n -- (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun w => f w - g w) := sorry_proof\n\n\n", "theoremStatement": "@[fun_trans]\ntheorem HSub.hSub.arg_a0a1.parDistribDeriv_rule (f g : W \u2192 \ud835\udc9f'(X,Y)) :\n -- (hf : DistribDifferentiable f) (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => f w - g w)\n =\n fun w dw =>\n let dy := parDistribDeriv f w dw\n let dz := parDistribDeriv g w dw\n dy - dz ", "theoremName": "SciLean.HSub.hSub.arg_a0a1.parDistribDeriv_rule", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "a33d3993", "date": "2024-04-09"}, "file": "scilean/SciLean/Core/Distribution/ParametricDistribDeriv.lean", "module": "SciLean.Core.Distribution.ParametricDistribDeriv", "jsonFile": "SciLean.Core.Distribution.ParametricDistribDeriv.jsonl", "positionMetadata": {"lineInFile": 322, "tokenPositionInFile": 10793, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 17, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.Rand.Condition\n\nimport SciLean.Core.Distribution.ParametricDistribDeriv\n\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nnamespace SciLean\n\n\nopen MeasureTheory\nvariable\n {R} [RealScalar R]\n {X Z} [MeasurableSpace X] [MeasurableSpace Z]\n\n/-- Kullback\u2013Leibler divergence of `D\u2096\u2097(P\u2016Q)` -/\nnoncomputable\ndef KLDiv (P Q : Rand X) : R := P.\ud835\udd3c (fun x => Scalar.log (P.pdf R Q.\u2119 x))\n\n/-- Evidence Lower Bound\n\n```\n\ud835\udd3c_Q [log Q(Z) - log P(Z,X)]\n```\nreference: https://en.wikipedia.org/wiki/Variational_Bayesian_methods#Evidence_lower_bound\n -/\nnoncomputable\ndef ELBO {X Z} [MeasureSpace Z] [MeasureSpace X]\n (P : Rand (Z\u00d7X)) (Q : Rand Z) (x : X) : R :=\n - Q.\ud835\udd3c (fun z => Scalar.log (Q.pdf R volume z) - Scalar.log (P.pdf R volume (z,x)))\n\n\n/-- Express `Kullback\u2013Leibler divergence` as log evidence + ELBO -/\ntheorem kldiv_elbo\n {X Z} [MeasureSpace Z] [MeasureSpace X]\n (P : Rand (Z\u00d7X)) (Q : Rand Z) (x : X) :\n KLDiv Q (P.conditionSnd x)\n =\n (Scalar.log (P.snd.pdf R volume x)) - ELBO P Q x := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- KLDiv properties --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable\n {W} [Vec R W]\n [Vec R X]\n\n", "theoremStatement": "@[fun_trans]\ntheorem KLDiv.arg_P.cderiv_rule (P : W \u2192 Rand X) (Q : Rand X) :\n cderiv R (fun w => KLDiv (R:=R) (P w) Q)\n =\n fun w dw =>\n let dP := parDistribDeriv (fun w => (P w).\u2119.toDistribution (R:=R)) w dw\n dP.extAction' (fun x => Scalar.log ((P w).pdf R Q.\u2119 x) - 1) ", "theoremName": "SciLean.KLDiv.arg_P.cderiv_rule", "fileCreated": {"commit": "69a5a1aa", "date": "2024-04-10"}, "theoremCreated": {"commit": "a33d3993", "date": "2024-04-09"}, "file": "scilean/SciLean/Core/Rand/VariationalInference.lean", "module": "SciLean.Core.Rand.VariationalInference", "jsonFile": "SciLean.Core.Rand.VariationalInference.jsonl", "positionMetadata": {"lineInFile": 50, "tokenPositionInFile": 1414, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 17, "numPremises": 34, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.Rand.Condition", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation", "SciLean.Core.Distribution.ParametricDistribDeriv"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FloatAsReal\n-- import SciLean.Modules.Prob.DRand\n-- import SciLean.Modules.Prob.FDRand\n\nnamespace SciLean.Rand\n\nvariable\n {R} [RealScalar R]\n -- {X} [NormedAddCommGroup X] [NormedSpace \u211d X] [NormedSpace R X] [CompleteSpace X] [MeasurableSpace X]\n -- {Y} [NormedAddCommGroup Y] [NormedSpace \u211d Y] [NormedSpace R Y] [CompleteSpace Y] [MeasurableSpace Y]\n -- {Z} [NormedAddCommGroup Z] [NormedSpace \u211d Z] [NormedSpace R Z] [CompleteSpace Z] [MeasurableSpace Z]\n {X : Type _} [MeasurableSpace X] [AddCommGroup X] [Module \u211d X]\n {Y : Type _} [AddCommGroup Y] [Module \u211d Y] [MeasurableSpace Y]\n {Z : Type _} [AddCommGroup Z] [Module \u211d Z]\n\nopen Rand\n\n@[rand_pull_E]\ntheorem bind_pull_mean (x : Rand X) (f : X \u2192 Rand Y) :\n (x >>= (fun x' => pure (f x').mean)).mean\n =\n (x >>= f).mean := by simp[rand_push_E]\n\n@[rand_push_E]\ntheorem ite_push_E {c} [Decidable c] (t e : Rand X) (\u03c6 : X \u2192 Y):\n (if c then t else e).\ud835\udd3c \u03c6 = (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c6) := by\n if h : c then simp[h] else simp[h]\n\n-- I don't think this is a desirable `rand_pull_E` theorem as it duplicates the if statement\n-- @[rand_pull_E]\ntheorem ite_pull_E {c} [Decidable c] (t e : Rand X) (\u03c6 \u03c8 : X \u2192 Y):\n (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c8) = (if c then t else e).\ud835\udd3c (if c then \u03c6 else \u03c8) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_push_E]\ntheorem ite_push_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t else e).mean = (if c then t.mean else e.mean) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E]\ntheorem ite_pull_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t.mean else e.mean) = (if c then t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_t {c} [Decidable c] (t : Rand X) (e : X) :\n (if c then t.mean else e) = (if c then t else pure e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_f {c} [Decidable c] (t : X) (e : Rand X) :\n (if c then t else e.mean) = (if c then pure t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n-- this has messed up universes\n@[rand_pull_E]\ntheorem pull_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n (fun x => r.\ud835\udd3c (fun y => f x y))\n =\n r.\ud835\udd3c (fun y x => f x y) := sorry_proof\n\n-- this has messed up universes\n@[rand_push_E]\ntheorem push_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n r.\ud835\udd3c (fun y x => f x y)\n =\n (fun x => r.\ud835\udd3c (fun y => f x y)) := sorry_proof\n\n-- can't be simp as it has variable head\ntheorem pull_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n (f (r.\ud835\udd3c \u03c6)) = r.\ud835\udd3c (fun x => f (\u03c6 x)) := by sorry_proof -- have := hf; sorry_proof\n\n@[rand_push_E]\ntheorem push_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n r.\ud835\udd3c (fun x => f (\u03c6 x)) = (f (r.\ud835\udd3c \u03c6)) := by rw[pull_E_affine (hf:=hf)]\n\n@[rand_pull_E]\ntheorem pull_mean_add (x y : Rand X) :\n x.mean + y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' + y' := sorry_proof\n\n", "theoremStatement": "@[rand_pull_E]\ntheorem pull_mean_add_1 (x : Rand X) (y : X) :\n x.mean + y\n =\n Rand.mean do\n let x' \u2190 x\n return x' + y ", "theoremName": "SciLean.Rand.pull_mean_add_1", "fileCreated": {"commit": "c09d1780", "date": "2024-02-24"}, "theoremCreated": {"commit": "23206ae3", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Rand/PushPullExpectation.lean", "module": "SciLean.Core.Rand.PushPullExpectation", "jsonFile": "SciLean.Core.Rand.PushPullExpectation.jsonl", "positionMetadata": {"lineInFile": 90, "tokenPositionInFile": 3095, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.Objects.IsomorphicType.RealToFloat", "SciLean.Core.FunctionTransformations.Isomorph.RealToFloat", "SciLean.Core.Objects.IsReal", "SciLean.Core.FloatAsReal"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FloatAsReal\n-- import SciLean.Modules.Prob.DRand\n-- import SciLean.Modules.Prob.FDRand\n\nnamespace SciLean.Rand\n\nvariable\n {R} [RealScalar R]\n -- {X} [NormedAddCommGroup X] [NormedSpace \u211d X] [NormedSpace R X] [CompleteSpace X] [MeasurableSpace X]\n -- {Y} [NormedAddCommGroup Y] [NormedSpace \u211d Y] [NormedSpace R Y] [CompleteSpace Y] [MeasurableSpace Y]\n -- {Z} [NormedAddCommGroup Z] [NormedSpace \u211d Z] [NormedSpace R Z] [CompleteSpace Z] [MeasurableSpace Z]\n {X : Type _} [MeasurableSpace X] [AddCommGroup X] [Module \u211d X]\n {Y : Type _} [AddCommGroup Y] [Module \u211d Y] [MeasurableSpace Y]\n {Z : Type _} [AddCommGroup Z] [Module \u211d Z]\n\nopen Rand\n\n@[rand_pull_E]\ntheorem bind_pull_mean (x : Rand X) (f : X \u2192 Rand Y) :\n (x >>= (fun x' => pure (f x').mean)).mean\n =\n (x >>= f).mean := by simp[rand_push_E]\n\n@[rand_push_E]\ntheorem ite_push_E {c} [Decidable c] (t e : Rand X) (\u03c6 : X \u2192 Y):\n (if c then t else e).\ud835\udd3c \u03c6 = (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c6) := by\n if h : c then simp[h] else simp[h]\n\n-- I don't think this is a desirable `rand_pull_E` theorem as it duplicates the if statement\n-- @[rand_pull_E]\ntheorem ite_pull_E {c} [Decidable c] (t e : Rand X) (\u03c6 \u03c8 : X \u2192 Y):\n (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c8) = (if c then t else e).\ud835\udd3c (if c then \u03c6 else \u03c8) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_push_E]\ntheorem ite_push_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t else e).mean = (if c then t.mean else e.mean) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E]\ntheorem ite_pull_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t.mean else e.mean) = (if c then t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_t {c} [Decidable c] (t : Rand X) (e : X) :\n (if c then t.mean else e) = (if c then t else pure e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_f {c} [Decidable c] (t : X) (e : Rand X) :\n (if c then t else e.mean) = (if c then pure t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n-- this has messed up universes\n@[rand_pull_E]\ntheorem pull_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n (fun x => r.\ud835\udd3c (fun y => f x y))\n =\n r.\ud835\udd3c (fun y x => f x y) := sorry_proof\n\n-- this has messed up universes\n@[rand_push_E]\ntheorem push_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n r.\ud835\udd3c (fun y x => f x y)\n =\n (fun x => r.\ud835\udd3c (fun y => f x y)) := sorry_proof\n\n-- can't be simp as it has variable head\ntheorem pull_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n (f (r.\ud835\udd3c \u03c6)) = r.\ud835\udd3c (fun x => f (\u03c6 x)) := by sorry_proof -- have := hf; sorry_proof\n\n@[rand_push_E]\ntheorem push_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n r.\ud835\udd3c (fun x => f (\u03c6 x)) = (f (r.\ud835\udd3c \u03c6)) := by rw[pull_E_affine (hf:=hf)]\n\n@[rand_pull_E]\ntheorem pull_mean_add (x y : Rand X) :\n x.mean + y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' + y' := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_add_1 (x : Rand X) (y : X) :\n x.mean + y\n =\n Rand.mean do\n let x' \u2190 x\n return x' + y := sorry_proof\n\n", "theoremStatement": "@[rand_pull_E]\ntheorem pull_mean_add_2 (x : X) (y : Rand X) :\n x + y.mean\n =\n Rand.mean do\n let y' \u2190 y\n return x + y' ", "theoremName": "SciLean.Rand.pull_mean_add_2", "fileCreated": {"commit": "c09d1780", "date": "2024-02-24"}, "theoremCreated": {"commit": "23206ae3", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Rand/PushPullExpectation.lean", "module": "SciLean.Core.Rand.PushPullExpectation", "jsonFile": "SciLean.Core.Rand.PushPullExpectation.jsonl", "positionMetadata": {"lineInFile": 98, "tokenPositionInFile": 3248, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.Objects.IsomorphicType.RealToFloat", "SciLean.Core.FunctionTransformations.Isomorph.RealToFloat", "SciLean.Core.Objects.IsReal", "SciLean.Core.FloatAsReal"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Rand.Rand\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FloatAsReal\n-- import SciLean.Modules.Prob.DRand\n-- import SciLean.Modules.Prob.FDRand\n\nnamespace SciLean.Rand\n\nvariable\n {R} [RealScalar R]\n -- {X} [NormedAddCommGroup X] [NormedSpace \u211d X] [NormedSpace R X] [CompleteSpace X] [MeasurableSpace X]\n -- {Y} [NormedAddCommGroup Y] [NormedSpace \u211d Y] [NormedSpace R Y] [CompleteSpace Y] [MeasurableSpace Y]\n -- {Z} [NormedAddCommGroup Z] [NormedSpace \u211d Z] [NormedSpace R Z] [CompleteSpace Z] [MeasurableSpace Z]\n {X : Type _} [MeasurableSpace X] [AddCommGroup X] [Module \u211d X]\n {Y : Type _} [AddCommGroup Y] [Module \u211d Y] [MeasurableSpace Y]\n {Z : Type _} [AddCommGroup Z] [Module \u211d Z]\n\nopen Rand\n\n@[rand_pull_E]\ntheorem bind_pull_mean (x : Rand X) (f : X \u2192 Rand Y) :\n (x >>= (fun x' => pure (f x').mean)).mean\n =\n (x >>= f).mean := by simp[rand_push_E]\n\n@[rand_push_E]\ntheorem ite_push_E {c} [Decidable c] (t e : Rand X) (\u03c6 : X \u2192 Y):\n (if c then t else e).\ud835\udd3c \u03c6 = (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c6) := by\n if h : c then simp[h] else simp[h]\n\n-- I don't think this is a desirable `rand_pull_E` theorem as it duplicates the if statement\n-- @[rand_pull_E]\ntheorem ite_pull_E {c} [Decidable c] (t e : Rand X) (\u03c6 \u03c8 : X \u2192 Y):\n (if c then t.\ud835\udd3c \u03c6 else e.\ud835\udd3c \u03c8) = (if c then t else e).\ud835\udd3c (if c then \u03c6 else \u03c8) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_push_E]\ntheorem ite_push_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t else e).mean = (if c then t.mean else e.mean) := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E]\ntheorem ite_pull_mean {c} [Decidable c] (t e : Rand X) :\n (if c then t.mean else e.mean) = (if c then t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_t {c} [Decidable c] (t : Rand X) (e : X) :\n (if c then t.mean else e) = (if c then t else pure e).mean := by\n if h : c then simp[h] else simp[h]\n\n@[rand_pull_E mid-1]\ntheorem ite_pull_mean_f {c} [Decidable c] (t : X) (e : Rand X) :\n (if c then t else e.mean) = (if c then pure t else e).mean := by\n if h : c then simp[h] else simp[h]\n\n-- this has messed up universes\n@[rand_pull_E]\ntheorem pull_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n (fun x => r.\ud835\udd3c (fun y => f x y))\n =\n r.\ud835\udd3c (fun y x => f x y) := sorry_proof\n\n-- this has messed up universes\n@[rand_push_E]\ntheorem push_E_lambda (r : Rand Y) (f : X \u2192 Y \u2192 Z) :\n r.\ud835\udd3c (fun y x => f x y)\n =\n (fun x => r.\ud835\udd3c (fun y => f x y)) := sorry_proof\n\n-- can't be simp as it has variable head\ntheorem pull_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n (f (r.\ud835\udd3c \u03c6)) = r.\ud835\udd3c (fun x => f (\u03c6 x)) := by sorry_proof -- have := hf; sorry_proof\n\n@[rand_push_E]\ntheorem push_E_affine (r : Rand X) (\u03c6 : X \u2192 Y)\n (f : Y \u2192 Z) (hf : IsAffineMap \u211d f := by fun_prop) :\n r.\ud835\udd3c (fun x => f (\u03c6 x)) = (f (r.\ud835\udd3c \u03c6)) := by rw[pull_E_affine (hf:=hf)]\n\n@[rand_pull_E]\ntheorem pull_mean_add (x y : Rand X) :\n x.mean + y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' + y' := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_add_1 (x : Rand X) (y : X) :\n x.mean + y\n =\n Rand.mean do\n let x' \u2190 x\n return x' + y := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_add_2 (x : X) (y : Rand X) :\n x + y.mean\n =\n Rand.mean do\n let y' \u2190 y\n return x + y' := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_sub (x y : Rand X) :\n x.mean - y.mean\n =\n Rand.mean do\n let x' \u2190 x\n let y' \u2190 y\n return x' - y' := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_smul [Module R X] (r : R) (x : Rand X) :\n r \u2022 x.mean\n =\n Rand.mean do\n let x' \u2190 x\n return r \u2022 x' := sorry_proof\n\n@[rand_pull_E]\ntheorem pull_mean_mul (r : R) (x : Rand R) :\n r * x.mean\n =\n Rand.mean do\n let x' \u2190 x\n return r * x' := sorry_proof\n\n", "theoremStatement": "@[rand_pull_E]\ntheorem pull_mean_div (x : Rand R) (y : R) :\n x.mean / y\n =\n Rand.mean do\n let x' \u2190 x\n return x' / y", "theoremName": "SciLean.Rand.pull_mean_div", "fileCreated": {"commit": "c09d1780", "date": "2024-02-24"}, "theoremCreated": {"commit": "23206ae3", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Rand/PushPullExpectation.lean", "module": "SciLean.Core.Rand.PushPullExpectation", "jsonFile": "SciLean.Core.Rand.PushPullExpectation.jsonl", "positionMetadata": {"lineInFile": 131, "tokenPositionInFile": 3885, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 10, "numPremises": 34, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Erased", "Mathlib.Control.Monad.Basic", "Mathlib.Init.Data.Int.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Control.ULiftable", "Mathlib.Algebra.NeZero", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Control.Random", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Core.Objects.Vec", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Rand.SimpAttr", "Mathlib.MeasureTheory.Measure.GiryMonad", "SciLean.Core.Rand.Rand", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Core.FunctionPropositions.Diffeomorphism", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.Objects.IsomorphicType.RealToFloat", "SciLean.Core.FunctionTransformations.Isomorph.RealToFloat", "SciLean.Core.Objects.IsReal", "SciLean.Core.FloatAsReal"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem smul_restrict (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (r \u2022 T).restrict A = r \u2022 (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem smul_restrict' (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n r \u2022 (T.restrict A) = (r \u2022 T).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem neg_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (- T).restrict A = - (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem neg_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n - (T.restrict A) = (- T).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_push]\ntheorem finset_sum_restrict {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_pull]\ntheorem finset_sum_restrict' {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem indextype_sum_restrict {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem indextype_sum_restrict' {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem iteD_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (ifD A then T else 0) = T.restrict A := by rfl\n\n@[simp,ftrans_simp]\ntheorem iteD_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (ifD A then 0 else T) = T.restrict A\u1d9c := sorry_proof\n\n\n", "theoremStatement": "@[action_push]\ntheorem Distribution.extAction_iteD' (A B : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((iteD A t e).restrict B).extAction \u03c6 L =\n (t.restrict B).extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n (e.restrict B).extAction (fun x => if x \u2209 A then \u03c6 x else 0) L ", "theoremName": "SciLean.Distribution.extAction_iteD'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "283fa584", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 358, "tokenPositionInFile": 13318, "theoremPositionInFile": 46}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= by sorry_proof", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 17}} +{"srcContext": "import SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nset_option linter.unusedVariables false\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Function space --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nstructure SmoothLinearMap (X Y : Type _) [Vec K X] [Vec K Y] where\n toFun : X \u2192 Y\n is_smooth_linear_map : IsSmoothLinearMap K toFun\n\ninstance : FunLike (SmoothLinearMap K X Y) X Y where\n coe f := f.toFun\n coe_injective' := sorry_proof\n\nmacro X:term:25 \" \u22b8[\" K:term \"] \" Y:term:26 : term =>\n `(SmoothLinearMap $K $X $Y)\n\nmacro X:term:25 \" \u22b8 \" Y:term:26 : term =>\n `(SmoothLinearMap defaultScalar% $X $Y)\n\n@[app_unexpander SmoothLinearMap] def unexpandSmoothLinearMap : Lean.PrettyPrinter.Unexpander\n | `($(_) $R $X $Y) => `($X \u22b8[$R] $Y)\n | _ => throw ()\n\n\n@[fun_prop]\ntheorem SmoothLinearMap_apply_right (f : X \u22b8[K] Y) : IsSmoothLinearMap K (fun x => f x) := f.2\n\n\n-- Lambda function notation ----------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk f hf x = f x := by rfl\n\n@[simp]\ntheorem SmoothLinearMap.eta_reduce (f : SmoothLinearMap K X Y) :\n (mk f.1 f.2) = f := by rfl\n\n@[ext]\ntheorem SmoothLinearMap.ext (f g : X \u22b8[K] Y) : (\u2200 x, f x = g x) \u2192 f = g := sorry_proof\n\nvariable (K)\ndef SmoothLinearMap.mk' (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) : X \u22b8[K] Y := \u27e8f,hf\u27e9\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk'_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk' K f hf x = f x := by rfl\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8[\" K:term \"] \" b:term : term =>\n `(SmoothLinearMap.mk' $K (fun $x => $b) (by fun_prop))\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8 \" b:term : term =>\n `(SmoothLinearMap.mk' defaultScalar% (fun $x => $b) (by fun_prop))\n\n@[app_unexpander SmoothLinearMap.mk'] def unexpandSmoothLinearMapMk' : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $R $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8[$R] $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | _ => throw ()\n | _ => throw ()\n\n@[app_unexpander SmoothLinearMap.mk] def unexpandSmoothLinearMapMk : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8 $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | _ => throw ()\n | _ => throw ()\n\n\n-- Algebra ---------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\ninstance : Add (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x + g x\u27e9\ninstance : Sub (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x - g x\u27e9\ninstance : Neg (X \u22b8[K] Y) := \u27e8fun f => fun x \u22b8[K] - f x\u27e9\ninstance : SMul K (X \u22b8[K] Y) := \u27e8fun r f => fun x \u22b8[K] r \u2022 f x\u27e9\ninstance : Zero (X \u22b8[K] Y) := \u27e8fun x \u22b8[K] 0\u27e9\n\nsection AlgebraSimps\n\nvariable (f g : X \u22b8[K] Y) (x : X) (r : K)\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.add_apply : (f + g) x = f x + g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.sub_apply : (f - g) x = f x - g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.neg_apply : (- f) x = - f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.smul_apply : (r \u2022 f) x = r \u2022 f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.zero_apply : (0 : X\u22b8[K]Y) x = 0 := by sorry_proof\n\n", "theoremStatement": "@[simp,ftrans_simp]\ntheorem SmoothLinearMap.apply_zero (f : X \u22b8[K] Y) : f 0 = 0 ", "theoremName": "SciLean.SmoothLinearMap.apply_zero", "fileCreated": {"commit": "77a8f8fc", "date": "2024-02-22"}, "theoremCreated": {"commit": "ebd9c756", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/FunctionSpaces/SmoothLinearMap.lean", "module": "SciLean.Core.FunctionSpaces.SmoothLinearMap", "jsonFile": "SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl", "positionMetadata": {"lineInFile": 123, "tokenPositionInFile": 3976, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 15, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.Prod", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap"]}, "proofMetadata": {"hasProof": false, "proof": ":= by sorry_proof", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 17}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n", "theoremStatement": "@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A ", "theoremName": "SciLean.add_restrict'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 303, "tokenPositionInFile": 11453, "theoremPositionInFile": 33}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n", "theoremStatement": "@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A ", "theoremName": "SciLean.sub_restrict'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 311, "tokenPositionInFile": 11728, "theoremPositionInFile": 35}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem smul_restrict (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (r \u2022 T).restrict A = r \u2022 (T.restrict A) := sorry_proof\n\n", "theoremStatement": "@[restrict_pull]\ntheorem smul_restrict' (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n r \u2022 (T.restrict A) = (r \u2022 T).restrict A ", "theoremName": "SciLean.smul_restrict'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 319, "tokenPositionInFile": 12001, "theoremPositionInFile": 37}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem smul_restrict (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (r \u2022 T).restrict A = r \u2022 (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem smul_restrict' (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n r \u2022 (T.restrict A) = (r \u2022 T).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem neg_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (- T).restrict A = - (T.restrict A) := sorry_proof\n\n", "theoremStatement": "@[restrict_pull]\ntheorem neg_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n - (T.restrict A) = (- T).restrict A ", "theoremName": "SciLean.neg_restrict'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 327, "tokenPositionInFile": 12259, "theoremPositionInFile": 39}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 17, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem smul_restrict (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (r \u2022 T).restrict A = r \u2022 (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem smul_restrict' (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n r \u2022 (T.restrict A) = (r \u2022 T).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem neg_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (- T).restrict A = - (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem neg_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n - (T.restrict A) = (- T).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_push]\ntheorem finset_sum_restrict {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\nopen BigOperators in\n", "theoremStatement": "@[restrict_pull]\ntheorem finset_sum_restrict' {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A ", "theoremName": "SciLean.finset_sum_restrict'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 337, "tokenPositionInFile": 12583, "theoremPositionInFile": 41}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem smul_restrict (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (r \u2022 T).restrict A = r \u2022 (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem smul_restrict' (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n r \u2022 (T.restrict A) = (r \u2022 T).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem neg_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (- T).restrict A = - (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem neg_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n - (T.restrict A) = (- T).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_push]\ntheorem finset_sum_restrict {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_pull]\ntheorem finset_sum_restrict' {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem indextype_sum_restrict {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\n", "theoremStatement": "@[restrict_pull]\ntheorem indextype_sum_restrict' {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A ", "theoremName": "SciLean.indextype_sum_restrict'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 345, "tokenPositionInFile": 12904, "theoremPositionInFile": 43}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 16, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem smul_restrict (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (r \u2022 T).restrict A = r \u2022 (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem smul_restrict' (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n r \u2022 (T.restrict A) = (r \u2022 T).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem neg_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (- T).restrict A = - (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem neg_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n - (T.restrict A) = (- T).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_push]\ntheorem finset_sum_restrict {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_pull]\ntheorem finset_sum_restrict' {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem indextype_sum_restrict {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem indextype_sum_restrict' {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem iteD_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (ifD A then T else 0) = T.restrict A := by rfl\n\n@[simp,ftrans_simp]\ntheorem iteD_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (ifD A then 0 else T) = T.restrict A\u1d9c := sorry_proof\n\n\n@[action_push]\ntheorem Distribution.extAction_iteD' (A B : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((iteD A t e).restrict B).extAction \u03c6 L =\n (t.restrict B).extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n (e.restrict B).extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributiona product --------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable {X\u2081} [Vec R X\u2081] {X\u2082} [Vec R X\u2082] {Y\u2081} [Vec R Y\u2081] {Y\u2082} [Vec R Y\u2082]\n\n-- can we extended to vector valued distributions?\nnoncomputable\ndef Distribution.prod (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f' (X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 Z) : \ud835\udc9f'(X,Z) :=\n \u27e8fun \u03c6 => T.extAction (fun x\u2081 => S x\u2081 \u27e8fun x\u2082 => \u03c6 (p x\u2081 x\u2082), sorry_proof\u27e9) L, sorry_proof\u27e9\n\n@[simp, ftrans_simp]\ntheorem Distribution.prod_restrict (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f'(X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 Z) (A : Set X) :\n (prod p T S L).restrict A = prod p (T.restrict (A.preimage1 p)) (fun x\u2081 => (S x\u2081).restrict (p x\u2081 \u207b\u00b9' A)) L := sorry_proof\n\n@[action_push]\ntheorem Distribution.prod'_extAction (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f'(X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 Z) (K : Z \u22b8 R \u22b8 Z) (\u03c6 : X \u2192 R) :\n (prod p T S L).extAction \u03c6 K\n =\n T.extAction (fun x\u2081 => (S x\u2081).extAction (fun x\u2082 => \u03c6 (p x\u2081 x\u2082)) (fun y\u2082 \u22b8 fun r \u22b8 r \u2022 y\u2082)) (fun y\u2081 \u22b8 fun y\u2082 \u22b8 K (L y\u2081 y\u2082) 1) := sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.prod'_extAction' (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f'(X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 U) (\u03c6 : X \u2192 V) (K : U \u22b8 V \u22b8 W) :\n-- (prod p T S L).extAction \u03c6 K\n-- =\n-- T.extAction (fun x\u2081 => (S x\u2081).extAction (fun x\u2082 => \u03c6 (p x\u2081 x\u2082)) (sorry : Y\u2082 \u22b8 V \u22b8 Y\u2082\u2297V)) (fun y\u2081 \u22b8 fun yv \u22b8 ) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Post Composition --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nnoncomputable\ndef Distribution.postComp (T : \ud835\udc9f'(X,Y)) (f : Y \u22b8 Z) : \ud835\udc9f'(X,Z) := fun \u03c6 \u22b8 f (T \u03c6)\n\n-- @[pp_dot]\n-- noncomputable\n-- abbrev Distribution.postExtAction (T : \ud835\udc9f'(X,\ud835\udc9f'(Y,U))) (\u03c6 : Y \u2192 V) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(X,W) :=\n-- T.postComp (fun u \u22b8 u.extAction \u03c6 L)\n\n@[fun_prop]\ntheorem Distribution.postComp.arg_T.IsSmoothLinarMap_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (f : Y \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).postComp f) := by unfold postComp; sorry_proof\n\n@[simp, ftrans_simp]\ntheorem postComp_id (u : \ud835\udc9f'(X,Y)) :\n (u.postComp (fun y \u22b8 y)) = u := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem postComp_comp (x : \ud835\udc9f'(X,U)) (g : U \u22b8 V) (f : V \u22b8 W) :\n (x.postComp g).postComp f\n =\n x.postComp (fun u \u22b8 f (g u)) := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem postComp_assoc (x : \ud835\udc9f'(X,U)) (y : U \u22b8 \ud835\udc9f'(Y,V)) (f : V \u22b8 W) (\u03c6 : Y \u2192 R) :\n (x.postComp y).postComp (fun T \u22b8 T.postComp f)\n =\n (x.postComp (fun u \u22b8 (y u).postComp f)) := sorry_proof\n\n@[action_push]\ntheorem postComp_extAction (x : \ud835\udc9f'(X,U)) (f : U \u22b8 V) (\u03c6 : X \u2192 W) (L : V \u22b8 W \u22b8 Z) :\n (x.postComp y).extAction \u03c6 L\n =\n (x.extAction \u03c6 (fun u \u22b8 fun w \u22b8 L (f u) w)) := sorry_proof\n\n@[action_push]\ntheorem postComp_restrict_extAction (x : \ud835\udc9f'(X,U)) (f : U \u22b8 V) (A : Set X) (\u03c6 : X \u2192 W) (L : V \u22b8 W \u22b8 Z) :\n ((x.postComp f).restrict A).extAction \u03c6 L\n =\n ((x.restrict A).extAction \u03c6 (fun u \u22b8 fun w \u22b8 (L (f u) w))) := sorry_proof\n\n\n-- @[simp, ftrans_simp, action_push]\n-- theorem Distribution.zero_postExtAction (\u03c6 : Y \u2192 R) : (0 : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))).postExtAction \u03c6 = 0 := by sorry_proof\n\n-- -- todo: this needs some integrability condition\n-- @[action_push]\n-- theorem Distribution.add_postExtAction (T T' : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (T + T').postExtAction \u03c6 = T.postExtAction \u03c6 + T'.postExtAction \u03c6 := by sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.sub_postExtAction (T T' : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (T - T').postExtAction \u03c6 = T.postExtAction \u03c6 - T'.postExtAction \u03c6 := by sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.smul_postExtAction (r : R) (T : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (r \u2022 T).postExtAction \u03c6 = r \u2022 T.postExtAction \u03c6 := by sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.neg_postExtAction (T : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (- T).postExtAction \u03c6 = - T.postExtAction \u03c6 := by sorry_proof\n\n-- open BigOperators in\n-- @[action_push]\n-- theorem Distribution.fintype_sum_postExtAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (\u2211 i, T i).postExtAction \u03c6 = \u2211 i, (T i).postExtAction \u03c6 := by sorry_proof\n\n\n-- @[action_push]\n-- theorem Distribution.ifD_postExtAction (T T' : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (A : Set X) (\u03c6 : Y \u2192 R) :\n-- (ifD A then T else T').postExtAction \u03c6 = ifD A then T.postExtAction \u03c6 else T'.postExtAction \u03c6 := by sorry_proof\n\n\n-- -- @[action_push]\n-- -- theorem Distribution.indextype_sum_postExtAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- -- (\u2211 i, T i).postExtAction \u03c6 = \u2211 i, (T i).postExtAction \u03c6 := by sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Functions as distributions ----------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[coe, fun_trans]\nnoncomputable\ndef _root_.Function.toDistribution (f : X \u2192 Y) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8 \u222b' x, \u03c6 x \u2022 f x\n\ndef Distribution.IsFunction (T : \ud835\udc9f'(X,Y)) : Prop :=\n \u2203 (f : X \u2192 Y), \u2200 (\u03c6 : \ud835\udc9f X),\n T \u03c6 = \u222b' x, \u03c6 x \u2022 f x\n\nnoncomputable\ndef Distribution.toFunction (T : \ud835\udc9f'(X,Y)) : X \u2192 Y :=\n if h : T.IsFunction then\n choose h\n else\n 0\n\n@[action_push]\ntheorem Function.toDistribution_action (f : X \u2192 Y) (\u03c6 : \ud835\udc9f X) :\n f.toDistribution \u03c6 = \u222b' x, \u03c6 x \u2022 f x := by rfl\n\n@[action_push]\ntheorem Function.toDistribution_extAction (f : X \u2192 Y) (\u03c6 : X \u2192 R) :\n f.toDistribution.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n =\n \u222b' x, \u03c6 x \u2022 f x := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Function.toDistribution_zero :\n Function.toDistribution (fun (_ : X) => 0) = (0 : \ud835\udc9f'(X,Y)) := by sorry_proof\n\n\n", "theoremStatement": "@[fun_trans,toDistrib_push]\ntheorem HAdd.hAdd.arg_a0a1.toDistribution_rule (f g : X \u2192 Y) :\n (fun x => f x + g x).toDistribution (R:=R)\n =\n f.toDistribution + g.toDistribution ", "theoremName": "SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 511, "tokenPositionInFile": 19688, "theoremPositionInFile": 63}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 26, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.GiryMonad\nimport Mathlib.MeasureTheory.Decomposition.Lebesgue\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.FunctionSpaces\nimport SciLean.Core.Integral.CIntegral\nimport SciLean.Core.Distribution.TestFunction\nimport SciLean.Core.Distribution.SimpAttr\nimport SciLean.Util.SorryProof\nimport SciLean.Util.Limit\n\nopen MeasureTheory ENNReal Classical\n\nnamespace SciLean\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W] [Module \u211d W]\n {X} [Vec R X] -- [TopologicalSpace X] [space : TCOr (Vec R X) (DiscreteTopology X)]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z]\n {U} [Vec R U]\n {V} [Vec R V]\n\nset_default_scalar R\n\nvariable (R X Y)\nabbrev Distribution := (\ud835\udc9f X) \u22b8[R] Y\nvariable {R X Y}\n\n\nnotation \"\ud835\udc9f'\" X => Distribution defaultScalar% X defaultScalar%\nnotation \"\ud835\udc9f'\" \"(\" X \", \" Y \")\" => Distribution defaultScalar% X Y\n\n@[app_unexpander Distribution] def unexpandDistribution : Lean.PrettyPrinter.Unexpander\n | `($(_) $_ $X $Y) => `(\ud835\udc9f'($X,$Y))\n | _ => throw ()\n\n\n@[ext]\ntheorem Distribution.ext (x y : \ud835\udc9f'(X,Y)) :\n (\u2200 (\u03c6 : \ud835\udc9f X), x \u03c6 = y \u03c6)\n \u2192\n x = y := by\n\n apply SmoothLinearMap.ext\n\n\n----------------------------------------------------------------------------------------------------\n-- Algebra -----------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- instance : Zero (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Add (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Sub (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Neg (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : SMul R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : SMul \u211d (\ud835\udc9f'(X,Y)) := \u27e8fun r f => \u27e8fun \u03c6 => r \u2022 (f \u03c6), sorry_proof\u27e9\u27e9\n\n-- instance : UniformSpace (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\n-- instance : Vec R (\ud835\udc9f'(X,Y)) := by unfold Distribution; infer_instance\ninstance [Module \u211d Y] : Module \u211d (\ud835\udc9f'(X,Y)) := Module.mkSorryProofs\n\n\n----------------------------------------------------------------------------------------------------\n-- Extended action ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nopen BigOperators in\n@[pp_dot]\nnoncomputable\ndef Distribution.extAction (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 Z) (L : Y \u22b8 Z \u22b8 W) : W :=\n if h : \u2203 (z\u2099 : \u2115 \u2192 Z) (\u03c6\u2099 : \u2115 \u2192 \ud835\udc9f X), \u2200 x, \u2211' i, \u03c6\u2099 i x \u2022 z\u2099 i = \u03c6 x then\n let z\u2099 := Classical.choose h\n let \u03c6\u2099 := (Classical.choose_spec h).choose\n \u2211' i, L (T (\u03c6\u2099 i)) (z\u2099 i)\n else\n 0\n\nnamespace Distribution\nscoped notation \"\u27ea\" T \", \" \u03c6 \"\u27eb[\" L \"]\" => Distribution.extAction T \u03c6 L\nend Distribution\n\n\nnoncomputable\nabbrev Distribution.extAction' (T : \ud835\udc9f'(X,Y)) (\u03c6 : X \u2192 R) : Y := T.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nnoncomputable\nabbrev Distribution.integrate (T : \ud835\udc9f'(X,Y)) : Y := T.extAction' (fun _ => 1)\n\n@[fun_prop]\ntheorem TestFunction.apply_IsSmoothLinearMap : IsSmoothLinearMap R fun (\u03c6 : \ud835\udc9f X) => (\u03c6 : X \u2192 R) := sorry_proof\n\ntheorem Distribution.mk_extAction (T : (X \u2192 R) \u2192 Y) (hT : IsSmoothLinearMap R (fun \u03c6 : \ud835\udc9f X => T \u03c6)) (\u03c6 : X \u2192 R) :\n Distribution.extAction (SmoothLinearMap.mk' R (fun (\u03c6 : \ud835\udc9f X) => T \u03c6) hT : Distribution _ _ _) \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y) = T \u03c6 := sorry_proof\n\n\n-- This is definitely not true as stated, what kind of condistions do we need on `\u03c6` and `T`?\n@[fun_prop]\ntheorem Distribution.extAction.arg_\u03c6.IsSmoothLinearMap (T : \ud835\udc9f'(X,U)) (\u03c6 : W \u2192 X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (h\u03c6 : IsSmoothLinearMap R \u03c6) :\n IsSmoothLinearMap R (fun w => T.extAction (\u03c6 w) L) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.extAction.arg_T.IsSmoothLinearMap (T : W \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).extAction \u03c6 L) := sorry_proof\n\n\n-- open Lean Meta in\n-- /-- Simproc to apply `Distribution.mk_extAction` theorem -/\n-- simproc_decl Distribution.mk_extAction_simproc (Distribution.extAction (Distribution.mk (SmoothLinearMap.mk _ _)) _) := fun e => do\n\n-- let \u03c6 := e.appArg!\n-- let T := e.appFn!.appArg!\n\n-- let .lam xName xType xBody xBi := T.appArg!.appFn!.appArg!\n-- | return .continue\n-- let hT := T.appArg!.appArg!\n\n-- withLocalDecl xName xBi xType fun x => do\n-- let R := xType.getArg! 0\n-- let X := xType.getArg! 2\n-- withLocalDecl `\u03c6' xBi (\u2190 mkArrow X R) fun \u03c6' => do\n-- let b := xBody.instantiate1 x\n-- let b := b.replace (fun e' =>\n-- if e'.isAppOf ``DFunLike.coe &&\n-- 5 \u2264 e'.getAppNumArgs &&\n-- e'.getArg! 4 == x then\n-- .some (mkAppN \u03c6' e'.getAppArgs[5:])\n-- else\n-- .none)\n\n-- if b.containsFVar x.fvarId! then\n-- return .continue\n\n-- let T \u2190 mkLambdaFVars #[\u03c6'] b\n-- let prf \u2190 mkAppM ``Distribution.mk_extAction #[T, hT, \u03c6]\n-- return .visit {expr := T.beta #[\u03c6], proof? := prf}\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Monadic structure -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n-- def dirac (x : X) : Distribution X := fun \u03c6 => \u03c6 x\n\n-- instance : Monad (Distribution R) where\n-- pure := fun x => \u27e8fun \u03c6 => \u03c6 x\u27e9\n-- bind := fun x f => \u27e8fun \u03c6 => \u27eax, fun x' => \u27ea(f x'), \u03c6\u27eb\u27eb\u27e9\n\n-- instance : LawfulMonad (Distribution R) where\n-- bind_pure_comp := by intros; rfl\n-- bind_map := by intros; rfl\n-- pure_bind := by intros; rfl\n-- bind_assoc := by intros; rfl\n-- map_const := by intros; rfl\n-- id_map := by intros; rfl\n-- seqLeft_eq := by intros; rfl\n-- seqRight_eq := by intros; rfl\n-- pure_seq := by intros; rfl\n\ndef dirac (x : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 \u03c6 x\n\nopen Notation\nnoncomputable\ndef Distribution.bind (x' : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(Y,W) :=\n fun \u03c6 \u22b8 x'.extAction (fun x => (f x).extAction \u03c6 (fun v \u22b8 fun r \u22b8 r \u2022 v)) L\n\n\n----------------------------------------------------------------------------------------------------\n-- Basic identities --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[simp, ftrans_simp]\ntheorem action_dirac (x : X) (\u03c6 : \ud835\udc9f X) : dirac x \u03c6 = \u03c6 x := by simp[dirac]\n\n@[simp, ftrans_simp]\ntheorem action_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : \ud835\udc9f Y) :\n x.bind f L \u03c6 = x.extAction (fun x' => (f x').extAction' \u03c6) L := by\n simp[Distribution.bind]\n\n\n-- @[simp, ftrans_simp]\n-- theorem extAction_bind (x : \ud835\udc9f'(X,U)) (f : X \u2192 \ud835\udc9f'(Y,V)) (L : U \u22b8 V \u22b8 W) (\u03c6 : Y \u2192 Z) (K : W \u22b8 Z \u22b8 W') :\n-- (x.bind f L).extAction \u03c6 K = x.extAction (fun x' => (f x').extAction \u03c6 (sorry : V \u22b8 Z \u22b8 V\u2297Z)) (sorry : U \u22b8 (V\u2297Z) \u22b8 W') := by\n-- simp [Distribution.bind]\n\n\n----------------------------------------------------------------------------------------------------\n-- Arithmetics -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nsection Arithmetics\n\n@[simp, ftrans_simp, action_push]\ntheorem Distribution.zero_extAction (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) : (0 : \ud835\udc9f'(X,U)).extAction \u03c6 L = 0 := by\n unfold extAction; simp\n\n\n-- todo: this needs some integrability condition\n@[action_push]\ntheorem Distribution.add_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((T + T') : \ud835\udc9f'(X,U)).extAction \u03c6 L = T.extAction \u03c6 L + T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.sub_extAction (T T' : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (T - T').extAction \u03c6 L = T.extAction \u03c6 L - T'.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.smul_extAction (r : R) (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (r \u2022 T).extAction \u03c6 L = r \u2022 T.extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.neg_extAction (T : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (- T).extAction \u03c6 L = - T.extAction \u03c6 L := by sorry_proof\n\nopen BigOperators in\n@[action_push]\ntheorem Distribution.fintype_sum_extAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\n@[action_push]\ntheorem Distribution.indextype_sum_extAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (\u2211 i, T i).extAction \u03c6 L = \u2211 i, (T i).extAction \u03c6 L := by sorry_proof\n\nend Arithmetics\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributional if statement ---------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable [MeasureSpace X]\n\nopen Classical Notation in\nnoncomputable\ndef iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2208 A then 0 else \u03c6 x) (fun y \u22b8 fun r \u22b8 r \u2022 y)\n\nopen Lean.Parser Term in\nsyntax withPosition(\"ifD \" term \" then \"\n ppDedent(ppLine ppSpace ppSpace) term ppDedent(ppLine)\n \"else\"\n ppDedent(ppLine ppSpace ppSpace) term) : term\n\nmacro_rules\n | `(ifD $A then $t else $e) => `(iteD $A $t $e)\n\nopen Lean Parser in\n@[app_unexpander iteD]\ndef unexpandIteD : Lean.PrettyPrinter.Unexpander\n | `($(_) $A $t $e) => `(ifD $A then $t else $e)\n | _ => throw ()\n\n@[action_push]\ntheorem Distribution.action_iteD (A : Set X) (t e : \ud835\udc9f'(X,Y)) (\u03c6 : \ud835\udc9f X) :\n iteD A t e \u03c6 =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) (fun y \u22b8 fun r \u22b8 r \u2022 y) := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Distribution.iteD_same (A : Set X) (u : \ud835\udc9f'(X,Y)) :\n iteD A u u = u := by sorry_proof\n\n@[action_push]\ntheorem Distribution.extAction_iteD (A : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n (iteD A t e).extAction \u03c6 L =\n t.extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n e.extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n@[fun_prop]\ntheorem iteD.arg_te.IsSmoothLinearMap_rule (A : Set X) (t e : W \u2192 \ud835\udc9f'(X,Y))\n (ht : IsSmoothLinearMap R t) (he : IsSmoothLinearMap R e) :\n IsSmoothLinearMap R (fun w => iteD A (t w) (e w)) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Set restriction ---------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[pp_dot]\nnoncomputable\ndef Distribution.restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) : \ud835\udc9f'(X,Y) :=\n ifD A then T else 0\n\n@[simp,ftrans_simp]\ntheorem restrict_univ (T : \ud835\udc9f'(X,Y)) :\n T.restrict Set.univ = T := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem zero_restrict (A : Set X) :\n (0 : \ud835\udc9f'(X,Y)).restrict A = 0 := sorry_proof\n\n@[restrict_push]\ntheorem add_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T + S).restrict A = T.restrict A + S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem add_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A + S.restrict A = (T + S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem sub_restrict (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n (T - S).restrict A = T.restrict A - S.restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem sub_restrict' (T S : \ud835\udc9f'(X,Y)) (A : Set X) :\n T.restrict A - S.restrict A = (T - S).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem smul_restrict (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (r \u2022 T).restrict A = r \u2022 (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem smul_restrict' (r : R) (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n r \u2022 (T.restrict A) = (r \u2022 T).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem neg_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (- T).restrict A = - (T.restrict A) := sorry_proof\n\n@[restrict_pull]\ntheorem neg_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n - (T.restrict A) = (- T).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_push]\ntheorem finset_sum_restrict {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\nopen BigOperators in\n@[restrict_pull]\ntheorem finset_sum_restrict' {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,Y)) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A := sorry_proof\n\n@[restrict_push]\ntheorem indextype_sum_restrict {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n (\u2211 i, T i).restrict A = \u2211 i, (T i).restrict A := sorry_proof\n\n@[restrict_pull]\ntheorem indextype_sum_restrict' {I} [IndexType I] (T : I \u2192 \ud835\udc9f' X) (A : Set X) :\n \u2211 i, (T i).restrict A = (\u2211 i, T i).restrict A := sorry_proof\n\n@[simp,ftrans_simp]\ntheorem iteD_restrict (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (ifD A then T else 0) = T.restrict A := by rfl\n\n@[simp,ftrans_simp]\ntheorem iteD_restrict' (T : \ud835\udc9f'(X,Y)) (A : Set X) :\n (ifD A then 0 else T) = T.restrict A\u1d9c := sorry_proof\n\n\n@[action_push]\ntheorem Distribution.extAction_iteD' (A B : Set X) (t e : \ud835\udc9f'(X,U)) (\u03c6 : X \u2192 V) (L : U \u22b8 V \u22b8 W) :\n ((iteD A t e).restrict B).extAction \u03c6 L =\n (t.restrict B).extAction (fun x => if x \u2208 A then \u03c6 x else 0) L +\n (e.restrict B).extAction (fun x => if x \u2209 A then \u03c6 x else 0) L := by sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Distributiona product --------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nvariable {X\u2081} [Vec R X\u2081] {X\u2082} [Vec R X\u2082] {Y\u2081} [Vec R Y\u2081] {Y\u2082} [Vec R Y\u2082]\n\n-- can we extended to vector valued distributions?\nnoncomputable\ndef Distribution.prod (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f' (X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 Z) : \ud835\udc9f'(X,Z) :=\n \u27e8fun \u03c6 => T.extAction (fun x\u2081 => S x\u2081 \u27e8fun x\u2082 => \u03c6 (p x\u2081 x\u2082), sorry_proof\u27e9) L, sorry_proof\u27e9\n\n@[simp, ftrans_simp]\ntheorem Distribution.prod_restrict (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f'(X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 Z) (A : Set X) :\n (prod p T S L).restrict A = prod p (T.restrict (A.preimage1 p)) (fun x\u2081 => (S x\u2081).restrict (p x\u2081 \u207b\u00b9' A)) L := sorry_proof\n\n@[action_push]\ntheorem Distribution.prod'_extAction (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f'(X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 Z) (K : Z \u22b8 R \u22b8 Z) (\u03c6 : X \u2192 R) :\n (prod p T S L).extAction \u03c6 K\n =\n T.extAction (fun x\u2081 => (S x\u2081).extAction (fun x\u2082 => \u03c6 (p x\u2081 x\u2082)) (fun y\u2082 \u22b8 fun r \u22b8 r \u2022 y\u2082)) (fun y\u2081 \u22b8 fun y\u2082 \u22b8 K (L y\u2081 y\u2082) 1) := sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.prod'_extAction' (p : X\u2081 \u2192 X\u2082 \u2192 X) (T : \ud835\udc9f'(X\u2081,Y\u2081)) (S : X\u2081 \u2192 \ud835\udc9f'(X\u2082,Y\u2082)) (L : Y\u2081 \u22b8 Y\u2082 \u22b8 U) (\u03c6 : X \u2192 V) (K : U \u22b8 V \u22b8 W) :\n-- (prod p T S L).extAction \u03c6 K\n-- =\n-- T.extAction (fun x\u2081 => (S x\u2081).extAction (fun x\u2082 => \u03c6 (p x\u2081 x\u2082)) (sorry : Y\u2082 \u22b8 V \u22b8 Y\u2082\u2297V)) (fun y\u2081 \u22b8 fun yv \u22b8 ) := sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Post Composition --------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\nnoncomputable\ndef Distribution.postComp (T : \ud835\udc9f'(X,Y)) (f : Y \u22b8 Z) : \ud835\udc9f'(X,Z) := fun \u03c6 \u22b8 f (T \u03c6)\n\n-- @[pp_dot]\n-- noncomputable\n-- abbrev Distribution.postExtAction (T : \ud835\udc9f'(X,\ud835\udc9f'(Y,U))) (\u03c6 : Y \u2192 V) (L : U \u22b8 V \u22b8 W) : \ud835\udc9f'(X,W) :=\n-- T.postComp (fun u \u22b8 u.extAction \u03c6 L)\n\n@[fun_prop]\ntheorem Distribution.postComp.arg_T.IsSmoothLinarMap_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (f : Y \u22b8 Z)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).postComp f) := by unfold postComp; sorry_proof\n\n@[simp, ftrans_simp]\ntheorem postComp_id (u : \ud835\udc9f'(X,Y)) :\n (u.postComp (fun y \u22b8 y)) = u := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem postComp_comp (x : \ud835\udc9f'(X,U)) (g : U \u22b8 V) (f : V \u22b8 W) :\n (x.postComp g).postComp f\n =\n x.postComp (fun u \u22b8 f (g u)) := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem postComp_assoc (x : \ud835\udc9f'(X,U)) (y : U \u22b8 \ud835\udc9f'(Y,V)) (f : V \u22b8 W) (\u03c6 : Y \u2192 R) :\n (x.postComp y).postComp (fun T \u22b8 T.postComp f)\n =\n (x.postComp (fun u \u22b8 (y u).postComp f)) := sorry_proof\n\n@[action_push]\ntheorem postComp_extAction (x : \ud835\udc9f'(X,U)) (f : U \u22b8 V) (\u03c6 : X \u2192 W) (L : V \u22b8 W \u22b8 Z) :\n (x.postComp y).extAction \u03c6 L\n =\n (x.extAction \u03c6 (fun u \u22b8 fun w \u22b8 L (f u) w)) := sorry_proof\n\n@[action_push]\ntheorem postComp_restrict_extAction (x : \ud835\udc9f'(X,U)) (f : U \u22b8 V) (A : Set X) (\u03c6 : X \u2192 W) (L : V \u22b8 W \u22b8 Z) :\n ((x.postComp f).restrict A).extAction \u03c6 L\n =\n ((x.restrict A).extAction \u03c6 (fun u \u22b8 fun w \u22b8 (L (f u) w))) := sorry_proof\n\n\n-- @[simp, ftrans_simp, action_push]\n-- theorem Distribution.zero_postExtAction (\u03c6 : Y \u2192 R) : (0 : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))).postExtAction \u03c6 = 0 := by sorry_proof\n\n-- -- todo: this needs some integrability condition\n-- @[action_push]\n-- theorem Distribution.add_postExtAction (T T' : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (T + T').postExtAction \u03c6 = T.postExtAction \u03c6 + T'.postExtAction \u03c6 := by sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.sub_postExtAction (T T' : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (T - T').postExtAction \u03c6 = T.postExtAction \u03c6 - T'.postExtAction \u03c6 := by sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.smul_postExtAction (r : R) (T : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (r \u2022 T).postExtAction \u03c6 = r \u2022 T.postExtAction \u03c6 := by sorry_proof\n\n-- @[action_push]\n-- theorem Distribution.neg_postExtAction (T : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (- T).postExtAction \u03c6 = - T.postExtAction \u03c6 := by sorry_proof\n\n-- open BigOperators in\n-- @[action_push]\n-- theorem Distribution.fintype_sum_postExtAction {I} [Fintype I] (T : I \u2192 \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- (\u2211 i, T i).postExtAction \u03c6 = \u2211 i, (T i).postExtAction \u03c6 := by sorry_proof\n\n\n-- @[action_push]\n-- theorem Distribution.ifD_postExtAction (T T' : \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (A : Set X) (\u03c6 : Y \u2192 R) :\n-- (ifD A then T else T').postExtAction \u03c6 = ifD A then T.postExtAction \u03c6 else T'.postExtAction \u03c6 := by sorry_proof\n\n\n-- -- @[action_push]\n-- -- theorem Distribution.indextype_sum_postExtAction {I} [IndexType I] (T : I \u2192 \ud835\udc9f'(X,\ud835\udc9f'(Y,Z))) (\u03c6 : Y \u2192 R) :\n-- -- (\u2211 i, T i).postExtAction \u03c6 = \u2211 i, (T i).postExtAction \u03c6 := by sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Functions as distributions ----------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[coe, fun_trans]\nnoncomputable\ndef _root_.Function.toDistribution (f : X \u2192 Y) : \ud835\udc9f'(X,Y) :=\n fun \u03c6 \u22b8 \u222b' x, \u03c6 x \u2022 f x\n\ndef Distribution.IsFunction (T : \ud835\udc9f'(X,Y)) : Prop :=\n \u2203 (f : X \u2192 Y), \u2200 (\u03c6 : \ud835\udc9f X),\n T \u03c6 = \u222b' x, \u03c6 x \u2022 f x\n\nnoncomputable\ndef Distribution.toFunction (T : \ud835\udc9f'(X,Y)) : X \u2192 Y :=\n if h : T.IsFunction then\n choose h\n else\n 0\n\n@[action_push]\ntheorem Function.toDistribution_action (f : X \u2192 Y) (\u03c6 : \ud835\udc9f X) :\n f.toDistribution \u03c6 = \u222b' x, \u03c6 x \u2022 f x := by rfl\n\n@[action_push]\ntheorem Function.toDistribution_extAction (f : X \u2192 Y) (\u03c6 : X \u2192 R) :\n f.toDistribution.extAction \u03c6 (fun y \u22b8 fun r \u22b8 r \u2022 y)\n =\n \u222b' x, \u03c6 x \u2022 f x := sorry_proof\n\n@[simp, ftrans_simp]\ntheorem Function.toDistribution_zero :\n Function.toDistribution (fun (_ : X) => 0) = (0 : \ud835\udc9f'(X,Y)) := by sorry_proof\n\n\n@[fun_trans,toDistrib_push]\ntheorem HAdd.hAdd.arg_a0a1.toDistribution_rule (f g : X \u2192 Y) :\n (fun x => f x + g x).toDistribution (R:=R)\n =\n f.toDistribution + g.toDistribution := sorry_proof\n\n", "theoremStatement": "@[toDistrib_pull]\ntheorem HAdd.hAdd.arg_a0a1.toDistribution_rule' (f g : X \u2192 Y) :\n f.toDistribution + g.toDistribution\n =\n (fun x => f x + g x).toDistribution (R:=R) ", "theoremName": "SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule'", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/Basic.lean", "module": "SciLean.Core.Distribution.Basic", "jsonFile": "SciLean.Core.Distribution.Basic.jsonl", "positionMetadata": {"lineInFile": 517, "tokenPositionInFile": 19888, "theoremPositionInFile": 64}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 26, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Distribution.Basic\nimport SciLean.Core.FunctionTransformations\nimport SciLean.Core.FunctionPropositions\nimport SciLean.Core.Notation\n\n\nopen MeasureTheory\n\nnamespace SciLean\n\nopen Distribution\n\nvariable\n {R} [RealScalar R]\n {W} [Vec R W]\n {X} [Vec R X] [MeasureSpace X]\n {Y} [Vec R Y] [Module \u211d Y]\n {Z} [Vec R Z] [Module \u211d Z]\n {U} [Vec R U] -- [Module \u211d U]\n {V} [Vec R V] -- [Module \u211d U]\n\n\nset_default_scalar R\n\n\nnoncomputable\ndef diracDeriv (x dx : X) : \ud835\udc9f' X := fun \u03c6 \u22b8 cderiv R \u03c6 x dx\n\n@[fun_prop]\ndef DistribDifferentiableAt (f : X \u2192 \ud835\udc9f'(Y,Z)) (x : X) :=\n \u2200 (\u03c6 : X \u2192 \ud835\udc9f Y), CDifferentiableAt R \u03c6 x \u2192 CDifferentiableAt R (fun x => f x (\u03c6 x)) x\n\n\ntheorem distribDifferentiableAt_const_test_fun\n {f : X \u2192 \ud835\udc9f'(Y,Z)} {x : X}\n (hf : DistribDifferentiableAt f x)\n {\u03c6 : \ud835\udc9f Y} :\n CDifferentiableAt R (fun x => f x \u03c6) x := by\n apply hf\n fun_prop\n\n\n@[fun_prop]\ndef DistribDifferentiable (f : X \u2192 \ud835\udc9f'(Y,Z)) :=\n \u2200 x, DistribDifferentiableAt f x\n\n\n-- TODO:\n-- probably change the definition of `parDistribDeriv` to:\n-- \u27e8\u27e8fun \u03c6 =>\n-- if h : DistribDifferentiableAt f x then\n-- \u2202 (x':=x;dx), \u27eaf x', \u03c6\u27eb\n-- else\n-- 0 , sorry_proof\u27e9\u27e9\n-- I believe in that case the function is indeed linear in \u03c6\n\nopen Classical in\n@[fun_trans]\nnoncomputable\ndef parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) : \ud835\udc9f'(Y,Z) :=\n \u27e8fun \u03c6 => \u2202 (x':=x;dx), f x' \u03c6, sorry_proof\u27e9\n\n\n@[simp, ftrans_simp]\ntheorem action_parDistribDeriv (f : X \u2192 \ud835\udc9f'(Y,Z)) (x dx : X) (\u03c6 : \ud835\udc9f Y) :\n parDistribDeriv f x dx \u03c6 = \u2202 (x':=x;dx), f x' \u03c6 := rfl\n\n\n----------------------------------------------------------------------------------------------------\n-- Const rule --------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.const_rule (T : \ud835\udc9f'(X,Y)) :\n DistribDifferentiable (fun _ : W => T) := by\n intro _ \u03c6 h\u03c6; simp; fun_prop\n\n@[fun_trans]\ntheorem parDistribDeriv.const_rule (T : \ud835\udc9f'(X,Y)) :\n parDistribDeriv (fun _ : W => T)\n =\n fun w dw =>\n 0 := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Pure --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem dirac.arg_xy.DistribDiffrentiable_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n DistribDifferentiable (R:=R) (fun w => dirac (x w)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp [action_dirac, dirac]\n fun_prop\n\n\n@[fun_trans]\ntheorem dirac.arg_x.parDistribDeriv_rule\n (x : W \u2192 X) (hx : CDifferentiable R x) :\n parDistribDeriv (R:=R) (fun w => dirac (x w))\n =\n fun w dw =>\n let xdx := fwdDeriv R x w dw\n diracDeriv xdx.1 xdx.2 := by --= (dpure (R:=R) ydy.1 ydy.2) := by\n funext w dw; ext \u03c6\n unfold parDistribDeriv dirac diracDeriv\n simp [pure, fwdDeriv, DistribDifferentiableAt]\n fun_trans\n\n\n----------------------------------------------------------------------------------------------------\n-- Composition -------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem DistribDiffrentiable.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n DistribDifferentiable (fun x => f (g x)) := by\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n apply CDifferentiable.comp_rule (K:=R) (f:=fun xy : X\u00d7Y => f xy.2 (\u03c6 xy.1)) (g:=fun x => (x, g x))\n (hg:=by fun_prop)\n intro x\n sorry_proof -- is this even true ?\n\n\n@[fun_trans]\ntheorem parDistribDeriv.comp_rule\n (f : Y \u2192 \ud835\udc9f'(Z,U)) (g : X \u2192 Y)\n (hf : DistribDifferentiable f) (hg : CDifferentiable R g) :\n parDistribDeriv (fun x => f (g x))\n =\n fun x dx =>\n let ydy := fwdDeriv R g x dx\n parDistribDeriv f ydy.1 ydy.2 := by\n\n funext x dx; ext \u03c6\n unfold parDistribDeriv\n simp[hg]\n sorry_proof\n\n\n----------------------------------------------------------------------------------------------------\n-- Bind --------------------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n\n-- The assumptions here are definitely not right!!!\n-- I think `f` has to be `deg`\n@[fun_prop]\ntheorem Bind.bind.arg_fx.DistribDifferentiable_rule\n (f : X \u2192 Y \u2192 \ud835\udc9f'(Z,V)) (g : X \u2192 \ud835\udc9f'(Y,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (x,y) => f x y)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n DistribDifferentiable (fun x => (g x).bind (f x) L) := by\n\n intro x\n unfold DistribDifferentiableAt\n intro \u03c6 h\u03c6\n simp\n sorry_proof\n\n\n@[fun_trans]\ntheorem Bind.bind.arg_fx.parDistribDiff_rule\n (f : W \u2192 X \u2192 \ud835\udc9f'(Y,V)) (g : W \u2192 \ud835\udc9f'(X,U)) (L : U \u22b8 V \u22b8 W)\n (hf : DistribDifferentiable (fun (w,x) => f w x)) -- `f` has to be nice enough to accomodate action of `g`\n (hg : DistribDifferentiable g) :\n parDistribDeriv (fun w => (g w).bind (f w) L)\n =\n fun w dw =>\n ((parDistribDeriv g w dw).bind (f x \u00b7 ) L)\n +\n ((g w).bind (fun x => parDistribDeriv (f \u00b7 x) w dw) L) := sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n-- Move these around -------------------------------------------------------------------------------\n----------------------------------------------------------------------------------------------------\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : IsSmoothLinearMap R T) :\n IsSmoothLinearMap R (fun w => (T w).restrict A) := sorry_proof\n\n@[fun_prop]\ntheorem Distribution.restrict.arg_T.IsSmoothLinearMap_rule_simple (A : Set X) :\n IsSmoothLinearMap R (fun (T : \ud835\udc9f'(X,Y)) => T.restrict A) := sorry_proof\n\n\n", "theoremStatement": "@[fun_trans]\ntheorem Distribution.restrict.arg_T.parDistribDeriv_rule (T : W \u2192 \ud835\udc9f'(X,Y)) (A : Set X)\n (hT : DistribDifferentiable T) :\n parDistribDeriv (fun w => (T w).restrict A)\n =\n fun w dw =>\n (parDistribDeriv T w dw).restrict A ", "theoremName": "SciLean.Distribution.restrict.arg_T.parDistribDeriv_rule", "fileCreated": {"commit": "a95bab92", "date": "2024-03-20"}, "theoremCreated": {"commit": "d43df004", "date": "2024-04-05"}, "file": "scilean/SciLean/Core/Distribution/ParametricDistribDeriv.lean", "module": "SciLean.Core.Distribution.ParametricDistribDeriv", "jsonFile": "SciLean.Core.Distribution.ParametricDistribDeriv.jsonl", "positionMetadata": {"lineInFile": 201, "tokenPositionInFile": 6208, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.TryThis", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "SciLean.Util.SorryProof", "SciLean.Core.FunctionPropositions.Bijective", "SciLean.Core.Objects.Vec", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.Meta.ToAnyPoint", "SciLean.Core.FunctionPropositions.ContCDiff", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab", "SciLean.Core.FunctionTransformations.CDeriv", "SciLean.Core.FunctionPropositions.Diffeomorphism", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Tactic.FunProp.Differentiable", "SciLean.Core.FunctionPropositions.Differentiable", "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "SciLean.Core.FunctionPropositions.HasAdjDiff", "SciLean.Core.FunctionPropositions.IsAffineMap", "SciLean.Core.FunctionPropositions.IsContinuousLinearMap", "SciLean.Core.FunctionPropositions", "SciLean.Core.FunctionSpaces.SmoothLinearMap", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.Analysis.InnerProductSpace.Adjoint", "SciLean.Mathlib.Analysis.InnerProductSpace.Prod", "SciLean.Notation", "SciLean.Core.FunctionTransformations.Adjoint", "SciLean.Core.FunctionTransformations.FDeriv", "SciLean.Core.FunctionTransformations.FwdDeriv", "SciLean.Core.FunctionTransformations.FwdFDeriv", "SciLean.Core.FunctionTransformations.InvFun", "SciLean.Tactic.FTrans.Simp", "SciLean.Core.Objects.IsomorphicType", "SciLean.Core.FunctionTransformations.Isomorph", "SciLean.Core.FunctionTransformations.Preimage", "SciLean.Core.FunctionTransformations.SemiAdjoint", "SciLean.Core.FunctionTransformations.RevCDeriv", "SciLean.Data.StructType.Basic", "SciLean.Data.StructType.Algebra", "SciLean.Core.FunctionTransformations.RevDeriv", "SciLean.Core.FunctionTransformations.RevFDeriv", "SciLean.Core.FunctionTransformations", "SciLean.Lean.Meta.Structure", "SciLean.Tactic.LetNormalize2", "SciLean.Tactic.Autodiff", "SciLean.Core.Notation.CDeriv", "SciLean.Core.FunctionSpaces.ContCDiffMap", "SciLean.Core.FunctionSpaces.ContCDiffMapFD", "SciLean.Core.FunctionSpaces", "SciLean.Core.Integral.CIntegral", "SciLean.Core.Distribution.TestFunction", "SciLean.Core.Distribution.SimpAttr", "SciLean.Util.Limit", "SciLean.Core.Distribution.Basic", "SciLean.Core.Notation.Gradient", "SciLean.Core.Notation.FwdDeriv", "SciLean.Core.Notation.RevCDeriv", "SciLean.Core.Notation"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.Objects.SemiInnerProductSpace\nimport SciLean.Core.FunctionPropositions.IsLinearMap\nimport SciLean.Core.FunctionPropositions.CDifferentiable\nimport SciLean.Core.FunctionPropositions.IsSmoothLinearMap\nimport SciLean.Core.Simp\nimport SciLean.Core.Meta.GenerateLinearMapSimp\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nset_option linter.unusedVariables false\n\nopen LeanColls\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [SemiInnerProductSpace K X]\n {Y : Type _} [SemiInnerProductSpace K Y]\n {Z : Type _} [SemiInnerProductSpace K Z]\n {W : Type _} [SemiInnerProductSpace K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, SemiInnerProductSpace K (E i)]\n\n@[fun_prop]\nstructure HasSemiAdjoint (f : X \u2192 Y) : Prop where\n semiAdjoint_exists : \u2203 f' : Y \u2192 X,\n \u2200 x y, TestFunction x \u2192 \u27eay, f x\u27eb[K] = \u27eaf' y, x\u27eb[K]\n -- at some point I was convinced that these conditions are important\n -- maybe add: \u2200 x, TestFunction x \u2192 TestFunction (f x) - I think this is important for proving linearity of f'\n -- maybe add: \u2200 y, TestFunction y \u2192 TestFunction (f' y)\n -- Right now we have no use for functions that have semiAdjoint and are not differentiable\n -- so we just assume that all are differentiable\n is_differentiable : CDifferentiable K f\n\n/-- Generalization of adjoint of linear map `f : X \u2192 Y`.\n\nIf `f : X \u2192 Y` is linear map between Hilbert spaces then `semiAdjoint K f = adjoint K f`.\n\n`semiAdjoint` is a generalization of adjoint to spaces that are not necessarily complete\nand might have inner product defined only on a dense subset, see `SemiInnerProductSpace`\nfor more information.\n -/\n@[fun_trans]\nnoncomputable\ndef semiAdjoint (f : X \u2192 Y) (y : Y) : X :=\n match Classical.dec (HasSemiAdjoint K f) with\n | isTrue h => Classical.choose h.semiAdjoint_exists y\n | isFalse _ => 0\n\n\n-- Basic identities ------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n/-- `semiAdjoint K f \u00b7` is always linear because either `f` has adjoint and is linear or\n`semiAdjoint K f \u00b7` is zero function and thus linear too. -/\n@[fun_prop]\ntheorem semiAdjoint.arg_y.IsLinearMap_rule (f : X \u2192 Y) :\n IsLinearMap K (fun y => semiAdjoint K f y) := sorry_proof\n\n#generate_linear_map_simps SciLean.semiAdjoint.arg_y.IsLinearMap_rule\n\n@[fun_prop]\ntheorem semiAdjoint.arg_y.CDifferentiable_rule (f : X \u2192 Y) :\n CDifferentiable K (fun y => semiAdjoint K f y) := sorry_proof\n\n", "theoremStatement": "@[fun_prop]\ntheorem semiAdjoint.arg_y.IsSmoothLinearMap_rule (f : X \u2192 Y) :\n IsSmoothLinearMap K (fun y => semiAdjoint K f y) ", "theoremName": "SciLean.semiAdjoint.arg_y.IsSmoothLinearMap_rule", "fileCreated": {"commit": "7346b918", "date": "2023-08-07"}, "theoremCreated": {"commit": "bf02c94e", "date": "2024-04-01"}, "file": "scilean/SciLean/Core/FunctionPropositions/HasSemiAdjoint.lean", "module": "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "jsonFile": "SciLean.Core.FunctionPropositions.HasSemiAdjoint.jsonl", "positionMetadata": {"lineInFile": 68, "tokenPositionInFile": 2561, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 19, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab"]}, "proofMetadata": {"hasProof": true, "proof": ":= by constructor; fun_prop; fun_prop", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 37}} +{"srcContext": "import SciLean.Core.Objects.SemiInnerProductSpace\nimport SciLean.Core.FunctionPropositions.IsLinearMap\nimport SciLean.Core.FunctionPropositions.CDifferentiable\nimport SciLean.Core.FunctionPropositions.IsSmoothLinearMap\nimport SciLean.Core.Simp\nimport SciLean.Core.Meta.GenerateLinearMapSimp\n\nimport Mathlib.Tactic.FunTrans.Attr\nimport Mathlib.Tactic.FunTrans.Elab\n\nset_option linter.unusedVariables false\n\nopen LeanColls\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [SemiInnerProductSpace K X]\n {Y : Type _} [SemiInnerProductSpace K Y]\n {Z : Type _} [SemiInnerProductSpace K Z]\n {W : Type _} [SemiInnerProductSpace K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, SemiInnerProductSpace K (E i)]\n\n@[fun_prop]\nstructure HasSemiAdjoint (f : X \u2192 Y) : Prop where\n semiAdjoint_exists : \u2203 f' : Y \u2192 X,\n \u2200 x y, TestFunction x \u2192 \u27eay, f x\u27eb[K] = \u27eaf' y, x\u27eb[K]\n -- at some point I was convinced that these conditions are important\n -- maybe add: \u2200 x, TestFunction x \u2192 TestFunction (f x) - I think this is important for proving linearity of f'\n -- maybe add: \u2200 y, TestFunction y \u2192 TestFunction (f' y)\n -- Right now we have no use for functions that have semiAdjoint and are not differentiable\n -- so we just assume that all are differentiable\n is_differentiable : CDifferentiable K f\n\n/-- Generalization of adjoint of linear map `f : X \u2192 Y`.\n\nIf `f : X \u2192 Y` is linear map between Hilbert spaces then `semiAdjoint K f = adjoint K f`.\n\n`semiAdjoint` is a generalization of adjoint to spaces that are not necessarily complete\nand might have inner product defined only on a dense subset, see `SemiInnerProductSpace`\nfor more information.\n -/\n@[fun_trans]\nnoncomputable\ndef semiAdjoint (f : X \u2192 Y) (y : Y) : X :=\n match Classical.dec (HasSemiAdjoint K f) with\n | isTrue h => Classical.choose h.semiAdjoint_exists y\n | isFalse _ => 0\n\n\n-- Basic identities ------------------------------------------------------------\n--------------------------------------------------------------------------------\n\n/-- `semiAdjoint K f \u00b7` is always linear because either `f` has adjoint and is linear or\n`semiAdjoint K f \u00b7` is zero function and thus linear too. -/\n@[fun_prop]\ntheorem semiAdjoint.arg_y.IsLinearMap_rule (f : X \u2192 Y) :\n IsLinearMap K (fun y => semiAdjoint K f y) := sorry_proof\n\n#generate_linear_map_simps SciLean.semiAdjoint.arg_y.IsLinearMap_rule\n\n@[fun_prop]\ntheorem semiAdjoint.arg_y.CDifferentiable_rule (f : X \u2192 Y) :\n CDifferentiable K (fun y => semiAdjoint K f y) := sorry_proof\n\n@[fun_prop]\ntheorem semiAdjoint.arg_y.IsSmoothLinearMap_rule (f : X \u2192 Y) :\n IsSmoothLinearMap K (fun y => semiAdjoint K f y) := by constructor; fun_prop; fun_prop\n\n\n-- Do we need joint smoothness in `w` and `x` for `f` ???\n", "theoremStatement": "@[fun_prop]\ntheorem semiAdjoint.arg_f.IsSmoothLinearMap_rule (f : W \u2192 X \u2192 Y)\n (hf\u2081 : \u2200 x, IsSmoothLinearMap K (f \u00b7 x)) (hf\u2082 : \u2200 w, HasSemiAdjoint K (f w \u00b7)) :\n IsSmoothLinearMap K (fun w => semiAdjoint K (f w)) ", "theoremName": "SciLean.semiAdjoint.arg_f.IsSmoothLinearMap_rule", "fileCreated": {"commit": "7346b918", "date": "2023-08-07"}, "theoremCreated": {"commit": "bf02c94e", "date": "2024-04-01"}, "file": "scilean/SciLean/Core/FunctionPropositions/HasSemiAdjoint.lean", "module": "SciLean.Core.FunctionPropositions.HasSemiAdjoint", "jsonFile": "SciLean.Core.FunctionPropositions.HasSemiAdjoint.jsonl", "positionMetadata": {"lineInFile": 74, "tokenPositionInFile": 2787, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap", "Mathlib.Tactic.FunTrans.Decl", "Mathlib.Tactic.FunTrans.Theorems", "Mathlib.Tactic.FunTrans.Attr", "Mathlib.Tactic.FunTrans.Types", "Mathlib.Tactic.FunTrans.Core", "Mathlib.Tactic.FunTrans.Elab"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nset_option linter.unusedVariables false\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Function space --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nstructure SmoothLinearMap (X Y : Type _) [Vec K X] [Vec K Y] where\n toFun : X \u2192 Y\n is_smooth_linear_map : IsSmoothLinearMap K toFun\n\ninstance : FunLike (SmoothLinearMap K X Y) X Y where\n coe f := f.toFun\n coe_injective' := sorry_proof\n\nmacro X:term:25 \" \u22b8[\" K:term \"] \" Y:term:26 : term =>\n `(SmoothLinearMap $K $X $Y)\n\nmacro X:term:25 \" \u22b8 \" Y:term:26 : term =>\n `(SmoothLinearMap defaultScalar% $X $Y)\n\n@[app_unexpander SmoothLinearMap] def unexpandSmoothLinearMap : Lean.PrettyPrinter.Unexpander\n | `($(_) $R $X $Y) => `($X \u22b8[$R] $Y)\n | _ => throw ()\n\n\n@[fun_prop]\ntheorem SmoothLinearMap_apply_right (f : X \u22b8[K] Y) : IsSmoothLinearMap K (fun x => f x) := f.2\n\n\n-- Lambda function notation ----------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk f hf x = f x := by rfl\n\n@[simp]\ntheorem SmoothLinearMap.eta_reduce (f : SmoothLinearMap K X Y) :\n (mk f.1 f.2) = f := by rfl\n\n@[ext]\ntheorem SmoothLinearMap.ext (f g : X \u22b8[K] Y) : (\u2200 x, f x = g x) \u2192 f = g := sorry_proof\n\nvariable (K)\ndef SmoothLinearMap.mk' (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) : X \u22b8[K] Y := \u27e8f,hf\u27e9\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk'_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk' K f hf x = f x := by rfl\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8[\" K:term \"] \" b:term : term =>\n `(SmoothLinearMap.mk' $K (fun $x => $b) (by fun_prop))\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8 \" b:term : term =>\n `(SmoothLinearMap.mk' defaultScalar% (fun $x => $b) (by fun_prop))\n\n@[app_unexpander SmoothLinearMap.mk'] def unexpandSmoothLinearMapMk' : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $R $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8[$R] $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | _ => throw ()\n | _ => throw ()\n\n@[app_unexpander SmoothLinearMap.mk] def unexpandSmoothLinearMapMk : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8 $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | _ => throw ()\n | _ => throw ()\n\n\n-- Algebra ---------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\ninstance : Add (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x + g x\u27e9\ninstance : Sub (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x - g x\u27e9\ninstance : Neg (X \u22b8[K] Y) := \u27e8fun f => fun x \u22b8[K] - f x\u27e9\ninstance : SMul K (X \u22b8[K] Y) := \u27e8fun r f => fun x \u22b8[K] r \u2022 f x\u27e9\ninstance : Zero (X \u22b8[K] Y) := \u27e8fun x \u22b8[K] 0\u27e9\n\nsection AlgebraSimps\n\nvariable (f g : X \u22b8[K] Y) (x : X) (r : K)\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.add_apply : (f + g) x = f x + g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.sub_apply : (f - g) x = f x - g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.neg_apply : (- f) x = - f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.smul_apply : (r \u2022 f) x = r \u2022 f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.zero_apply : (0 : X\u22b8[K]Y) x = 0 := by sorry_proof\n\n@[simp,ftrans_simp]\ntheorem SmoothLinearMap.apply_zero (f : X \u22b8[K] Y) : f 0 = 0 := by sorry_proof\n\nend AlgebraSimps\n\ninstance : UniformSpace (X \u22b8[K] Y) := sorry\ninstance : Vec K (X \u22b8[K] Y) := Vec.mkSorryProofs\n\nopen BigOperators in\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.fintype_sum_apply {I} [Fintype I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x := by sorry_proof\n\n", "theoremStatement": "@[simp, ftrans_simp]\ntheorem SmoothLinearMap.indextype_sum_apply {I} [IndexType I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x ", "theoremName": "SciLean.SmoothLinearMap.indextype_sum_apply", "fileCreated": {"commit": "77a8f8fc", "date": "2024-02-22"}, "theoremCreated": {"commit": "b87d1bf9", "date": "2024-04-01"}, "file": "scilean/SciLean/Core/FunctionSpaces/SmoothLinearMap.lean", "module": "SciLean.Core.FunctionSpaces.SmoothLinearMap", "jsonFile": "SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl", "positionMetadata": {"lineInFile": 136, "tokenPositionInFile": 4365, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.Prod", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap"]}, "proofMetadata": {"hasProof": false, "proof": ":= by sorry_proof", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 17}} +{"srcContext": "import SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nset_option linter.unusedVariables false\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Function space --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nstructure SmoothLinearMap (X Y : Type _) [Vec K X] [Vec K Y] where\n toFun : X \u2192 Y\n is_smooth_linear_map : IsSmoothLinearMap K toFun\n\ninstance : FunLike (SmoothLinearMap K X Y) X Y where\n coe f := f.toFun\n coe_injective' := sorry_proof\n\nmacro X:term:25 \" \u22b8[\" K:term \"] \" Y:term:26 : term =>\n `(SmoothLinearMap $K $X $Y)\n\nmacro X:term:25 \" \u22b8 \" Y:term:26 : term =>\n `(SmoothLinearMap defaultScalar% $X $Y)\n\n@[app_unexpander SmoothLinearMap] def unexpandSmoothLinearMap : Lean.PrettyPrinter.Unexpander\n | `($(_) $R $X $Y) => `($X \u22b8[$R] $Y)\n | _ => throw ()\n\n\n@[fun_prop]\ntheorem SmoothLinearMap_apply_right (f : X \u22b8[K] Y) : IsSmoothLinearMap K (fun x => f x) := f.2\n\n\n-- Lambda function notation ----------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk f hf x = f x := by rfl\n\n@[simp]\ntheorem SmoothLinearMap.eta_reduce (f : SmoothLinearMap K X Y) :\n (mk f.1 f.2) = f := by rfl\n\n@[ext]\ntheorem SmoothLinearMap.ext (f g : X \u22b8[K] Y) : (\u2200 x, f x = g x) \u2192 f = g := sorry_proof\n\nvariable (K)\ndef SmoothLinearMap.mk' (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) : X \u22b8[K] Y := \u27e8f,hf\u27e9\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk'_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk' K f hf x = f x := by rfl\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8[\" K:term \"] \" b:term : term =>\n `(SmoothLinearMap.mk' $K (fun $x => $b) (by fun_prop))\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8 \" b:term : term =>\n `(SmoothLinearMap.mk' defaultScalar% (fun $x => $b) (by fun_prop))\n\n@[app_unexpander SmoothLinearMap.mk'] def unexpandSmoothLinearMapMk' : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $R $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8[$R] $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | _ => throw ()\n | _ => throw ()\n\n@[app_unexpander SmoothLinearMap.mk] def unexpandSmoothLinearMapMk : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8 $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | _ => throw ()\n | _ => throw ()\n\n\n-- Algebra ---------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\ninstance : Add (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x + g x\u27e9\ninstance : Sub (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x - g x\u27e9\ninstance : Neg (X \u22b8[K] Y) := \u27e8fun f => fun x \u22b8[K] - f x\u27e9\ninstance : SMul K (X \u22b8[K] Y) := \u27e8fun r f => fun x \u22b8[K] r \u2022 f x\u27e9\ninstance : Zero (X \u22b8[K] Y) := \u27e8fun x \u22b8[K] 0\u27e9\n\nsection AlgebraSimps\n\nvariable (f g : X \u22b8[K] Y) (x : X) (r : K)\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.add_apply : (f + g) x = f x + g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.sub_apply : (f - g) x = f x - g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.neg_apply : (- f) x = - f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.smul_apply : (r \u2022 f) x = r \u2022 f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.zero_apply : (0 : X\u22b8[K]Y) x = 0 := by sorry_proof\n\n@[simp,ftrans_simp]\ntheorem SmoothLinearMap.apply_zero (f : X \u22b8[K] Y) : f 0 = 0 := by sorry_proof\n\nend AlgebraSimps\n\ninstance : UniformSpace (X \u22b8[K] Y) := sorry\ninstance : Vec K (X \u22b8[K] Y) := Vec.mkSorryProofs\n\nopen BigOperators in\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.fintype_sum_apply {I} [Fintype I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.indextype_sum_apply {I} [IndexType I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x := by sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n\n\n", "theoremStatement": "@[fun_prop]\ntheorem SmoothLinearMap.mk'.arg_f.IsSmoothLinearMap_rule\n (f : W \u2192 X \u2192 Y)\n (hf : CDifferentiable K (fun (w,x) => f w x))\n (hf\u2081 : \u2200 x, IsSmoothLinearMap K (f \u00b7 x)) (hf\u2082 : \u2200 w, IsSmoothLinearMap K (f w \u00b7)) :\n IsSmoothLinearMap K (fun w => (fun x \u22b8[K] f w x)) ", "theoremName": "SciLean.SmoothLinearMap.mk'.arg_f.IsSmoothLinearMap_rule", "fileCreated": {"commit": "77a8f8fc", "date": "2024-02-22"}, "theoremCreated": {"commit": "b87d1bf9", "date": "2024-04-01"}, "file": "scilean/SciLean/Core/FunctionSpaces/SmoothLinearMap.lean", "module": "SciLean.Core.FunctionSpaces.SmoothLinearMap", "jsonFile": "SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl", "positionMetadata": {"lineInFile": 145, "tokenPositionInFile": 4631, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 11, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.Prod", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nset_option linter.unusedVariables false\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Function space --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nstructure SmoothLinearMap (X Y : Type _) [Vec K X] [Vec K Y] where\n toFun : X \u2192 Y\n is_smooth_linear_map : IsSmoothLinearMap K toFun\n\ninstance : FunLike (SmoothLinearMap K X Y) X Y where\n coe f := f.toFun\n coe_injective' := sorry_proof\n\nmacro X:term:25 \" \u22b8[\" K:term \"] \" Y:term:26 : term =>\n `(SmoothLinearMap $K $X $Y)\n\nmacro X:term:25 \" \u22b8 \" Y:term:26 : term =>\n `(SmoothLinearMap defaultScalar% $X $Y)\n\n@[app_unexpander SmoothLinearMap] def unexpandSmoothLinearMap : Lean.PrettyPrinter.Unexpander\n | `($(_) $R $X $Y) => `($X \u22b8[$R] $Y)\n | _ => throw ()\n\n\n@[fun_prop]\ntheorem SmoothLinearMap_apply_right (f : X \u22b8[K] Y) : IsSmoothLinearMap K (fun x => f x) := f.2\n\n\n-- Lambda function notation ----------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk f hf x = f x := by rfl\n\n@[simp]\ntheorem SmoothLinearMap.eta_reduce (f : SmoothLinearMap K X Y) :\n (mk f.1 f.2) = f := by rfl\n\n@[ext]\ntheorem SmoothLinearMap.ext (f g : X \u22b8[K] Y) : (\u2200 x, f x = g x) \u2192 f = g := sorry_proof\n\nvariable (K)\ndef SmoothLinearMap.mk' (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) : X \u22b8[K] Y := \u27e8f,hf\u27e9\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk'_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk' K f hf x = f x := by rfl\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8[\" K:term \"] \" b:term : term =>\n `(SmoothLinearMap.mk' $K (fun $x => $b) (by fun_prop))\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8 \" b:term : term =>\n `(SmoothLinearMap.mk' defaultScalar% (fun $x => $b) (by fun_prop))\n\n@[app_unexpander SmoothLinearMap.mk'] def unexpandSmoothLinearMapMk' : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $R $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8[$R] $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | _ => throw ()\n | _ => throw ()\n\n@[app_unexpander SmoothLinearMap.mk] def unexpandSmoothLinearMapMk : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8 $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | _ => throw ()\n | _ => throw ()\n\n\n-- Algebra ---------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\ninstance : Add (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x + g x\u27e9\ninstance : Sub (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x - g x\u27e9\ninstance : Neg (X \u22b8[K] Y) := \u27e8fun f => fun x \u22b8[K] - f x\u27e9\ninstance : SMul K (X \u22b8[K] Y) := \u27e8fun r f => fun x \u22b8[K] r \u2022 f x\u27e9\ninstance : Zero (X \u22b8[K] Y) := \u27e8fun x \u22b8[K] 0\u27e9\n\nsection AlgebraSimps\n\nvariable (f g : X \u22b8[K] Y) (x : X) (r : K)\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.add_apply : (f + g) x = f x + g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.sub_apply : (f - g) x = f x - g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.neg_apply : (- f) x = - f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.smul_apply : (r \u2022 f) x = r \u2022 f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.zero_apply : (0 : X\u22b8[K]Y) x = 0 := by sorry_proof\n\n@[simp,ftrans_simp]\ntheorem SmoothLinearMap.apply_zero (f : X \u22b8[K] Y) : f 0 = 0 := by sorry_proof\n\nend AlgebraSimps\n\ninstance : UniformSpace (X \u22b8[K] Y) := sorry\ninstance : Vec K (X \u22b8[K] Y) := Vec.mkSorryProofs\n\nopen BigOperators in\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.fintype_sum_apply {I} [Fintype I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.indextype_sum_apply {I} [IndexType I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x := by sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n\n\n@[fun_prop]\ntheorem SmoothLinearMap.mk'.arg_f.IsSmoothLinearMap_rule\n (f : W \u2192 X \u2192 Y)\n (hf : CDifferentiable K (fun (w,x) => f w x))\n (hf\u2081 : \u2200 x, IsSmoothLinearMap K (f \u00b7 x)) (hf\u2082 : \u2200 w, IsSmoothLinearMap K (f w \u00b7)) :\n IsSmoothLinearMap K (fun w => (fun x \u22b8[K] f w x)) := sorry_proof\n\n", "theoremStatement": "@[fun_prop]\ntheorem SmoothLinearMap_apply_left\n (f : W \u2192 X \u22b8[K] Y) (x : X) (hf : IsSmoothLinearMap K f) :\n IsSmoothLinearMap K fun w => (f w) x ", "theoremName": "SciLean.SmoothLinearMap_apply_left", "fileCreated": {"commit": "77a8f8fc", "date": "2024-02-22"}, "theoremCreated": {"commit": "b87d1bf9", "date": "2024-04-01"}, "file": "scilean/SciLean/Core/FunctionSpaces/SmoothLinearMap.lean", "module": "SciLean.Core.FunctionSpaces.SmoothLinearMap", "jsonFile": "SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl", "positionMetadata": {"lineInFile": 152, "tokenPositionInFile": 4928, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.Prod", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} +{"srcContext": "import SciLean.Core.FunctionPropositions.IsSmoothLinearMap\n\nset_option linter.unusedVariables false\n\nnamespace SciLean\n\nvariable\n (K : Type _) [RCLike K]\n {X : Type _} [Vec K X]\n {Y : Type _} [Vec K Y]\n {Z : Type _} [Vec K Z]\n {W : Type _} [Vec K W]\n {\u03b9 : Type _} [IndexType \u03b9] [LawfulIndexType \u03b9] [DecidableEq \u03b9]\n {E : \u03b9 \u2192 Type _} [\u2200 i, Vec K (E i)]\n\n\n-- Function space --------------------------------------------------------------\n--------------------------------------------------------------------------------\n\nstructure SmoothLinearMap (X Y : Type _) [Vec K X] [Vec K Y] where\n toFun : X \u2192 Y\n is_smooth_linear_map : IsSmoothLinearMap K toFun\n\ninstance : FunLike (SmoothLinearMap K X Y) X Y where\n coe f := f.toFun\n coe_injective' := sorry_proof\n\nmacro X:term:25 \" \u22b8[\" K:term \"] \" Y:term:26 : term =>\n `(SmoothLinearMap $K $X $Y)\n\nmacro X:term:25 \" \u22b8 \" Y:term:26 : term =>\n `(SmoothLinearMap defaultScalar% $X $Y)\n\n@[app_unexpander SmoothLinearMap] def unexpandSmoothLinearMap : Lean.PrettyPrinter.Unexpander\n | `($(_) $R $X $Y) => `($X \u22b8[$R] $Y)\n | _ => throw ()\n\n\n@[fun_prop]\ntheorem SmoothLinearMap_apply_right (f : X \u22b8[K] Y) : IsSmoothLinearMap K (fun x => f x) := f.2\n\n\n-- Lambda function notation ----------------------------------------------------\n--------------------------------------------------------------------------------\n\nvariable {K}\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk f hf x = f x := by rfl\n\n@[simp]\ntheorem SmoothLinearMap.eta_reduce (f : SmoothLinearMap K X Y) :\n (mk f.1 f.2) = f := by rfl\n\n@[ext]\ntheorem SmoothLinearMap.ext (f g : X \u22b8[K] Y) : (\u2200 x, f x = g x) \u2192 f = g := sorry_proof\n\nvariable (K)\ndef SmoothLinearMap.mk' (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) : X \u22b8[K] Y := \u27e8f,hf\u27e9\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.mk'_eval (x : X) (f : X \u2192 Y) (hf : IsSmoothLinearMap K f) :\n mk' K f hf x = f x := by rfl\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8[\" K:term \"] \" b:term : term =>\n `(SmoothLinearMap.mk' $K (fun $x => $b) (by fun_prop))\n\nopen Lean Parser Term in\nmacro \"fun \" x:funBinder \" \u22b8 \" b:term : term =>\n `(SmoothLinearMap.mk' defaultScalar% (fun $x => $b) (by fun_prop))\n\n@[app_unexpander SmoothLinearMap.mk'] def unexpandSmoothLinearMapMk' : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $R $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8[$R] $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8[$R] $b)\n | _ => throw ()\n | _ => throw ()\n\n@[app_unexpander SmoothLinearMap.mk] def unexpandSmoothLinearMapMk : Lean.PrettyPrinter.Unexpander\n\n | `($(_) $f:term $_:term) =>\n match f with\n | `(fun $x':ident => $b:term) => `(fun $x' \u22b8 $b)\n | `(fun ($x':ident : $ty) => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | `(fun $x':ident : $ty => $b:term) => `(fun ($x' : $ty) \u22b8 $b)\n | _ => throw ()\n | _ => throw ()\n\n\n-- Algebra ---------------------------------------------------------------------\n--------------------------------------------------------------------------------\n\ninstance : Add (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x + g x\u27e9\ninstance : Sub (X \u22b8[K] Y) := \u27e8fun f g => fun x \u22b8[K] f x - g x\u27e9\ninstance : Neg (X \u22b8[K] Y) := \u27e8fun f => fun x \u22b8[K] - f x\u27e9\ninstance : SMul K (X \u22b8[K] Y) := \u27e8fun r f => fun x \u22b8[K] r \u2022 f x\u27e9\ninstance : Zero (X \u22b8[K] Y) := \u27e8fun x \u22b8[K] 0\u27e9\n\nsection AlgebraSimps\n\nvariable (f g : X \u22b8[K] Y) (x : X) (r : K)\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.add_apply : (f + g) x = f x + g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.sub_apply : (f - g) x = f x - g x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.neg_apply : (- f) x = - f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.smul_apply : (r \u2022 f) x = r \u2022 f x := by rfl\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.zero_apply : (0 : X\u22b8[K]Y) x = 0 := by sorry_proof\n\n@[simp,ftrans_simp]\ntheorem SmoothLinearMap.apply_zero (f : X \u22b8[K] Y) : f 0 = 0 := by sorry_proof\n\nend AlgebraSimps\n\ninstance : UniformSpace (X \u22b8[K] Y) := sorry\ninstance : Vec K (X \u22b8[K] Y) := Vec.mkSorryProofs\n\nopen BigOperators in\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.fintype_sum_apply {I} [Fintype I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x := by sorry_proof\n\n@[simp, ftrans_simp]\ntheorem SmoothLinearMap.indextype_sum_apply {I} [IndexType I] (f : I \u2192 X\u22b8[K] Y) (x : X) :\n (\u2211 i, f i) x = \u2211 i, f i x := by sorry_proof\n\n\n\n----------------------------------------------------------------------------------------------------\n\n\n@[fun_prop]\ntheorem SmoothLinearMap.mk'.arg_f.IsSmoothLinearMap_rule\n (f : W \u2192 X \u2192 Y)\n (hf : CDifferentiable K (fun (w,x) => f w x))\n (hf\u2081 : \u2200 x, IsSmoothLinearMap K (f \u00b7 x)) (hf\u2082 : \u2200 w, IsSmoothLinearMap K (f w \u00b7)) :\n IsSmoothLinearMap K (fun w => (fun x \u22b8[K] f w x)) := sorry_proof\n\n@[fun_prop]\ntheorem SmoothLinearMap_apply_left\n (f : W \u2192 X \u22b8[K] Y) (x : X) (hf : IsSmoothLinearMap K f) :\n IsSmoothLinearMap K fun w => (f w) x := sorry_proof\n\n", "theoremStatement": "@[fun_prop]\ntheorem SmoothLinearMap.mk'.arg_f.CDifferentiable_rule\n (f : W \u2192 X \u22b8[K] Y) (g : W \u2192 X) (hf : CDifferentiable K f) (hg : CDifferentiable K g) :\n CDifferentiable K (fun w => f w (g w)) ", "theoremName": "SciLean.SmoothLinearMap.mk'.arg_f.CDifferentiable_rule", "fileCreated": {"commit": "77a8f8fc", "date": "2024-02-22"}, "theoremCreated": {"commit": "b87d1bf9", "date": "2024-04-01"}, "file": "scilean/SciLean/Core/FunctionSpaces/SmoothLinearMap.lean", "module": "SciLean.Core.FunctionSpaces.SmoothLinearMap", "jsonFile": "SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl", "positionMetadata": {"lineInFile": 157, "tokenPositionInFile": 5094, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Array.Init.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Init.Lemmas", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.ShowUnused", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Subsingleton", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Group.Hom.End", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Ring.Int", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Invertible.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.TryThis", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.Prod", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Algebra.Group.Center", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.Algebra.Group.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.Algebra.Ring.Center", "Mathlib.Algebra.Ring.Centralizer", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "LeanColls.MathlibUpstream", "LeanColls.Classes.Ops", "LeanColls.Classes.Bag", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.ProxyType", "LeanColls.Classes.Ops.Fold", "LeanColls.Data.Transformer.View", "LeanColls.Classes.IndexType.Basic", "LeanColls.Classes.Indexed.Basic", "LeanColls.Classes.Indexed.Notation", "LeanColls.Classes.IndexType.Instances", "LeanColls.Classes.Map", "LeanColls.Classes.Ops.Insert", "Mathlib.Data.List.Lemmas", "LeanColls.Data.List", "LeanColls.Classes.Seq", "LeanColls.Data.Transformer.FixSize", "LeanColls.Data.Transformer.Slice", "Mathlib.Data.Array.Lemmas", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "LeanColls.Data.Range", "LeanColls.Data.Array", "LeanColls.Data.AssocList", "LeanColls.Data.HashMap", "LeanColls.Data.RBMap", "LeanColls.Util.Cached", "LeanColls", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Center", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Centralizer", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Prod", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.RingTheory.Ideal.Colon", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Algebra.Subalgebra.Operations", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "SciLean.Util.SorryProof", "SciLean.Core.Objects.Vec", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "SciLean.Lean.Array", "SciLean.Lean.MergeMapDeclarationExtension", "Mathlib.Lean.Expr", "SciLean.Lean.Expr", "SciLean.Lean.Meta.Basic", "SciLean.Lean.ToSSA", "SciLean.Tactic.StructureDecomposition", "SciLean.Util.RewriteBy", "SciLean.Tactic.LetNormalize", "SciLean.Data.IndexType", "SciLean.Data.Function", "SciLean.Tactic.StructuralInverse", "SciLean.Data.ArraySet", "SciLean.Tactic.AnalyzeConstLambda", "SciLean.Tactic.FTrans.Init", "SciLean.Core.Objects.Scalar", "SciLean.Core.NotationOverField", "SciLean.Core.Objects.SemiInnerProductSpace", "SciLean.Core.Objects.FinVec", "SciLean.Tactic.FProp.Init", "SciLean.Tactic.AnalyzeLambda", "SciLean.Tactic.FProp.Basic", "SciLean.Tactic.FProp.Notation", "SciLean.Core.Simp", "SciLean.Core.Meta.GenerateLinearMapSimp", "SciLean.Core.FunctionPropositions.IsLinearMap", "SciLean.Core.FunctionPropositions.CDifferentiable", "SciLean.Core.FunctionPropositions.IsSmoothLinearMap"]}, "proofMetadata": {"hasProof": false, "proof": ":= sorry_proof", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 14}} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.BrunTitchmarsh.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.BrunTitchmarsh.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d28605790d6b75f2ffb79e52168ff3b8732dd0f0 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.BrunTitchmarsh.jsonl @@ -0,0 +1,35 @@ +{"name":"BrunTitchmarsh.card_range_filter_prime_isBigO","declaration":"theorem BrunTitchmarsh.card_range_filter_prime_isBigO : (fun N => ↑(Finset.filter Nat.Prime (Finset.range N)).card) =O[Filter.atTop] fun N => ↑N / Real.log ↑N"} +{"name":"BrunTitchmarsh.primesBetween_le_siftedSum_add","declaration":"theorem BrunTitchmarsh.primesBetween_le_siftedSum_add (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) (hz : 1 ≤ z) : ↑(BrunTitchmarsh.primesBetween x (x + y)) ≤ Sieve.siftedSum (BrunTitchmarsh.primeInterSieve x y z hz).toSieve + z"} +{"name":"BrunTitchmarsh.primesBetween_le","declaration":"theorem BrunTitchmarsh.primesBetween_le (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 1 < z) : ↑(BrunTitchmarsh.primesBetween x (x + y)) ≤ 2 * y / Real.log z + 6 * z * (1 + Real.log z) ^ 3"} +{"name":"BrunTitchmarsh.Nat.ceil_le_self_add_one","declaration":"theorem BrunTitchmarsh.Nat.ceil_le_self_add_one (x : ℝ) (hx : 0 ≤ x) : ↑⌈x⌉₊ ≤ x + 1"} +{"name":"BrunTitchmarsh.err_isBigO","declaration":"theorem BrunTitchmarsh.err_isBigO : (fun x => x ^ (1 / 2) * (1 + 1 / 2 * Real.log x) ^ 3) =O[Filter.atTop] fun x => x / Real.log x"} +{"name":"BrunTitchmarsh.card_isPrimePow_isBigO","declaration":"theorem BrunTitchmarsh.card_isPrimePow_isBigO : (fun N => ↑(Finset.filter IsPrimePow (Finset.range N)).card) =O[Filter.atTop] fun N => ↑N / Real.log ↑N"} +{"name":"BrunTitchmarsh.pow_half_mul_one_add_log_div_isBigO","declaration":"theorem BrunTitchmarsh.pow_half_mul_one_add_log_div_isBigO : (fun N => N ^ (1 / 2) * (1 + Real.log N / Real.log 2)) =O[Filter.atTop] fun N => N / Real.log N"} +{"name":"BrunTitchmarsh.ceil_approx","declaration":"theorem BrunTitchmarsh.ceil_approx (x : ℝ) (hx : 0 ≤ x) : ∃ C, |C| ≤ 1 ∧ ↑⌈x⌉₊ = x + C"} +{"name":"BrunTitchmarsh.Ioc_filter_dvd_eq","declaration":"theorem BrunTitchmarsh.Ioc_filter_dvd_eq (d : ℕ) (a : ℕ) (b : ℕ) (hd : d ≠ 0) : Finset.filter (fun x => d ∣ x) (Finset.Ioc a b) = Finset.image (fun x => x * d) (Finset.Ioc (a / d) (b / d))"} +{"name":"BrunTitchmarsh.primeInterSieve","declaration":"def BrunTitchmarsh.primeInterSieve (x : ℝ) (y : ℝ) (z : ℝ) (hz : 1 ≤ z) : SelbergSieve"} +{"name":"BrunTitchmarsh.nat_div_approx","declaration":"theorem BrunTitchmarsh.nat_div_approx (a : ℕ) (b : ℕ) : ∃ C, |C| ≤ 1 ∧ ↑(a / b) = ↑a / ↑b + C"} +{"name":"BrunTitchmarsh.card_range_filter_isPrimePow_le","declaration":"theorem BrunTitchmarsh.card_range_filter_isPrimePow_le : ∃ C, ∀ (N : ℕ), ↑(Finset.filter IsPrimePow (Finset.range N)).card ≤ C * (↑N / Real.log ↑N)"} +{"name":"BrunTitchmarsh.prime_or_pow","declaration":"theorem BrunTitchmarsh.prime_or_pow (N : ℕ) (n : ℕ) (hnN : n < N) (hnprime : IsPrimePow n) : Nat.Prime n ∨ ∃ m, ↑m < Real.sqrt ↑N ∧ ∃ k ≤ Nat.log 2 N, n = m ^ k"} +{"name":"BrunTitchmarsh.tmp","declaration":"theorem BrunTitchmarsh.tmp (N : ℕ) : ↑(Finset.filter Nat.Prime (Finset.range N)).card ≤\n 4 * (↑N / Real.log ↑N) + 6 * (↑N ^ (1 / 2) * (1 + 1 / 2 * Real.log ↑N) ^ 3)"} +{"name":"BrunTitchmarsh.boudingSum_ge","declaration":"theorem BrunTitchmarsh.boudingSum_ge (x : ℝ) (y : ℝ) (z : ℝ) (hz : 1 ≤ z) : SelbergSieve.selbergBoundingSum (BrunTitchmarsh.primeInterSieve x y z hz) ≥ Real.log z / 2"} +{"name":"BrunTitchmarsh.pows_small_primes_le","declaration":"theorem BrunTitchmarsh.pows_small_primes_le (N : ℕ) : ↑(Finset.image (fun p => p.1 ^ p.2) (Finset.Ico 1 ⌈Real.sqrt ↑N⌉₊ ×ˢ Finset.range (Nat.log 2 N + 1))).card ≤\n ↑N ^ (1 / 2) * (1 + Real.log ↑N / Real.log 2)"} +{"name":"BrunTitchmarsh.one_add_log_div_log_two_isBigO","declaration":"theorem BrunTitchmarsh.one_add_log_div_log_two_isBigO : (fun N => 1 + Real.log N / Real.log 2) =O[Filter.atTop] fun N => Real.log N"} +{"name":"BrunTitchmarsh.primeSieve_rem_sum_le","declaration":"theorem BrunTitchmarsh.primeSieve_rem_sum_le (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 1 ≤ z) : (Finset.sum (Nat.divisors (BrunTitchmarsh.primeInterSieve x y z hz).prodPrimes) fun d =>\n if ↑d ≤ z then\n 3 ^ ArithmeticFunction.cardDistinctFactors d * |Sieve.rem (BrunTitchmarsh.primeInterSieve x y z hz).toSieve d|\n else 0) ≤\n 5 * z * (1 + Real.log z) ^ 3"} +{"name":"BrunTitchmarsh.rpow_mul_rpow_log_isBigO_id_div_log","declaration":"theorem BrunTitchmarsh.rpow_mul_rpow_log_isBigO_id_div_log (k : ℝ) {r : ℝ} (hr : r < 1) : (fun x => x ^ r * Real.log x ^ k) =O[Filter.atTop] fun x => x / Real.log x"} +{"name":"BrunTitchmarsh.range_filter_isPrimePow_subset_union","declaration":"theorem BrunTitchmarsh.range_filter_isPrimePow_subset_union (N : ℕ) : Finset.filter IsPrimePow (Finset.range N) ⊆\n Finset.filter Nat.Prime (Finset.range N) ∪\n Finset.image (fun p => p.1 ^ p.2) (Finset.Ico 1 ⌈Real.sqrt ↑N⌉₊ ×ˢ Finset.range (Nat.log 2 N + 1))"} +{"name":"BrunTitchmarsh.siftedSum_le","declaration":"theorem BrunTitchmarsh.siftedSum_le (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 1 < z) : Sieve.siftedSum (BrunTitchmarsh.primeInterSieve x y z ⋯).toSieve ≤ 2 * y / Real.log z + 5 * z * (1 + Real.log z) ^ 3"} +{"name":"BrunTitchmarsh.rem_eq","declaration":"theorem BrunTitchmarsh.rem_eq (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) (hz : 1 ≤ z) (d : ℕ) (hd : d ≠ 0) : Sieve.rem (BrunTitchmarsh.primeInterSieve x y z hz).toSieve d = ↑(⌊x + y⌋₊ / d - (⌈x⌉₊ - 1) / d) - (↑d)⁻¹ * y"} +{"name":"BrunTitchmarsh.IsBigO.nat_Top_of_atTop","declaration":"theorem BrunTitchmarsh.IsBigO.nat_Top_of_atTop (f : ℕ → ℝ) (g : ℕ → ℝ) (h : f =O[Filter.atTop] g) (h0 : ∀ (n : ℕ), g n = 0 → f n = 0) : f =O[⊤] g"} +{"name":"BrunTitchmarsh.primesBetween","declaration":"def BrunTitchmarsh.primesBetween (a : ℝ) (b : ℝ) : ℕ"} +{"name":"BrunTitchmarsh.multSum_eq","declaration":"theorem BrunTitchmarsh.multSum_eq (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) (hz : 1 ≤ z) (d : ℕ) (hd : d ≠ 0) : Sieve.multSum (BrunTitchmarsh.primeInterSieve x y z hz).toSieve d = ↑(⌊x + y⌋₊ / d - (⌈x⌉₊ - 1) / d)"} +{"name":"BrunTitchmarsh.floor_approx","declaration":"theorem BrunTitchmarsh.floor_approx (x : ℝ) (hx : 0 ≤ x) : ∃ C, |C| ≤ 1 ∧ ↑⌊x⌋₊ = x + C"} +{"name":"BrunTitchmarsh.primesBetween_subset","declaration":"theorem BrunTitchmarsh.primesBetween_subset (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) : Finset.filter Nat.Prime (Finset.Icc ⌈x⌉₊ ⌊x + y⌋₊) ⊆\n Finset.filter (fun d => ∀ (p : ℕ), Nat.Prime p → ↑p ≤ z → ¬p ∣ d) (Finset.Icc ⌈x⌉₊ ⌊x + y⌋₊) ∪ Finset.Icc 1 ⌊z⌋₊"} +{"name":"BrunTitchmarsh.abs_rem_le","declaration":"theorem BrunTitchmarsh.abs_rem_le (x : ℝ) (y : ℝ) (z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 1 ≤ z) {d : ℕ} (hd : d ≠ 0) : |Sieve.rem (BrunTitchmarsh.primeInterSieve x y z hz).toSieve d| ≤ 5"} +{"name":"BrunTitchmarsh.floor_div_approx","declaration":"theorem BrunTitchmarsh.floor_div_approx (x : ℝ) (hx : 0 ≤ x) (d : ℕ) : ∃ C, |C| ≤ 2 ∧ ↑(⌊x⌋₊ / d) = x / ↑d + C"} +{"name":"BrunTitchmarsh.primesBetween_mono_right","declaration":"theorem BrunTitchmarsh.primesBetween_mono_right (a : ℝ) (b : ℝ) (c : ℝ) (hbc : b ≤ c) : BrunTitchmarsh.primesBetween a b ≤ BrunTitchmarsh.primesBetween a c"} +{"name":"BrunTitchmarsh.siftedSum_eq_card","declaration":"theorem BrunTitchmarsh.siftedSum_eq_card (x : ℝ) (y : ℝ) (z : ℝ) (hz : 1 ≤ z) : Sieve.siftedSum (BrunTitchmarsh.primeInterSieve x y z hz).toSieve =\n ↑(Finset.filter (fun d => ∀ (p : ℕ), Nat.Prime p → ↑p ≤ z → ¬p ∣ d) (Finset.Icc ⌈x⌉₊ ⌊x + y⌋₊)).card"} +{"name":"BrunTitchmarsh.primesBetween_one","declaration":"theorem BrunTitchmarsh.primesBetween_one (n : ℕ) : BrunTitchmarsh.primesBetween 1 ↑n = (Finset.filter Nat.Prime (Finset.range (n + 1))).card"} +{"name":"BrunTitchmarsh.card_Ioc_filter_dvd","declaration":"theorem BrunTitchmarsh.card_Ioc_filter_dvd (d : ℕ) (a : ℕ) (b : ℕ) (hd : d ≠ 0) : (Finset.filter (fun x => d ∣ x) (Finset.Ioc a b)).card = b / d - a / d"} +{"name":"BrunTitchmarsh.Nat.log_eq_floor_logb","declaration":"theorem BrunTitchmarsh.Nat.log_eq_floor_logb (b : ℕ) (n : ℕ) (hb : 1 < b) : Nat.log b n = ⌊Real.logb ↑b ↑n⌋₊"} +{"name":"BrunTitchmarsh.card_pows_aux","declaration":"theorem BrunTitchmarsh.card_pows_aux : (fun N =>\n ↑(Finset.image (fun p => p.1 ^ p.2)\n (Finset.Ico 1 ⌈Real.sqrt ↑N⌉₊ ×ˢ Finset.range (Nat.log 2 N + 1))).card) =O[Filter.atTop]\n fun N => ↑N / Real.log ↑N"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Consequences.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Consequences.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cb1f0624d5acc594c673a0566578a9858a8e46ed --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Consequences.jsonl @@ -0,0 +1,15 @@ +{"name":"primorial_bounds","declaration":"theorem primorial_bounds : ∃ E,\n (E =o[Filter.atTop] fun x => x) ∧\n ∀ (x : ℝ), ↑(Finset.prod (Finset.filter Nat.Prime (Finset.range ⌊x⌋₊)) fun p => p) = Real.exp (x + E x)"} +{"name":"mu_pnt","declaration":"theorem mu_pnt : (fun x => Finset.sum (Finset.range ⌊x⌋₊) fun n => ArithmeticFunction.moebius n) =o[Filter.atTop] fun x => x"} +{"name":"prime_between","declaration":"theorem prime_between {ε : ℝ} (hε : 0 < ε) : ∀ᶠ (x : ℝ) in Filter.atTop, ∃ p, Nat.Prime p ∧ x < ↑p ∧ ↑p < (1 + ε) * x"} +{"name":"pn_pn_plus_one","declaration":"theorem pn_pn_plus_one : ∃ c,\n (c =o[Filter.atTop] fun x => 1) ∧\n ∀ (n : ℕ), ↑(Nat.nth Nat.Prime (n + 1)) - ↑(Nat.nth Nat.Prime n) = c n * ↑(Nat.nth Nat.Prime n)"} +{"name":"pn_asymptotic","declaration":"theorem pn_asymptotic : ∃ c, (c =o[Filter.atTop] fun x => 1) ∧ ∀ (n : ℕ), ↑(Nat.nth Nat.Prime n) = (1 + c n) * ↑n * Real.log ↑n"} +{"name":"lambda_pnt","declaration":"theorem lambda_pnt : (fun x => Finset.sum (Finset.range ⌊x⌋₊) fun n => (-1) ^ ArithmeticFunction.cardFactors n) =o[Filter.atTop] fun x => x"} +{"name":"pi_asymp","declaration":"theorem pi_asymp : ∃ c,\n (c =o[Filter.atTop] fun x => 1) ∧\n ∀ (x : ℝ), ↑(Nat.primeCounting ⌊x⌋₊) = (1 + c x) * ∫ (t : ℝ) in Set.Icc 2 x, 1 / Real.log t"} +{"name":"mu_pnt_alt","declaration":"theorem mu_pnt_alt : (fun x => Finset.sum (Finset.range ⌊x⌋₊) fun n => ↑(ArithmeticFunction.moebius n) / ↑n) =o[Filter.atTop] fun x => 1"} +{"name":"finsum_range_eq_sum_range'","declaration":"theorem finsum_range_eq_sum_range' {R : Type u_1} [AddCommMonoid R] {f : ArithmeticFunction R} (x : ℝ) : (finsum fun n => finsum fun x => f n) = Finset.sum (Finset.Iic ⌊x⌋₊) fun n => f n"} +{"name":"primorial_bounds_finprod","declaration":"theorem primorial_bounds_finprod : ∃ E,\n (E =o[Filter.atTop] fun x => x) ∧\n ∀ (x : ℝ), ↑(finprod fun p => finprod fun x => finprod fun x => p) = Real.exp (x + E x)"} +{"name":"chebyshev_asymptotic_finsum","declaration":"theorem chebyshev_asymptotic_finsum : Asymptotics.IsEquivalent Filter.atTop (fun x => finsum fun p => finsum fun x => finsum fun x => Real.log ↑p) fun x => ↑x"} +{"name":"sum_mobius_div_self_le","declaration":"theorem sum_mobius_div_self_le (N : ℕ) : |Finset.sum (Finset.range N) fun n => ↑(ArithmeticFunction.moebius n) / ↑n| ≤ 1"} +{"name":"finsum_range_eq_sum_range","declaration":"theorem finsum_range_eq_sum_range {R : Type u_1} [AddCommMonoid R] {f : ArithmeticFunction R} (x : ℝ) : (finsum fun n => finsum fun x => f n) = Finset.sum (Finset.range ⌈x⌉₊) fun n => f n"} +{"name":"chebyshev_asymptotic","declaration":"theorem chebyshev_asymptotic : Asymptotics.IsEquivalent Filter.atTop\n (fun x => Finset.sum (Finset.filter Nat.Prime (Finset.range ⌈x⌉₊)) fun p => Real.log ↑p) fun x => x"} +{"name":"pi_alt","declaration":"theorem pi_alt : ∃ c, (c =o[Filter.atTop] fun x => 1) ∧ ∀ (x : ℝ), ↑(Nat.primeCounting ⌊x⌋₊) = (1 + c x) * x / Real.log x"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Fourier.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Fourier.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ce4a4c7b1435de07ff4663a13cceb1629f8f3a27 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Fourier.jsonl @@ -0,0 +1,26 @@ +{"name":"FS","declaration":"def FS (f : SchwartzMap ℝ ℂ) : SchwartzMap ℝ ℂ"} +{"name":"FS4","declaration":"theorem FS4 (f : SchwartzMap ℝ ℂ) : FS^[4] f = f"} +{"name":"fourierIntegral_deriv_aux2","declaration":"theorem fourierIntegral_deriv_aux2 (e : BoundedContinuousFunction ℝ ℂ) {f : ℝ → ℂ} (hf : MeasureTheory.Integrable f MeasureTheory.volume) : MeasureTheory.Integrable (⇑e * f) MeasureTheory.volume"} +{"name":"fourierIntegral_decay_aux","declaration":"theorem fourierIntegral_decay_aux (f : ℝ → ℂ) (k : ℕ) (h1 : ContDiff ℝ (↑k) f) (h2 : ∀ n ≤ k, MeasureTheory.Integrable (iteratedDeriv n f) MeasureTheory.volume) (x : ℝ) : ‖(2 * ↑Real.pi * Complex.I * ↑x) ^ k • Real.fourierIntegral f x‖ ≤ ∫ (y : ℝ), ‖iteratedDeriv k f y‖"} +{"name":"FS_toFun","declaration":"theorem FS_toFun (f : SchwartzMap ℝ ℂ) : ⇑(FS f) = Real.fourierIntegral ⇑f"} +{"name":"deriv_ofReal","declaration":"theorem deriv_ofReal : deriv Complex.ofReal' = fun x => 1"} +{"name":"nnnorm_eq_of_mem_circle","declaration":"theorem nnnorm_eq_of_mem_circle (z : ↥circle) : ‖↑z‖₊ = 1"} +{"name":"F_sub","declaration":"theorem F_sub {f : ℝ → ℂ} {g : ℝ → ℂ} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (hg : MeasureTheory.Integrable g MeasureTheory.volume) (x : ℝ) : Real.fourierIntegral (fun x => f x - g x) x = Real.fourierIntegral f x - Real.fourierIntegral g x"} +{"name":"F_neg","declaration":"theorem F_neg {f : ℝ → ℂ} {u : ℝ} : Real.fourierIntegral (fun x => -f x) u = -Real.fourierIntegral f u"} +{"name":"fourierIntegral_self_add_deriv_deriv","declaration":"theorem fourierIntegral_self_add_deriv_deriv (f : W21) (u : ℝ) : (1 + ↑u ^ 2) * Real.fourierIntegral f.toFun u =\n Real.fourierIntegral (fun u => f.toFun u - 1 / (4 * ↑Real.pi ^ 2) * deriv^[2] f.toFun u) u"} +{"name":"instCoeForAllRealForAllComplex","declaration":"def instCoeForAllRealForAllComplex {E : Type u_1} : Coe (E → ℝ) (E → ℂ)"} +{"name":"fourierIntegral_decay","declaration":"theorem fourierIntegral_decay (f : SchwartzMap ℝ ℂ) (k : ℕ) : ∃ C, ∀ (x : ℝ), ‖x‖ ^ k * ‖Real.fourierIntegral (⇑f) x‖ ≤ C"} +{"name":"nnnorm_circle_smul","declaration":"theorem nnnorm_circle_smul (z : ↥circle) (s : ℂ) : ‖z • s‖₊ = ‖s‖₊"} +{"name":"iteratedDeriv_schwartz","declaration":"theorem iteratedDeriv_schwartz (f : SchwartzMap ℝ ℂ) (n : ℕ) : iteratedDeriv n ⇑f = ⇑((⇑(SchwartzMap.derivCLM ℝ))^[n] f)"} +{"name":"F_add","declaration":"theorem F_add {f : ℝ → ℂ} {g : ℝ → ℂ} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (hg : MeasureTheory.Integrable g MeasureTheory.volume) (x : ℝ) : Real.fourierIntegral (fun x => f x + g x) x = Real.fourierIntegral f x + Real.fourierIntegral g x"} +{"name":"F_mul","declaration":"theorem F_mul {f : ℝ → ℂ} {c : ℂ} {u : ℝ} : Real.fourierIntegral (fun x => c * f x) u = c * Real.fourierIntegral f u"} +{"name":"MS_iterate","declaration":"theorem MS_iterate (a : ℂ) (f : SchwartzMap ℝ ℂ) (n : ℕ) : ⇑((MS a)^[n] f) = fun x => (a * ↑x) ^ n • f x"} +{"name":"MS_apply","declaration":"theorem MS_apply (a : ℂ) (f : SchwartzMap ℝ ℂ) (x : ℝ) : (MS a f) x = (a * ↑x) • f x"} +{"name":"schwarz_reduce","declaration":"theorem schwarz_reduce (f : ℝ → ℂ) (h1 : ContDiff ℝ ⊤ f) (h2 : ∀ (k n : ℕ), ∃ C, ∀ (x : ℝ), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ C) (x : ℝ) : { toFun := f, smooth' := h1, decay' := h2 } x = f x"} +{"name":"e_apply","declaration":"theorem e_apply (u : ℝ) (v : ℝ) : (e u) v = ↑(Real.fourierChar (-v * u))"} +{"name":"bla","declaration":"theorem bla (a : ℂ) (f : ℝ → ℂ) (n : ℕ) (hf : ContDiff ℝ (↑n) f) : (iteratedDeriv n fun x => a * ↑x * f x) = fun x => a * ↑x * iteratedDeriv n f x + ↑n * a * iteratedDeriv (n - 1) f x"} +{"name":"hasDerivAt_e","declaration":"theorem hasDerivAt_e {u : ℝ} {x : ℝ} : HasDerivAt (⇑(e u)) (-2 * ↑Real.pi * ↑u * Complex.I * (e u) x) x"} +{"name":"MS","declaration":"def MS (a : ℂ) (f : SchwartzMap ℝ ℂ) : SchwartzMap ℝ ℂ"} +{"name":"FS_apply","declaration":"theorem FS_apply (f : SchwartzMap ℝ ℂ) (x : ℝ) : (FS f) x = Real.fourierIntegral (⇑f) x"} +{"name":"fourierfourier","declaration":"theorem fourierfourier {f : ℝ → ℂ} (hfi : MeasureTheory.Integrable f MeasureTheory.volume) (hfi' : MeasureTheory.Integrable (Real.fourierIntegral f) MeasureTheory.volume) (hfc : Continuous f) (x : ℝ) : Real.fourierIntegral (Real.fourierIntegral f) x = f (-x)"} +{"name":"e","declaration":"def e (u : ℝ) : BoundedContinuousFunction ℝ ℂ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.HadamardFactorization.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.HadamardFactorization.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/pnt-declarations/PrimeNumberTheoremAnd.HoffsteinLockhart.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.HoffsteinLockhart.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..5ee3ccac18f8045040b068d41c7dd311a9d44b05 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics.jsonl @@ -0,0 +1,5 @@ +{"name":"Asymptotics.isLittleO_const_id_atTop2","declaration":"theorem Asymptotics.isLittleO_const_id_atTop2 {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] [LinearOrder F''] [NoMaxOrder F''] [ClosedIciTopology F''] [ProperSpace F''] (c : E'') : (fun _x => c) =o[Filter.atTop] id"} +{"name":"Filter.Eventually.natCast","declaration":"theorem Filter.Eventually.natCast {f : ℝ → Prop} (hf : ∀ᶠ (x : ℝ) in Filter.atTop, f x) : ∀ᶠ (n : ℕ) in Filter.atTop, f ↑n"} +{"name":"Asymptotics.isLittleO_const_id_atBot2","declaration":"theorem Asymptotics.isLittleO_const_id_atBot2 {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] [LinearOrder F''] [NoMinOrder F''] [ClosedIicTopology F''] [ProperSpace F''] (c : E'') : (fun _x => c) =o[Filter.atBot] id"} +{"name":"Asymptotics.isLittleO_const_id_cocompact","declaration":"theorem Asymptotics.isLittleO_const_id_cocompact {E'' : Type u_9} {F'' : Type u_10} [NormedAddCommGroup E''] [NormedAddCommGroup F''] [ProperSpace F''] (c : E'') : (fun _x => c) =o[Filter.cocompact F''] id"} +{"name":"Asymptotics.IsBigO.natCast","declaration":"theorem Asymptotics.IsBigO.natCast {E : Type u_3} [Norm E] {f : ℝ → E} {g : ℝ → E} (h : f =O[Filter.atTop] g) : (fun n => f ↑n) =O[Filter.atTop] fun n => g ↑n"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f264a3602f5bca567caeb56cb715edf08c558d9f --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta.jsonl @@ -0,0 +1 @@ +{"name":"Asymptotics.isTheta_bot","declaration":"theorem Asymptotics.isTheta_bot {α : Type u_1} {E : Type u_3} {F : Type u_5} [Norm E] [Norm F] {f : α → E} {g : α → F} : f =Θ[⊥] g"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..292fa09ab0a3f2b520278e3a9db6b6ebc2865aa8 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly.jsonl @@ -0,0 +1,14 @@ +{"name":"ContinuousOn.const_isBigOUniformlyOn_isCompact_rev","declaration":"/-- A family of constant functions `f (i, x) = C i` is uniformly Ω(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous with no zeros on `s`. -/\ntheorem ContinuousOn.const_isBigOUniformlyOn_isCompact_rev {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [TopologicalSpace ι] {C : ι → E} {c : F} [NormedAddGroup E] [SeminormedAddGroup F] (hf : ContinuousOn C s) (hs : IsCompact s) (hC : ∀ i ∈ s, C i ≠ 0) (l : Filter α) : (fun x => c) =O[Filter.principal s ×ˢ l] fun x =>\n match x with\n | (i, _x) => C i"} +{"name":"ContinuousOn.const_isThetaUniformlyOn_isCompact","declaration":"/-- A family of constant functions `f (i, x) = C i` is uniformly Θ(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous with no zeros on `s`. -/\ntheorem ContinuousOn.const_isThetaUniformlyOn_isCompact {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [TopologicalSpace ι] {C : ι → E} {c : F} [NormedAddGroup E] [SeminormedAddGroup F] (hf : ContinuousOn C s) (hs : IsCompact s) (hc : ‖c‖ ≠ 0) (hC : ∀ i ∈ s, C i ≠ 0) (l : Filter α) : (fun x =>\n match x with\n | (i, _x) => C i) =Θ[Filter.principal s ×ˢ l]\n fun x => c"} +{"name":"ContinuousOn.const_isBigOWithUniformlyOn_isCompact","declaration":"/-- A family of constant functions `f (i, x) = C i` is uniformly bounded w.r.t. `s` by\n`⨆ i ∈ s, ‖C i‖`, if `s` is compact and `C` is continuous. -/\ntheorem ContinuousOn.const_isBigOWithUniformlyOn_isCompact {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [TopologicalSpace ι] {C : ι → E} {c : F} [SeminormedAddGroup E] [Norm F] (hf : ContinuousOn C s) (hs : IsCompact s) (hc : ‖c‖ ≠ 0) (l : Filter α) : Asymptotics.IsBigOWith (sSup (norm '' (C '' s)) / ‖c‖) (Filter.principal s ×ˢ l)\n (fun x =>\n match x with\n | (i, _x) => C i)\n fun x => c"} +{"name":"Asymptotics.isLittleO_const_snd_atBot","declaration":"theorem Asymptotics.isLittleO_const_snd_atBot {α : Type u_1} {E : Type u_3} {F : Type u_4} [NormedAddCommGroup α] [LinearOrder α] [ProperSpace α] [NormedAddCommGroup F] [NoMinOrder α] [ClosedIicTopology α] (c : F) (lx : Filter E) : (fun x => c) =o[lx ×ˢ Filter.atBot] Prod.snd"} +{"name":"Asymptotics.isBigO_rev_of_isBigOUniformly_rev","declaration":"/-- If f = Ω(g) uniformly on `s`, then f_i = Ω(g) for any i.` -/\ntheorem Asymptotics.isBigO_rev_of_isBigOUniformly_rev {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [Norm E] [Norm F] {f : ι × α → E} {g : α → F} {l : Filter α} (h : (g ∘ Prod.snd) =O[Filter.principal s ×ˢ l] f) {i : ι} (hi : i ∈ s) : g =O[l] fun x => f (i, x)"} +{"name":"Asymptotics.IsBigO.eventually_integrableOn","declaration":"/-- Let `f : X x Y → Z`. If as y → l, f(x, y) = O(g(y)) uniformly on `s : Set X` of finite measure,\nthen f is eventually (as y → l) integrable along `s`. -/\ntheorem Asymptotics.IsBigO.eventually_integrableOn {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [MeasurableSpace ι] [NormedAddCommGroup E] {f : ι × α → E} {g : α → F} {μ : MeasureTheory.Measure ι} {l : Filter α} [Norm F] (hf : f =O[Filter.principal s ×ˢ l] fun x =>\n match x with\n | (_i, x) => g x) (hfm : ∀ᶠ (x : α) in l, MeasureTheory.AEStronglyMeasurable (fun i => f (i, x)) (μ.restrict s)) (hs : MeasurableSet s) (hμ : ↑↑μ s < ⊤) : ∀ᶠ (x : α) in l, MeasureTheory.IntegrableOn (fun i => f (i, x)) s μ"} +{"name":"Asymptotics.isLittleO_const_snd_atTop","declaration":"theorem Asymptotics.isLittleO_const_snd_atTop {α : Type u_1} {E : Type u_3} {F : Type u_4} [NormedAddCommGroup α] [LinearOrder α] [ProperSpace α] [NormedAddCommGroup F] [NoMaxOrder α] [ClosedIciTopology α] (c : F) (lx : Filter E) : (fun x => c) =o[lx ×ˢ Filter.atTop] Prod.snd"} +{"name":"Asymptotics.isLittleO_const_fst_atBot","declaration":"theorem Asymptotics.isLittleO_const_fst_atBot {α : Type u_1} {E : Type u_3} {F : Type u_4} [NormedAddCommGroup α] [LinearOrder α] [ProperSpace α] [NormedAddCommGroup F] [NoMinOrder α] [ClosedIicTopology α] (c : F) (ly : Filter E) : (fun x => c) =o[Filter.atBot ×ˢ ly] Prod.fst"} +{"name":"ContinuousOn.const_isBigOWithUniformlyOn_isCompact_rev","declaration":"/-- A family of constant functions `f (i, x) = C i` is uniformly bounded below w.r.t. `s` by\n`⊓ i ∈ s, ‖C i‖`, if `s` is compact and `C` is continuous. -/\ntheorem ContinuousOn.const_isBigOWithUniformlyOn_isCompact_rev {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [TopologicalSpace ι] {C : ι → E} {c : F} [NormedAddGroup E] [SeminormedAddGroup F] (hf : ContinuousOn C s) (hs : IsCompact s) (hC : ∀ i ∈ s, C i ≠ 0) (l : Filter α) : Asymptotics.IsBigOWith (‖c‖ / sInf (norm '' (C '' s))) (Filter.principal s ×ˢ l) (fun x => c) fun x =>\n match x with\n | (i, _x) => C i"} +{"name":"Asymptotics.isBigO_of_isBigOUniformly","declaration":"/-- If f = O(g) uniformly on `s`, then f_i = O(g) for any i.` -/\ntheorem Asymptotics.isBigO_of_isBigOUniformly {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [Norm E] [Norm F] {f : ι × α → E} {g : α → F} {l : Filter α} (h : f =O[Filter.principal s ×ˢ l] (g ∘ Prod.snd)) {i : ι} (hi : i ∈ s) : (fun x => f (i, x)) =O[l] g"} +{"name":"ContinuousOn.const_isBigOUniformlyOn_isCompact","declaration":"/-- A family of constant functions `f (i, x) = C i` is uniformly O(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous. -/\ntheorem ContinuousOn.const_isBigOUniformlyOn_isCompact {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [TopologicalSpace ι] {C : ι → E} {c : F} [SeminormedAddGroup E] [Norm F] (hf : ContinuousOn C s) (hs : IsCompact s) (hc : ‖c‖ ≠ 0) (l : Filter α) : (fun x =>\n match x with\n | (i, _x) => C i) =O[Filter.principal s ×ˢ l]\n fun x => c"} +{"name":"Asymptotics.isTheta_of_isThetaUniformly","declaration":"/-- If f = Θ(g) uniformly on `s`, then f_i = Θ(g) for any i.` -/\ntheorem Asymptotics.isTheta_of_isThetaUniformly {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [Norm E] [Norm F] {f : ι × α → E} {g : α → F} {l : Filter α} (h : f =Θ[Filter.principal s ×ˢ l] (g ∘ Prod.snd)) {i : ι} (hi : i ∈ s) : (fun x => f (i, x)) =Θ[l] g"} +{"name":"Asymptotics.IsBigO.set_integral_isBigO","declaration":"/-- Let `f : X x Y → Z`. If as y → l, f(x, y) = O(g(y)) uniformly on `s : Set X` of finite measure,\nthen the integral of f along s is O(g(y)). -/\ntheorem Asymptotics.IsBigO.set_integral_isBigO {α : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {s : Set ι} [MeasurableSpace ι] [NormedAddCommGroup E] {f : ι × α → E} {g : α → F} {μ : MeasureTheory.Measure ι} {l : Filter α} [NormedSpace ℝ E] [NormedAddCommGroup F] (hf : f =O[Filter.principal s ×ˢ l] fun x =>\n match x with\n | (_i, x) => g x) (hs : MeasurableSet s) (hμ : ↑↑μ s < ⊤) : (fun x => ∫ (i : ι) in s, f (i, x) ∂μ) =O[l] g"} +{"name":"Asymptotics.isLittleO_const_fst_atTop","declaration":"theorem Asymptotics.isLittleO_const_fst_atTop {α : Type u_1} {E : Type u_3} {F : Type u_4} [NormedAddCommGroup α] [LinearOrder α] [ProperSpace α] [NormedAddCommGroup F] [NoMaxOrder α] [ClosedIciTopology α] (c : F) (ly : Filter E) : (fun x => c) =o[Filter.atTop ×ˢ ly] Prod.fst"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..129bcfda863dc27d9752c8495c6f68755391e707 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts.jsonl @@ -0,0 +1,7 @@ +{"name":"integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1","declaration":"theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1 {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] {μ : MeasureTheory.Measure E} [MeasureTheory.SigmaFinite μ] {f : E × ℝ → F} {f' : E × ℝ → F} {g : E × ℝ → G} {g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B (f' x)) (g x)) (MeasureTheory.Measure.prod μ MeasureTheory.volume)) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) (g' x)) (MeasureTheory.Measure.prod μ MeasureTheory.volume)) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) (MeasureTheory.Measure.prod μ MeasureTheory.volume)) (hf : ∀ (x : E × ℝ), HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ (x : E × ℝ), HasLineDerivAt ℝ g (g' x) x (0, 1)) : ∫ (x : E × ℝ), (B (f x)) (g' x) ∂MeasureTheory.Measure.prod μ MeasureTheory.volume =\n -∫ (x : E × ℝ), (B (f' x)) (g x) ∂MeasureTheory.Measure.prod μ MeasureTheory.volume"} +{"name":"integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable","declaration":"/-- **Integration by parts for Fréchet derivatives**\nVersion with two scalar functions: `∫ f * g' = - ∫ f' * g` when `f * g'` and `f' * g` and `f * g`\nare integrable, where `f'` and `g'` are the derivatives of `f` and `g` in a given direction `v`. -/\ntheorem integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [MeasureTheory.Measure.IsAddHaarMeasure μ] {𝕜 : Type u_5} [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] {f : E → 𝕜} {g : E → 𝕜} {v : E} (hf'g : MeasureTheory.Integrable (fun x => (fderiv ℝ f x) v * g x) μ) (hfg' : MeasureTheory.Integrable (fun x => f x * (fderiv ℝ g x) v) μ) (hfg : MeasureTheory.Integrable (fun x => f x * g x) μ) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) : ∫ (x : E), f x * (fderiv ℝ g x) v ∂μ = -∫ (x : E), (fderiv ℝ f x) v * g x ∂μ"} +{"name":"integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable","declaration":"/-- **Integration by parts for line derivatives**\nVersion with a general bilinear form `B`.\nIf `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are derivatives\nof `f` and `g` in a given direction `v`, then `∫ B f g' = - ∫ B f' g`. -/\ntheorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [MeasureTheory.Measure.IsAddHaarMeasure μ] {f : E → F} {f' : E → F} {g : E → G} {g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B (f' x)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) (g' x)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : ∀ (x : E), HasLineDerivAt ℝ f (f' x) x v) (hg : ∀ (x : E), HasLineDerivAt ℝ g (g' x) x v) : ∫ (x : E), (B (f x)) (g' x) ∂μ = -∫ (x : E), (B (f' x)) (g x) ∂μ"} +{"name":"integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable","declaration":"/-- **Integration by parts for Fréchet derivatives**\nVersion with a scalar function: `∫ f • g' = - ∫ f' • g` when `f • g'` and `f' • g` and `f • g`\nare integrable, where `f'` and `g'` are the derivatives of `f` and `g` in a given direction `v`. -/\ntheorem integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable {E : Type u_1} {G : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup G] [NormedSpace ℝ G] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [MeasureTheory.Measure.IsAddHaarMeasure μ] {𝕜 : Type u_5} [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 G] [IsScalarTower ℝ 𝕜 G] {f : E → 𝕜} {g : E → G} {v : E} (hf'g : MeasureTheory.Integrable (fun x => (fderiv ℝ f x) v • g x) μ) (hfg' : MeasureTheory.Integrable (fun x => f x • (fderiv ℝ g x) v) μ) (hfg : MeasureTheory.Integrable (fun x => f x • g x) μ) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) : ∫ (x : E), f x • (fderiv ℝ g x) v ∂μ = -∫ (x : E), (fderiv ℝ f x) v • g x ∂μ"} +{"name":"integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2","declaration":"theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2 {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure (E × ℝ)} [MeasureTheory.Measure.IsAddHaarMeasure μ] {f : E × ℝ → F} {f' : E × ℝ → F} {g : E × ℝ → G} {g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B (f' x)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) (g' x)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : ∀ (x : E × ℝ), HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ (x : E × ℝ), HasLineDerivAt ℝ g (g' x) x (0, 1)) : ∫ (x : E × ℝ), (B (f x)) (g' x) ∂μ = -∫ (x : E × ℝ), (B (f' x)) (g x) ∂μ"} +{"name":"integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable","declaration":"/-- **Integration by parts for Fréchet derivatives**\nVersion with a general bilinear form `B`.\nIf `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are derivatives\nof `f` and `g` in a given direction `v`, then `∫ B f g' = - ∫ B f' g`. -/\ntheorem integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [MeasureTheory.Measure.IsAddHaarMeasure μ] {f : E → F} {f' : E → E →L[ℝ] F} {g : E → G} {g' : E → E →L[ℝ] G} {v : E} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B ((f' x) v)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) ((g' x) v)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : ∀ (x : E), HasFDerivAt f (f' x) x) (hg : ∀ (x : E), HasFDerivAt g (g' x) x) : ∫ (x : E), (B (f x)) ((g' x) v) ∂μ = -∫ (x : E), (B ((f' x) v)) (g x) ∂μ"} +{"name":"integral_bilinear_fderiv_right_eq_neg_left_of_integrable","declaration":"/-- **Integration by parts for Fréchet derivatives**\nVersion with a general bilinear form `B`.\nIf `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are the derivatives\nof `f` and `g` in a given direction `v`, then `∫ B f g' = - ∫ B f' g`. -/\ntheorem integral_bilinear_fderiv_right_eq_neg_left_of_integrable {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [MeasureTheory.Measure.IsAddHaarMeasure μ] {f : E → F} {g : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B ((fderiv ℝ f x) v)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) ((fderiv ℝ g x) v)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) : ∫ (x : E), (B (f x)) ((fderiv ℝ g x) v) ∂μ = -∫ (x : E), (B ((fderiv ℝ f x) v)) (g x) ∂μ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..08c626a40beae3b32c535ba230a6aa9e2146ac7a --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv.jsonl @@ -0,0 +1,14 @@ +{"name":"Real.hasFDerivAt_fourierChar_neg_bilinear_left","declaration":"theorem Real.hasFDerivAt_fourierChar_neg_bilinear_left {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) (w : W) : HasFDerivAt (fun v => ↑(Real.fourierChar (-(L v) w)))\n ((-2 * ↑Real.pi * Complex.I * ↑(Real.fourierChar (-(L v) w))) •\n ContinuousLinearMap.comp Complex.ofRealCLM ((ContinuousLinearMap.flip L) w))\n v"} +{"name":"Real.fderiv_fourierChar_neg_bilinear_right_apply","declaration":"theorem Real.fderiv_fourierChar_neg_bilinear_right_apply {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) (w : W) (y : W) : (fderiv ℝ (fun w => ↑(Real.fourierChar (-(L v) w))) w) y =\n -2 * ↑Real.pi * Complex.I * ↑((L v) y) * ↑(Real.fourierChar (-(L v) w))"} +{"name":"VectorFourier.fourierIntegral_iteratedFDeriv","declaration":"/-- The Fourier integral of the `n`-th derivative of a function is obtained by multiplying the\nFourier integral of the original function by `(2πI L w ⬝ )^n`. -/\ntheorem VectorFourier.fourierIntegral_iteratedFDeriv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {μ : MeasureTheory.Measure V} [MeasureTheory.Measure.IsAddHaarMeasure μ] {N : ℕ∞} (hf : ContDiff ℝ N f) (h'f : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (iteratedFDeriv ℝ n f) μ) {n : ℕ} (hn : ↑n ≤ N) : VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₂ L) (iteratedFDeriv ℝ n f) = fun w =>\n VectorFourier.fourierPowSMulRight (-ContinuousLinearMap.flip L)\n (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₂ L) f) w n"} +{"name":"Real.fourierIntegral_iteratedDeriv","declaration":"theorem Real.fourierIntegral_iteratedDeriv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} {N : ℕ∞} {n : ℕ} (hf : ContDiff ℝ N f) (h'f : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (iteratedDeriv n f) MeasureTheory.volume) (hn : ↑n ≤ N) : Real.fourierIntegral (iteratedDeriv n f) = fun x => (2 * ↑Real.pi * Complex.I * ↑x) ^ n • Real.fourierIntegral f x"} +{"name":"Real.fourierIntegral_fderiv","declaration":"/-- The Fourier integral of the Fréchet derivative of a function is obtained by multiplying the\nFourier integral of the original function by `2πI ⟪v, w⟫`. -/\ntheorem Real.fourierIntegral_fderiv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : Differentiable ℝ f) (hf' : MeasureTheory.Integrable (fderiv ℝ f) MeasureTheory.volume) : Real.fourierIntegral (fderiv ℝ f) = VectorFourier.fourierSMulRight (-innerSL ℝ) (Real.fourierIntegral f)"} +{"name":"Real.differentiable_fourierChar","declaration":"theorem Real.differentiable_fourierChar : Differentiable ℝ fun x => ↑(Real.fourierChar x)"} +{"name":"Real.deriv_fourierChar","declaration":"theorem Real.deriv_fourierChar (x : ℝ) : deriv (fun x => ↑(Real.fourierChar x)) x = 2 * ↑Real.pi * Complex.I * ↑(Real.fourierChar x)"} +{"name":"Real.differentiable_fourierChar_neg_bilinear_right","declaration":"theorem Real.differentiable_fourierChar_neg_bilinear_right {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) : Differentiable ℝ fun w => ↑(Real.fourierChar (-(L v) w))"} +{"name":"Real.hasFDerivAt_fourierChar_neg_bilinear_right","declaration":"theorem Real.hasFDerivAt_fourierChar_neg_bilinear_right {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) (w : W) : HasFDerivAt (fun w => ↑(Real.fourierChar (-(L v) w)))\n ((-2 * ↑Real.pi * Complex.I * ↑(Real.fourierChar (-(L v) w))) • ContinuousLinearMap.comp Complex.ofRealCLM (L v)) w"} +{"name":"Real.differentiable_fourierChar_neg_bilinear_left","declaration":"theorem Real.differentiable_fourierChar_neg_bilinear_left {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (w : W) : Differentiable ℝ fun v => ↑(Real.fourierChar (-(L v) w))"} +{"name":"Real.fourierIntegral_deriv","declaration":"/-- The Fourier integral of the Fréchet derivative of a function is obtained by multiplying the\nFourier integral of the original function by `2πI x`. -/\ntheorem Real.fourierIntegral_deriv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℝ → E} (hf : MeasureTheory.Integrable f MeasureTheory.volume) (h'f : Differentiable ℝ f) (hf' : MeasureTheory.Integrable (deriv f) MeasureTheory.volume) : Real.fourierIntegral (deriv f) = fun x => (2 * ↑Real.pi * Complex.I * ↑x) • Real.fourierIntegral f x"} +{"name":"VectorFourier.fourierIntegral_fderiv","declaration":"/-- The Fourier integral of the derivative of a function is obtained by multiplying the Fourier\nintegral of the original function by `-L w v`. -/\ntheorem VectorFourier.fourierIntegral_fderiv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {μ : MeasureTheory.Measure V} [MeasureTheory.Measure.IsAddHaarMeasure μ] (hf : MeasureTheory.Integrable f μ) (h'f : Differentiable ℝ f) (hf' : MeasureTheory.Integrable (fderiv ℝ f) μ) : VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₂ L) (fderiv ℝ f) =\n VectorFourier.fourierSMulRight (-ContinuousLinearMap.flip L)\n (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₂ L) f)"} +{"name":"Real.fderiv_fourierChar_neg_bilinear_left_apply","declaration":"theorem Real.fderiv_fourierChar_neg_bilinear_left_apply {V : Type u_1} {W : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] (L : V →L[ℝ] W →L[ℝ] ℝ) (v : V) (y : V) (w : W) : (fderiv ℝ (fun v => ↑(Real.fourierChar (-(L v) w))) v) y =\n -2 * ↑Real.pi * Complex.I * ↑((L y) w) * ↑(Real.fourierChar (-(L v) w))"} +{"name":"Real.fourierIntegral_iteratedFDeriv","declaration":"/-- The Fourier integral of the `n`-th derivative of a function is obtained by multiplying the\nFourier integral of the original function by `(2πI L w ⬝ )^n`. -/\ntheorem Real.fourierIntegral_iteratedFDeriv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {f : V → E} {N : ℕ∞} (hf : ContDiff ℝ N f) (h'f : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (iteratedFDeriv ℝ n f) MeasureTheory.volume) {n : ℕ} (hn : ↑n ≤ N) : Real.fourierIntegral (iteratedFDeriv ℝ n f) = fun w =>\n VectorFourier.fourierPowSMulRight (-innerSL ℝ) (Real.fourierIntegral f) w n"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6730be99aca71eafcf02d27c8921955792ad25d8 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal.jsonl @@ -0,0 +1,17 @@ +{"name":"Nat.card_finMulAntidiagonal_pi","declaration":"theorem Nat.card_finMulAntidiagonal_pi (d : ℕ) (n : ℕ) (hn : Squarefree n) : (Finset.pi (List.toFinset (Nat.factors n)) fun x => Finset.univ).card = (Nat.finMulAntidiagonal d n).card"} +{"name":"Nat.ne_zero_of_mem_finMulAntidiagonal","declaration":"theorem Nat.ne_zero_of_mem_finMulAntidiagonal {d : ℕ} {n : ℕ} {f : Fin d → ℕ} (hf : f ∈ Nat.finMulAntidiagonal d n) (i : Fin d) : f i ≠ 0"} +{"name":"Nat.dvd_of_mem_finMulAntidiagonal","declaration":"theorem Nat.dvd_of_mem_finMulAntidiagonal {n : ℕ} {d : ℕ} {f : Fin d → ℕ} (hf : f ∈ Nat.finMulAntidiagonal d n) (i : Fin d) : f i ∣ n"} +{"name":"Nat.card_lcm_eq","declaration":"theorem Nat.card_lcm_eq {n : ℕ} (hn : Squarefree n) : (Finset.filter\n (fun x =>\n match x with\n | (x, y) => Nat.lcm x y = n)\n (Nat.divisors n ×ˢ Nat.divisors n)).card =\n 3 ^ ArithmeticFunction.cardDistinctFactors n"} +{"name":"Nat.filter_primeFactors","declaration":"theorem Nat.filter_primeFactors {m : ℕ} {n : ℕ} (hmn : m ∣ n) (hn : n ≠ 0) : Finset.filter (fun p => p ∣ m) n.primeFactors = m.primeFactors"} +{"name":"Nat.finMulAntidiagonal_zero","declaration":"theorem Nat.finMulAntidiagonal_zero {d : ℕ} : Nat.finMulAntidiagonal d 0 = ∅"} +{"name":"Nat.finMulAntidiagonal.aux","declaration":"/-- Auxiliary construction for `finMulAntidiagonal` that bundles a proof of lawfulness\n(`mem_finMulAntidiagonal`), as this is needed to invoke `disjiUnion`. Using `Finset.disjiUnion` makes\nthis computationally much more efficient than using `Finset.biUnion`. -/\ndef Nat.finMulAntidiagonal.aux (d : ℕ) (n : ℕ) : { s // ∀ (f : Fin d → ℕ), f ∈ s ↔ (Finset.prod Finset.univ fun i => f i) = n ∧ n ≠ 0 }"} +{"name":"Nat.finMulAntidiagonal_univ_eq","declaration":"theorem Nat.finMulAntidiagonal_univ_eq {d : ℕ} {m : ℕ} {n : ℕ} (hmn : m ∣ n) (hn : n ≠ 0) : Nat.finMulAntidiagonal d m =\n Finset.filter (fun f => (Finset.prod Finset.univ fun i => f i) = m) (Fintype.piFinset fun x => Nat.divisors n)"} +{"name":"Nat.image_apply_finMulAntidiagonal","declaration":"theorem Nat.image_apply_finMulAntidiagonal {d : ℕ} {n : ℕ} {i : Fin d} (hd : d ≠ 1) : Finset.image (fun f => f i) (Nat.finMulAntidiagonal d n) = Nat.divisors n"} +{"name":"Nat.card_finMulAntidiagonal","declaration":"theorem Nat.card_finMulAntidiagonal {d : ℕ} {n : ℕ} (hn : Squarefree n) : (Nat.finMulAntidiagonal d n).card = d ^ ArithmeticFunction.cardDistinctFactors n"} +{"name":"Nat.finMulAntidiagonal","declaration":"def Nat.finMulAntidiagonal (d : ℕ) (n : ℕ) : Finset (Fin d → ℕ)"} +{"name":"Nat.mem_finMulAntidiagonal","declaration":"theorem Nat.mem_finMulAntidiagonal {d : ℕ} {n : ℕ} {f : Fin d → ℕ} : f ∈ Nat.finMulAntidiagonal d n ↔ (Finset.prod Finset.univ fun i => f i) = n ∧ n ≠ 0"} +{"name":"Nat.finMulAntidiagonal_one","declaration":"theorem Nat.finMulAntidiagonal_one {d : ℕ} : Nat.finMulAntidiagonal d 1 = {fun x => 1}"} +{"name":"Nat.prod_eq_of_mem_finMulAntidiagonal","declaration":"theorem Nat.prod_eq_of_mem_finMulAntidiagonal {d : ℕ} {n : ℕ} {f : Fin d → ℕ} (hf : f ∈ Nat.finMulAntidiagonal d n) : (Finset.prod Finset.univ fun i => f i) = n"} +{"name":"Nat.image_piFinTwoEquiv","declaration":"theorem Nat.image_piFinTwoEquiv {n : ℕ} : Finset.image (⇑(piFinTwoEquiv fun x => ℕ)) (Nat.finMulAntidiagonal 2 n) = Nat.divisorsAntidiagonal n"} +{"name":"Nat.finMulAntidiagonal_exists_unique_prime_dvd","declaration":"theorem Nat.finMulAntidiagonal_exists_unique_prime_dvd {d : ℕ} {n : ℕ} {p : ℕ} (hn : Squarefree n) (hp : p ∈ Nat.factors n) (f : Fin d → ℕ) (hf : f ∈ Nat.finMulAntidiagonal d n) : ∃! i, p ∣ f i"} +{"name":"Nat.finMulAntidiagonal_empty_of_ne_one","declaration":"theorem Nat.finMulAntidiagonal_empty_of_ne_one {n : ℕ} (hn : n ≠ 1) : Nat.finMulAntidiagonal 0 n = ∅"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c5e51d58b8268faf0be30f6febfa7dbadb60959f --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space.jsonl @@ -0,0 +1,2 @@ +{"name":"ContinuousLinearEquiv.integrable_comp_iff","declaration":"theorem ContinuousLinearEquiv.integrable_comp_iff {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {𝕜 : Type u_6} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {H : Type u_7} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : α → H} (L : H ≃L[𝕜] E) : MeasureTheory.Integrable (fun a => L (φ a)) μ ↔ MeasureTheory.Integrable φ μ"} +{"name":"LinearIsometryEquiv.integrable_comp_iff","declaration":"theorem LinearIsometryEquiv.integrable_comp_iff {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {𝕜 : Type u_6} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {H : Type u_7} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : α → H} (L : H ≃ₗᵢ[𝕜] E) : MeasureTheory.Integrable (fun a => L (φ a)) μ ↔ MeasureTheory.Integrable φ μ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e2a78cc81e0766aa3ad97c4d2a806a74c64b397e --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable.jsonl @@ -0,0 +1,5 @@ +{"name":"integrable_iff_integrableAtFilter_atBot_atTop","declaration":"theorem integrable_iff_integrableAtFilter_atBot_atTop {X : Type u_1} {E : Type u_2} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [LinearOrder X] [CompactIccSpace X] : MeasureTheory.Integrable f μ ↔\n (MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.IntegrableAtFilter f Filter.atTop μ) ∧\n MeasureTheory.LocallyIntegrable f μ"} +{"name":"integrableOn_Ici_iff_integrableAtFilter_atTop","declaration":"theorem integrableOn_Ici_iff_integrableAtFilter_atTop {X : Type u_1} {E : Type u_2} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {a : X} [LinearOrder X] [CompactIccSpace X] : MeasureTheory.IntegrableOn f (Set.Ici a) μ ↔\n MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Ici a) μ"} +{"name":"integrable_iff_integrableAtFilter_atBot","declaration":"theorem integrable_iff_integrableAtFilter_atBot {X : Type u_1} {E : Type u_2} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [LinearOrder X] [OrderTop X] [CompactIccSpace X] : MeasureTheory.Integrable f μ ↔ MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.LocallyIntegrable f μ"} +{"name":"integrable_iff_integrableAtFilter_atTop","declaration":"theorem integrable_iff_integrableAtFilter_atTop {X : Type u_1} {E : Type u_2} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [LinearOrder X] [OrderBot X] [CompactIccSpace X] : MeasureTheory.Integrable f μ ↔ MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧ MeasureTheory.LocallyIntegrable f μ"} +{"name":"integrableOn_Iic_iff_integrableAtFilter_atBot","declaration":"theorem integrableOn_Iic_iff_integrableAtFilter_atBot {X : Type u_1} {E : Type u_2} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {a : X} [LinearOrder X] [CompactIccSpace X] : MeasureTheory.IntegrableOn f (Set.Iic a) μ ↔\n MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Iic a) μ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c424d4df42fa438ff9bc117ba82fe2d44225efa0 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics.jsonl @@ -0,0 +1,3 @@ +{"name":"MeasureTheory.Asymptotics.IsBigO.set_integral_isBigO","declaration":"/-- Let `f : X x Y → Z`. If as y → l, f(x, y) = O(g(y)) uniformly on `s : Set X` of finite measure,\nthen the integral of f along s is O(g(y)). -/\ntheorem MeasureTheory.Asymptotics.IsBigO.set_integral_isBigO {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] {g : α → F} {l : Filter α} {ι : Type u_4} [MeasurableSpace ι] {f : ι × α → E} {s : Set ι} {μ : MeasureTheory.Measure ι} [NormedSpace ℝ E] [NormedAddCommGroup F] (hf : f =O[Filter.principal s ×ˢ l] fun x =>\n match x with\n | (_i, x) => g x) (hs : MeasurableSet s) (hμ : ↑↑μ s < ⊤) : (fun x => ∫ (i : ι) in s, f (i, x) ∂μ) =O[l] g"} +{"name":"MeasureTheory.Asymptotics.IsBigO.eventually_integrableOn","declaration":"/-- Let `f : X x Y → Z`. If as y → l, f(x, y) = O(g(y)) uniformly on `s : Set X` of finite measure,\nthen f is eventually (as y → l) integrable along `s`. -/\ntheorem MeasureTheory.Asymptotics.IsBigO.eventually_integrableOn {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] {g : α → F} {l : Filter α} {ι : Type u_4} [MeasurableSpace ι] {f : ι × α → E} {s : Set ι} {μ : MeasureTheory.Measure ι} [Norm F] (hf : f =O[Filter.principal s ×ˢ l] fun x =>\n match x with\n | (_i, x) => g x) (hfm : ∀ᶠ (x : α) in l, MeasureTheory.AEStronglyMeasurable (fun i => f (i, x)) (μ.restrict s)) (hs : MeasurableSet s) (hμ : ↑↑μ s < ⊤) : ∀ᶠ (x : α) in l, MeasureTheory.IntegrableOn (fun i => f (i, x)) s μ"} +{"name":"MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atTop_of_norm_eq_norm_neg","declaration":"/-- If `f` is locally integrable, `‖f(-x)‖ = ‖f(x)‖`, and `f =O[atTop] g`, for some\n`g` integrable at `atTop`, then `f` is integrable. -/\ntheorem MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atTop_of_norm_eq_norm_neg {α : Type u_1} {E : Type u_2} {F : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [SecondCountableTopology α] [LinearOrderedAddCommGroup α] [CompactIccSpace α] [Filter.IsMeasurablyGenerated Filter.atTop] [MeasurableNeg α] [MeasureTheory.Measure.IsNegInvariant μ] (hf : MeasureTheory.LocallyIntegrable f μ) (hsymm : norm ∘ f =ᶠ[MeasureTheory.Measure.ae μ] norm ∘ f ∘ Neg.neg) (ho : f =O[Filter.atTop] g) (hg : MeasureTheory.IntegrableAtFilter g Filter.atTop μ) : MeasureTheory.Integrable f μ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..2a6dbfcef67473cb8d69252be62795542ec0df71 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn.jsonl @@ -0,0 +1 @@ +{"name":"IntegrableAtFilter.sup_iff","declaration":"theorem IntegrableAtFilter.sup_iff {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [NormedAddCommGroup β] {f : α → β} {μ : MeasureTheory.Measure α} {l : Filter α} {l' : Filter α} : MeasureTheory.IntegrableAtFilter f (l ⊔ l') μ ↔\n MeasureTheory.IntegrableAtFilter f l μ ∧ MeasureTheory.IntegrableAtFilter f l' μ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.ArithmeticFunction.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.ArithmeticFunction.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7c377b38bceff0db06f2f0a0aaeee90aa07c49aa --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.ArithmeticFunction.jsonl @@ -0,0 +1,2 @@ +{"name":"ArithmeticFunction.sum_Icc_mul_zeta","declaration":"theorem ArithmeticFunction.sum_Icc_mul_zeta {R : Type u_1} [Semiring R] (f : ArithmeticFunction R) (N : ℕ) : (Finset.sum (Finset.Icc 1 N) fun d => (f * ↑ArithmeticFunction.zeta) d) =\n Finset.sum (Finset.Icc 1 N) fun d => (N / d) • f d"} +{"name":"ArithmeticFunction.sum_range_mul_zeta","declaration":"theorem ArithmeticFunction.sum_range_mul_zeta {R : Type u_1} [Semiring R] (f : ArithmeticFunction R) (N : ℕ) : (Finset.sum (Finset.range (N + 1)) fun d => (f * ↑ArithmeticFunction.zeta) d) =\n Finset.sum (Finset.range (N + 1)) fun d => (N / d) • f d"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b4d4c5ed91f39f382036f62fc061d24bd4255701 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults.jsonl @@ -0,0 +1,24 @@ +{"name":"Aux.primeDivisors_nonempty","declaration":"theorem Aux.primeDivisors_nonempty (n : ℕ) (hn : 2 ≤ n) : n.primeFactors.Nonempty"} +{"name":"Aux.div_mult_of_dvd_squarefree","declaration":"theorem Aux.div_mult_of_dvd_squarefree (f : ArithmeticFunction ℝ) (h_mult : ArithmeticFunction.IsMultiplicative f) (l : ℕ) (d : ℕ) (hdl : d ∣ l) (hl : Squarefree l) (hd : f d ≠ 0) : f l / f d = f (l / d)"} +{"name":"Aux.inv_antitoneOn_pos","declaration":"theorem Aux.inv_antitoneOn_pos {R : Type u_1} [LinearOrderedField R] : AntitoneOn (fun x => x⁻¹) (Set.Ioi 0)"} +{"name":"Aux.gcd_dvd_mul","declaration":"theorem Aux.gcd_dvd_mul (m : ℕ) (n : ℕ) : Nat.gcd m n ∣ m * n"} +{"name":"Aux.sum_over_dvd_ite","declaration":"theorem Aux.sum_over_dvd_ite {α : Type u_1} [Ring α] {P : ℕ} (hP : P ≠ 0) {n : ℕ} (hn : n ∣ P) {f : ℕ → α} : (Finset.sum (Nat.divisors n) fun d => f d) = Finset.sum (Nat.divisors P) fun d => if d ∣ n then f d else 0"} +{"name":"Aux.sum_inv_le_log","declaration":"theorem Aux.sum_inv_le_log (n : ℕ) (hn : 1 ≤ n) : (Finset.sum (Finset.Icc 1 n) fun d => (↑d)⁻¹) ≤ 1 + Real.log ↑n"} +{"name":"Aux.inv_sub_antitoneOn_gt","declaration":"theorem Aux.inv_sub_antitoneOn_gt {R : Type u_1} [LinearOrderedField R] (c : R) : AntitoneOn (fun x => (x - c)⁻¹) (Set.Ioi c)"} +{"name":"Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow","declaration":"theorem Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow {P : ℕ} {k : ℕ} (x : ℝ) (hx : 1 ≤ x) (hP : Squarefree P) : (Finset.sum (Nat.divisors P) fun d => if ↑d ≤ x then ↑k ^ ArithmeticFunction.cardDistinctFactors d / ↑d else 0) ≤\n (1 + Real.log x) ^ k"} +{"name":"ArithmeticFunction.IsMultiplicative.prod_factors_of_mult","declaration":"theorem ArithmeticFunction.IsMultiplicative.prod_factors_of_mult (f : ArithmeticFunction ℝ) (h_mult : ArithmeticFunction.IsMultiplicative f) {l : ℕ} (hl : Squarefree l) : (Finset.prod l.primeFactors fun a => f a) = f l"} +{"name":"ArithmeticFunction.IsMultiplicative.mult_lcm_eq_of_ne_zero","declaration":"theorem ArithmeticFunction.IsMultiplicative.mult_lcm_eq_of_ne_zero {R : Type u_1} [CommGroupWithZero R] (f : ArithmeticFunction R) (h_mult : ArithmeticFunction.IsMultiplicative f) (x : ℕ) (y : ℕ) (hf : f (Nat.gcd x y) ≠ 0) : f (Nat.lcm x y) = f x * f y / f (Nat.gcd x y)"} +{"name":"Aux.moebius_inv_dvd_lower_bound_real","declaration":"theorem Aux.moebius_inv_dvd_lower_bound_real {P : ℕ} (hP : Squarefree P) (l : ℕ) (m : ℕ) (hm : m ∣ P) : (Finset.sum (Nat.divisors P) fun d => if l ∣ d ∧ d ∣ m then ↑(ArithmeticFunction.moebius d) else 0) =\n if l = m then ↑(ArithmeticFunction.moebius l) else 0"} +{"name":"Aux.log_add_one_le_sum_inv","declaration":"theorem Aux.log_add_one_le_sum_inv (n : ℕ) : Real.log ↑(n + 1) ≤ Finset.sum (Finset.Icc 1 n) fun d => (↑d)⁻¹"} +{"name":"Aux.conv_lambda_sq_larger_sum","declaration":"theorem Aux.conv_lambda_sq_larger_sum (f : ℕ → ℕ → ℕ → ℝ) (n : ℕ) : (Finset.sum (Nat.divisors n) fun d =>\n Finset.sum (Nat.divisors d) fun d1 =>\n Finset.sum (Nat.divisors d) fun d2 => if d = Nat.lcm d1 d2 then f d1 d2 d else 0) =\n Finset.sum (Nat.divisors n) fun d =>\n Finset.sum (Nat.divisors n) fun d1 =>\n Finset.sum (Nat.divisors n) fun d2 => if d = Nat.lcm d1 d2 then f d1 d2 d else 0"} +{"name":"Aux.log_le_sum_inv","declaration":"theorem Aux.log_le_sum_inv (y : ℝ) (hy : 1 ≤ y) : Real.log y ≤ Finset.sum (Finset.Icc 1 ⌊y⌋₊) fun d => (↑d)⁻¹"} +{"name":"Aux.sum_inv_le_log_real","declaration":"theorem Aux.sum_inv_le_log_real (y : ℝ) (hy : 1 ≤ y) : (Finset.sum (Finset.Icc 1 ⌊y⌋₊) fun d => (↑d)⁻¹) ≤ 1 + Real.log y"} +{"name":"Aux.inv_sub_antitoneOn_Icc","declaration":"theorem Aux.inv_sub_antitoneOn_Icc {R : Type u_1} [LinearOrderedField R] (a : R) (b : R) (c : R) (ha : c < a) : AntitoneOn (fun x => (x - c)⁻¹) (Set.Icc a b)"} +{"name":"Aux.sum_intro","declaration":"theorem Aux.sum_intro {α : Type u_1} {M : Type u_2} [AddCommMonoid M] [DecidableEq α] (s : Finset α) {f : α → M} (d : α) (hd : d ∈ s) : f d = Finset.sum s fun k => if k = d then f k else 0"} +{"name":"Aux.moebius_inv_dvd_lower_bound'","declaration":"theorem Aux.moebius_inv_dvd_lower_bound' {P : ℕ} (hP : Squarefree P) (l : ℕ) (m : ℕ) (hm : m ∣ P) : (Finset.sum (Nat.divisors P) fun d => if l ∣ d ∧ d ∣ m then ArithmeticFunction.moebius d else 0) =\n if l = m then ArithmeticFunction.moebius l else 0"} +{"name":"Aux.sum_pow_cardDistinctFactors_le_self_mul_log_pow","declaration":"theorem Aux.sum_pow_cardDistinctFactors_le_self_mul_log_pow {P : ℕ} {h : ℕ} (x : ℝ) (hx : 1 ≤ x) (hP : Squarefree P) : (Finset.sum (Nat.divisors P) fun d => if ↑d ≤ x then ↑h ^ ArithmeticFunction.cardDistinctFactors d else 0) ≤\n x * (1 + Real.log x) ^ h"} +{"name":"Aux.ite_sum_zero","declaration":"theorem Aux.ite_sum_zero {p : Prop} [Decidable p] (s : Finset ℕ) (f : ℕ → ℝ) : (if p then Finset.sum s fun x => f x else 0) = Finset.sum s fun x => if p then f x else 0"} +{"name":"Aux.multiplicative_zero_of_zero_dvd","declaration":"theorem Aux.multiplicative_zero_of_zero_dvd (f : ArithmeticFunction ℝ) (h_mult : ArithmeticFunction.IsMultiplicative f) {m : ℕ} {n : ℕ} (h_sq : Squarefree n) (hmn : m ∣ n) (h_zero : f m = 0) : f n = 0"} +{"name":"Aux.inv_antitoneOn_Icc","declaration":"theorem Aux.inv_antitoneOn_Icc {R : Type u_1} [LinearOrderedField R] (a : R) (b : R) (ha : 0 < a) : AntitoneOn (fun x => x⁻¹) (Set.Icc a b)"} +{"name":"Aux.moebius_inv_dvd_lower_bound","declaration":"theorem Aux.moebius_inv_dvd_lower_bound (l : ℕ) (m : ℕ) (hm : Squarefree m) : (Finset.sum (Nat.divisors m) fun d => if l ∣ d then ArithmeticFunction.moebius d else 0) =\n if l = m then ArithmeticFunction.moebius l else 0"} +{"name":"Aux.Nat.le_prod","declaration":"theorem Aux.Nat.le_prod {ι : Type u_1} [DecidableEq ι] {f : ι → ℕ} {s : Finset ι} {i : ι} (hi : i ∈ s) (hf : ∀ i ∈ s, f i ≠ 0) : f i ≤ Finset.prod s fun j => f j"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f80e64a488fd871c7c2dfc3b3f769844b065892b --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic.jsonl @@ -0,0 +1,53 @@ +{"name":"Sieve.squarefree_of_mem_divisors_prodPrimes","declaration":"theorem Sieve.squarefree_of_mem_divisors_prodPrimes (s : Sieve) {d : ℕ} (hd : d ∈ Nat.divisors s.prodPrimes) : Squarefree d"} +{"name":"Sieve.nu_ne_zero_of_mem_divisors_prodPrimes","declaration":"theorem Sieve.nu_ne_zero_of_mem_divisors_prodPrimes (s : Sieve) {d : ℕ} (hd : d ∈ Nat.divisors s.prodPrimes) : s.nu d ≠ 0"} +{"name":"Sieve.siftedSum_as_delta","declaration":"theorem Sieve.siftedSum_as_delta (s : Sieve) : Sieve.siftedSum s = Finset.sum s.support fun d => s.weights d * Sieve.delta (Nat.gcd s.prodPrimes d)"} +{"name":"Sieve.support","declaration":"def Sieve.support (self : Sieve) : Finset ℕ"} +{"name":"Sieve.errSum","declaration":"def Sieve.errSum (s : Sieve) (μPlus : ℕ → ℝ) : ℝ"} +{"name":"Sieve.LowerBoundSieve","declaration":"structure Sieve.LowerBoundSieve : Type"} +{"name":"Sieve.nu","declaration":"def Sieve.nu (self : Sieve) : ArithmeticFunction ℝ"} +{"name":"Sieve.LowerBoundSieve.μMinus","declaration":"def Sieve.LowerBoundSieve.μMinus (self : Sieve.LowerBoundSieve) : ℕ → ℝ"} +{"name":"Sieve.LowerBoundSieve.hμMinus","declaration":"def Sieve.LowerBoundSieve.hμMinus (self : Sieve.LowerBoundSieve) : Sieve.LowerMoebius self.μMinus"} +{"name":"Sieve.conv_selbergTerms_eq_selbergTerms_mul_nu","declaration":"theorem Sieve.conv_selbergTerms_eq_selbergTerms_mul_nu (s : Sieve) {d : ℕ} (hd : d ∣ s.prodPrimes) : (Finset.sum (Nat.divisors s.prodPrimes) fun l => if l ∣ d then (Sieve.selbergTerms s) l else 0) =\n (Sieve.selbergTerms s) d * (s.nu d)⁻¹"} +{"name":"Sieve.lambdaSquared_mainSum_eq_diag_quad_form","declaration":"theorem Sieve.lambdaSquared_mainSum_eq_diag_quad_form (s : Sieve) (w : ℕ → ℝ) : Sieve.mainSum s (Sieve.lambdaSquared w) =\n Finset.sum (Nat.divisors s.prodPrimes) fun l =>\n 1 / (Sieve.selbergTerms s) l *\n (Finset.sum (Nat.divisors s.prodPrimes) fun d => if l ∣ d then s.nu d * w d else 0) ^ 2"} +{"name":"Sieve.multSum_eq_main_err","declaration":"theorem Sieve.multSum_eq_main_err (s : Sieve) (d : ℕ) : Sieve.multSum s d = s.nu d * s.totalMass + Sieve.rem s d"} +{"name":"Sieve.selbergTerms_apply","declaration":"def Sieve.selbergTerms_apply (s : Sieve) (d : ℕ) : (Sieve.selbergTerms s) d = s.nu d * Finset.prod d.primeFactors fun p => 1 / (1 - s.nu p)"} +{"name":"Sieve.rem","declaration":"def Sieve.rem (s : Sieve) (d : ℕ) : ℝ"} +{"name":"Sieve.delta","declaration":"def Sieve.delta (n : ℕ) : ℝ"} +{"name":"Sieve.weights_nonneg","declaration":"def Sieve.weights_nonneg (self : Sieve) (n : ℕ) : 0 ≤ self.weights n"} +{"name":"Sieve.lambdaSquared_mainSum_eq_quad_form","declaration":"theorem Sieve.lambdaSquared_mainSum_eq_quad_form (s : Sieve) (w : ℕ → ℝ) : Sieve.mainSum s (Sieve.lambdaSquared w) =\n Finset.sum (Nat.divisors s.prodPrimes) fun d1 =>\n Finset.sum (Nat.divisors s.prodPrimes) fun d2 => s.nu d1 * w d1 * s.nu d2 * w d2 * (s.nu (Nat.gcd d1 d2))⁻¹"} +{"name":"Sieve.upper_bound_of_UpperBoundSieve","declaration":"theorem Sieve.upper_bound_of_UpperBoundSieve (s : Sieve) (μPlus : Sieve.UpperBoundSieve) : Sieve.siftedSum s ≤ Finset.sum (Nat.divisors s.prodPrimes) fun d => μPlus.μPlus d * Sieve.multSum s d"} +{"name":"Sieve.lambdaSquared","declaration":"def Sieve.lambdaSquared (weights : ℕ → ℝ) : ℕ → ℝ"} +{"name":"Sieve.squarefree_of_dvd_prodPrimes","declaration":"theorem Sieve.squarefree_of_dvd_prodPrimes (s : Sieve) {d : ℕ} (hd : d ∣ s.prodPrimes) : Squarefree d"} +{"name":"Sieve.UpperBoundSieve.hμPlus","declaration":"def Sieve.UpperBoundSieve.hμPlus (self : Sieve.UpperBoundSieve) : Sieve.UpperMoebius self.μPlus"} +{"name":"Sieve.prodPrimes_ne_zero","declaration":"theorem Sieve.prodPrimes_ne_zero (s : Sieve) : s.prodPrimes ≠ 0"} +{"name":"Sieve.lbToμMinus","declaration":"instance Sieve.lbToμMinus : CoeFun Sieve.LowerBoundSieve fun x => ℕ → ℝ"} +{"name":"Sieve.totalMass","declaration":"def Sieve.totalMass (self : Sieve) : ℝ"} +{"name":"Sieve.UpperBoundSieve.mk","declaration":"ctor Sieve.UpperBoundSieve.mk (μPlus : ℕ → ℝ) (hμPlus : Sieve.UpperMoebius μPlus) : Sieve.UpperBoundSieve"} +{"name":"Sieve.upperMoebius_of_lambda_sq","declaration":"theorem Sieve.upperMoebius_of_lambda_sq (weights : ℕ → ℝ) (hw : weights 1 = 1) : Sieve.UpperMoebius (Sieve.lambdaSquared weights)"} +{"name":"Sieve.nu_lt_self_of_dvd_prodPrimes","declaration":"theorem Sieve.nu_lt_self_of_dvd_prodPrimes (s : Sieve) (d : ℕ) (hdP : d ∣ s.prodPrimes) (hd_ne_one : d ≠ 1) : s.nu d < 1"} +{"name":"Sieve.nu_lt_one_of_prime","declaration":"def Sieve.nu_lt_one_of_prime (self : Sieve) (p : ℕ) : Nat.Prime p → p ∣ self.prodPrimes → self.nu p < 1"} +{"name":"Sieve.ubToμPlus","declaration":"instance Sieve.ubToμPlus : CoeFun Sieve.UpperBoundSieve fun x => ℕ → ℝ"} +{"name":"Sieve.lambdaSquared_eq_zero_of_support","declaration":"theorem Sieve.lambdaSquared_eq_zero_of_support (w : ℕ → ℝ) (y : ℝ) (hw : ∀ (d : ℕ), ¬↑d ^ 2 ≤ y → w d = 0) (d : ℕ) (hd : ¬↑d ≤ y) : Sieve.lambdaSquared w d = 0"} +{"name":"Sieve.selbergTerms","declaration":"def Sieve.selbergTerms (s : Sieve) : ArithmeticFunction ℝ"} +{"name":"Sieve.weights","declaration":"def Sieve.weights (self : Sieve) : ℕ → ℝ"} +{"name":"Sieve.siftedSum_le_mainSum_errSum_of_UpperBoundSieve","declaration":"theorem Sieve.siftedSum_le_mainSum_errSum_of_UpperBoundSieve (s : Sieve) (μPlus : Sieve.UpperBoundSieve) : Sieve.siftedSum s ≤ s.totalMass * Sieve.mainSum s μPlus.μPlus + Sieve.errSum s μPlus.μPlus"} +{"name":"Sieve.mainSum","declaration":"def Sieve.mainSum (s : Sieve) (μPlus : ℕ → ℝ) : ℝ"} +{"name":"Sieve.LowerBoundSieve.mk","declaration":"ctor Sieve.LowerBoundSieve.mk (μMinus : ℕ → ℝ) (hμMinus : Sieve.LowerMoebius μMinus) : Sieve.LowerBoundSieve"} +{"name":"Sieve.selbergTerms_pos","declaration":"theorem Sieve.selbergTerms_pos (s : Sieve) (l : ℕ) (hl : l ∣ s.prodPrimes) : 0 < (Sieve.selbergTerms s) l"} +{"name":"Sieve.UpperBoundSieve","declaration":"structure Sieve.UpperBoundSieve : Type"} +{"name":"Sieve.prodPrimes","declaration":"def Sieve.prodPrimes (self : Sieve) : ℕ"} +{"name":"Sieve.selbergTerms_mult","declaration":"theorem Sieve.selbergTerms_mult (s : Sieve) : ArithmeticFunction.IsMultiplicative (Sieve.selbergTerms s)"} +{"name":"Sieve.siftedSum","declaration":"def Sieve.siftedSum (s : Sieve) : ℝ"} +{"name":"Sieve.nu_pos_of_prime","declaration":"def Sieve.nu_pos_of_prime (self : Sieve) (p : ℕ) : Nat.Prime p → p ∣ self.prodPrimes → 0 < self.nu p"} +{"name":"Sieve.nu_pos_of_dvd_prodPrimes","declaration":"theorem Sieve.nu_pos_of_dvd_prodPrimes (s : Sieve) {d : ℕ} (hd : d ∣ s.prodPrimes) : 0 < s.nu d"} +{"name":"Sieve.nu_mult","declaration":"def Sieve.nu_mult (self : Sieve) : ArithmeticFunction.IsMultiplicative self.nu"} +{"name":"Sieve.nu_ne_zero","declaration":"theorem Sieve.nu_ne_zero (s : Sieve) {d : ℕ} (hd : d ∣ s.prodPrimes) : s.nu d ≠ 0"} +{"name":"Sieve.UpperBoundSieve.μPlus","declaration":"def Sieve.UpperBoundSieve.μPlus (self : Sieve.UpperBoundSieve) : ℕ → ℝ"} +{"name":"Sieve.nu_eq_conv_one_div_selbergTerms","declaration":"theorem Sieve.nu_eq_conv_one_div_selbergTerms (s : Sieve) (d : ℕ) (hdP : d ∣ s.prodPrimes) : (s.nu d)⁻¹ = Finset.sum (Nat.divisors s.prodPrimes) fun l => if l ∣ d then 1 / (Sieve.selbergTerms s) l else 0"} +{"name":"Sieve.LowerMoebius","declaration":"def Sieve.LowerMoebius (μMinus : ℕ → ℝ) : Prop"} +{"name":"Sieve.UpperMoebius","declaration":"def Sieve.UpperMoebius (μ_plus : ℕ → ℝ) : Prop"} +{"name":"Sieve.prodPrimes_squarefree","declaration":"def Sieve.prodPrimes_squarefree (self : Sieve) : Squarefree self.prodPrimes"} +{"name":"Sieve.one_div_selbergTerms_eq_conv_moebius_nu","declaration":"theorem Sieve.one_div_selbergTerms_eq_conv_moebius_nu (s : Sieve) (l : ℕ) (hl : Squarefree l) (hnu_nonzero : s.nu l ≠ 0) : 1 / (Sieve.selbergTerms s) l = Finset.sum (Nat.divisors l) fun d => ↑(ArithmeticFunction.moebius (l / d)) * (s.nu d)⁻¹"} +{"name":"Sieve.multSum","declaration":"def Sieve.multSum (s : Sieve) (d : ℕ) : ℝ"} +{"name":"Sieve.mk","declaration":"ctor Sieve.mk (support : Finset ℕ) (prodPrimes : ℕ) (prodPrimes_squarefree : Squarefree prodPrimes) (weights : ℕ → ℝ) (weights_nonneg : ∀ (n : ℕ), 0 ≤ weights n) (totalMass : ℝ) (nu : ArithmeticFunction ℝ) (nu_mult : ArithmeticFunction.IsMultiplicative nu) (nu_pos_of_prime : ∀ (p : ℕ), Nat.Prime p → p ∣ prodPrimes → 0 < nu p) (nu_lt_one_of_prime : ∀ (p : ℕ), Nat.Prime p → p ∣ prodPrimes → nu p < 1) : Sieve"} +{"name":"Sieve","declaration":"structure Sieve : Type"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0422f919e1d47292bb3975219284deb616c3a828 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg.jsonl @@ -0,0 +1,29 @@ +{"name":"SelbergSieve.selbergBoundingSum_pos","declaration":"theorem SelbergSieve.selbergBoundingSum_pos (s : SelbergSieve) : 0 < SelbergSieve.selbergBoundingSum s"} +{"name":"SelbergSieve.selberg_bound_muPlus","declaration":"theorem SelbergSieve.selberg_bound_muPlus (s : SelbergSieve) (n : ℕ) (hn : n ∈ Nat.divisors s.prodPrimes) : |SelbergSieve.selbergMuPlus s n| ≤ 3 ^ ArithmeticFunction.cardDistinctFactors n"} +{"name":"SelbergSieve.selbergBoundingSum_nonneg","declaration":"theorem SelbergSieve.selbergBoundingSum_nonneg (s : SelbergSieve) : 0 ≤ SelbergSieve.selbergBoundingSum s"} +{"name":"SelbergSieve.selbergWeights_mul_mu_nonneg","declaration":"theorem SelbergSieve.selbergWeights_mul_mu_nonneg (s : SelbergSieve) (d : ℕ) (hdP : d ∣ s.prodPrimes) : 0 ≤ SelbergSieve.selbergWeights s d * ↑(ArithmeticFunction.moebius d)"} +{"name":"SelbergSieve.selbergWeights_eq_zero_of_not_dvd","declaration":"theorem SelbergSieve.selbergWeights_eq_zero_of_not_dvd (s : SelbergSieve) {d : ℕ} (hd : ¬d ∣ s.prodPrimes) : SelbergSieve.selbergWeights s d = 0"} +{"name":"SelbergSieve.selberg_bound_simple_mainSum","declaration":"theorem SelbergSieve.selberg_bound_simple_mainSum (s : SelbergSieve) : Sieve.mainSum s.toSieve (SelbergSieve.selbergMuPlus s) = (SelbergSieve.selbergBoundingSum s)⁻¹"} +{"name":"SelbergSieve.selbergWeights_eq_dvds_sum","declaration":"theorem SelbergSieve.selbergWeights_eq_dvds_sum (s : SelbergSieve) (d : ℕ) : s.nu d * SelbergSieve.selbergWeights s d =\n (SelbergSieve.selbergBoundingSum s)⁻¹ * ↑(ArithmeticFunction.moebius d) *\n Finset.sum (Nat.divisors s.prodPrimes) fun l =>\n if d ∣ l ∧ ↑l ^ 2 ≤ s.level then (Sieve.selbergTerms s.toSieve) l else 0"} +{"name":"SelbergSieve.level","declaration":"def SelbergSieve.level (self : SelbergSieve) : ℝ"} +{"name":"SelbergSieve.selberg_bound_weights","declaration":"theorem SelbergSieve.selberg_bound_weights (s : SelbergSieve) (d : ℕ) : |SelbergSieve.selbergWeights s d| ≤ 1"} +{"name":"SelbergSieve.abs_moebius_eq_one_of_squarefree","declaration":"theorem SelbergSieve.abs_moebius_eq_one_of_squarefree {l : ℕ} (hl : Squarefree l) : |ArithmeticFunction.moebius l| = 1"} +{"name":"SelbergSieve.selbergWeights","declaration":"def SelbergSieve.selbergWeights (s : SelbergSieve) : ℕ → ℝ"} +{"name":"SelbergSieve.selbergWeights_eq_zero","declaration":"theorem SelbergSieve.selbergWeights_eq_zero (s : SelbergSieve) (d : ℕ) (hd : ¬↑d ^ 2 ≤ s.level) : SelbergSieve.selbergWeights s d = 0"} +{"name":"SelbergSieve.selbergUbSieve","declaration":"def SelbergSieve.selbergUbSieve (s : SelbergSieve) : Sieve.UpperBoundSieve"} +{"name":"SelbergSieve","declaration":"structure SelbergSieve : Type"} +{"name":"SelbergSieve.mainSum_eq_diag_quad_form","declaration":"theorem SelbergSieve.mainSum_eq_diag_quad_form (s : SelbergSieve) : Sieve.mainSum s.toSieve (SelbergSieve.selbergMuPlus s) =\n Finset.sum (Nat.divisors s.prodPrimes) fun l =>\n 1 / (Sieve.selbergTerms s.toSieve) l *\n (Finset.sum (Nat.divisors s.prodPrimes) fun d => if l ∣ d then s.nu d * SelbergSieve.selbergWeights s d else 0) ^\n 2"} +{"name":"SelbergSieve.selbergBoundingSum_ge","declaration":"theorem SelbergSieve.selbergBoundingSum_ge (s : SelbergSieve) {d : ℕ} (hdP : d ∣ s.prodPrimes) : SelbergSieve.selbergBoundingSum s ≥\n SelbergSieve.selbergWeights s d * ↑(ArithmeticFunction.moebius d) * SelbergSieve.selbergBoundingSum s"} +{"name":"SelbergSieve.moebius_sq_eq_one_of_squarefree","declaration":"/-- These two are in Mathlib per #10672 -/\ntheorem SelbergSieve.moebius_sq_eq_one_of_squarefree {l : ℕ} (hl : Squarefree l) : ArithmeticFunction.moebius l ^ 2 = 1"} +{"name":"SelbergSieve.weight_one_of_selberg","declaration":"theorem SelbergSieve.weight_one_of_selberg (s : SelbergSieve) : SelbergSieve.selbergWeights s 1 = 1"} +{"name":"SelbergSieve.selbergBoundingSum_ne_zero","declaration":"theorem SelbergSieve.selbergBoundingSum_ne_zero (s : SelbergSieve) : SelbergSieve.selbergBoundingSum s ≠ 0"} +{"name":"SelbergSieve.selbergMuPlus","declaration":"def SelbergSieve.selbergMuPlus (s : SelbergSieve) : ℕ → ℝ"} +{"name":"SelbergSieve.selberg_bound_simple","declaration":"theorem SelbergSieve.selberg_bound_simple (s : SelbergSieve) : Sieve.siftedSum s.toSieve ≤\n s.totalMass / SelbergSieve.selbergBoundingSum s +\n Finset.sum (Nat.divisors s.prodPrimes) fun d =>\n if ↑d ≤ s.level then 3 ^ ArithmeticFunction.cardDistinctFactors d * |Sieve.rem s.toSieve d| else 0"} +{"name":"SelbergSieve.one_le_level","declaration":"def SelbergSieve.one_le_level (self : SelbergSieve) : 1 ≤ self.level"} +{"name":"SelbergSieve.selbergμPlus_eq_zero","declaration":"theorem SelbergSieve.selbergμPlus_eq_zero (s : SelbergSieve) (d : ℕ) (hd : ¬↑d ≤ s.level) : SelbergSieve.selbergMuPlus s d = 0"} +{"name":"SelbergSieve.sum_mul_subst","declaration":"theorem SelbergSieve.sum_mul_subst (k : ℕ) (n : ℕ) {f : ℕ → ℝ} (h : ∀ (l : ℕ), l ∣ n → ¬k ∣ l → f l = 0) : (Finset.sum (Nat.divisors n) fun l => f l) = Finset.sum (Nat.divisors n) fun m => if k * m ∣ n then f (k * m) else 0"} +{"name":"SelbergSieve.selbergWeights_diagonalisation","declaration":"theorem SelbergSieve.selbergWeights_diagonalisation (s : SelbergSieve) (l : ℕ) (hl : l ∈ Nat.divisors s.prodPrimes) : (Finset.sum (Nat.divisors s.prodPrimes) fun d => if l ∣ d then s.nu d * SelbergSieve.selbergWeights s d else 0) =\n if ↑l ^ 2 ≤ s.level then\n (Sieve.selbergTerms s.toSieve) l * ↑(ArithmeticFunction.moebius l) * (SelbergSieve.selbergBoundingSum s)⁻¹\n else 0"} +{"name":"SelbergSieve.eq_gcd_mul_of_dvd_of_coprime","declaration":"theorem SelbergSieve.eq_gcd_mul_of_dvd_of_coprime {k : ℕ} {d : ℕ} {m : ℕ} (hkd : k ∣ d) (hmd : Nat.Coprime m d) (hk : k ≠ 0) : k = Nat.gcd d (k * m)"} +{"name":"SelbergSieve.mk","declaration":"ctor SelbergSieve.mk (toSieve : Sieve) (level : ℝ) (one_le_level : 1 ≤ level) : SelbergSieve"} +{"name":"SelbergSieve.selbergBoundingSum","declaration":"def SelbergSieve.selbergBoundingSum (s : SelbergSieve) : ℝ"} +{"name":"SelbergSieve.selberg_bound_simple_errSum","declaration":"theorem SelbergSieve.selberg_bound_simple_errSum (s : SelbergSieve) : Sieve.errSum s.toSieve (SelbergSieve.selbergMuPlus s) ≤\n Finset.sum (Nat.divisors s.prodPrimes) fun d =>\n if ↑d ≤ s.level then 3 ^ ArithmeticFunction.cardDistinctFactors d * |Sieve.rem s.toSieve d| else 0"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1933878556cc67a5c10d60a7e860ec39ac16e7a8 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds.jsonl @@ -0,0 +1,22 @@ +{"name":"Sieve.CompletelyMultiplicative.pdiv","declaration":"theorem Sieve.CompletelyMultiplicative.pdiv {f : ArithmeticFunction ℝ} {g : ArithmeticFunction ℝ} (hf : Sieve.CompletelyMultiplicative f) (hg : Sieve.CompletelyMultiplicative g) : Sieve.CompletelyMultiplicative (ArithmeticFunction.pdiv f g)"} +{"name":"Sieve.zeta_lt_self_of_prime","declaration":"theorem Sieve.zeta_lt_self_of_prime (p : ℕ) : Nat.Prime p → ↑ArithmeticFunction.zeta p < ↑p"} +{"name":"Sieve.boundingSum_ge_log","declaration":"theorem Sieve.boundingSum_ge_log (s : SelbergSieve) (hnu : s.nu = ArithmeticFunction.pdiv ↑ArithmeticFunction.zeta ↑ArithmeticFunction.id) (hP : ∀ (p : ℕ), Nat.Prime p → ↑p ≤ s.level → p ∣ s.prodPrimes) : SelbergSieve.selbergBoundingSum s ≥ Real.log s.level / 2"} +{"name":"Sieve.CompletelyMultiplicative","declaration":"def Sieve.CompletelyMultiplicative (f : ArithmeticFunction ℝ) : Prop"} +{"name":"Sieve.CompletelyMultiplicative.apply_pow","declaration":"theorem Sieve.CompletelyMultiplicative.apply_pow (f : ArithmeticFunction ℝ) (hf : Sieve.CompletelyMultiplicative f) (a : ℕ) (n : ℕ) : f (a ^ n) = f a ^ n"} +{"name":"Sieve.Nat.squarefree_dvd_pow","declaration":"theorem Sieve.Nat.squarefree_dvd_pow (a : ℕ) (b : ℕ) (N : ℕ) (ha : Squarefree a) (hab : a ∣ b ^ N) : a ∣ b"} +{"name":"Sieve.CompletelyMultiplicative.isMultiplicative","declaration":"theorem Sieve.CompletelyMultiplicative.isMultiplicative {f : ArithmeticFunction ℝ} (hf : Sieve.CompletelyMultiplicative f) : ArithmeticFunction.IsMultiplicative f"} +{"name":"Sieve.prod_factors_one_div_compMult_ge","declaration":"theorem Sieve.prod_factors_one_div_compMult_ge (M : ℕ) (f : ArithmeticFunction ℝ) (hf : Sieve.CompletelyMultiplicative f) (hf_nonneg : ∀ (n : ℕ), 0 ≤ f n) (d : ℕ) (hd : Squarefree d) (hf_size : ∀ (n : ℕ), Nat.Prime n → n ∣ d → f n < 1) : (f d * Finset.prod d.primeFactors fun p => 1 / (1 - f p)) ≥\n Finset.prod d.primeFactors fun p => Finset.sum (Finset.Icc 1 M) fun n => f (p ^ n)"} +{"name":"Sieve.sqrt_le_self","declaration":"theorem Sieve.sqrt_le_self (x : ℝ) (hx : 1 ≤ x) : Real.sqrt x ≤ x"} +{"name":"Sieve.selbergBoundingSum_ge_sum_div","declaration":"theorem Sieve.selbergBoundingSum_ge_sum_div (s : SelbergSieve) (hP : ∀ (p : ℕ), Nat.Prime p → ↑p ≤ s.level → p ∣ s.prodPrimes) (hnu : Sieve.CompletelyMultiplicative s.nu) (hnu_nonneg : ∀ (n : ℕ), 0 ≤ s.nu n) (hnu_lt : ∀ (p : ℕ), Nat.Prime p → p ∣ s.prodPrimes → s.nu p < 1) : SelbergSieve.selbergBoundingSum s ≥ Finset.sum (Finset.Icc 1 ⌊Real.sqrt s.level⌋₊) fun m => s.nu m"} +{"name":"Sieve.primorial_squarefree","declaration":"theorem Sieve.primorial_squarefree (n : ℕ) : Squarefree (primorial n)"} +{"name":"Sieve.CompletelyMultiplicative.pmul","declaration":"theorem Sieve.CompletelyMultiplicative.pmul (f : ArithmeticFunction ℝ) (g : ArithmeticFunction ℝ) (hf : Sieve.CompletelyMultiplicative f) (hg : Sieve.CompletelyMultiplicative g) : Sieve.CompletelyMultiplicative (ArithmeticFunction.pmul f g)"} +{"name":"Sieve.prod_primes_dvd_of_dvd","declaration":"theorem Sieve.prod_primes_dvd_of_dvd (P : ℕ) {s : Finset ℕ} (h : ∀ p ∈ s, p ∣ P) (h' : ∀ p ∈ s, Nat.Prime p) : (Finset.prod s fun p => p) ∣ P"} +{"name":"Sieve.prod_factors_sum_pow_compMult","declaration":"theorem Sieve.prod_factors_sum_pow_compMult (M : ℕ) (hM : M ≠ 0) (f : ArithmeticFunction ℝ) (hf : Sieve.CompletelyMultiplicative f) (d : ℕ) (hd : Squarefree d) : (Finset.prod d.primeFactors fun p => Finset.sum (Finset.Icc 1 M) fun n => f (p ^ n)) =\n Finset.sum (Finset.filter (fun x => d ∣ x) (Nat.divisors (d ^ M))) fun m => f m"} +{"name":"Sieve.zeta_pos_of_prime","declaration":"theorem Sieve.zeta_pos_of_prime (p : ℕ) : Nat.Prime p → 0 < ↑ArithmeticFunction.zeta p"} +{"name":"Sieve.prime_dvd_primorial_iff","declaration":"theorem Sieve.prime_dvd_primorial_iff (n : ℕ) (p : ℕ) (hp : Nat.Prime p) : p ∣ primorial n ↔ p ≤ n"} +{"name":"Sieve.boundingSum_ge_sum","declaration":"theorem Sieve.boundingSum_ge_sum (s : SelbergSieve) (hnu : s.nu = ArithmeticFunction.pdiv ↑ArithmeticFunction.zeta ↑ArithmeticFunction.id) (hP : ∀ (p : ℕ), Nat.Prime p → ↑p ≤ s.level → p ∣ s.prodPrimes) : SelbergSieve.selbergBoundingSum s ≥ Finset.sum (Finset.Icc 1 ⌊Real.sqrt s.level⌋₊) fun m => 1 / ↑m"} +{"name":"Sieve.siftedSum_eq","declaration":"theorem Sieve.siftedSum_eq (s : SelbergSieve) (hw : ∀ i ∈ s.support, s.weights i = 1) (z : ℝ) (hz : 1 ≤ z) (hP : s.prodPrimes = primorial ⌊z⌋₊) : Sieve.siftedSum s.toSieve = ↑(Finset.filter (fun d => ∀ (p : ℕ), Nat.Prime p → ↑p ≤ z → ¬p ∣ d) s.support).card"} +{"name":"Sieve.CompletelyMultiplicative.id","declaration":"theorem Sieve.CompletelyMultiplicative.id : Sieve.CompletelyMultiplicative ↑ArithmeticFunction.id"} +{"name":"Sieve.prodDistinctPrimes_squarefree","declaration":"theorem Sieve.prodDistinctPrimes_squarefree (s : Finset ℕ) (h : ∀ p ∈ s, Nat.Prime p) : Squarefree (Finset.prod s fun p => p)"} +{"name":"Sieve.CompletelyMultiplicative.zeta","declaration":"theorem Sieve.CompletelyMultiplicative.zeta : Sieve.CompletelyMultiplicative ↑ArithmeticFunction.zeta"} +{"name":"Sieve.rem_sum_le_of_const","declaration":"theorem Sieve.rem_sum_le_of_const (s : SelbergSieve) (C : ℝ) (hrem : ∀ d > 0, |Sieve.rem s.toSieve d| ≤ C) : (Finset.sum (Nat.divisors s.prodPrimes) fun d =>\n if ↑d ≤ s.level then 3 ^ ArithmeticFunction.cardDistinctFactors d * |Sieve.rem s.toSieve d| else 0) ≤\n C * s.level * (1 + Real.log s.level) ^ 3"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..11b51caf5af962deb3d110edf5a0f3e3e95008f5 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence.jsonl @@ -0,0 +1,2 @@ +{"name":"TendstoUniformlyOnFilter.tendsto_of_eventually_tendsto","declaration":"theorem TendstoUniformlyOnFilter.tendsto_of_eventually_tendsto {ι : Type u_1} {α : Type u_2} {β : Type u_3} [UniformSpace β] {l : Filter ι} [Filter.NeBot l] {p : Filter α} {F : ι → α → β} {f : α → β} {L : ι → β} {ℓ : β} (h1 : TendstoUniformlyOnFilter F f l p) (h2 : ∀ᶠ (i : ι) in l, Filter.Tendsto (F i) p (nhds (L i))) (h3 : Filter.Tendsto L l (nhds ℓ)) : Filter.Tendsto f p (nhds ℓ)"} +{"name":"TendstoUniformly.tendsto_of_eventually_tendsto","declaration":"theorem TendstoUniformly.tendsto_of_eventually_tendsto {ι : Type u_1} {α : Type u_2} {β : Type u_3} [UniformSpace β] {l : Filter ι} [Filter.NeBot l] {p : Filter α} {F : ι → α → β} {f : α → β} {L : ι → β} {ℓ : β} (h1 : TendstoUniformly F f l) (h2 : ∀ᶠ (i : ι) in l, Filter.Tendsto (F i) p (nhds (L i))) (h3 : Filter.Tendsto L l (nhds ℓ)) : Filter.Tendsto f p (nhds ℓ)"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.MediumPNT.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.MediumPNT.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..105de1d2a5c473f1a2fa57e5e569aac64b945e58 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.MediumPNT.jsonl @@ -0,0 +1,8 @@ +{"name":"ChebyshevPsi","declaration":"def ChebyshevPsi (x : ℝ) : ℝ"} +{"name":"mellintransform2","declaration":"def mellintransform2 : Lean.ParserDescr"} +{"name":"SmoothedChebyshev","declaration":"def SmoothedChebyshev (ψ : ℝ → ℝ) (ε : ℝ) (X : ℝ) : ℂ"} +{"name":"LogDerivativeDirichlet","declaration":"theorem LogDerivativeDirichlet (s : ℂ) (hs : 1 < s.re) : -deriv riemannZeta s / riemannZeta s = ∑' (n : ℕ), ↑(ArithmeticFunction.vonMangoldt n) / ↑n ^ s"} +{"name":"SmoothedChebyshevClose","declaration":"theorem SmoothedChebyshevClose {ψ : ℝ → ℝ} (ε : ℝ) (ε_pos : 0 < ε) (suppΨ : Function.support ψ ⊆ Set.Icc (1 / 2) 2) (Ψnonneg : ∀ x > 0, 0 ≤ ψ x) (mass_one : ∫ (x : ℝ) in Set.Ioi 0, ψ x / x = 1) (X : ℝ) : (fun X => ‖SmoothedChebyshev ψ ε X - ↑(ChebyshevPsi X)‖) =O[Filter.atTop] fun X => ε * X * Real.log X"} +{"name":"SmoothedChebyshevDirichlet","declaration":"theorem SmoothedChebyshevDirichlet {ψ : ℝ → ℝ} (diffΨ : ContDiff ℝ 1 ψ) (ε : ℝ) (εpos : 0 < ε) (suppΨ : Function.support ψ ⊆ Set.Icc (1 / 2) 2) (X : ℝ) (X_pos : 0 < X) : SmoothedChebyshev ψ ε X = ↑(∑' (n : ℕ), ArithmeticFunction.vonMangoldt n * Smooth1 ψ ε (↑n / X))"} +{"name":"MediumPNT","declaration":"/-- *** Prime Number Theorem (Medium Strength) *** The `ChebyshevPsi` function is asymptotic to `x`. -/\ntheorem MediumPNT : ∃ c, ∃ (_ : c > 0), (ChebyshevPsi - id) =O[Filter.atTop] fun x => x * Real.exp (-c * Real.log x ^ (1 / 18))"} +{"name":"SmoothedChebyshevIntegrand","declaration":"def SmoothedChebyshevIntegrand (ψ : ℝ → ℝ) (ε : ℝ) (X : ℝ) : ℂ → ℂ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.MellinCalculus.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.MellinCalculus.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cfa4c38cd43413a25803bb63a875e976919c9c03 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.MellinCalculus.jsonl @@ -0,0 +1,84 @@ +{"name":"Filter.TendstoAtZero_of_support_in_Icc","declaration":"theorem Filter.TendstoAtZero_of_support_in_Icc {𝕂 : Type u_1} [RCLike 𝕂] {a : ℝ} {b : ℝ} (f : ℝ → 𝕂) (ha : 0 < a) (fSupp : Function.support f ⊆ Set.Icc a b) : Filter.Tendsto f (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)"} +{"name":"MellinOfPsi_aux","declaration":"theorem MellinOfPsi_aux {Ψ : ℝ → ℝ} (diffΨ : ContDiff ℝ 1 Ψ) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) {s : ℂ} (hs : s ≠ 0) : ∫ (x : ℝ) in Set.Ioi 0, ↑(Ψ x) * ↑x ^ (s - 1) = -(1 / s) * ∫ (x : ℝ) in Set.Ioi 0, ↑(deriv Ψ x) * ↑x ^ s"} +{"name":"MellinInversion_aux1","declaration":"theorem MellinInversion_aux1 {f : ℝ → ℂ} {s : ℂ} (s_ne_zero : s ≠ 0) (fDiff : DifferentiableOn ℝ f (Set.Ioi 0)) (hfs : Filter.Tendsto (fun x => f x * ↑x ^ s) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)) (hfinf : Filter.Tendsto (fun x => f x * ↑x ^ s) Filter.atTop (nhds 0)) : ∫ (x : ℝ) in Set.Ioi 0, f x * ↑x ^ s / ↑x = -∫ (x : ℝ) in Set.Ioi 0, deriv f x * ↑x ^ s / s"} +{"name":"MellinConvolutionTransform","declaration":"theorem MellinConvolutionTransform (f : ℝ → ℂ) (g : ℝ → ℂ) (s : ℂ) (hf : MeasureTheory.IntegrableOn (Function.uncurry fun x y => f y * g (x / y) / ↑y * ↑x ^ (s - 1)) (Set.Ioi 0 ×ˢ Set.Ioi 0)\n MeasureTheory.volume) : MellinTransform (MellinConvolution f g) s = MellinTransform f s * MellinTransform g s"} +{"name":"Function.support_mul_subset_of_subset","declaration":"theorem Function.support_mul_subset_of_subset {𝕂 : Type u_1} [RCLike 𝕂] {s : Set ℝ} {f : ℝ → 𝕂} {g : ℝ → 𝕂} (fSupp : Function.support f ⊆ s) : Function.support (f * g) ⊆ s"} +{"name":"Complex.hasDerivAt_ofReal","declaration":"theorem Complex.hasDerivAt_ofReal (x : ℝ) : HasDerivAt Complex.ofReal' 1 x"} +{"name":"SetIntegral.integral_eq_integral_inter_of_support_subset_Icc","declaration":"theorem SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a : ℝ} {b : ℝ} {μ : MeasureTheory.Measure ℝ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set ℝ} {f : ℝ → E} (h : Function.support f ⊆ Set.Icc a b) (hs : Set.Icc a b ⊆ s) : ∫ (x : ℝ) in s, f x ∂μ = ∫ (x : ℝ) in Set.Icc a b, f x ∂μ"} +{"name":"MellinInverseTransform_eq","declaration":"theorem MellinInverseTransform_eq (σ : ℝ) (f : ℂ → ℂ) : MellinInverseTransform f σ = mellinInv σ f"} +{"name":"intervalIntegral.norm_integral_le_of_norm_le_const'","declaration":"theorem intervalIntegral.norm_integral_le_of_norm_le_const' {a : ℝ} {b : ℝ} {C : ℝ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} (hab : a ≤ b) (h : ∀ x ∈ Set.Icc a b, ‖f x‖ ≤ C) : ‖∫ (x : ℝ) in a..b, f x‖ ≤ C * |b - a|"} +{"name":"MellinOf1","declaration":"theorem MellinOf1 (s : ℂ) (h : s.re > 0) : MellinTransform (fun x => if 0 < x ∧ x ≤ 1 then 1 else 0) s = 1 / s"} +{"name":"MellinOfDeltaSpikeAt1","declaration":"theorem MellinOfDeltaSpikeAt1 (Ψ : ℝ → ℝ) {ε : ℝ} (εpos : ε > 0) : MellinTransform (fun x => ↑(DeltaSpike Ψ ε x)) 1 = MellinTransform (fun x => ↑(Ψ x)) ↑ε"} +{"name":"MellinInversion_aux3","declaration":"theorem MellinInversion_aux3 {f : ℝ → ℂ} (σ : ℝ) (σ_ne_zero : σ ≠ 0) (σ_ne_negOne : σ ≠ -1) (fInt : MeasureTheory.IntegrableOn (fun x => f x * ↑x ^ ↑σ) (Set.Ioi 0) MeasureTheory.volume) : MeasureTheory.IntegrableOn\n (fun x =>\n match x with\n | (x, t) => f x * ↑x ^ (↑σ + ↑t * Complex.I) / ((↑σ + ↑t * Complex.I) * (↑σ + ↑t * Complex.I + 1)))\n (Set.prod (Set.Ioi 0) Set.univ) MeasureTheory.volume"} +{"name":"Function.support_ofReal","declaration":"theorem Function.support_ofReal {f : ℝ → ℝ} : (Function.support fun x => ↑(f x)) = Function.support f"} +{"name":"DeltaSpike","declaration":"def DeltaSpike (Ψ : ℝ → ℝ) (ε : ℝ) : ℝ → ℝ"} +{"name":"Smooth1Properties_above","declaration":"theorem Smooth1Properties_above {Ψ : ℝ → ℝ} (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) {ε : ℝ} (hε : ε ∈ Set.Ioo 0 1) : ∃ c, 0 < c ∧ ∀ (x : ℝ), 1 + c * ε ≤ x → Smooth1 Ψ ε x = 0"} +{"name":"Smooth1Properties_above_aux2","declaration":"theorem Smooth1Properties_above_aux2 {x : ℝ} {y : ℝ} {ε : ℝ} (hε : ε ∈ Set.Ioo 0 1) (hy : y ∈ Set.Ioc 0 1) (hx2 : 2 ^ ε < x) : 2 < (x / y) ^ (1 / ε)"} +{"name":"Smooth1LeOne","declaration":"theorem Smooth1LeOne {Ψ : ℝ → ℝ} (Ψnonneg : ∀ x > 0, 0 ≤ Ψ x) (mass_one : ∫ (x : ℝ) in Set.Ioi 0, Ψ x / x = 1) {ε : ℝ} (εpos : 0 < ε) (x : ℝ) : 0 < x → Smooth1 Ψ ε x ≤ 1"} +{"name":"Smooth1Properties_below","declaration":"theorem Smooth1Properties_below {Ψ : ℝ → ℝ} (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) (ε : ℝ) (εpos : 0 < ε) (mass_one : ∫ (x : ℝ) in Set.Ioi 0, Ψ x / x = 1) : ∃ c, 0 < c ∧ ∀ (x : ℝ), 0 < x → x ≤ 1 - c * ε → Smooth1 Ψ ε x = 1"} +{"name":"DifferentiableAt.ofReal_comp_iff","declaration":"theorem DifferentiableAt.ofReal_comp_iff {z : ℝ} {f : ℝ → ℝ} : DifferentiableAt ℝ (fun y => ↑(f y)) z ↔ DifferentiableAt ℝ f z"} +{"name":"DifferentiableAt.comp_ofReal","declaration":"theorem DifferentiableAt.comp_ofReal {e : ℂ → ℂ} {z : ℝ} (hf : DifferentiableAt ℂ e ↑z) : DifferentiableAt ℝ (fun x => e ↑x) z"} +{"name":"DeltaSpikeContinuous","declaration":"theorem DeltaSpikeContinuous {Ψ : ℝ → ℝ} {ε : ℝ} (εpos : 0 < ε) (diffΨ : ContDiff ℝ 1 Ψ) : Continuous fun x => DeltaSpike Ψ ε x"} +{"name":"PartialIntegration_of_support_in_Icc","declaration":"theorem PartialIntegration_of_support_in_Icc {a : ℝ} {b : ℝ} (f : ℝ → ℂ) (g : ℝ → ℂ) (ha : 0 < a) (h : a ≤ b) (fSupp : Function.support f ⊆ Set.Icc a b) (fDiff : DifferentiableOn ℝ f (Set.Ioi 0)) (gDiff : DifferentiableOn ℝ g (Set.Ioi 0)) (fderivCont : ContinuousOn (deriv f) (Set.Ioi 0)) (gderivCont : ContinuousOn (deriv g) (Set.Ioi 0)) : ∫ (x : ℝ) in Set.Ioi 0, f x * deriv g x = -∫ (x : ℝ) in Set.Ioi 0, deriv f x * g x"} +{"name":"Smooth1Properties_below_aux","declaration":"theorem Smooth1Properties_below_aux {x : ℝ} {ε : ℝ} (hx : x ≤ 1 - Real.log 2 * ε) (εpos : 0 < ε) : x < 2 ^ (-ε)"} +{"name":"Differentiable.ofReal_comp_iff","declaration":"theorem Differentiable.ofReal_comp_iff {f : ℝ → ℝ} : (Differentiable ℝ fun y => ↑(f y)) ↔ Differentiable ℝ f"} +{"name":"MeasureTheory.integral_comp_mul_right_I0i_haar_real","declaration":"theorem MeasureTheory.integral_comp_mul_right_I0i_haar_real (f : ℝ → ℝ) {a : ℝ} (ha : 0 < a) : ∫ (y : ℝ) in Set.Ioi 0, f (y * a) / y = ∫ (y : ℝ) in Set.Ioi 0, f y / y"} +{"name":"MeasureTheory.set_integral_integral_swap","declaration":"theorem MeasureTheory.set_integral_integral_swap {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite ν] [NormedSpace ℝ E] [MeasureTheory.SigmaFinite μ] (f : α → β → E) {s : Set α} {t : Set β} (hf : MeasureTheory.IntegrableOn (Function.uncurry f) (s ×ˢ t) (MeasureTheory.Measure.prod μ ν)) : ∫ (x : α) in s, ∫ (y : β) in t, f x y ∂ν ∂μ = ∫ (y : β) in t, ∫ (x : α) in s, f x y ∂μ ∂ν"} +{"name":"MeasureTheory.integral_comp_rpow_I0i_haar_real","declaration":"theorem MeasureTheory.integral_comp_rpow_I0i_haar_real (f : ℝ → ℝ) {p : ℝ} (hp : p ≠ 0) : ∫ (y : ℝ) in Set.Ioi 0, |p| * f (y ^ p) / y = ∫ (y : ℝ) in Set.Ioi 0, f y / y"} +{"name":"mellintransform","declaration":"def mellintransform : Lean.ParserDescr"} +{"name":"MellinInverseTransform","declaration":"def MellinInverseTransform (F : ℂ → ℂ) (σ : ℝ) (x : ℝ) : ℂ"} +{"name":"DeltaSpikeSupport_aux","declaration":"theorem DeltaSpikeSupport_aux {Ψ : ℝ → ℝ} {ε : ℝ} (εpos : 0 < ε) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) : (Function.support fun x => if x < 0 then 0 else DeltaSpike Ψ ε x) ⊆ Set.Icc (2 ^ (-ε)) (2 ^ ε)"} +{"name":"MellinOfDeltaSpike","declaration":"theorem MellinOfDeltaSpike (Ψ : ℝ → ℝ) {ε : ℝ} (εpos : ε > 0) (s : ℂ) : MellinTransform (fun x => ↑(DeltaSpike Ψ ε x)) s = MellinTransform (fun x => ↑(Ψ x)) (↑ε * s)"} +{"name":"DifferentiableAt.ofReal_comp","declaration":"theorem DifferentiableAt.ofReal_comp {z : ℝ} {f : ℝ → ℝ} (hf : DifferentiableAt ℝ f z) : DifferentiableAt ℝ (fun y => ↑(f y)) z"} +{"name":"MeasureTheory.integral_comp_div_I0i_haar","declaration":"theorem MeasureTheory.integral_comp_div_I0i_haar {𝕂 : Type u_1} [RCLike 𝕂] (f : ℝ → 𝕂) {a : ℝ} (ha : 0 < a) : ∫ (y : ℝ) in Set.Ioi 0, f (a / y) / ↑y = ∫ (y : ℝ) in Set.Ioi 0, f y / ↑y"} +{"name":"Filter.BigO_zero_atZero_of_support_in_Icc","declaration":"theorem Filter.BigO_zero_atZero_of_support_in_Icc {𝕂 : Type u_1} [RCLike 𝕂] {a : ℝ} {b : ℝ} (f : ℝ → 𝕂) (ha : 0 < a) (fSupp : Function.support f ⊆ Set.Icc a b) : f =O[nhdsWithin 0 (Set.Ioi 0)] fun x => 0"} +{"name":"MellinOfSmooth1a","declaration":"theorem MellinOfSmooth1a (Ψ : ℝ → ℝ) (diffΨ : ContDiff ℝ 1 Ψ) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) {ε : ℝ} (εpos : 0 < ε) {s : ℂ} (hs : 0 < s.re) : MellinTransform (fun x => ↑(Smooth1 Ψ ε x)) s = 1 / s * MellinTransform (fun x => ↑(Ψ x)) (↑ε * s)"} +{"name":"Function.support_deriv_subset_Icc","declaration":"theorem Function.support_deriv_subset_Icc {𝕂 : Type u_1} [RCLike 𝕂] {a : ℝ} {b : ℝ} {f : ℝ → 𝕂} (fSupp : Function.support f ⊆ Set.Icc a b) : Function.support (deriv f) ⊆ Set.Icc a b"} +{"name":"MeasureTheory.integral_comp_mul_left_I0i_haar","declaration":"theorem MeasureTheory.integral_comp_mul_left_I0i_haar {𝕂 : Type u_1} [RCLike 𝕂] (f : ℝ → 𝕂) {a : ℝ} (ha : 0 < a) : ∫ (y : ℝ) in Set.Ioi 0, f (a * y) / ↑y = ∫ (y : ℝ) in Set.Ioi 0, f y / ↑y"} +{"name":"Smooth1LeOne_aux","declaration":"theorem Smooth1LeOne_aux {x : ℝ} {ε : ℝ} {Ψ : ℝ → ℝ} (xpos : 0 < x) (εpos : 0 < ε) (mass_one : ∫ (x : ℝ) in Set.Ioi 0, Ψ x / x = 1) : ∫ (y : ℝ) in Set.Ioi 0, Ψ ((x / y) ^ (1 / ε)) / ε / y = 1"} +{"name":"IntervalIntegral.integral_eq_integral_of_support_subset_Icc","declaration":"theorem IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a : ℝ} {b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} (h : Function.support f ⊆ Set.Icc a b) : ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ"} +{"name":"DeltaSpikeMass","declaration":"theorem DeltaSpikeMass {Ψ : ℝ → ℝ} (mass_one : ∫ (x : ℝ) in Set.Ioi 0, Ψ x / x = 1) {ε : ℝ} (εpos : 0 < ε) : ∫ (x : ℝ) in Set.Ioi 0, DeltaSpike Ψ ε x / x = 1"} +{"name":"MellinTransform","declaration":"def MellinTransform (f : ℝ → ℂ) (s : ℂ) : ℂ"} +{"name":"deriv.comp_ofReal","declaration":"theorem deriv.comp_ofReal {e : ℂ → ℂ} {z : ℝ} (hf : DifferentiableAt ℂ e ↑z) : deriv (fun x => e ↑x) z = deriv e ↑z"} +{"name":"MellinInversion_aux2","declaration":"theorem MellinInversion_aux2 {f : ℝ → ℂ} (s : ℂ) (fDiff : DifferentiableOn ℝ f (Set.Ioi 0)) (fDiff2 : DifferentiableOn ℝ (deriv f) (Set.Ioi 0)) (hfs : Filter.Tendsto (fun x => deriv f x * ↑x ^ s) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)) (hfinf : Filter.Tendsto (fun x => deriv f x * ↑x ^ s) Filter.atTop (nhds 0)) : ∫ (x : ℝ) in Set.Ioi 0, deriv f x * ↑x ^ s = -∫ (x : ℝ) in Set.Ioi 0, deriv (deriv f) x * ↑x ^ (s + 1) / (s + 1)"} +{"name":"PerronInverseMellin_lt","declaration":"theorem PerronInverseMellin_lt {t : ℝ} {x : ℝ} (tpos : 0 < t) (t_lt_x : t < x) {σ : ℝ} (σpos : 0 < σ) : MellinInverseTransform (Perron.f t) σ x = 0"} +{"name":"Complex.differentiable_ofReal","declaration":"theorem Complex.differentiable_ofReal : Differentiable ℝ Complex.ofReal'"} +{"name":"Differentiable.ofReal_comp","declaration":"theorem Differentiable.ofReal_comp {f : ℝ → ℝ} (hf : Differentiable ℝ f) : Differentiable ℝ fun y => ↑(f y)"} +{"name":"MellinOfSmooth1b","declaration":"theorem MellinOfSmooth1b {Ψ : ℝ → ℝ} (diffΨ : ContDiff ℝ 1 Ψ) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) {σ₁ : ℝ} {σ₂ : ℝ} (σ₁pos : 0 < σ₁) (ε : ℝ) (εpos : 0 < ε) : (fun s => ‖MellinTransform (fun x => ↑(Smooth1 Ψ ε x)) s‖) =O[Filter.principal {s | σ₁ ≤ s.re ∧ s.re ≤ σ₂}] fun s =>\n 1 / (ε * ‖s‖ ^ 2)"} +{"name":"MellinInversion_aux4","declaration":"theorem MellinInversion_aux4 {f : ℝ → ℂ} (σ : ℝ) (σ_ne_zero : σ ≠ 0) (σ_ne_negOne : σ ≠ -1) (fInt : MeasureTheory.IntegrableOn (fun x => f x * ↑x ^ ↑σ) (Set.Ioi 0) MeasureTheory.volume) : VerticalIntegral (fun s => ∫ (x : ℝ) in Set.Ioi 0, f x * ↑x ^ (s + 1) / (s * (s + 1))) σ =\n ∫ (x : ℝ) in Set.Ioi 0, VerticalIntegral (fun s => f x * ↑x ^ (s + 1) / (s * (s + 1))) σ"} +{"name":"MellinOfSmooth1c","declaration":"theorem MellinOfSmooth1c {Ψ : ℝ → ℝ} (diffΨ : ContDiff ℝ 1 Ψ) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) (mass_one : ∫ (x : ℝ) in Set.Ioi 0, Ψ x / x = 1) : (fun ε => MellinTransform (fun x => ↑(Smooth1 Ψ ε x)) 1 - 1) =O[nhdsWithin 0 (Set.Ioi 0)] id"} +{"name":"Function.support_id","declaration":"theorem Function.support_id : (Function.support fun x => x) = Set.Iio 0 ∪ Set.Ioi 0"} +{"name":"Smooth1","declaration":"def Smooth1 (Ψ : ℝ → ℝ) (ε : ℝ) : ℝ → ℝ"} +{"name":"PartialIntegration","declaration":"/-- *Need differentiability, and decay at `0` and `∞`* -/\ntheorem PartialIntegration (f : ℝ → ℂ) (g : ℝ → ℂ) (fDiff : DifferentiableOn ℝ f (Set.Ioi 0)) (gDiff : DifferentiableOn ℝ g (Set.Ioi 0)) (fDerivgInt : MeasureTheory.IntegrableOn (f * deriv g) (Set.Ioi 0) MeasureTheory.volume) (gDerivfInt : MeasureTheory.IntegrableOn (deriv f * g) (Set.Ioi 0) MeasureTheory.volume) (lim_at_zero : Filter.Tendsto (f * g) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)) (lim_at_inf : Filter.Tendsto (f * g) Filter.atTop (nhds 0)) : ∫ (x : ℝ) in Set.Ioi 0, f x * deriv g x = -∫ (x : ℝ) in Set.Ioi 0, deriv f x * g x"} +{"name":"MellinConvNonNeg_of_NonNeg","declaration":"theorem MellinConvNonNeg_of_NonNeg {f : ℝ → ℝ} {g : ℝ → ℝ} (f_nonneg : ∀ x > 0, 0 ≤ f x) (g_nonneg : ∀ x > 0, 0 ≤ g x) {x : ℝ} (xpos : 0 < x) : 0 ≤ MellinConvolution f g x"} +{"name":"DeltaSpikeSupport'","declaration":"theorem DeltaSpikeSupport' {Ψ : ℝ → ℝ} {ε : ℝ} {x : ℝ} (εpos : 0 < ε) (xnonneg : 0 ≤ x) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) : DeltaSpike Ψ ε x ≠ 0 → x ∈ Set.Icc (2 ^ (-ε)) (2 ^ ε)"} +{"name":"MellinConvolutionSymmetric","declaration":"theorem MellinConvolutionSymmetric {𝕂 : Type u_1} [RCLike 𝕂] (f : ℝ → 𝕂) (g : ℝ → 𝕂) {x : ℝ} (xpos : 0 < x) : MellinConvolution f g x = MellinConvolution g f x"} +{"name":"deriv.ofReal_comp","declaration":"theorem deriv.ofReal_comp {z : ℝ} {f : ℝ → ℝ} : deriv (fun y => ↑(f y)) z = ↑(deriv f z)"} +{"name":"SetIntegral.integral_eq_integral_inter_of_support_subset","declaration":"theorem SetIntegral.integral_eq_integral_inter_of_support_subset {μ : MeasureTheory.Measure ℝ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set ℝ} {t : Set ℝ} {f : ℝ → E} (h : Function.support f ⊆ t) (ht : MeasurableSet t) : ∫ (x : ℝ) in s, f x ∂μ = ∫ (x : ℝ) in s ∩ t, f x ∂μ"} +{"name":"Filter.TendstoAtTop_of_support_in_Icc","declaration":"theorem Filter.TendstoAtTop_of_support_in_Icc {𝕂 : Type u_1} [RCLike 𝕂] {a : ℝ} {b : ℝ} (f : ℝ → 𝕂) (fSupp : Function.support f ⊆ Set.Icc a b) : Filter.Tendsto f Filter.atTop (nhds 0)"} +{"name":"HasDerivAt.of_hasDerivAt_ofReal_comp","declaration":"theorem HasDerivAt.of_hasDerivAt_ofReal_comp {z : ℝ} {f : ℝ → ℝ} {u : ℂ} (hf : HasDerivAt (fun y => ↑(f y)) u z) : ∃ u', u = ↑u' ∧ HasDerivAt f u' z"} +{"name":"Filter.BigO_zero_atTop_of_support_in_Icc","declaration":"theorem Filter.BigO_zero_atTop_of_support_in_Icc {𝕂 : Type u_1} [RCLike 𝕂] {a : ℝ} {b : ℝ} (f : ℝ → 𝕂) (fSupp : Function.support f ⊆ Set.Icc a b) : f =O[Filter.atTop] fun x => 0"} +{"name":"Complex.differentiableAt_ofReal","declaration":"theorem Complex.differentiableAt_ofReal (x : ℝ) : DifferentiableAt ℝ Complex.ofReal' x"} +{"name":"MellinConvolution","declaration":"def MellinConvolution {𝕂 : Type u_1} [RCLike 𝕂] (f : ℝ → 𝕂) (g : ℝ → 𝕂) (x : ℝ) : 𝕂"} +{"name":"MellinInversion","declaration":"theorem MellinInversion (σ : ℝ) {f : ℝ → ℂ} {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f ↑σ) (hFf : Complex.VerticalIntegrable (mellin f) σ MeasureTheory.volume) (hfx : ContinuousAt f x) : MellinInverseTransform (MellinTransform f) σ x = f x"} +{"name":"MellinOfPsi","declaration":"theorem MellinOfPsi {Ψ : ℝ → ℝ} (diffΨ : ContDiff ℝ 1 Ψ) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) {σ₁ : ℝ} {σ₂ : ℝ} (σ₁pos : 0 < σ₁) : (fun s => ‖MellinTransform (fun x => ↑(Ψ x)) s‖) =O[Filter.principal {s | σ₁ ≤ s.re ∧ s.re ≤ σ₂}] fun s => 1 / ‖s‖"} +{"name":"Function.support_of_along_fiber_subset_subset","declaration":"theorem Function.support_of_along_fiber_subset_subset {α : Type u_2} {β : Type u_3} {M : Type u_4} [Zero M] {f : α × β → M} {s : Set α} {t : Set β} (hx : ∀ (y : β), (Function.support fun x => f (x, y)) ⊆ s) (hy : ∀ (x : α), (Function.support fun y => f (x, y)) ⊆ t) : Function.support f ⊆ s ×ˢ t"} +{"name":"SmoothExistence","declaration":"theorem SmoothExistence : ∃ Ψ,\n ContDiff ℝ ⊤ Ψ ∧ (∀ (x : ℝ), 0 ≤ Ψ x) ∧ Function.support Ψ ⊆ Set.Icc (1 / 2) 2 ∧ ∫ (x : ℝ) in Set.Ici 0, Ψ x / x = 1"} +{"name":"Smooth1Nonneg","declaration":"theorem Smooth1Nonneg {Ψ : ℝ → ℝ} (Ψnonneg : ∀ x > 0, 0 ≤ Ψ x) {ε : ℝ} {x : ℝ} (xpos : 0 < x) (εpos : 0 < ε) : 0 ≤ Smooth1 Ψ ε x"} +{"name":"Smooth1Properties_above_aux","declaration":"theorem Smooth1Properties_above_aux {x : ℝ} {ε : ℝ} (hx : 1 + 2 * Real.log 2 * ε ≤ x) (hε : ε ∈ Set.Ioo 0 1) : 2 ^ ε < x"} +{"name":"DeltaSpikeSupport","declaration":"theorem DeltaSpikeSupport {Ψ : ℝ → ℝ} {ε : ℝ} {x : ℝ} (εpos : 0 < ε) (xnonneg : 0 ≤ x) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) : x ∉ Set.Icc (2 ^ (-ε)) (2 ^ ε) → DeltaSpike Ψ ε x = 0"} +{"name":"MeasureTheory.integral_comp_mul_right_I0i_haar","declaration":"theorem MeasureTheory.integral_comp_mul_right_I0i_haar {𝕂 : Type u_1} [RCLike 𝕂] (f : ℝ → 𝕂) {a : ℝ} (ha : 0 < a) : ∫ (y : ℝ) in Set.Ioi 0, f (y * a) / ↑y = ∫ (y : ℝ) in Set.Ioi 0, f y / ↑y"} +{"name":"Differentiable.comp_ofReal","declaration":"theorem Differentiable.comp_ofReal {e : ℂ → ℂ} (h : Differentiable ℂ e) : Differentiable ℝ fun x => e ↑x"} +{"name":"Complex.deriv_ofReal","declaration":"theorem Complex.deriv_ofReal (x : ℝ) : deriv Complex.ofReal' x = 1"} +{"name":"Complex.ofReal_rpow","declaration":"theorem Complex.ofReal_rpow {x : ℝ} (h : x > 0) (y : ℝ) : ↑(x ^ y) = ↑x ^ ↑y"} +{"name":"MellinOfDeltaSpikeAt1_asymp","declaration":"theorem MellinOfDeltaSpikeAt1_asymp {Ψ : ℝ → ℝ} (diffΨ : ContDiff ℝ 1 Ψ) (suppΨ : Function.support Ψ ⊆ Set.Icc (1 / 2) 2) (mass_one : ∫ (x : ℝ) in Set.Ioi 0, Ψ x / x = 1) : (fun ε => MellinTransform (fun x => ↑(Ψ x)) ↑ε - 1) =O[nhdsWithin 0 (Set.Ioi 0)] id"} +{"name":"Smooth1Properties_estimate","declaration":"theorem Smooth1Properties_estimate {ε : ℝ} (εpos : 0 < ε) : (1 - 2 ^ (-ε)) / ε < Real.log 2"} +{"name":"deriv.ofReal_comp'","declaration":"theorem deriv.ofReal_comp' {f : ℝ → ℝ} : (deriv fun x => ↑(f x)) = fun x => ↑(deriv f x)"} +{"name":"MeasureTheory.integral_comp_inv_I0i_haar","declaration":"theorem MeasureTheory.integral_comp_inv_I0i_haar {𝕂 : Type u_1} [RCLike 𝕂] (f : ℝ → 𝕂) : ∫ (y : ℝ) in Set.Ioi 0, f (1 / y) / ↑y = ∫ (y : ℝ) in Set.Ioi 0, f y / ↑y"} +{"name":"DeltaSpikeNonNeg_of_NonNeg","declaration":"theorem DeltaSpikeNonNeg_of_NonNeg {Ψ : ℝ → ℝ} (Ψnonneg : ∀ x > 0, 0 ≤ Ψ x) {x : ℝ} {ε : ℝ} (xpos : 0 < x) (εpos : 0 < ε) : 0 ≤ DeltaSpike Ψ ε x"} +{"name":"PerronInverseMellin_gt","declaration":"theorem PerronInverseMellin_gt {t : ℝ} {x : ℝ} (xpos : 0 < x) (x_lt_t : x < t) {σ : ℝ} (σpos : 0 < σ) : MellinInverseTransform (Perron.f t) σ x = 1 - ↑x / ↑t"} +{"name":"mem_within_strip","declaration":"theorem mem_within_strip (σ₁ : ℝ) (σ₂ : ℝ) : {s | σ₁ ≤ s.re ∧ s.re ≤ σ₂} ∈ Filter.principal {s | σ₁ ≤ s.re ∧ s.re ≤ σ₂}"} +{"name":"Function.support_abs","declaration":"theorem Function.support_abs {𝕂 : Type u_1} [RCLike 𝕂] {α : Type u_2} (f : α → 𝕂) : (Function.support fun x => ‖f x‖) = Function.support f"} +{"name":"deriv.comp_ofReal'","declaration":"theorem deriv.comp_ofReal' {e : ℂ → ℂ} (hf : Differentiable ℂ e) : (deriv fun x => e ↑x) = fun x => deriv e ↑x"} +{"name":"MellinTransform_eq","declaration":"theorem MellinTransform_eq : MellinTransform = mellin"} +{"name":"DeltaSpikeOfRealContinuous","declaration":"theorem DeltaSpikeOfRealContinuous {Ψ : ℝ → ℝ} {ε : ℝ} (εpos : 0 < ε) (diffΨ : ContDiff ℝ 1 Ψ) : Continuous fun x => ↑(DeltaSpike Ψ ε x)"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.PerronFormula.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.PerronFormula.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b837d5f514f2dcd61b6c514515752a1f82d15658 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.PerronFormula.jsonl @@ -0,0 +1,48 @@ +{"name":"Perron.residuePull1","declaration":"theorem Perron.residuePull1 {x : ℝ} {σ : ℝ} (x_gt_one : 1 < x) (σ_pos : 0 < σ) : VerticalIntegral' (Perron.f x) σ = 1 + VerticalIntegral' (Perron.f x) (-1 / 2)"} +{"name":"Complex.cpow_neg_eq_inv_pow_ofReal_pos","declaration":"theorem Complex.cpow_neg_eq_inv_pow_ofReal_pos {a : ℝ} (ha : 0 < a) (r : ℂ) : ↑a ^ (-r) = (↑a)⁻¹ ^ r"} +{"name":"limitOfConstantLeft","declaration":"theorem limitOfConstantLeft {a : ℝ → ℂ} {σ : ℝ} (σlt : σ ≤ -3 / 2) (ha : ∀ (σ' σ'' : ℝ), σ' ≤ -3 / 2 → σ'' ≤ -3 / 2 → a σ' = a σ'') (ha' : Filter.Tendsto a Filter.atBot (nhds 0)) : a σ = 0"} +{"name":"Perron.formulaLtOne","declaration":"theorem Perron.formulaLtOne {x : ℝ} {σ : ℝ} (xpos : 0 < x) (x_lt_one : x < 1) (σ_pos : 0 < σ) : VerticalIntegral (Perron.f x) σ = 0"} +{"name":"Perron.sPlusOneNeZero","declaration":"theorem Perron.sPlusOneNeZero {s : ℂ} (s_ne_neg_one : s ≠ -1) : s + 1 ≠ 0"} +{"name":"Perron.diffBddAtZero","declaration":"theorem Perron.diffBddAtZero {x : ℝ} (xpos : 0 < x) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n BddAbove ((norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - 1 / s) '' (Square 0 c \\ {0}))"} +{"name":"Perron.vertIntBoundLeft","declaration":"theorem Perron.vertIntBoundLeft {x : ℝ} (xpos : 0 < x) : ∃ C, ∀ σ < -3 / 2, ‖VerticalIntegral' (Perron.f x) σ‖ ≤ C * x ^ σ"} +{"name":"Perron.tendsto_zero_Upper","declaration":"theorem Perron.tendsto_zero_Upper {x : ℝ} (xpos : 0 < x) (σ' : ℝ) (σ'' : ℝ) : Filter.Tendsto (fun t => ∫ (σ : ℝ) in σ'..σ'', Perron.f x (↑σ + ↑t * Complex.I)) Filter.atTop (nhds 0)"} +{"name":"Perron.horizontal_integral_isBigO","declaration":"theorem Perron.horizontal_integral_isBigO {x : ℝ} (xpos : 0 < x) (σ' : ℝ) (σ'' : ℝ) (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsLocallyFiniteMeasure μ] : (fun y => ∫ (σ : ℝ) in σ'..σ'', Perron.f x (↑σ + ↑y * Complex.I) ∂μ) =O[Filter.atBot ⊔ Filter.atTop] fun y => 1 / y ^ 2"} +{"name":"Perron.integralPosAux'_of_le","declaration":"theorem Perron.integralPosAux'_of_le (c₁ : ℝ) (c₂ : ℝ) (c₁_pos : 0 < c₁) (hle : c₁ ≤ c₂) : 0 < ∫ (t : ℝ), 1 / (Real.sqrt (c₁ + t ^ 2) * Real.sqrt (c₂ + t ^ 2))"} +{"name":"Perron.HolomorphicOn.lowerUIntegral_eq_zero","declaration":"theorem Perron.HolomorphicOn.lowerUIntegral_eq_zero {f : ℂ → ℂ} {σ : ℝ} {σ' : ℝ} {T : ℝ} (hσ : σ ≤ σ') (hf : HolomorphicOn f {z | σ ≤ z.re ∧ z.re ≤ σ' ∧ z.im ≤ -T}) (hbot : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atBot (nhds 0)) (hleft : MeasureTheory.Integrable (fun y => f (↑σ + ↑y * Complex.I)) MeasureTheory.volume) (hright : MeasureTheory.Integrable (fun y => f (↑σ' + ↑y * Complex.I)) MeasureTheory.volume) : LowerUIntegral f σ σ' T = 0"} +{"name":"RectangleIntegral_tendsTo_UpperU","declaration":"theorem RectangleIntegral_tendsTo_UpperU {σ : ℝ} {σ' : ℝ} {T : ℝ} {f : ℂ → ℂ} (htop : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atTop (nhds 0)) (hleft : MeasureTheory.Integrable (fun y => f (↑σ + ↑y * Complex.I)) MeasureTheory.volume) (hright : MeasureTheory.Integrable (fun y => f (↑σ' + ↑y * Complex.I)) MeasureTheory.volume) : Filter.Tendsto (fun U => RectangleIntegral f (↑σ + Complex.I * ↑T) (↑σ' + Complex.I * ↑U)) Filter.atTop\n (nhds (UpperUIntegral f σ σ' T))"} +{"name":"Perron.bddAbove_square_of_tendsto","declaration":"theorem Perron.bddAbove_square_of_tendsto {β : Type u_2} [LinearOrder β] [NoMaxOrder β] [TopologicalSpace β] [ClosedIciTopology β] {y : β} {f : ℂ → β} {x : ℂ} (hf : Filter.Tendsto f (nhdsWithin x {x}ᶜ) (nhds y)) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0), BddAbove (f '' (Square x c \\ {x}))"} +{"name":"Perron.f_mul_eq_f","declaration":"theorem Perron.f_mul_eq_f {x : ℝ} {t : ℝ} (tpos : 0 < t) (xpos : 0 < x) (s : ℂ) : Perron.f t s * ↑x ^ (-s) = Perron.f (t / x) s"} +{"name":"Perron.contourPull","declaration":"theorem Perron.contourPull {x : ℝ} {σ' : ℝ} {σ'' : ℝ} (xpos : 0 < x) (hσ0 : 0 ∉ Set.uIcc σ' σ'') (hσ1 : -1 ∉ Set.uIcc σ' σ'') : VerticalIntegral (Perron.f x) σ' = VerticalIntegral (Perron.f x) σ''"} +{"name":"Perron.residueAtNegOne","declaration":"theorem Perron.residueAtNegOne {x : ℝ} (xpos : 0 < x) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n RectangleIntegral' (Perron.f x) (-↑c - ↑c * Complex.I - 1) (↑c + ↑c * Complex.I - 1) = -↑x⁻¹"} +{"name":"RectangleIntegral_tendsTo_LowerU","declaration":"theorem RectangleIntegral_tendsTo_LowerU {σ : ℝ} {σ' : ℝ} {T : ℝ} {f : ℂ → ℂ} (hbot : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atBot (nhds 0)) (hleft : MeasureTheory.Integrable (fun y => f (↑σ + ↑y * Complex.I)) MeasureTheory.volume) (hright : MeasureTheory.Integrable (fun y => f (↑σ' + ↑y * Complex.I)) MeasureTheory.volume) : Filter.Tendsto (fun U => RectangleIntegral f (↑σ - Complex.I * ↑U) (↑σ' - Complex.I * ↑T)) Filter.atTop\n (nhds (-LowerUIntegral f σ σ' T))"} +{"name":"Perron.integralPosAux'","declaration":"theorem Perron.integralPosAux' (c₁ : ℝ) (c₂ : ℝ) (c₁_pos : 0 < c₁) (c₂_pos : 0 < c₂) : 0 < ∫ (t : ℝ), 1 / (Real.sqrt (c₁ + t ^ 2) * Real.sqrt (c₂ + t ^ 2))"} +{"name":"RectangleIntegral_tendsTo_VerticalIntegral","declaration":"theorem RectangleIntegral_tendsTo_VerticalIntegral {σ : ℝ} {σ' : ℝ} {f : ℂ → ℂ} (hbot : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atBot (nhds 0)) (htop : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atTop (nhds 0)) (hleft : MeasureTheory.Integrable (fun y => f (↑σ + ↑y * Complex.I)) MeasureTheory.volume) (hright : MeasureTheory.Integrable (fun y => f (↑σ' + ↑y * Complex.I)) MeasureTheory.volume) : Filter.Tendsto (fun T => RectangleIntegral f (↑σ - Complex.I * ↑T) (↑σ' + Complex.I * ↑T)) Filter.atTop\n (nhds (VerticalIntegral f σ' - VerticalIntegral f σ))"} +{"name":"Perron.Integrable.one_div_const_add_sq","declaration":"theorem Perron.Integrable.one_div_const_add_sq (c : ℝ) (hc : 0 < c) : MeasureTheory.Integrable (fun t => 1 / (c + t ^ 2)) MeasureTheory.volume"} +{"name":"Perron.map_conj","declaration":"theorem Perron.map_conj {x : ℝ} (hx : 0 ≤ x) (s : ℂ) : Perron.f x ((starRingEnd ℂ) s) = (starRingEnd ℂ) (Perron.f x s)"} +{"name":"verticalIntegral_sub_verticalIntegral_eq_squareIntegral","declaration":"theorem verticalIntegral_sub_verticalIntegral_eq_squareIntegral {σ : ℝ} {σ' : ℝ} {f : ℂ → ℂ} {p : ℂ} (hσ : σ < p.re ∧ p.re < σ') (hf : HolomorphicOn f (Set.Icc σ σ' ×ℂ Set.univ \\ {p})) (hbot : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atBot (nhds 0)) (htop : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atTop (nhds 0)) (hleft : MeasureTheory.Integrable (fun y => f (↑σ + ↑y * Complex.I)) MeasureTheory.volume) (hright : MeasureTheory.Integrable (fun y => f (↑σ' + ↑y * Complex.I)) MeasureTheory.volume) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n VerticalIntegral f σ' - VerticalIntegral f σ =\n RectangleIntegral f (-↑c - ↑c * Complex.I + p) (↑c + ↑c * Complex.I + p)"} +{"name":"Perron.integral_one_div_const_add_sq_pos","declaration":"theorem Perron.integral_one_div_const_add_sq_pos (c : ℝ) (hc : 0 < c) : 0 < ∫ (t : ℝ), 1 / (c + t ^ 2)"} +{"name":"zeroTendstoDiff","declaration":"theorem zeroTendstoDiff (L₁ : ℂ) (L₂ : ℂ) (f : ℝ → ℂ) (h : ∀ᶠ (T : ℝ) in Filter.atTop, f T = 0) (h' : Filter.Tendsto f Filter.atTop (nhds (L₂ - L₁))) : L₁ = L₂"} +{"name":"instMeasurableDiv₂RealMeasurableSpaceToDivInstLinearOrderedField","declaration":"instance instMeasurableDiv₂RealMeasurableSpaceToDivInstLinearOrderedField : MeasurableDiv₂ ℝ"} +{"name":"limitOfConstant","declaration":"theorem limitOfConstant {a : ℝ → ℂ} {σ : ℝ} (σpos : 0 < σ) (ha : ∀ (σ' σ'' : ℝ), 0 < σ' → 0 < σ'' → a σ' = a σ'') (ha' : Filter.Tendsto a Filter.atTop (nhds 0)) : a σ = 0"} +{"name":"Perron.formulaGtOne","declaration":"theorem Perron.formulaGtOne {x : ℝ} {σ : ℝ} (x_gt_one : 1 < x) (σ_pos : 0 < σ) : VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) σ = 1 - 1 / ↑x"} +{"name":"Perron.isTheta_uniformlyOn_uIoc","declaration":"theorem Perron.isTheta_uniformlyOn_uIoc {x : ℝ} (xpos : 0 < x) (σ' : ℝ) (σ'' : ℝ) : (fun x_1 =>\n match x_1 with\n | (σ, y) => Perron.f x (↑σ + ↑y * Complex.I)) =Θ[Filter.principal (Ι σ' σ'') ×ˢ (Filter.atBot ⊔ Filter.atTop)]\n fun x =>\n match x with\n | (σ, y) => 1 / y ^ 2"} +{"name":"Perron.diffBddAtNegOne","declaration":"theorem Perron.diffBddAtNegOne {x : ℝ} (xpos : 0 < x) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n BddAbove ((norm ∘ fun s => ↑x ^ s / (s * (s + 1)) - -↑x⁻¹ / (s + 1)) '' (Square (-1) c \\ {-1}))"} +{"name":"tendsto_rpow_atTop_nhds_zero_of_norm_gt_one","declaration":"theorem tendsto_rpow_atTop_nhds_zero_of_norm_gt_one {x : ℝ} (x_gt_one : 1 < x) (C : ℝ) : Filter.Tendsto (fun σ => x ^ σ * C) Filter.atBot (nhds 0)"} +{"name":"Perron.isTheta_uniformlyOn_uIcc","declaration":"theorem Perron.isTheta_uniformlyOn_uIcc {x : ℝ} (xpos : 0 < x) (σ' : ℝ) (σ'' : ℝ) : (fun x_1 =>\n match x_1 with\n | (σ, y) =>\n Perron.f x (↑σ + ↑y * Complex.I)) =Θ[Filter.principal (Set.uIcc σ' σ'') ×ˢ (Filter.atBot ⊔ Filter.atTop)]\n ((fun y => 1 / y ^ 2) ∘ Prod.snd)"} +{"name":"Perron.integralPosAux","declaration":"theorem Perron.integralPosAux : 0 < ∫ (t : ℝ), 1 / (Real.sqrt (1 + t ^ 2) * Real.sqrt (2 + t ^ 2))"} +{"name":"verticalIntegral_eq_verticalIntegral","declaration":"theorem verticalIntegral_eq_verticalIntegral {σ : ℝ} {σ' : ℝ} {f : ℂ → ℂ} (hf : HolomorphicOn f (Set.uIcc σ σ' ×ℂ Set.univ)) (hbot : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atBot (nhds 0)) (htop : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atTop (nhds 0)) (hleft : MeasureTheory.Integrable (fun y => f (↑σ + ↑y * Complex.I)) MeasureTheory.volume) (hright : MeasureTheory.Integrable (fun y => f (↑σ' + ↑y * Complex.I)) MeasureTheory.volume) : VerticalIntegral f σ = VerticalIntegral f σ'"} +{"name":"tendsto_rpow_atTop_nhds_zero_of_norm_lt_one","declaration":"theorem tendsto_rpow_atTop_nhds_zero_of_norm_lt_one {x : ℝ} (xpos : 0 < x) (x_lt_one : x < 1) (C : ℝ) : Filter.Tendsto (fun σ => x ^ σ * C) Filter.atTop (nhds 0)"} +{"name":"Complex.cpow_inv_ofReal_pos","declaration":"theorem Complex.cpow_inv_ofReal_pos {a : ℝ} (ha : 0 ≤ a) (r : ℂ) : (↑a ^ r)⁻¹ = (↑a)⁻¹ ^ r"} +{"name":"Perron.keyIdentity","declaration":"theorem Perron.keyIdentity (x : ℝ) {s : ℂ} (s_ne_zero : s ≠ 0) (s_ne_neg_one : s ≠ -1) : ↑x ^ s / (s * (s + 1)) = ↑x ^ s / s - ↑x ^ s / (s + 1)"} +{"name":"Complex.cpow_eq_exp_log_ofReal","declaration":"theorem Complex.cpow_eq_exp_log_ofReal (x : ℝ) (hx : 0 < x) (y : ℂ) : ↑x ^ y = Complex.exp (↑(Real.log x) * y)"} +{"name":"Perron.isHolomorphicOn","declaration":"theorem Perron.isHolomorphicOn {x : ℝ} (xpos : 0 < x) : HolomorphicOn (Perron.f x) {0, -1}ᶜ"} +{"name":"Perron.f","declaration":"def Perron.f (x : ℝ) (s : ℂ) : ℂ"} +{"name":"Perron.HolomorphicOn.upperUIntegral_eq_zero","declaration":"theorem Perron.HolomorphicOn.upperUIntegral_eq_zero {f : ℂ → ℂ} {σ : ℝ} {σ' : ℝ} {T : ℝ} (hσ : σ ≤ σ') (hf : HolomorphicOn f {z | σ ≤ z.re ∧ z.re ≤ σ' ∧ T ≤ z.im}) (htop : Filter.Tendsto (fun y => ∫ (x : ℝ) in σ..σ', f (↑x + ↑y * Complex.I)) Filter.atTop (nhds 0)) (hleft : MeasureTheory.Integrable (fun y => f (↑σ + ↑y * Complex.I)) MeasureTheory.volume) (hright : MeasureTheory.Integrable (fun y => f (↑σ' + ↑y * Complex.I)) MeasureTheory.volume) : UpperUIntegral f σ σ' T = 0"} +{"name":"Perron.residuePull2","declaration":"theorem Perron.residuePull2 {x : ℝ} (x_gt_one : 1 < x) : VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) (-1 / 2) =\n -1 / ↑x + VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) (-3 / 2)"} +{"name":"Perron.vertIntBound","declaration":"theorem Perron.vertIntBound {x : ℝ} {σ : ℝ} (xpos : 0 < x) (σ_gt_one : 1 < σ) : ‖VerticalIntegral (Perron.f x) σ‖ ≤ x ^ σ * ∫ (t : ℝ), 1 / (Real.sqrt (1 + t ^ 2) * Real.sqrt (2 + t ^ 2))"} +{"name":"Filter.Tendsto.eventually_bddAbove","declaration":"theorem Filter.Tendsto.eventually_bddAbove {α : Type u_1} {β : Type u_2} [LinearOrder β] [NoMaxOrder β] [TopologicalSpace β] [ClosedIciTopology β] {y : β} {l : Filter α} {f : α → β} (hf : Filter.Tendsto f l (nhds y)) : ∀ᶠ (s : Set α) in Filter.smallSets l, BddAbove (f '' s)"} +{"name":"Perron.residueAtZero","declaration":"theorem Perron.residueAtZero {x : ℝ} (xpos : 0 < x) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0), RectangleIntegral' (Perron.f x) (-↑c - ↑c * Complex.I) (↑c + ↑c * Complex.I) = 1"} +{"name":"Perron.tendsto_zero_Lower","declaration":"theorem Perron.tendsto_zero_Lower {x : ℝ} (xpos : 0 < x) (σ' : ℝ) (σ'' : ℝ) : Filter.Tendsto (fun t => ∫ (σ : ℝ) in σ'..σ'', Perron.f x (↑σ + ↑t * Complex.I)) Filter.atBot (nhds 0)"} +{"name":"Perron.contourPull3","declaration":"theorem Perron.contourPull3 {x : ℝ} {σ' : ℝ} {σ'' : ℝ} (x_gt_one : 1 < x) (σ'le : σ' ≤ -3 / 2) (σ''le : σ'' ≤ -3 / 2) : VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) σ' = VerticalIntegral' (fun s => ↑x ^ s / (s * (s + 1))) σ''"} +{"name":"Perron.isTheta","declaration":"theorem Perron.isTheta {x : ℝ} {σ : ℝ} (xpos : 0 < x) : ((fun y => Perron.f x (↑σ + ↑y * Complex.I)) =Θ[Filter.atBot] fun y => 1 / y ^ 2) ∧\n (fun y => Perron.f x (↑σ + ↑y * Complex.I)) =Θ[Filter.atTop] fun y => 1 / y ^ 2"} +{"name":"Perron.isIntegrable","declaration":"theorem Perron.isIntegrable {x : ℝ} {σ : ℝ} (xpos : 0 < x) (σ_ne_zero : σ ≠ 0) (σ_ne_neg_one : σ ≠ -1) : MeasureTheory.Integrable (fun t => Perron.f x (↑σ + ↑t * Complex.I)) MeasureTheory.volume"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Rectangle.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Rectangle.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..eb66fe097a87c087c962356d5eb5b36fd2cc9f12 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Rectangle.jsonl @@ -0,0 +1,51 @@ +{"name":"mem_Rect","declaration":"theorem mem_Rect {z : ℂ} {w : ℂ} (zRe_lt_wRe : z.re ≤ w.re) (zIm_lt_wIm : z.im ≤ w.im) (p : ℂ) : p ∈ Rectangle z w ↔ z.re ≤ p.re ∧ p.re ≤ w.re ∧ z.im ≤ p.im ∧ p.im ≤ w.im"} +{"name":"mapsTo_rectangle_left_re","declaration":"theorem mapsTo_rectangle_left_re (z : ℂ) (w : ℂ) : Set.MapsTo (fun y => ↑z.re + ↑y * Complex.I) (Set.uIcc z.im w.im) (Rectangle z w)"} +{"name":"mapsTo_rectangleBorder_right_re","declaration":"theorem mapsTo_rectangleBorder_right_re (z : ℂ) (w : ℂ) : Set.MapsTo (fun y => ↑w.re + ↑y * Complex.I) (Set.uIcc z.im w.im) (RectangleBorder z w)"} +{"name":"Square_apply","declaration":"theorem Square_apply {c : ℝ} (p : ℂ) (cpos : c > 0) : Square p c = Set.Icc (-c + p.re) (c + p.re) ×ℂ Set.Icc (-c + p.im) (c + p.im)"} +{"name":"Set.ne_left_of_mem_uIoo","declaration":"theorem Set.ne_left_of_mem_uIoo {a : ℝ} {b : ℝ} {c : ℝ} (hc : c ∈ Set.uIoo a b) : c ≠ a"} +{"name":"square_subset_square","declaration":"theorem square_subset_square {p : ℂ} {c₁ : ℝ} {c₂ : ℝ} (hc₁ : 0 < c₁) (hc : c₁ ≤ c₂) : Square p c₁ ⊆ Square p c₂"} +{"name":"Rectangle.symm","declaration":"theorem Rectangle.symm {z : ℂ} {w : ℂ} : Rectangle z w = Rectangle w z"} +{"name":"segment_reProdIm_segment_eq_convexHull","declaration":"/-- The axis-parallel complex rectangle with opposite corners `z` and `w` is complex product\nof two intervals, which is also the convex hull of the four corners. Golfed from mathlib4\\#9598.-/\ntheorem segment_reProdIm_segment_eq_convexHull (z : ℂ) (w : ℂ) : Set.uIcc z.re w.re ×ℂ Set.uIcc z.im w.im = (convexHull ℝ) {z, ↑z.re + ↑w.im * Complex.I, ↑w.re + ↑z.im * Complex.I, w}"} +{"name":"RectangleBorder","declaration":"/-- A `RectangleBorder` has corners `z` and `w`. -/\ndef RectangleBorder (z : ℂ) (w : ℂ) : Set ℂ"} +{"name":"horizontalSegment_eq","declaration":"/-- A real segment `[a₁, a₂]` translated by `b * I` is the complex line segment.\nGolfed from mathlib\\#9598.-/\ntheorem horizontalSegment_eq (a₁ : ℝ) (a₂ : ℝ) (b : ℝ) : (fun x => ↑x + ↑b * Complex.I) '' Set.uIcc a₁ a₂ = Set.uIcc a₁ a₂ ×ℂ {b}"} +{"name":"mapsTo_rectangle_left_im_NoP","declaration":"theorem mapsTo_rectangle_left_im_NoP (z : ℂ) (w : ℂ) {p : ℂ} (pNotOnBorder : p ∉ RectangleBorder z w) : Set.MapsTo (fun x => ↑x + ↑z.im * Complex.I) (Set.uIcc z.re w.re) (Rectangle z w \\ {p})"} +{"name":"mapsTo_rectangleBorder_right_im","declaration":"theorem mapsTo_rectangleBorder_right_im (z : ℂ) (w : ℂ) : Set.MapsTo (fun x => ↑x + ↑w.im * Complex.I) (Set.uIcc z.re w.re) (RectangleBorder z w)"} +{"name":"rectangle_subset_punctured_rect","declaration":"theorem rectangle_subset_punctured_rect {z₀ : ℂ} {z₁ : ℂ} {z₂ : ℂ} {z₃ : ℂ} {p : ℂ} (hz : z₀.re ≤ z₁.re ∧ z₁.re ≤ z₂.re ∧ z₂.re ≤ z₃.re ∧ z₀.im ≤ z₁.im ∧ z₁.im ≤ z₂.im ∧ z₂.im ≤ z₃.im) (hp : p.re < z₁.re ∧ p.re < z₂.re ∨ p.im < z₁.im ∧ p.im < z₂.im ∨ z₁.re < p.re ∧ z₂.re < p.re ∨ z₁.im < p.im ∧ z₂.im < p.im) : Rectangle z₁ z₂ ⊆ Rectangle z₀ z₃ \\ {p}"} +{"name":"rectangleBorder_subset_rectangle","declaration":"theorem rectangleBorder_subset_rectangle (z : ℂ) (w : ℂ) : RectangleBorder z w ⊆ Rectangle z w"} +{"name":"RectSubRect'","declaration":"theorem RectSubRect' {z₀ : ℂ} {z₁ : ℂ} {z₂ : ℂ} {z₃ : ℂ} (x₀_le_x₁ : z₀.re ≤ z₁.re) (x₁_le_x₂ : z₁.re ≤ z₂.re) (x₂_le_x₃ : z₂.re ≤ z₃.re) (y₀_le_y₁ : z₀.im ≤ z₁.im) (y₁_le_y₂ : z₁.im ≤ z₂.im) (y₂_le_y₃ : z₂.im ≤ z₃.im) : Rectangle z₁ z₂ ⊆ Rectangle z₀ z₃"} +{"name":"mapsTo_rectangleBorder_left_im","declaration":"theorem mapsTo_rectangleBorder_left_im (z : ℂ) (w : ℂ) : Set.MapsTo (fun x => ↑x + ↑z.im * Complex.I) (Set.uIcc z.re w.re) (RectangleBorder z w)"} +{"name":"rectangle_mem_nhds_iff","declaration":"theorem rectangle_mem_nhds_iff {z : ℂ} {w : ℂ} {p : ℂ} : Rectangle z w ∈ nhds p ↔ p ∈ Set.uIoo z.re w.re ×ℂ Set.uIoo z.im w.im"} +{"name":"reProdIm_subset_iff'","declaration":"/-- If `s ⊆ s₁ ⊆ ℝ` and `t ⊆ t₁ ⊆ ℝ`, then `s × t ⊆ s₁ × t₁` in `ℂ`. -/\ntheorem reProdIm_subset_iff' {s : Set ℝ} {s₁ : Set ℝ} {t : Set ℝ} {t₁ : Set ℝ} : s ×ℂ t ⊆ s₁ ×ℂ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅"} +{"name":"ContinuousLinearEquiv.coe_toLinearEquiv_symm","declaration":"theorem ContinuousLinearEquiv.coe_toLinearEquiv_symm {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_3) [TopologicalSpace M] [AddCommMonoid M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] (e : M ≃SL[σ] M₂) : ⇑(LinearEquiv.symm e.toLinearEquiv) = ⇑(ContinuousLinearEquiv.symm e)"} +{"name":"SmallSquareInRectangle","declaration":"theorem SmallSquareInRectangle {z : ℂ} {w : ℂ} {p : ℂ} (pInRectInterior : Rectangle z w ∈ nhds p) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0), Square p c ⊆ Rectangle z w"} +{"name":"Set.uIoo","declaration":"def Set.uIoo {α : Type u_1} [Lattice α] (a : α) (b : α) : Set α"} +{"name":"rectangle_disjoint_singleton","declaration":"/-- Note: try using `by simp` for `h`. -/\ntheorem rectangle_disjoint_singleton {z : ℂ} {w : ℂ} {p : ℂ} (h : p.re < z.re ∧ p.re < w.re ∨ p.im < z.im ∧ p.im < w.im ∨ z.re < p.re ∧ w.re < p.re ∨ z.im < p.im ∧ w.im < p.im) : Disjoint (Rectangle z w) {p}"} +{"name":"mapsTo_rectangleBorder_left_re","declaration":"theorem mapsTo_rectangleBorder_left_re (z : ℂ) (w : ℂ) : Set.MapsTo (fun y => ↑z.re + ↑y * Complex.I) (Set.uIcc z.im w.im) (RectangleBorder z w)"} +{"name":"verticalSegment_eq","declaration":"/-- A vertical segment `[b₁, b₂]` translated by `a` is the complex line segment.\nGolfed from mathlib\\#9598.-/\ntheorem verticalSegment_eq (a : ℝ) (b₁ : ℝ) (b₂ : ℝ) : (fun y => ↑a + ↑y * Complex.I) '' Set.uIcc b₁ b₂ = {a} ×ℂ Set.uIcc b₁ b₂"} +{"name":"square_neg","declaration":"theorem square_neg (p : ℂ) (c : ℝ) : Square p (-c) = Square p c"} +{"name":"reProdIm_subset_iff","declaration":"/-- The inequality `s × t ⊆ s₁ × t₁` holds in `ℂ` iff it holds in `ℝ × ℝ`. -/\ntheorem reProdIm_subset_iff {s : Set ℝ} {s₁ : Set ℝ} {t : Set ℝ} {t₁ : Set ℝ} : s ×ℂ t ⊆ s₁ ×ℂ t₁ ↔ s ×ˢ t ⊆ s₁ ×ˢ t₁"} +{"name":"uIoo_of_le","declaration":"theorem uIoo_of_le {α : Type u_1} [Lattice α] {a : α} {b : α} (h : a ≤ b) : Set.uIoo a b = Set.Ioo a b"} +{"name":"preimage_equivRealProdCLM_reProdIm","declaration":"theorem preimage_equivRealProdCLM_reProdIm (s : Set ℝ) (t : Set ℝ) : ⇑(ContinuousLinearEquiv.symm Complex.equivRealProdCLM) ⁻¹' (s ×ℂ t) = s ×ˢ t"} +{"name":"left_mem_rect","declaration":"theorem left_mem_rect (z : ℂ) (w : ℂ) : z ∈ Rectangle z w"} +{"name":"mapsTo_rectangle_left_im","declaration":"theorem mapsTo_rectangle_left_im (z : ℂ) (w : ℂ) : Set.MapsTo (fun x => ↑x + ↑z.im * Complex.I) (Set.uIcc z.re w.re) (Rectangle z w)"} +{"name":"rect_subset_iff","declaration":"theorem rect_subset_iff {z : ℂ} {w : ℂ} {z' : ℂ} {w' : ℂ} : Rectangle z' w' ⊆ Rectangle z w ↔ z' ∈ Rectangle z w ∧ w' ∈ Rectangle z w"} +{"name":"Set.ne_right_of_mem_uIoo","declaration":"theorem Set.ne_right_of_mem_uIoo {a : ℝ} {b : ℝ} {c : ℝ} (hc : c ∈ Set.uIoo a b) : c ≠ b"} +{"name":"Complex.nhds_hasBasis_square","declaration":"theorem Complex.nhds_hasBasis_square (p : ℂ) : Filter.HasBasis (nhds p) (fun x => 0 < x) fun x => Square p x"} +{"name":"right_mem_rect","declaration":"theorem right_mem_rect (z : ℂ) (w : ℂ) : w ∈ Rectangle z w"} +{"name":"rectangleBorder_disjoint_singleton","declaration":"theorem rectangleBorder_disjoint_singleton {z : ℂ} {w : ℂ} {p : ℂ} (h : p.re ≠ z.re ∧ p.re ≠ w.re ∧ p.im ≠ z.im ∧ p.im ≠ w.im) : Disjoint (RectangleBorder z w) {p}"} +{"name":"Set.left_not_mem_uIoo","declaration":"theorem Set.left_not_mem_uIoo {a : ℝ} {b : ℝ} : a ∉ Set.uIoo a b"} +{"name":"square_mem_nhds","declaration":"theorem square_mem_nhds (p : ℂ) {c : ℝ} (hc : c ≠ 0) : Square p c ∈ nhds p"} +{"name":"RectSubRect","declaration":"theorem RectSubRect {x₀ : ℝ} {x₁ : ℝ} {x₂ : ℝ} {x₃ : ℝ} {y₀ : ℝ} {y₁ : ℝ} {y₂ : ℝ} {y₃ : ℝ} (x₀_le_x₁ : x₀ ≤ x₁) (x₁_le_x₂ : x₁ ≤ x₂) (x₂_le_x₃ : x₂ ≤ x₃) (y₀_le_y₁ : y₀ ≤ y₁) (y₁_le_y₂ : y₁ ≤ y₂) (y₂_le_y₃ : y₂ ≤ y₃) : Rectangle (↑x₁ + ↑y₁ * Complex.I) (↑x₂ + ↑y₂ * Complex.I) ⊆ Rectangle (↑x₀ + ↑y₀ * Complex.I) (↑x₃ + ↑y₃ * Complex.I)"} +{"name":"mapsTo_rectangle_right_re","declaration":"theorem mapsTo_rectangle_right_re (z : ℂ) (w : ℂ) : Set.MapsTo (fun y => ↑w.re + ↑y * Complex.I) (Set.uIcc z.im w.im) (Rectangle z w)"} +{"name":"preimage_equivRealProd_prod","declaration":"/-- The preimage under `equivRealProd` of `s ×ˢ t` is `s ×ℂ t`. -/\ntheorem preimage_equivRealProd_prod (s : Set ℝ) (t : Set ℝ) : ⇑Complex.equivRealProd ⁻¹' s ×ˢ t = s ×ℂ t"} +{"name":"Square","declaration":"def Square (p : ℂ) (c : ℝ) : Set ℂ"} +{"name":"rectangle_in_convex","declaration":"/-- If the four corners of a rectangle are contained in a convex set `U`, then the whole\nrectangle is. Golfed from mathlib4\\#9598.-/\ntheorem rectangle_in_convex {U : Set ℂ} (U_convex : Convex ℝ U) {z : ℂ} {w : ℂ} (hz : z ∈ U) (hw : w ∈ U) (hzw : ↑z.re + ↑w.im * Complex.I ∈ U) (hwz : ↑w.re + ↑z.im * Complex.I ∈ U) : Rectangle z w ⊆ U"} +{"name":"not_mem_rectangleBorder_of_rectangle_mem_nhds","declaration":"theorem not_mem_rectangleBorder_of_rectangle_mem_nhds {z : ℂ} {w : ℂ} {p : ℂ} (hp : Rectangle z w ∈ nhds p) : p ∉ RectangleBorder z w"} +{"name":"mapsTo_rectangle_right_im_NoP","declaration":"theorem mapsTo_rectangle_right_im_NoP (z : ℂ) (w : ℂ) {p : ℂ} (pNotOnBorder : p ∉ RectangleBorder z w) : Set.MapsTo (fun x => ↑x + ↑w.im * Complex.I) (Set.uIcc z.re w.re) (Rectangle z w \\ {p})"} +{"name":"mapsTo_rectangle_left_re_NoP","declaration":"theorem mapsTo_rectangle_left_re_NoP (z : ℂ) (w : ℂ) {p : ℂ} (pNotOnBorder : p ∉ RectangleBorder z w) : Set.MapsTo (fun y => ↑z.re + ↑y * Complex.I) (Set.uIcc z.im w.im) (Rectangle z w \\ {p})"} +{"name":"rectangleBorder_subset_punctured_rect","declaration":"theorem rectangleBorder_subset_punctured_rect {z₀ : ℂ} {z₁ : ℂ} {z₂ : ℂ} {z₃ : ℂ} {p : ℂ} (hz : z₀.re ≤ z₁.re ∧ z₁.re ≤ z₂.re ∧ z₂.re ≤ z₃.re ∧ z₀.im ≤ z₁.im ∧ z₁.im ≤ z₂.im ∧ z₂.im ≤ z₃.im) (hp : p.re ≠ z₁.re ∧ p.re ≠ z₂.re ∧ p.im ≠ z₁.im ∧ p.im ≠ z₂.im) : RectangleBorder z₁ z₂ ⊆ Rectangle z₀ z₃ \\ {p}"} +{"name":"Rectangle.symm_re","declaration":"theorem Rectangle.symm_re {z : ℂ} {w : ℂ} : Rectangle (↑w.re + ↑z.im * Complex.I) (↑z.re + ↑w.im * Complex.I) = Rectangle z w"} +{"name":"mapsTo_rectangle_right_re_NoP","declaration":"theorem mapsTo_rectangle_right_re_NoP (z : ℂ) (w : ℂ) {p : ℂ} (pNotOnBorder : p ∉ RectangleBorder z w) : Set.MapsTo (fun y => ↑w.re + ↑y * Complex.I) (Set.uIcc z.im w.im) (Rectangle z w \\ {p})"} +{"name":"Set.right_not_mem_uIoo","declaration":"theorem Set.right_not_mem_uIoo {a : ℝ} {b : ℝ} : b ∉ Set.uIoo a b"} +{"name":"mapsTo_rectangle_right_im","declaration":"theorem mapsTo_rectangle_right_im (z : ℂ) (w : ℂ) : Set.MapsTo (fun x => ↑x + ↑w.im * Complex.I) (Set.uIcc z.re w.re) (Rectangle z w)"} +{"name":"Rectangle","declaration":"/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z : ℂ) (w : ℂ) : Set ℂ"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d211005716fc3ef834bb40b07ccd17305cf7c287 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.ResidueCalcOnRectangles.jsonl @@ -0,0 +1,54 @@ +{"name":"ContinuousOn.rectangleBorderNoPIntegrable","declaration":"theorem ContinuousOn.rectangleBorderNoPIntegrable {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {z : ℂ} {w : ℂ} {p : ℂ} (hf : ContinuousOn f (Rectangle z w \\ {p})) (pNotOnBorder : p ∉ RectangleBorder z w) : RectangleBorderIntegrable f z w"} +{"name":"RectanglePullToNhdOfPole''","declaration":"theorem RectanglePullToNhdOfPole'' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} [CompleteSpace E] {z : ℂ} {w : ℂ} {p : ℂ} (zRe_le_wRe : z.re ≤ w.re) (zIm_le_wIm : z.im ≤ w.im) (pInRectInterior : Rectangle z w ∈ nhds p) (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n RectangleIntegral' f z w = RectangleIntegral' f (-↑c - Complex.I * ↑c + p) (↑c + Complex.I * ↑c + p)"} +{"name":"ResidueTheoremAtOrigin'","declaration":"theorem ResidueTheoremAtOrigin' {z : ℂ} {w : ℂ} {c : ℂ} (h1 : z.re < 0) (h2 : z.im < 0) (h3 : 0 < w.re) (h4 : 0 < w.im) : RectangleIntegral (fun s => c / s) z w = 2 * Complex.I * ↑Real.pi * c"} +{"name":"LowerUIntegral","declaration":"def LowerUIntegral {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (σ : ℝ) (σ' : ℝ) (T : ℝ) : E"} +{"name":"HolomorphicOn.rectangleBorderIntegrable","declaration":"theorem HolomorphicOn.rectangleBorderIntegrable {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {z : ℂ} {w : ℂ} (hf : HolomorphicOn f (Rectangle z w)) : RectangleBorderIntegrable f z w"} +{"name":"RectangleIntegralHSplit'","declaration":"theorem RectangleIntegralHSplit' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {a : ℝ} {x₀ : ℝ} {x₁ : ℝ} {y₀ : ℝ} {y₁ : ℝ} (ha : a ∈ Set.uIcc x₀ x₁) (hf : RectangleBorderIntegrable f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I)) : RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I) =\n RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑a + ↑y₁ * Complex.I) +\n RectangleIntegral f (↑a + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I)"} +{"name":"RectangleIntegral_congr","declaration":"theorem RectangleIntegral_congr {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {g : ℂ → E} {z : ℂ} {w : ℂ} (h : Set.EqOn f g (RectangleBorder z w)) : RectangleIntegral f z w = RectangleIntegral g z w"} +{"name":"integral_const_div_sq_add_sq","declaration":"theorem integral_const_div_sq_add_sq {x₁ : ℝ} {x₂ : ℝ} {y : ℝ} (hy : y ≠ 0) : ∫ (x : ℝ) in x₁..x₂, y / (x ^ 2 + y ^ 2) = Real.arctan (x₂ / y) - Real.arctan (x₁ / y)"} +{"name":"RectangleBorderIntegrable","declaration":"def RectangleBorderIntegrable {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (z : ℂ) (w : ℂ) : Prop"} +{"name":"HIntegral_symm","declaration":"theorem HIntegral_symm {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {x₁ : ℝ} {x₂ : ℝ} {y : ℝ} : HIntegral f x₁ x₂ y = -HIntegral f x₂ x₁ y"} +{"name":"RectangleIntegral'","declaration":"/-- A `RectangleIntegral'` of a function `f` is one over a rectangle determined by\n`z` and `w` in `ℂ`, divided by `2 * π * I`. -/\ndef RectangleIntegral' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) : E"} +{"name":"RectangleBorderIntegrable.add","declaration":"theorem RectangleBorderIntegrable.add {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {w : ℂ} {f : ℂ → E} {g : ℂ → E} (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) : RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w"} +{"name":"RectangleIntegral.const_smul","declaration":"theorem RectangleIntegral.const_smul {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) (c : ℂ) : RectangleIntegral (fun s => c • f s) z w = c • RectangleIntegral f z w"} +{"name":"ResidueTheoremAtOrigin_aux1c'","declaration":"theorem ResidueTheoremAtOrigin_aux1c' (a : ℝ) (b : ℝ) : let f := fun y => (↑y - Complex.I)⁻¹;\nIntervalIntegrable f MeasureTheory.volume a b"} +{"name":"ResidueTheoremAtOrigin_aux2c'","declaration":"theorem ResidueTheoremAtOrigin_aux2c' (a : ℝ) (b : ℝ) : let f := fun y => (-1 + ↑y * Complex.I)⁻¹;\nIntervalIntegrable f MeasureTheory.volume a b"} +{"name":"RectangleIntegral'_congr","declaration":"theorem RectangleIntegral'_congr {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {g : ℂ → E} {z : ℂ} {w : ℂ} (h : Set.EqOn f g (RectangleBorder z w)) : RectangleIntegral' f z w = RectangleIntegral' g z w"} +{"name":"ContinuousOn.rectangleBorder_integrable","declaration":"theorem ContinuousOn.rectangleBorder_integrable {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {z : ℂ} {w : ℂ} (hf : ContinuousOn f (RectangleBorder z w)) : RectangleBorderIntegrable f z w"} +{"name":"integral_const_div_self_add_im","declaration":"theorem integral_const_div_self_add_im {A : ℂ} {x₁ : ℝ} {x₂ : ℝ} {y : ℝ} (hy : y ≠ 0) : ∫ (x : ℝ) in x₁..x₂, A / (↑x + ↑y * Complex.I) =\n A * (↑(Real.log (x₂ ^ 2 + y ^ 2)) / 2 - ↑(Real.log (x₁ ^ 2 + y ^ 2)) / 2) -\n A * Complex.I * (↑(Real.arctan (x₂ / y)) - ↑(Real.arctan (x₁ / y)))"} +{"name":"VerticalIntegral'","declaration":"def VerticalIntegral' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (σ : ℝ) : E"} +{"name":"Complex.inv_re_add_im","declaration":"theorem Complex.inv_re_add_im {x : ℝ} {y : ℝ} : (↑x + ↑y * Complex.I)⁻¹ = (↑x - Complex.I * ↑y) / (↑x ^ 2 + ↑y ^ 2)"} +{"name":"RectanglePullToNhdOfPole","declaration":"/-- Given `f` holomorphic on a rectangle `z` and `w` except at a point `p`, the integral of `f` over\nthe rectangle with corners `z` and `w` is the same as the integral of `f` over a small square\ncentered at `p`. -/\ntheorem RectanglePullToNhdOfPole {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} [CompleteSpace E] {z : ℂ} {w : ℂ} {p : ℂ} (zRe_lt_wRe : z.re ≤ w.re) (zIm_lt_wIm : z.im ≤ w.im) (hp : Rectangle z w ∈ nhds p) (fHolo : HolomorphicOn f (Rectangle z w \\ {p})) : ∀ᶠ (c : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n RectangleIntegral f z w = RectangleIntegral f (-↑c - Complex.I * ↑c + p) (↑c + Complex.I * ↑c + p)"} +{"name":"RectangleIntegralVSplit","declaration":"theorem RectangleIntegralVSplit {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {b : ℝ} {x₀ : ℝ} {x₁ : ℝ} {y₀ : ℝ} {y₁ : ℝ} (f_int_y₀_b_left : IntervalIntegrable (fun y => f (↑x₀ + ↑y * Complex.I)) MeasureTheory.volume y₀ b) (f_int_b_y₁_left : IntervalIntegrable (fun y => f (↑x₀ + ↑y * Complex.I)) MeasureTheory.volume b y₁) (f_int_y₀_b_right : IntervalIntegrable (fun y => f (↑x₁ + ↑y * Complex.I)) MeasureTheory.volume y₀ b) (f_int_b_y₁_right : IntervalIntegrable (fun y => f (↑x₁ + ↑y * Complex.I)) MeasureTheory.volume b y₁) : RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I) =\n RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑b * Complex.I) +\n RectangleIntegral f (↑x₀ + ↑b * Complex.I) (↑x₁ + ↑y₁ * Complex.I)"} +{"name":"VIntegral","declaration":"def VIntegral {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (x : ℝ) (y₁ : ℝ) (y₂ : ℝ) : E"} +{"name":"continuous_self_div_sq_add_sq","declaration":"theorem continuous_self_div_sq_add_sq {y : ℝ} (hy : y ≠ 0) : Continuous fun x => x / (x ^ 2 + y ^ 2)"} +{"name":"UpperUIntegral","declaration":"def UpperUIntegral {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (σ : ℝ) (σ' : ℝ) (T : ℝ) : E"} +{"name":"HolomorphicOn.rectangleBorderIntegrable'","declaration":"theorem HolomorphicOn.rectangleBorderIntegrable' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {z : ℂ} {w : ℂ} {p : ℂ} (hf : HolomorphicOn f (Rectangle z w \\ {p})) (hp : Rectangle z w ∈ nhds p) : RectangleBorderIntegrable f z w"} +{"name":"existsDifferentiableOn_of_bddAbove","declaration":"theorem existsDifferentiableOn_of_bddAbove {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} [CompleteSpace E] {s : Set ℂ} {c : ℂ} (hc : s ∈ nhds c) (hd : HolomorphicOn f (s \\ {c})) (hb : BddAbove (norm ∘ f '' (s \\ {c}))) : ∃ g, HolomorphicOn g s ∧ Set.EqOn f g (s \\ {c})"} +{"name":"verticalIntegral_split_three","declaration":"theorem verticalIntegral_split_three {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {σ : ℝ} (a : ℝ) (b : ℝ) (hf : MeasureTheory.Integrable (fun t => f (↑σ + ↑t * Complex.I)) MeasureTheory.volume) : VerticalIntegral f σ =\n (Complex.I • ∫ (t : ℝ) in Set.Iic a, f (↑σ + ↑t * Complex.I)) + VIntegral f σ a b +\n Complex.I • ∫ (t : ℝ) in Set.Ici b, f (↑σ + ↑t * Complex.I)"} +{"name":"ResidueTheoremAtOrigin","declaration":"theorem ResidueTheoremAtOrigin : RectangleIntegral' (fun s => 1 / s) (-1 - Complex.I) (1 + Complex.I) = 1"} +{"name":"rectangleIntegral_symm","declaration":"theorem rectangleIntegral_symm {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) : RectangleIntegral f z w = RectangleIntegral f w z"} +{"name":"rectangleIntegral_symm_re","declaration":"theorem rectangleIntegral_symm_re {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) : RectangleIntegral f (↑w.re + ↑z.im * Complex.I) (↑z.re + ↑w.im * Complex.I) = -RectangleIntegral f z w"} +{"name":"RectangleIntegralVSplit'","declaration":"theorem RectangleIntegralVSplit' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {b : ℝ} {x₀ : ℝ} {x₁ : ℝ} {y₀ : ℝ} {y₁ : ℝ} (hb : b ∈ Set.uIcc y₀ y₁) (hf : RectangleBorderIntegrable f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I)) : RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I) =\n RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑b * Complex.I) +\n RectangleIntegral f (↑x₀ + ↑b * Complex.I) (↑x₁ + ↑y₁ * Complex.I)"} +{"name":"HIntegral'","declaration":"def HIntegral' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (x₁ : ℝ) (x₂ : ℝ) (y : ℝ) : E"} +{"name":"VIntegral_symm","declaration":"theorem VIntegral_symm {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {x : ℝ} {y₁ : ℝ} {y₂ : ℝ} : VIntegral f x y₁ y₂ = -VIntegral f x y₂ y₁"} +{"name":"ResidueTheoremAtOrigin_aux2c","declaration":"theorem ResidueTheoremAtOrigin_aux2c (a : ℝ) (b : ℝ) : let f := fun y => (1 + ↑y * Complex.I)⁻¹;\nIntervalIntegrable f MeasureTheory.volume a b"} +{"name":"RectangleIntegralHSplit","declaration":"/-- Given `x₀ a x₁ : ℝ`, and `y₀ y₁ : ℝ` and a function `f : ℂ → ℂ` so that\nboth `(t : ℝ) ↦ f(t + y₀ * I)` and `(t : ℝ) ↦ f(t + y₁ * I)` are integrable over both\n`t ∈ Icc x₀ a` and `t ∈ Icc a x₁`, we have that\n`RectangleIntegral f (x₀ + y₀ * I) (x₁ + y₁ * I)` is the sum of\n`RectangleIntegral f (x₀ + y₀ * I) (a + y₁ * I)` and\n`RectangleIntegral f (a + y₀ * I) (x₁ + y₁ * I)`.\n-/\ntheorem RectangleIntegralHSplit {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {a : ℝ} {x₀ : ℝ} {x₁ : ℝ} {y₀ : ℝ} {y₁ : ℝ} (f_int_x₀_a_bot : IntervalIntegrable (fun x => f (↑x + ↑y₀ * Complex.I)) MeasureTheory.volume x₀ a) (f_int_a_x₁_bot : IntervalIntegrable (fun x => f (↑x + ↑y₀ * Complex.I)) MeasureTheory.volume a x₁) (f_int_x₀_a_top : IntervalIntegrable (fun x => f (↑x + ↑y₁ * Complex.I)) MeasureTheory.volume x₀ a) (f_int_a_x₁_top : IntervalIntegrable (fun x => f (↑x + ↑y₁ * Complex.I)) MeasureTheory.volume a x₁) : RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I) =\n RectangleIntegral f (↑x₀ + ↑y₀ * Complex.I) (↑a + ↑y₁ * Complex.I) +\n RectangleIntegral f (↑a + ↑y₀ * Complex.I) (↑x₁ + ↑y₁ * Complex.I)"} +{"name":"RectangleIntegral.translate'","declaration":"theorem RectangleIntegral.translate' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) (p : ℂ) : RectangleIntegral' (fun s => f (s - p)) z w = RectangleIntegral' f (z - p) (w - p)"} +{"name":"HIntegral","declaration":"def HIntegral {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (x₁ : ℝ) (x₂ : ℝ) (y : ℝ) : E"} +{"name":"integral_const_div_re_add_self","declaration":"theorem integral_const_div_re_add_self {A : ℂ} {x : ℝ} {y₁ : ℝ} {y₂ : ℝ} (hx : x ≠ 0) : ∫ (y : ℝ) in y₁..y₂, A / (↑x + ↑y * Complex.I) =\n A / Complex.I * (↑(Real.log (y₂ ^ 2 + (-x) ^ 2)) / 2 - ↑(Real.log (y₁ ^ 2 + (-x) ^ 2)) / 2) -\n A / Complex.I * Complex.I * (↑(Real.arctan (y₂ / -x)) - ↑(Real.arctan (y₁ / -x)))"} +{"name":"RectangleIntegral","declaration":"/-- A `RectangleIntegral` of a function `f` is one over a rectangle determined by\n`z` and `w` in `ℂ`. -/\ndef RectangleIntegral {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) : E"} +{"name":"RectanglePullToNhdOfPole'","declaration":"theorem RectanglePullToNhdOfPole' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} [CompleteSpace E] {z₀ : ℂ} {z₁ : ℂ} {z₂ : ℂ} {z₃ : ℂ} {p : ℂ} (h_orientation : z₀.re ≤ z₃.re ∧ z₀.im ≤ z₃.im ∧ z₁.re ≤ z₂.re ∧ z₁.im ≤ z₂.im) (hp : Rectangle z₁ z₂ ∈ nhds p) (hz : Rectangle z₁ z₂ ⊆ Rectangle z₀ z₃) (fHolo : HolomorphicOn f (Rectangle z₀ z₃ \\ {p})) : RectangleIntegral f z₀ z₃ = RectangleIntegral f z₁ z₂"} +{"name":"VerticalIntegral","declaration":"def VerticalIntegral {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (σ : ℝ) : E"} +{"name":"ContinuousOn.rectangleBorderIntegrable","declaration":"theorem ContinuousOn.rectangleBorderIntegrable {E : Type u_1} [NormedAddCommGroup E] {f : ℂ → E} {z : ℂ} {w : ℂ} (hf : ContinuousOn f (Rectangle z w)) : RectangleBorderIntegrable f z w"} +{"name":"RectangleIntegral.translate","declaration":"theorem RectangleIntegral.translate {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) (p : ℂ) : RectangleIntegral (fun s => f (s - p)) z w = RectangleIntegral f (z - p) (w - p)"} +{"name":"ResidueTheoremOnRectangleWithSimplePole","declaration":"theorem ResidueTheoremOnRectangleWithSimplePole {f : ℂ → ℂ} {g : ℂ → ℂ} {z : ℂ} {w : ℂ} {p : ℂ} {A : ℂ} (zRe_le_wRe : z.re ≤ w.re) (zIm_le_wIm : z.im ≤ w.im) (pInRectInterior : Rectangle z w ∈ nhds p) (gHolo : HolomorphicOn g (Rectangle z w)) (principalPart : Set.EqOn (f - fun s => A / (s - p)) g (Rectangle z w \\ {p})) : RectangleIntegral' f z w = A"} +{"name":"HolomorphicOn.vanishesOnRectangle","declaration":"theorem HolomorphicOn.vanishesOnRectangle {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {z : ℂ} {w : ℂ} [CompleteSpace E] {U : Set ℂ} (f_holo : HolomorphicOn f U) (hU : Rectangle z w ⊆ U) : RectangleIntegral f z w = 0"} +{"name":"HolomorphicOn","declaration":"/-- A function is `HolomorphicOn` a set if it is complex differentiable on that set. -/\ndef HolomorphicOn {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (s : Set ℂ) : Prop"} +{"name":"VIntegral'","declaration":"def VIntegral' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (x : ℝ) (y₁ : ℝ) (y₂ : ℝ) : E"} +{"name":"RectangleIntegral.const_mul'","declaration":"theorem RectangleIntegral.const_mul' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (w : ℂ) (c : ℂ) : RectangleIntegral' (fun s => c • f s) z w = c • RectangleIntegral' f z w"} +{"name":"ResidueTheoremAtOrigin_aux1c","declaration":"theorem ResidueTheoremAtOrigin_aux1c (a : ℝ) (b : ℝ) : let f := fun y => (↑y + Complex.I)⁻¹;\nIntervalIntegrable f MeasureTheory.volume a b"} +{"name":"ResidueTheoremInRectangle","declaration":"theorem ResidueTheoremInRectangle {z : ℂ} {w : ℂ} {p : ℂ} {c : ℂ} (zRe_le_wRe : z.re ≤ w.re) (zIm_le_wIm : z.im ≤ w.im) (pInRectInterior : Rectangle z w ∈ nhds p) : RectangleIntegral' (fun s => c / (s - p)) z w = c"} +{"name":"DiffVertRect_eq_UpperLowerUs","declaration":"theorem DiffVertRect_eq_UpperLowerUs {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {σ : ℝ} {σ' : ℝ} {T : ℝ} (f_int_σ : MeasureTheory.Integrable (fun t => f (↑σ + ↑t * Complex.I)) MeasureTheory.volume) (f_int_σ' : MeasureTheory.Integrable (fun t => f (↑σ' + ↑t * Complex.I)) MeasureTheory.volume) : VerticalIntegral f σ' - VerticalIntegral f σ - RectangleIntegral f (↑σ - Complex.I * ↑T) (↑σ' + Complex.I * ↑T) =\n UpperUIntegral f σ σ' T - LowerUIntegral f σ σ' T"} +{"name":"sq_add_sq_ne_zero","declaration":"theorem sq_add_sq_ne_zero {x : ℝ} {y : ℝ} (hy : y ≠ 0) : x ^ 2 + y ^ 2 ≠ 0"} +{"name":"integral_self_div_sq_add_sq","declaration":"theorem integral_self_div_sq_add_sq {x₁ : ℝ} {x₂ : ℝ} {y : ℝ} (hy : y ≠ 0) : ∫ (x : ℝ) in x₁..x₂, x / (x ^ 2 + y ^ 2) = Real.log (x₂ ^ 2 + y ^ 2) / 2 - Real.log (x₁ ^ 2 + y ^ 2) / 2"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Sobolev.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Sobolev.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..efd8f833938a7a731812feb8bd6861f579ecfc67 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Sobolev.jsonl @@ -0,0 +1,67 @@ +{"name":"CS.mk","declaration":"ctor CS.mk {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (toFun : ℝ → E) (h1 : ContDiff ℝ (↑n) toFun) (h2 : HasCompactSupport toFun) : CS n E"} +{"name":"W1.deriv","declaration":"def W1.deriv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : W1 (n + 1) E) : W1 n E"} +{"name":"funscale","declaration":"def funscale {E : Type u_2} (g : ℝ → E) (R : ℝ) (x : ℝ) : E"} +{"name":"W21.instCoeSchwartzMapRealComplexNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpaceInstNormedAddCommGroupComplexComplexToRealInstRCLikeComplexW21","declaration":"instance W21.instCoeSchwartzMapRealComplexNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpaceInstNormedAddCommGroupComplexComplexToRealInstRCLikeComplexW21 : Coe (SchwartzMap ℝ ℂ) W21"} +{"name":"W1.integrable","declaration":"def W1.integrable {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : W1 n E) ⦃k : ℕ⦄ : k ≤ n → MeasureTheory.Integrable (iteratedDeriv k self.toFun) MeasureTheory.volume"} +{"name":"CS.bounded","declaration":"theorem CS.bounded {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {f : CS n E} : ∃ C, ∀ (v : ℝ), ‖f.toFun v‖ ≤ C"} +{"name":"W21.ofCS2","declaration":"def W21.ofCS2 (f : CS 2 ℂ) : W21"} +{"name":"trunc.instCoeTruncCSOfNatNatInstOfNatNatRealNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpace","declaration":"instance trunc.instCoeTruncCSOfNatNatInstOfNatNatRealNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpace : Coe trunc (CS 2 ℝ)"} +{"name":"W21.norm","declaration":"def W21.norm (f : ℝ → ℂ) : ℝ"} +{"name":"trunc.mk","declaration":"ctor trunc.mk (toCS : CS 2 ℝ) (h3 : Set.indicator (Set.Icc (-1) 1) 1 ≤ toCS.toFun) (h4 : toCS.toFun ≤ Set.indicator (Set.Ioo (-2) 2) 1) : trunc"} +{"name":"W1.continuous","declaration":"theorem W1.continuous {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : W1 n E) : Continuous f.toFun"} +{"name":"W21.hf''","declaration":"theorem W21.hf'' (f : W21) : MeasureTheory.Integrable (deriv (deriv f.toFun)) MeasureTheory.volume"} +{"name":"trunc.nonneg","declaration":"theorem trunc.nonneg (g : trunc) (x : ℝ) : 0 ≤ g.toFun x"} +{"name":"CS.ext","declaration":"theorem CS.ext {n : ℕ} {E : Type u_2} : ∀ {inst : NormedAddCommGroup E} {inst_1 : NormedSpace ℝ E} (x y : CS n E), x.toFun = y.toFun → x = y"} +{"name":"W1","declaration":"structure W1 (n : ℕ) (E : Type u_2) [NormedAddCommGroup E] [NormedSpace ℝ E] : Type u_2"} +{"name":"W21.hf","declaration":"theorem W21.hf (f : W21) : MeasureTheory.Integrable f.toFun MeasureTheory.volume"} +{"name":"CS.smul","declaration":"def CS.smul {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (R : ℝ) (f : CS n E) : CS n E"} +{"name":"W21.instNormW21","declaration":"instance W21.instNormW21 : Norm W21"} +{"name":"W1.differentiable","declaration":"theorem W1.differentiable {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : W1 (n + 1) E) : Differentiable ℝ f.toFun"} +{"name":"W1.integrable_iteratedDeriv_Schwarz","declaration":"theorem W1.integrable_iteratedDeriv_Schwarz {n : ℕ} {f : SchwartzMap ℝ ℂ} : MeasureTheory.Integrable (iteratedDeriv n ⇑f) MeasureTheory.volume"} +{"name":"CS.neg","declaration":"def CS.neg {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : CS n E) : CS n E"} +{"name":"trunc.le_one","declaration":"theorem trunc.le_one (g : trunc) (x : ℝ) : g.toFun x ≤ 1"} +{"name":"CS.deriv","declaration":"def CS.deriv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : CS (n + 1) E) : CS n E"} +{"name":"CS.toFun","declaration":"def CS.toFun {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : CS n E) : ℝ → E"} +{"name":"CS.hasDerivAt_scale","declaration":"theorem CS.hasDerivAt_scale {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : CS (n + 1) E) (R : ℝ) (x : ℝ) : HasDerivAt (CS.scale f R).toFun (R⁻¹ • deriv f.toFun (R⁻¹ • x)) x"} +{"name":"CS.deriv_scale","declaration":"theorem CS.deriv_scale {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {R : ℝ} {f : CS (n + 1) E} : CS.deriv (CS.scale f R) = R⁻¹ • CS.scale (CS.deriv f) R"} +{"name":"CS.smul_apply","declaration":"theorem CS.smul_apply {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {f : CS n E} {R : ℝ} {x : ℝ} : (R • f).toFun x = R • f.toFun x"} +{"name":"W1.of_Schwartz","declaration":"def W1.of_Schwartz {n : ℕ} (f : SchwartzMap ℝ ℂ) : W1 n ℂ"} +{"name":"trunc.zero_at","declaration":"theorem trunc.zero_at {g : trunc} : g.toFun 0 = 1"} +{"name":"CS.deriv_scale'","declaration":"theorem CS.deriv_scale' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {R : ℝ} {v : ℝ} {f : CS (n + 1) E} : (CS.deriv (CS.scale f R)).toFun v = R⁻¹ • (CS.deriv f).toFun (R⁻¹ • v)"} +{"name":"CS.scale","declaration":"def CS.scale {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (g : CS n E) (R : ℝ) : CS n E"} +{"name":"CS.instNegCS","declaration":"instance CS.instNegCS {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} : Neg (CS n E)"} +{"name":"W21.instHMulCSOfNatNatInstOfNatNatComplexInstNormedAddCommGroupComplexToNormedSpaceRealRCLikeComplexToRealInnerProductSpaceInstRCLikeComplexW21","declaration":"instance W21.instHMulCSOfNatNatInstOfNatNatComplexInstNormedAddCommGroupComplexToNormedSpaceRealRCLikeComplexToRealInnerProductSpaceInstRCLikeComplexW21 : HMul (CS 2 ℂ) W21 (CS 2 ℂ)"} +{"name":"W21.instCoeCSOfNatNatInstOfNatNatComplexInstNormedAddCommGroupComplexToNormedSpaceRealRCLikeComplexToRealInnerProductSpaceInstRCLikeComplexW21","declaration":"instance W21.instCoeCSOfNatNatInstOfNatNatComplexInstNormedAddCommGroupComplexToNormedSpaceRealRCLikeComplexToRealInnerProductSpaceInstRCLikeComplexW21 : Coe (CS 2 ℂ) W21"} +{"name":"trunc.h4","declaration":"def trunc.h4 (self : trunc) : self.toFun ≤ Set.indicator (Set.Ioo (-2) 2) 1"} +{"name":"W1.mk","declaration":"ctor W1.mk {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (toFun : ℝ → E) (smooth : ContDiff ℝ (↑n) toFun) (integrable : ∀ ⦃k : ℕ⦄, k ≤ n → MeasureTheory.Integrable (iteratedDeriv k toFun) MeasureTheory.volume) : W1 n E"} +{"name":"CS.ext_iff","declaration":"theorem CS.ext_iff {n : ℕ} {E : Type u_2} : ∀ {inst : NormedAddCommGroup E} {inst_1 : NormedSpace ℝ E} (x y : CS n E), x = y ↔ x.toFun = y.toFun"} +{"name":"CS.h2","declaration":"def CS.h2 {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : CS n E) : HasCompactSupport self.toFun"} +{"name":"W21.norm_nonneg","declaration":"theorem W21.norm_nonneg {f : ℝ → ℂ} : 0 ≤ W21.norm f"} +{"name":"trunc.instCoeFunTruncForAllReal","declaration":"instance trunc.instCoeFunTruncForAllReal : CoeFun trunc fun x => ℝ → ℝ"} +{"name":"W1.toFun","declaration":"def W1.toFun {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : W1 n E) : ℝ → E"} +{"name":"W21.hf'","declaration":"theorem W21.hf' (f : W21) : MeasureTheory.Integrable (deriv f.toFun) MeasureTheory.volume"} +{"name":"W21.instHMulCSOfNatNatInstOfNatNatRealNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpaceW21ComplexInstNormedAddCommGroupComplexComplexToRealInstRCLikeComplex","declaration":"instance W21.instHMulCSOfNatNatInstOfNatNatRealNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpaceW21ComplexInstNormedAddCommGroupComplexComplexToRealInstRCLikeComplex : HMul (CS 2 ℝ) W21 (CS 2 ℂ)"} +{"name":"CS.continuous","declaration":"theorem CS.continuous {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : CS n E) : Continuous f.toFun"} +{"name":"CS.deriv_smul","declaration":"theorem CS.deriv_smul {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {R : ℝ} {f : CS (n + 1) E} : CS.deriv (R • f) = R • CS.deriv f"} +{"name":"CS.tendsto_scale","declaration":"theorem CS.tendsto_scale {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : CS n E) (x : ℝ) : Filter.Tendsto (fun R => (CS.scale f R).toFun x) Filter.atTop (nhds (f.toFun 0))"} +{"name":"CS.hasDerivAt","declaration":"theorem CS.hasDerivAt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : CS (n + 1) E) (x : ℝ) : HasDerivAt f.toFun ((CS.deriv f).toFun x) x"} +{"name":"contDiff_ofReal","declaration":"theorem contDiff_ofReal : ContDiff ℝ ⊤ Complex.ofReal'"} +{"name":"W1.hasDerivAt","declaration":"theorem W1.hasDerivAt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : W1 (n + 1) E) (x : ℝ) : HasDerivAt f.toFun ((W1.deriv f).toFun x) x"} +{"name":"CS.instCoeFunCSForAllReal","declaration":"instance CS.instCoeFunCSForAllReal {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} : CoeFun (CS n E) fun x => ℝ → E"} +{"name":"trunc.zero","declaration":"theorem trunc.zero (g : trunc) : g.toFun =ᶠ[nhds 0] 1"} +{"name":"CS.h1","declaration":"def CS.h1 {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : CS n E) : ContDiff ℝ (↑n) self.toFun"} +{"name":"tendsto_funscale","declaration":"theorem tendsto_funscale {E : Type u_1} [NormedAddCommGroup E] {f : ℝ → E} (hf : ContinuousAt f 0) (x : ℝ) : Filter.Tendsto (fun R => funscale f R x) Filter.atTop (nhds (f 0))"} +{"name":"W21","declaration":"def W21 : Type"} +{"name":"W1.iteratedDeriv_sub","declaration":"theorem W1.iteratedDeriv_sub {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {f : ℝ → E} {g : ℝ → E} (hf : ContDiff ℝ (↑n) f) (hg : ContDiff ℝ (↑n) g) : iteratedDeriv n (f - g) = iteratedDeriv n f - iteratedDeriv n g"} +{"name":"trunc.h3","declaration":"def trunc.h3 (self : trunc) : Set.indicator (Set.Icc (-1) 1) 1 ≤ self.toFun"} +{"name":"W1.instCoeFunW1ForAllReal","declaration":"instance W1.instCoeFunW1ForAllReal {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} : CoeFun (W1 n E) fun x => ℝ → E"} +{"name":"CS.neg_apply","declaration":"theorem CS.neg_apply {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {f : CS n E} {x : ℝ} : (-f).toFun x = -f.toFun x"} +{"name":"W1.smooth","declaration":"def W1.smooth {n : ℕ} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (self : W1 n E) : ContDiff ℝ (↑n) self.toFun"} +{"name":"trunc","declaration":"structure trunc : Type"} +{"name":"CS.instCoeCSRealNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpaceComplexInstNormedAddCommGroupComplexComplexToRealInstRCLikeComplex","declaration":"instance CS.instCoeCSRealNormedAddCommGroupToNormedSpaceRCLikeInnerProductSpaceComplexInstNormedAddCommGroupComplexComplexToRealInstRCLikeComplex {n : ℕ} : Coe (CS n ℝ) (CS n ℂ)"} +{"name":"W21_approximation","declaration":"theorem W21_approximation (f : W21) (g : trunc) : Filter.Tendsto (fun R => ‖f - W21.ofCS2 (CS.scale g.toCS R * f)‖) Filter.atTop (nhds 0)"} +{"name":"W1.instSubW1","declaration":"instance W1.instSubW1 {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} : Sub (W1 n E)"} +{"name":"W1.sub","declaration":"def W1.sub {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} (f : W1 n E) (g : W1 n E) : W1 n E"} +{"name":"CS.instHSMulRealCS","declaration":"instance CS.instHSMulRealCS {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} : HSMul ℝ (CS n E) (CS n E)"} +{"name":"CS","declaration":"structure CS (n : ℕ) (E : Type u_2) [NormedAddCommGroup E] [NormedSpace ℝ E] : Type u_2"} +{"name":"CS.deriv_apply","declaration":"theorem CS.deriv_apply {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} {f : CS (n + 1) E} {x : ℝ} : (CS.deriv f).toFun x = deriv f.toFun x"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.StrongPNT.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.StrongPNT.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/pnt-declarations/PrimeNumberTheoremAnd.Wiener.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.Wiener.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..9b4daefc30fc71e57808cbea094a4d30037e53e4 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.Wiener.jsonl @@ -0,0 +1,175 @@ +{"name":"WienerIkeharaInterval_discrete'","declaration":"theorem WienerIkeharaInterval_discrete' {A : ℝ} {a : ℝ} {b : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hcheby : cheby fun n => ↑(f n)) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) (ha : 0 < a) (hb : a ≤ b) : Filter.Tendsto (fun N => (Finset.sum (Finset.Ico ⌈a * ↑N⌉₊ ⌈b * ↑N⌉₊) fun n => f n) / ↑N) Filter.atTop\n (nhds (A * (b - a)))"} +{"name":"hh_antitone","declaration":"theorem hh_antitone {a : ℝ} (ha : a ∈ Set.Ioo (-1) 1) : AntitoneOn (hh a) (Set.Ioi 0)"} +{"name":"wiener_ikehara_smooth'","declaration":"theorem wiener_ikehara_smooth' {A : ℝ} {Ψ : ℝ → ℂ} {G : ℂ → ℂ} {f : ℕ → ℂ} (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (hcheby : cheby f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}) (hsmooth : ContDiff ℝ ⊤ Ψ) (hsupp : HasCompactSupport Ψ) (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) : Filter.Tendsto (fun x => (∑' (n : ℕ), f n * Ψ (↑n / x)) / ↑x) Filter.atTop (nhds (↑A * ∫ (y : ℝ) in Set.Ioi 0, Ψ y))"} +{"name":"log_isbigo_log_div","declaration":"theorem log_isbigo_log_div {d : ℝ} (hb : 0 < d) : (fun n => Real.log n) =O[Filter.atTop] fun n => Real.log (n / d)"} +{"name":"wiener_ikehara_smooth_real","declaration":"theorem wiener_ikehara_smooth_real {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} {Ψ : ℝ → ℝ} (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hcheby : cheby fun n => ↑(f n)) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) (hsmooth : ContDiff ℝ ⊤ Ψ) (hsupp : HasCompactSupport Ψ) (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) : Filter.Tendsto (fun x => (∑' (n : ℕ), f n * Ψ (↑n / x)) / x) Filter.atTop (nhds (A * ∫ (y : ℝ) in Set.Ioi 0, Ψ y))"} +{"name":"pp'","declaration":"def pp' (a : ℝ) (x : ℝ) : ℝ"} +{"name":"one_div_sub_one","declaration":"theorem one_div_sub_one (n : ℕ) : 1 / ↑(n - 1) ≤ 2 / ↑n"} +{"name":"summation_by_parts'","declaration":"theorem summation_by_parts' {E : Type u_1} [Ring E] {a : ℕ → E} {b : ℕ → E} {n : ℕ} : cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n"} +{"name":"cheby","declaration":"def cheby (f : ℕ → ℂ) : Prop"} +{"name":"continuous_LSeries_aux","declaration":"theorem continuous_LSeries_aux {σ' : ℝ} {f : ℕ → ℂ} (hf : Summable (nterm f σ')) : Continuous fun x => LSeries f (↑σ' + ↑x * Complex.I)"} +{"name":"limiting_cor_schwartz","declaration":"theorem limiting_cor_schwartz {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℂ} (ψ : SchwartzMap ℝ ℂ) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (hcheby : cheby f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}) : Filter.Tendsto\n (fun x =>\n ∑' (n : ℕ), f n / ↑n * Real.fourierIntegral (⇑ψ) (1 / (2 * Real.pi) * Real.log (↑n / x)) -\n ↑A * ∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral (⇑ψ) (u / (2 * Real.pi)))\n Filter.atTop (nhds 0)"} +{"name":"comp_exp_support2","declaration":"theorem comp_exp_support2 {Ψ : ℝ → ℂ} (hsupp : HasCompactSupport Ψ) : ∀ᶠ (x : ℝ) in Filter.atTop, (Ψ ∘ Real.exp) x = 0"} +{"name":"hf_coe1","declaration":"theorem hf_coe1 {σ' : ℝ} {f : ℕ → ℂ} (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (hσ : 1 < σ') : ∑' (i : ℕ), ↑‖LSeries.term f (↑σ') i‖₊ ≠ ⊤"} +{"name":"gg_of_hh","declaration":"theorem gg_of_hh {x : ℝ} (hx : x ≠ 0) (i : ℝ) : gg x i = x⁻¹ * hh (1 / (2 * Real.pi)) (i / x)"} +{"name":"mem_Icc_iff_div","declaration":"theorem mem_Icc_iff_div {n : ℕ} {a : ℝ} {b : ℝ} {x : ℝ} (hb : 0 ≤ b) (hx : 0 < x) : n ∈ Finset.Icc ⌈a * x⌉₊ ⌊b * x⌋₊ ↔ ↑n / x ∈ Set.Icc a b"} +{"name":"decay_bounds","declaration":"theorem decay_bounds {A : ℝ} {u : ℝ} (ψ : CS 2 ℂ) (hA : ∀ (t : ℝ), ‖ψ.toFun t‖ ≤ A / (1 + t ^ 2)) (hA' : ∀ (t : ℝ), ‖deriv^[2] ψ.toFun t‖ ≤ A / (1 + t ^ 2)) : ‖Real.fourierIntegral ψ.toFun u‖ ≤ (Real.pi + 1 / (4 * Real.pi)) * A / (1 + u ^ 2)"} +{"name":"nnabla_bound_aux","declaration":"theorem nnabla_bound_aux {x : ℝ} (hx : 0 < x) : (nnabla fun n => 1 / (n * ((2 * Real.pi) ^ 2 + Real.log (n / x) ^ 2))) =O[Filter.atTop] fun n =>\n 1 / (Real.log n ^ 2 * n ^ 2)"} +{"name":"Asymptotics.IsBigO.sq","declaration":"theorem Asymptotics.IsBigO.sq {α : Type u_1} [Preorder α] {f : α → ℝ} {g : α → ℝ} (h : f =O[Filter.atTop] g) : (fun n => f n ^ 2) =O[Filter.atTop] fun n => g n ^ 2"} +{"name":"second_fourier_integrable_aux2","declaration":"theorem second_fourier_integrable_aux2 {x : ℝ} {t : ℝ} {σ' : ℝ} (hσ : 1 < σ') : MeasureTheory.IntegrableOn (fun u => Complex.exp ((1 - ↑σ' - ↑t * Complex.I) * ↑u)) (Set.Ioi (-Real.log x))\n MeasureTheory.volume"} +{"name":"cancel_main'","declaration":"theorem cancel_main' {C : ℝ} {f : ℕ → ℝ} {g : ℕ → ℝ} (hf : 0 ≤ f) (hf0 : f 0 = 0) (hg : 0 ≤ g) (hf' : ∀ (n : ℕ), cumsum f n ≤ C * ↑n) (hg' : Antitone g) (n : ℕ) : cumsum (f * g) n ≤ C * cumsum g n"} +{"name":"hh_integral","declaration":"theorem hh_integral {a : ℝ} {b : ℝ} {c : ℝ} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : ∫ (t : ℝ) in Set.Ioi 0, a * hh b (t / c) = a * c / b * Real.pi"} +{"name":"second_fourier_integrable_aux1a","declaration":"theorem second_fourier_integrable_aux1a {x : ℝ} {σ' : ℝ} (hσ : 1 < σ') : MeasureTheory.IntegrableOn (fun x => Complex.exp (-(↑x * (↑σ' - 1)))) (Set.Ici (-Real.log x)) MeasureTheory.volume"} +{"name":"pp_deriv_eq","declaration":"theorem pp_deriv_eq (a : ℝ) : deriv (pp a) = pp' a"} +{"name":"summable_iff_bounded'","declaration":"theorem summable_iff_bounded' {u : ℕ → ℝ} (hu : ∀ᶠ (n : ℕ) in Filter.atTop, 0 ≤ u n) : Summable u ↔ Filter.BoundedAtFilter Filter.atTop (cumsum u)"} +{"name":"comp_exp_support1","declaration":"theorem comp_exp_support1 {Ψ : ℝ → ℂ} (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) : ∀ᶠ (x : ℝ) in Filter.atBot, Ψ (Real.exp x) = 0"} +{"name":"limiting_cor_W21","declaration":"theorem limiting_cor_W21 {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℂ} (ψ : W21) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (hcheby : cheby f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}) : Filter.Tendsto\n (fun x =>\n ∑' (n : ℕ), f n / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x)) -\n ↑A * ∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral ψ.toFun (u / (2 * Real.pi)))\n Filter.atTop (nhds 0)"} +{"name":"WI_sum_Iab_le'","declaration":"theorem WI_sum_Iab_le' {a : ℝ} {b : ℝ} {f : ℕ → ℝ} (hpos : 0 ≤ f) {C : ℝ} (hcheby : chebyWith C fun n => ↑(f n)) (hb : 0 < b) : ∀ᶠ (x : ℝ) in Filter.atTop, (∑' (n : ℕ), f n * Set.indicator (Set.Ico a b) 1 (↑n / x)) / x ≤ C * 2 * b"} +{"name":"summation_by_parts","declaration":"theorem summation_by_parts {E : Type u_1} [Ring E] {a : ℕ → E} {A : ℕ → E} {b : ℕ → E} (ha : a = nabla A) {n : ℕ} : cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => b (i + 1) - b i) n"} +{"name":"auto_cheby","declaration":"theorem auto_cheby {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) : cheby fun n => ↑(f n)"} +{"name":"ge_of_eventually_nhdsWithin","declaration":"theorem ge_of_eventually_nhdsWithin {a : ℝ} {b : ℝ} (h : ∀ᶠ (c : ℝ) in nhdsWithin b (Set.Iio b), c ≤ a) : b ≤ a"} +{"name":"nterm_eq_norm_term","declaration":"theorem nterm_eq_norm_term {n : ℕ} {σ' : ℝ} {f : ℕ → ℂ} : nterm f σ' n = ‖LSeries.term f (↑σ') n‖"} +{"name":"decay_bounds_cor_aux","declaration":"theorem decay_bounds_cor_aux (ψ : CS 2 ℂ) : ∃ C, ∀ (u : ℝ), ‖ψ.toFun u‖ ≤ C / (1 + u ^ 2)"} +{"name":"second_fourier","declaration":"theorem second_fourier {ψ : ℝ → ℂ} (hcont : Continuous ψ) (hsupp : MeasureTheory.Integrable ψ MeasureTheory.volume) {x : ℝ} {σ' : ℝ} (hx : 0 < x) (hσ : 1 < σ') : ∫ (u : ℝ) in Set.Ici (-Real.log x), ↑(Real.exp (-u * (σ' - 1))) * Real.fourierIntegral ψ (u / (2 * Real.pi)) =\n ↑(x ^ (σ' - 1)) * ∫ (t : ℝ), 1 / (↑σ' + ↑t * Complex.I - 1) * ψ t * ↑x ^ (↑t * Complex.I)"} +{"name":"limiting_fourier_lim2","declaration":"theorem limiting_fourier_lim2 {x : ℝ} (A : ℝ) (ψ : W21) (hx : 1 ≤ x) : Filter.Tendsto\n (fun σ' =>\n ↑A * ↑(x ^ (1 - σ')) *\n ∫ (u : ℝ) in Set.Ici (-Real.log x),\n ↑(Real.exp (-u * (σ' - 1))) * Real.fourierIntegral ψ.toFun (u / (2 * Real.pi)))\n (nhdsWithin 1 (Set.Ioi 1))\n (nhds (↑A * ∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral ψ.toFun (u / (2 * Real.pi))))"} +{"name":"pp_pos","declaration":"theorem pp_pos {a : ℝ} (ha : a ∈ Set.Ioo (-1) 1) (x : ℝ) : 0 < pp a x"} +{"name":"first_fourier_aux2","declaration":"theorem first_fourier_aux2 {x : ℝ} {y : ℝ} {σ' : ℝ} {ψ : ℝ → ℂ} {f : ℕ → ℂ} (hx : 0 < x) (n : ℕ) : LSeries.term f (↑σ') n * Real.fourierChar (-(y * (1 / (2 * Real.pi) * Real.log (↑n / x)))) • ψ y =\n LSeries.term f (↑σ' + ↑y * Complex.I) n • (ψ y * ↑x ^ (↑y * Complex.I))"} +{"name":"limiting_cor","declaration":"theorem limiting_cor {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℂ} (ψ : CS 2 ℂ) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (hcheby : cheby f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}) : Filter.Tendsto\n (fun x =>\n ∑' (n : ℕ), f n / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x)) -\n ↑A * ∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral ψ.toFun (u / (2 * Real.pi)))\n Filter.atTop (nhds 0)"} +{"name":"bound_I2","declaration":"theorem bound_I2 (x : ℝ) (ψ : W21) : ‖∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral ψ.toFun (u / (2 * Real.pi))‖ ≤\n W21.norm ψ.toFun * (2 * Real.pi ^ 2)"} +{"name":"summable_fourier","declaration":"theorem summable_fourier {f : ℕ → ℂ} (x : ℝ) (hx : 0 < x) (ψ : W21) (hcheby : cheby f) : Summable fun i => ‖f i / ↑i * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑i / x))‖"} +{"name":"decay_bounds_aux","declaration":"theorem decay_bounds_aux {A : ℝ} {f : ℝ → ℂ} (hf : MeasureTheory.AEStronglyMeasurable f MeasureTheory.volume) (h : ∀ (t : ℝ), ‖f t‖ ≤ A * (1 + t ^ 2)⁻¹) : ∫ (t : ℝ), ‖f t‖ ≤ Real.pi * A"} +{"name":"hh_le","declaration":"theorem hh_le (a : ℝ) (t : ℝ) (ht : 0 ≤ t) : |hh a t| ≤ t⁻¹"} +{"name":"WienerIkeharaTheorem''","declaration":"theorem WienerIkeharaTheorem'' {A : ℝ} {F : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hG : ContinuousOn F {s | 1 ≤ s.re}) (hG' : Set.EqOn F (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) : Filter.Tendsto (fun N => cumsum f N / ↑N) Filter.atTop (nhds A)"} +{"name":"smooth_urysohn_support_Ioo","declaration":"theorem smooth_urysohn_support_Ioo {a : ℝ} {b : ℝ} {c : ℝ} {d : ℝ} (h1 : a < b) (h3 : c < d) : ∃ Ψ,\n ContDiff ℝ ⊤ Ψ ∧\n HasCompactSupport Ψ ∧\n Set.indicator (Set.Icc b c) 1 ≤ Ψ ∧ Ψ ≤ Set.indicator (Set.Ioo a d) 1 ∧ Function.support Ψ = Set.Ioo a d"} +{"name":"summable_inv_mul_log_sq","declaration":"theorem summable_inv_mul_log_sq : Summable fun n => (↑n * Real.log ↑n ^ 2)⁻¹"} +{"name":"hh_integrable","declaration":"theorem hh_integrable {a : ℝ} {b : ℝ} {c : ℝ} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : MeasureTheory.IntegrableOn (fun t => a * hh b (t / c)) (Set.Ici 0) MeasureTheory.volume"} +{"name":"dirichlet_test'","declaration":"theorem dirichlet_test' {a : ℕ → ℝ} {b : ℕ → ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hAb : Filter.BoundedAtFilter Filter.atTop (shift (cumsum a) * b)) (hbb : ∀ᶠ (n : ℕ) in Filter.atTop, b (n + 1) ≤ b n) (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b)"} +{"name":"Finset.sum_shift_back","declaration":"theorem Finset.sum_shift_back {E : Type u_1} [Ring E] {u : ℕ → E} {n : ℕ} : cumsum u (n + 1) = cumsum u n + u n"} +{"name":"log_mul_add_isBigO_log","declaration":"theorem log_mul_add_isBigO_log {a : ℝ} (ha : 0 < a) (b : ℝ) : (fun x => Real.log (a * x + b)) =O[Filter.atTop] Real.log"} +{"name":"toSchwartz_apply","declaration":"theorem toSchwartz_apply (f : ℝ → ℂ) {h1 : ContDiff ℝ ⊤ f} {h2 : ∀ (k n : ℕ), ∃ C, ∀ (x : ℝ), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ C} {x : ℝ} : { toFun := f, smooth' := h1, decay' := h2 } x = f x"} +{"name":"nnabla","declaration":"def nnabla {α : Type u_1} {E : Type u_2} [OfNat α 1] [Add α] [Sub E] (u : α → E) (n : α) : E"} +{"name":"Finset.sum_shift_back'","declaration":"theorem Finset.sum_shift_back' {E : Type u_1} [Ring E] {u : ℕ → E} : shift (cumsum u) = cumsum u + u"} +{"name":"exists_antitone_of_eventually","declaration":"theorem exists_antitone_of_eventually {u : ℕ → ℝ} (hu : ∀ᶠ (n : ℕ) in Filter.atTop, u (n + 1) ≤ u n) : ∃ v, Set.range v ⊆ Set.range u ∧ Antitone v ∧ v =ᶠ[Filter.atTop] u"} +{"name":"nnabla_bound_aux2","declaration":"theorem nnabla_bound_aux2 (a : ℝ) {b : ℝ} (hb : 0 < b) : ∀ᶠ (x : ℝ) in Filter.atTop, 0 < x * (a + Real.log (x / b) ^ 2)"} +{"name":"cheby.bigO","declaration":"theorem cheby.bigO {f : ℕ → ℂ} (h : cheby f) : (cumsum fun x => ‖f x‖) =O[Filter.atTop] Nat.cast"} +{"name":"WI_tendsto_aux'","declaration":"theorem WI_tendsto_aux' (a : ℝ) (b : ℝ) {A : ℝ} (hA : 0 < A) : Filter.Tendsto (fun c => b - a - c / A) (nhdsWithin (A * (b - a)) (Set.Iio (A * (b - a)))) (nhdsWithin 0 (Set.Ioi 0))"} +{"name":"one_div_two_pi_mem_Ioo","declaration":"theorem one_div_two_pi_mem_Ioo : 1 / (2 * Real.pi) ∈ Set.Ioo (-1) 1"} +{"name":"isLittleO_const_of_tendsto_atTop","declaration":"theorem isLittleO_const_of_tendsto_atTop {α : Type u_1} [Preorder α] (a : ℝ) {f : α → ℝ} (hf : Filter.Tendsto f Filter.atTop Filter.atTop) : (fun x => a) =o[Filter.atTop] f"} +{"name":"WienerIkeharaTheorem'","declaration":"theorem WienerIkeharaTheorem' {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hcheby : cheby fun n => ↑(f n)) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) : Filter.Tendsto (fun N => cumsum f N / ↑N) Filter.atTop (nhds A)"} +{"name":"summation_by_parts''","declaration":"theorem summation_by_parts'' {E : Type u_1} [Ring E] {a : ℕ → E} {b : ℕ → E} : shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b)"} +{"name":"second_fourier_aux","declaration":"theorem second_fourier_aux {x : ℝ} {t : ℝ} {σ' : ℝ} (hx : 0 < x) : -(Complex.exp (-((1 - ↑σ' - ↑t * Complex.I) * ↑(Real.log x))) / (1 - ↑σ' - ↑t * Complex.I)) =\n ↑(x ^ (σ' - 1)) * (↑σ' + ↑t * Complex.I - 1)⁻¹ * ↑x ^ (↑t * Complex.I)"} +{"name":"tendsto_S_S_zero","declaration":"theorem tendsto_S_S_zero {f : ℕ → ℝ} (hpos : 0 ≤ f) (hcheby : cheby fun n => ↑(f n)) : TendstoUniformlyOnFilter (S f) (S f 0) (nhdsWithin 0 (Set.Ioi 0)) Filter.atTop"} +{"name":"bound_I1","declaration":"theorem bound_I1 {f : ℕ → ℂ} (x : ℝ) (hx : 0 < x) (ψ : W21) (hcheby : cheby f) : ‖∑' (n : ℕ), f n / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x))‖ ≤\n W21.norm ψ.toFun • ∑' (i : ℕ), ‖f i‖ / ↑i * (1 + (1 / (2 * Real.pi) * Real.log (↑i / x)) ^ 2)⁻¹"} +{"name":"instCoeForAllRealForAllComplex_1","declaration":"def instCoeForAllRealForAllComplex_1 {E : Type u_1} : Coe (E → ℝ) (E → ℂ)"} +{"name":"tendsto_mul_add_atTop","declaration":"theorem tendsto_mul_add_atTop {a : ℝ} (ha : 0 < a) (b : ℝ) : Filter.Tendsto (fun x => a * x + b) Filter.atTop Filter.atTop"} +{"name":"bound_main","declaration":"theorem bound_main {f : ℕ → ℂ} {C : ℝ} (A : ℂ) (x : ℝ) (hx : 1 ≤ x) (ψ : W21) (hcheby : chebyWith C f) : ‖∑' (n : ℕ), f n / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x)) -\n A * ∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral ψ.toFun (u / (2 * Real.pi))‖ ≤\n W21.norm ψ.toFun * (C * (1 + 2 * Real.pi ^ 2) + ‖A‖ * (2 * Real.pi ^ 2))"} +{"name":"S","declaration":"def S {𝕜 : Type} [RCLike 𝕜] (f : ℕ → 𝕜) (ε : ℝ) (N : ℕ) : 𝕜"} +{"name":"second_fourier_integrable_aux1","declaration":"theorem second_fourier_integrable_aux1 {x : ℝ} {σ' : ℝ} {ψ : ℝ → ℂ} (hcont : Continuous ψ) (hsupp : MeasureTheory.Integrable ψ MeasureTheory.volume) (hσ : 1 < σ') : let ν := MeasureTheory.Measure.prod (MeasureTheory.volume.restrict (Set.Ici (-Real.log x))) MeasureTheory.volume;\nMeasureTheory.Integrable\n (Function.uncurry fun u a =>\n ↑(Real.exp (-u * (σ' - 1))) • ↑(Real.fourierChar (Multiplicative.ofAdd (-(a * (u / (2 * Real.pi)))))) • ψ a)\n ν"} +{"name":"pp","declaration":"def pp (a : ℝ) (x : ℝ) : ℝ"} +{"name":"pp_deriv","declaration":"theorem pp_deriv (a : ℝ) (x : ℝ) : HasDerivAt (pp a) (pp' a x) x"} +{"name":"nabla","declaration":"def nabla {α : Type u_1} {E : Type u_2} [OfNat α 1] [Add α] [Sub E] (u : α → E) (n : α) : E"} +{"name":"cumsum_succ","declaration":"theorem cumsum_succ {E : Type u_2} [AddCommMonoid E] {u : ℕ → E} (n : ℕ) : cumsum u (n + 1) = cumsum u n + u n"} +{"name":"log_add_one_sub_log_le","declaration":"theorem log_add_one_sub_log_le {x : ℝ} (hx : 0 < x) : nabla Real.log x ≤ x⁻¹"} +{"name":"residue_nonneg","declaration":"theorem residue_nonneg {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hcheby : cheby fun n => ↑(f n)) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) : 0 ≤ A"} +{"name":"hh","declaration":"def hh (a : ℝ) (t : ℝ) : ℝ"} +{"name":"limiting_fourier_lim2_aux","declaration":"theorem limiting_fourier_lim2_aux (x : ℝ) (C : ℝ) : MeasureTheory.Integrable (fun t => |x| * (C / (1 + (t / (2 * Real.pi)) ^ 2)))\n (MeasureTheory.volume.restrict (Set.Ici (-Real.log x)))"} +{"name":"Finset.sum_shift_front'","declaration":"theorem Finset.sum_shift_front' {E : Type u_1} [Ring E] {u : ℕ → E} : shift (cumsum u) = (fun x => u 0) + cumsum (shift u)"} +{"name":"first_fourier_aux2a","declaration":"theorem first_fourier_aux2a {n : ℕ} {x : ℝ} {y : ℝ} : 2 * ↑Real.pi * -(↑y * (1 / (2 * ↑Real.pi) * ↑(Real.log (↑n / x)))) = -(↑y * ↑(Real.log (↑n / x)))"} +{"name":"mem_Ico_iff_div","declaration":"theorem mem_Ico_iff_div {n : ℕ} {a : ℝ} {b : ℝ} {x : ℝ} (hx : 0 < x) : n ∈ Finset.Ico ⌈a * x⌉₊ ⌈b * x⌉₊ ↔ ↑n / x ∈ Set.Ico a b"} +{"name":"Filter.EventuallyEq.summable","declaration":"theorem Filter.EventuallyEq.summable {u : ℕ → ℝ} {v : ℕ → ℝ} (h : u =ᶠ[Filter.atTop] v) (hu : Summable v) : Summable u"} +{"name":"cumsum_nonneg","declaration":"theorem cumsum_nonneg {u : ℕ → ℝ} (hu : 0 ≤ u) : 0 ≤ cumsum u"} +{"name":"nnabla_mul","declaration":"theorem nnabla_mul {α : Type u_1} {E : Type u_2} [OfNat α 1] [Add α] [Ring E] {u : α → E} {c : E} : (nnabla fun n => c * u n) = c • nnabla u"} +{"name":"vonMangoldt_cheby","declaration":"theorem vonMangoldt_cheby : cheby fun n => ↑(ArithmeticFunction.vonMangoldt n)"} +{"name":"cancel_aux'","declaration":"theorem cancel_aux' {C : ℝ} {f : ℕ → ℝ} {g : ℕ → ℝ} (hf : 0 ≤ f) (hg : 0 ≤ g) (hf' : ∀ (n : ℕ), cumsum f n ≤ C * ↑n) (hg' : Antitone g) (n : ℕ) : (Finset.sum (Finset.range n) fun i => f i * g i) ≤\n C * ↑n * g (n - 1) + C * cumsum g (n - 1 - 1 + 1) - C * (↑(n - 1 - 1) + 1) * g (n - 1)"} +{"name":"decay_bounds_key","declaration":"theorem decay_bounds_key (f : W21) (u : ℝ) : ‖Real.fourierIntegral f.toFun u‖ ≤ ‖f‖ * (1 + u ^ 2)⁻¹"} +{"name":"continuous_FourierIntegral","declaration":"theorem continuous_FourierIntegral (ψ : W21) : Continuous (Real.fourierIntegral ψ.toFun)"} +{"name":"BoundedAtFilter.comp_add","declaration":"theorem BoundedAtFilter.comp_add {u : ℕ → ℝ} {N : ℕ} : (Filter.BoundedAtFilter Filter.atTop fun n => u (n + N)) ↔ Filter.BoundedAtFilter Filter.atTop u"} +{"name":"le_of_eventually_nhdsWithin","declaration":"theorem le_of_eventually_nhdsWithin {a : ℝ} {b : ℝ} (h : ∀ᶠ (c : ℝ) in nhdsWithin b (Set.Ioi b), a ≤ c) : a ≤ b"} +{"name":"nabla_cumsum","declaration":"theorem nabla_cumsum {E : Type u_2} [AddCommGroup E] {u : ℕ → E} : nabla (cumsum u) = u"} +{"name":"first_fourier","declaration":"theorem first_fourier {x : ℝ} {σ' : ℝ} {ψ : ℝ → ℂ} {f : ℕ → ℂ} (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (hcont : Continuous ψ) (hsupp : MeasureTheory.Integrable ψ MeasureTheory.volume) (hx : 0 < x) (hσ : 1 < σ') : ∑' (n : ℕ), LSeries.term f (↑σ') n * Real.fourierIntegral ψ (1 / (2 * Real.pi) * Real.log (↑n / x)) =\n ∫ (t : ℝ), LSeries f (↑σ' + ↑t * Complex.I) * ψ t * ↑x ^ (↑t * Complex.I)"} +{"name":"one_add_sq_pos","declaration":"theorem one_add_sq_pos (u : ℝ) : 0 < 1 + u ^ 2"} +{"name":"wiener_ikehara_smooth","declaration":"theorem wiener_ikehara_smooth {A : ℝ} {Ψ : ℝ → ℂ} {G : ℂ → ℂ} {f : ℕ → ℂ} (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (hcheby : cheby f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}) (hsmooth : ContDiff ℝ ⊤ Ψ) (hsupp : HasCompactSupport Ψ) (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) : Filter.Tendsto (fun x => (∑' (n : ℕ), f n * Ψ (↑n / x)) / ↑x - ↑A * ∫ (y : ℝ) in Set.Ioi 0, Ψ y) Filter.atTop (nhds 0)"} +{"name":"W21.integrable_fourier","declaration":"theorem W21.integrable_fourier {c : ℝ} (ψ : W21) (hc : c ≠ 0) : MeasureTheory.Integrable (fun u => Real.fourierIntegral ψ.toFun (u / c)) MeasureTheory.volume"} +{"name":"pp'_deriv","declaration":"theorem pp'_deriv (a : ℝ) (x : ℝ) : HasDerivAt (pp' a) (a ^ 2 * 2) x"} +{"name":"decay_bounds_W21","declaration":"theorem decay_bounds_W21 {A : ℝ} (f : W21) (hA : ∀ (t : ℝ), ‖f.toFun t‖ ≤ A / (1 + t ^ 2)) (hA' : ∀ (t : ℝ), ‖deriv (deriv f.toFun) t‖ ≤ A / (1 + t ^ 2)) (u : ℝ) : ‖Real.fourierIntegral f.toFun u‖ ≤ (Real.pi + 1 / (4 * Real.pi)) * A / (1 + u ^ 2)"} +{"name":"bound_sum_log'","declaration":"theorem bound_sum_log' {f : ℕ → ℂ} {C : ℝ} (hf : chebyWith C f) {x : ℝ} (hx : 1 ≤ x) : ∑' (i : ℕ), ‖f i‖ / ↑i * (1 + (1 / (2 * Real.pi) * Real.log (↑i / x)) ^ 2)⁻¹ ≤ C * (1 + 2 * Real.pi ^ 2)"} +{"name":"WienerIkeharaInterval","declaration":"theorem WienerIkeharaInterval {A : ℝ} {a : ℝ} {b : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hcheby : cheby fun n => ↑(f n)) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) (ha : 0 < a) (hb : a ≤ b) : Filter.Tendsto (fun x => (∑' (n : ℕ), f n * Set.indicator (Set.Ico a b) 1 (↑n / x)) / x) Filter.atTop\n (nhds (A * (b - a)))"} +{"name":"comp_exp_support","declaration":"theorem comp_exp_support {Ψ : ℝ → ℂ} (hsupp : HasCompactSupport Ψ) (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) : HasCompactSupport (Ψ ∘ Real.exp)"} +{"name":"limiting_fourier_lim1_aux","declaration":"theorem limiting_fourier_lim1_aux {x : ℝ} {f : ℕ → ℂ} (hcheby : cheby f) (hx : 0 < x) (C : ℝ) (hC : 0 ≤ C) : Summable fun n => ‖f n‖ / ↑n * (C / (1 + (1 / (2 * Real.pi) * Real.log (↑n / x)) ^ 2))"} +{"name":"toSchwartz","declaration":"def toSchwartz (f : ℝ → ℂ) (h1 : ContDiff ℝ ⊤ f) (h2 : HasCompactSupport f) : SchwartzMap ℝ ℂ"} +{"name":"smooth_urysohn","declaration":"theorem smooth_urysohn (a : ℝ) (b : ℝ) (c : ℝ) (d : ℝ) (h1 : a < b) (h3 : c < d) : ∃ Ψ, ContDiff ℝ ⊤ Ψ ∧ HasCompactSupport Ψ ∧ Set.indicator (Set.Icc b c) 1 ≤ Ψ ∧ Ψ ≤ Set.indicator (Set.Ioo a d) 1"} +{"name":"quadratic_pos","declaration":"theorem quadratic_pos (a : ℝ) (b : ℝ) (c : ℝ) (x : ℝ) (ha : 0 < a) (hΔ : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c"} +{"name":"hh'","declaration":"def hh' (a : ℝ) (t : ℝ) : ℝ"} +{"name":"summable_congr_ae","declaration":"theorem summable_congr_ae {u : ℕ → ℝ} {v : ℕ → ℝ} (huv : u =ᶠ[Filter.atTop] v) : Summable u ↔ Summable v"} +{"name":"tendsto_mul_ceil_div","declaration":"/-- A version of the *Wiener-Ikehara Tauberian Theorem*: If `f` is a nonnegative arithmetic\nfunction whose L-series has a simple pole at `s = 1` with residue `A` and otherwise extends\ncontinuously to the closed half-plane `re s ≥ 1`, then `∑ n < N, f n` is asymptotic to `A*N`. -/\ntheorem tendsto_mul_ceil_div : Filter.Tendsto (fun p => ↑⌈p.1 * ↑p.2⌉₊ / ↑p.2) (nhdsWithin 0 (Set.Ioi 0) ×ˢ Filter.atTop) (nhds 0)"} +{"name":"set_integral_ofReal","declaration":"theorem set_integral_ofReal {f : ℝ → ℝ} {s : Set ℝ} : ∫ (x : ℝ) in s, ↑(f x) = ↑(∫ (x : ℝ) in s, f x)"} +{"name":"le_floor_mul_iff","declaration":"theorem le_floor_mul_iff {n : ℕ} {b : ℝ} {x : ℝ} (hb : 0 ≤ b) (hx : 0 < x) : n ≤ ⌊b * x⌋₊ ↔ ↑n / x ≤ b"} +{"name":"hh_integrable_aux","declaration":"theorem hh_integrable_aux {a : ℝ} {b : ℝ} {c : ℝ} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : MeasureTheory.IntegrableOn (fun t => a * hh b (t / c)) (Set.Ici 0) MeasureTheory.volume ∧\n ∫ (t : ℝ) in Set.Ioi 0, a * hh b (t / c) = a * c / b * Real.pi"} +{"name":"gg","declaration":"def gg (x : ℝ) (i : ℝ) : ℝ"} +{"name":"exists_trunc","declaration":"def exists_trunc : trunc"} +{"name":"comp_exp_support0","declaration":"theorem comp_exp_support0 {Ψ : ℝ → ℂ} (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) : ∀ᶠ (x : ℝ) in nhds 0, Ψ x = 0"} +{"name":"limiting_fourier_aux","declaration":"theorem limiting_fourier_aux {A : ℝ} {x : ℝ} {G : ℂ → ℂ} {f : ℕ → ℂ} (hG' : Set.EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (ψ : CS 2 ℂ) (hx : 1 ≤ x) (σ' : ℝ) (hσ' : 1 < σ') : ∑' (n : ℕ), LSeries.term f (↑σ') n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x)) -\n ↑A * ↑(x ^ (1 - σ')) *\n ∫ (u : ℝ) in Set.Ici (-Real.log x),\n ↑(Real.exp (-u * (σ' - 1))) * Real.fourierIntegral ψ.toFun (u / (2 * Real.pi)) =\n ∫ (t : ℝ), G (↑σ' + ↑t * Complex.I) * ψ.toFun t * ↑x ^ (↑t * Complex.I)"} +{"name":"isLittleO_mul_add_sq","declaration":"theorem isLittleO_mul_add_sq (a : ℝ) (b : ℝ) : (fun x => a * x + b) =o[Filter.atTop] fun x => x ^ 2"} +{"name":"isBigO_log_mul_add","declaration":"theorem isBigO_log_mul_add {a : ℝ} (ha : 0 < a) (b : ℝ) : Real.log =O[Filter.atTop] fun x => Real.log (a * x + b)"} +{"name":"hh_continuous","declaration":"theorem hh_continuous (a : ℝ) : ContinuousOn (hh a) (Set.Ioi 0)"} +{"name":"nabla_log","declaration":"theorem nabla_log {b : ℝ} (hb : 0 < b) : (nabla fun x => Real.log (x / b)) =O[Filter.atTop] fun x => 1 / x"} +{"name":"cancel_aux","declaration":"theorem cancel_aux {C : ℝ} {f : ℕ → ℝ} {g : ℕ → ℝ} (hf : 0 ≤ f) (hg : 0 ≤ g) (hf' : ∀ (n : ℕ), cumsum f n ≤ C * ↑n) (hg' : Antitone g) (n : ℕ) : (Finset.sum (Finset.range n) fun i => f i * g i) ≤\n g (n - 1) * (C * ↑n) +\n (C * (↑(n - 1 - 1) + 1) * g 0 - C * (↑(n - 1 - 1) + 1) * g (n - 1) -\n ((n - 1 - 1) • (C * g 0) - Finset.sum (Finset.range (n - 1 - 1)) fun x => C * g (x + 1)))"} +{"name":"S_sub_S","declaration":"theorem S_sub_S {𝕜 : Type} [RCLike 𝕜] {f : ℕ → 𝕜} {ε : ℝ} {N : ℕ} (hε : ε ≤ 1) : S f 0 N - S f ε N = cumsum f ⌈ε * ↑N⌉₊ / ↑N"} +{"name":"WI_sum_Iab_le","declaration":"theorem WI_sum_Iab_le {a : ℝ} {b : ℝ} {x : ℝ} {f : ℕ → ℝ} (hpos : 0 ≤ f) {C : ℝ} (hcheby : chebyWith C fun n => ↑(f n)) (hb : 0 < b) (hxb : 2 / b < x) : (∑' (n : ℕ), f n * Set.indicator (Set.Ico a b) 1 (↑n / x)) / x ≤ C * 2 * b"} +{"name":"hh_deriv","declaration":"theorem hh_deriv (a : ℝ) {t : ℝ} (ht : t ≠ 0) : HasDerivAt (hh a) (hh' a t) t"} +{"name":"bound_sum_log","declaration":"theorem bound_sum_log {f : ℕ → ℂ} {C : ℝ} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : ℝ} (hx : 1 ≤ x) : ∑' (i : ℕ), ‖f i‖ / ↑i * (1 + (1 / (2 * Real.pi) * Real.log (↑i / x)) ^ 2)⁻¹ ≤\n C * (1 + ∫ (t : ℝ) in Set.Ioi 0, hh (1 / (2 * Real.pi)) t)"} +{"name":"bound_I1'","declaration":"theorem bound_I1' {f : ℕ → ℂ} {C : ℝ} (x : ℝ) (hx : 1 ≤ x) (ψ : W21) (hcheby : chebyWith C f) : ‖∑' (n : ℕ), f n / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x))‖ ≤\n W21.norm ψ.toFun * C * (1 + 2 * Real.pi ^ 2)"} +{"name":"log_add_div_isBigO_log","declaration":"theorem log_add_div_isBigO_log (a : ℝ) {b : ℝ} (hb : 0 < b) : (fun x => Real.log ((x + a) / b)) =O[Filter.atTop] fun x => Real.log x"} +{"name":"wiener_ikehara_smooth_aux","declaration":"theorem wiener_ikehara_smooth_aux {Ψ : ℝ → ℂ} (l0 : Continuous Ψ) (hsupp : HasCompactSupport Ψ) (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) (x : ℝ) (hx : 0 < x) : ∫ (u : ℝ) in Set.Ioi (-Real.log x), ↑(Real.exp u) * Ψ (Real.exp u) = ∫ (y : ℝ) in Set.Ioi (1 / x), Ψ y"} +{"name":"crude_upper_bound","declaration":"theorem crude_upper_bound {A : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (ψ : CS 2 ℂ) (hψpos : ∀ (y : ℝ), 0 ≤ (Real.fourierIntegral ψ.toFun y).re ∧ (Real.fourierIntegral ψ.toFun y).im = 0) : ∃ B,\n ∀ (x : ℝ),\n 0 < x → ‖∑' (n : ℕ), ↑(f n) / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x))‖ ≤ B"} +{"name":"Asymptotics.IsBigO.add_isLittleO_right","declaration":"theorem Asymptotics.IsBigO.add_isLittleO_right {f : ℝ → ℝ} {g : ℝ → ℝ} (h : g =o[Filter.atTop] f) : f =O[Filter.atTop] (f + g)"} +{"name":"sum_le_integral","declaration":"theorem sum_le_integral {x₀ : ℝ} {f : ℝ → ℝ} {n : ℕ} (hf : AntitoneOn f (Set.Ioc x₀ (x₀ + ↑n))) (hfi : MeasureTheory.IntegrableOn f (Set.Icc x₀ (x₀ + ↑n)) MeasureTheory.volume) : (Finset.sum (Finset.range n) fun i => f (x₀ + ↑(i + 1))) ≤ ∫ (x : ℝ) in x₀..x₀ + ↑n, f x"} +{"name":"summable_iff_bounded","declaration":"theorem summable_iff_bounded {u : ℕ → ℝ} (hu : 0 ≤ u) : Summable u ↔ Filter.BoundedAtFilter Filter.atTop (cumsum u)"} +{"name":"BoundedAtFilter.add_const","declaration":"theorem BoundedAtFilter.add_const {u : ℕ → ℝ} {c : ℝ} : (Filter.BoundedAtFilter Filter.atTop fun n => u n + c) ↔ Filter.BoundedAtFilter Filter.atTop u"} +{"name":"hh'_nonpos","declaration":"theorem hh'_nonpos {a : ℝ} {x : ℝ} (ha : a ∈ Set.Ioo (-1) 1) : hh' a x ≤ 0"} +{"name":"nnabla_bound","declaration":"theorem nnabla_bound (C : ℝ) {x : ℝ} (hx : 0 < x) : (nnabla fun n => C / (1 + (Real.log (n / x) / (2 * Real.pi)) ^ 2) / n) =O[Filter.atTop] fun n =>\n (n ^ 2 * Real.log n ^ 2)⁻¹"} +{"name":"gg_l1","declaration":"theorem gg_l1 {x : ℝ} (hx : 0 < x) (n : ℕ) : |gg x ↑n| ≤ 1 / ↑n"} +{"name":"interval_approx_inf","declaration":"theorem interval_approx_inf {a : ℝ} {b : ℝ} (ha : 0 < a) (hab : a < b) : ∀ᶠ (ε : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n ∃ ψ,\n ContDiff ℝ ⊤ ψ ∧\n HasCompactSupport ψ ∧\n closure (Function.support ψ) ⊆ Set.Ioi 0 ∧\n ψ ≤ Set.indicator (Set.Ico a b) 1 ∧ b - a - ε ≤ ∫ (y : ℝ) in Set.Ioi 0, ψ y"} +{"name":"WI_sum_le","declaration":"theorem WI_sum_le {x : ℝ} {f : ℕ → ℝ} {g₁ : ℝ → ℝ} {g₂ : ℝ → ℝ} (hf : 0 ≤ f) (hg : g₁ ≤ g₂) (hx : 0 < x) (hg₁ : HasCompactSupport g₁) (hg₂ : HasCompactSupport g₂) : (∑' (n : ℕ), f n * g₁ (↑n / x)) / x ≤ (∑' (n : ℕ), f n * g₂ (↑n / x)) / x"} +{"name":"lt_ceil_mul_iff","declaration":"theorem lt_ceil_mul_iff {n : ℕ} {b : ℝ} {x : ℝ} (hx : 0 < x) : n < ⌈b * x⌉₊ ↔ ↑n / x < b"} +{"name":"nabla_log_main","declaration":"theorem nabla_log_main : nabla Real.log =O[Filter.atTop] fun x => 1 / x"} +{"name":"ceil_mul_le_iff","declaration":"theorem ceil_mul_le_iff {n : ℕ} {a : ℝ} {x : ℝ} (hx : 0 < x) : ⌈a * x⌉₊ ≤ n ↔ a ≤ ↑n / x"} +{"name":"nnabla_mul_log_sq","declaration":"theorem nnabla_mul_log_sq (a : ℝ) {b : ℝ} (hb : 0 < b) : (nabla fun x => x * (a + Real.log (x / b) ^ 2)) =O[Filter.atTop] fun x => Real.log x ^ 2"} +{"name":"isBigO_pow_pow_of_le","declaration":"theorem isBigO_pow_pow_of_le {m : ℕ} {n : ℕ} (h : m ≤ n) : (fun x => x ^ m) =O[Filter.atTop] fun x => x ^ n"} +{"name":"limiting_fourier_lim1","declaration":"theorem limiting_fourier_lim1 {x : ℝ} {f : ℕ → ℂ} (hcheby : cheby f) (ψ : W21) (hx : 0 < x) : Filter.Tendsto\n (fun σ' => ∑' (n : ℕ), LSeries.term f (↑σ') n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x)))\n (nhdsWithin 1 (Set.Ioi 1))\n (nhds (∑' (n : ℕ), f n / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x))))"} +{"name":"chebyWith","declaration":"def chebyWith (C : ℝ) (f : ℕ → ℂ) : Prop"} +{"name":"nnabla_cast","declaration":"theorem nnabla_cast {E : Type u_2} (u : ℝ → E) [Sub E] : nnabla u ∘ Nat.cast = nnabla (u ∘ Nat.cast)"} +{"name":"WI_tendsto_aux","declaration":"theorem WI_tendsto_aux (a : ℝ) (b : ℝ) {A : ℝ} (hA : 0 < A) : Filter.Tendsto (fun c => c / A - (b - a)) (nhdsWithin (A * (b - a)) (Set.Ioi (A * (b - a)))) (nhdsWithin 0 (Set.Ioi 0))"} +{"name":"first_fourier_aux1","declaration":"theorem first_fourier_aux1 {ψ : ℝ → ℂ} (hψ : Continuous ψ) {x : ℝ} (n : ℕ) : Measurable fun u => ↑‖Real.fourierChar (-(u * (1 / (2 * Real.pi) * Real.log (↑n / x)))) • ψ u‖₊"} +{"name":"cumsum_zero","declaration":"theorem cumsum_zero {E : Type u_2} [AddCommMonoid E] {u : ℕ → E} : cumsum u 0 = 0"} +{"name":"nnabla_bound_aux1","declaration":"theorem nnabla_bound_aux1 (a : ℝ) {b : ℝ} (hb : 0 < b) : Filter.Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) Filter.atTop Filter.atTop"} +{"name":"neg_cumsum","declaration":"theorem neg_cumsum {E : Type u_2} [AddCommGroup E] {u : ℕ → E} : -cumsum u = cumsum (-u)"} +{"name":"neg_nabla","declaration":"theorem neg_nabla {α : Type u_1} {E : Type u_2} [OfNat α 1] [Add α] [Ring E] {u : α → E} : -nabla u = nnabla u"} +{"name":"cumsum","declaration":"def cumsum {E : Type u_2} [AddCommMonoid E] (u : ℕ → E) (n : ℕ) : E"} +{"name":"sum_telescopic","declaration":"theorem sum_telescopic (a : ℕ → ℝ) (n : ℕ) : (Finset.sum (Finset.range n) fun i => a (i + 1) - a i) = a n - a 0"} +{"name":"limiting_fourier_variant","declaration":"theorem limiting_fourier_variant {A : ℝ} {x : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (ψ : CS 2 ℂ) (hψpos : ∀ (y : ℝ), 0 ≤ (Real.fourierIntegral ψ.toFun y).re ∧ (Real.fourierIntegral ψ.toFun y).im = 0) (hx : 1 ≤ x) : ∑' (n : ℕ), ↑(f n) / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x)) -\n ↑A * ∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral ψ.toFun (u / (2 * Real.pi)) =\n ∫ (t : ℝ), G (1 + ↑t * Complex.I) * ψ.toFun t * ↑x ^ (↑t * Complex.I)"} +{"name":"WeakPNT","declaration":"theorem WeakPNT : Filter.Tendsto (fun N => cumsum (⇑ArithmeticFunction.vonMangoldt) N / ↑N) Filter.atTop (nhds 1)"} +{"name":"interval_approx_sup","declaration":"theorem interval_approx_sup {a : ℝ} {b : ℝ} (ha : 0 < a) (hab : a < b) : ∀ᶠ (ε : ℝ) in nhdsWithin 0 (Set.Ioi 0),\n ∃ ψ,\n ContDiff ℝ ⊤ ψ ∧\n HasCompactSupport ψ ∧\n closure (Function.support ψ) ⊆ Set.Ioi 0 ∧\n Set.indicator (Set.Ico a b) 1 ≤ ψ ∧ ∫ (y : ℝ) in Set.Ioi 0, ψ y ≤ b - a + ε"} +{"name":"gg_le_one","declaration":"theorem gg_le_one {x : ℝ} (i : ℕ) : gg x ↑i ≤ 1"} +{"name":"hh_nonneg","declaration":"theorem hh_nonneg (a : ℝ) {t : ℝ} (ht : 0 ≤ t) : 0 ≤ hh a t"} +{"name":"bound_sum_log0","declaration":"theorem bound_sum_log0 {f : ℕ → ℂ} {C : ℝ} (hf : chebyWith C f) {x : ℝ} (hx : 1 ≤ x) : ∑' (i : ℕ), ‖f i‖ / ↑i * (1 + (1 / (2 * Real.pi) * Real.log (↑i / x)) ^ 2)⁻¹ ≤\n C * (1 + ∫ (t : ℝ) in Set.Ioi 0, hh (1 / (2 * Real.pi)) t)"} +{"name":"pp'_deriv_eq","declaration":"theorem pp'_deriv_eq (a : ℝ) : deriv (pp' a) = fun x => a ^ 2 * 2"} +{"name":"WienerIkeharaInterval_discrete","declaration":"theorem WienerIkeharaInterval_discrete {A : ℝ} {a : ℝ} {b : ℝ} {G : ℂ → ℂ} {f : ℕ → ℝ} (hpos : 0 ≤ f) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')) (hcheby : cheby fun n => ↑(f n)) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}) (ha : 0 < a) (hb : a ≤ b) : Filter.Tendsto (fun x => (Finset.sum (Finset.Ico ⌈a * x⌉₊ ⌈b * x⌉₊) fun n => f n) / x) Filter.atTop (nhds (A * (b - a)))"} +{"name":"limiting_fourier","declaration":"theorem limiting_fourier {A : ℝ} {x : ℝ} {G : ℂ → ℂ} {f : ℕ → ℂ} (hcheby : cheby f) (hG : ContinuousOn G {s | 1 ≤ s.re}) (hG' : Set.EqOn G (fun s => LSeries f s - ↑A / (s - 1)) {s | 1 < s.re}) (hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm f σ')) (ψ : CS 2 ℂ) (hx : 1 ≤ x) : ∑' (n : ℕ), f n / ↑n * Real.fourierIntegral ψ.toFun (1 / (2 * Real.pi) * Real.log (↑n / x)) -\n ↑A * ∫ (u : ℝ) in Set.Ici (-Real.log x), Real.fourierIntegral ψ.toFun (u / (2 * Real.pi)) =\n ∫ (t : ℝ), G (1 + ↑t * Complex.I) * ψ.toFun t * ↑x ^ (↑t * Complex.I)"} +{"name":"limiting_cor_aux","declaration":"theorem limiting_cor_aux {f : ℝ → ℂ} : Filter.Tendsto (fun x => ∫ (t : ℝ), f t * ↑x ^ (↑t * Complex.I)) Filter.atTop (nhds 0)"} +{"name":"nterm","declaration":"def nterm (f : ℕ → ℂ) (σ' : ℝ) (n : ℕ) : ℝ"} +{"name":"WI_summable","declaration":"theorem WI_summable {x : ℝ} {f : ℕ → ℝ} {g : ℝ → ℝ} (hg : HasCompactSupport g) (hx : 0 < x) : Summable fun n => f n * g (↑n / x)"} +{"name":"nabla_mul","declaration":"theorem nabla_mul {α : Type u_1} {E : Type u_2} [OfNat α 1] [Add α] [Ring E] {u : α → E} {c : E} : (nabla fun n => c * u n) = c • nabla u"} +{"name":"continuous_multiplicative_ofAdd","declaration":"theorem continuous_multiplicative_ofAdd : Continuous ⇑Multiplicative.ofAdd"} +{"name":"log_sq_isbigo_mul","declaration":"theorem log_sq_isbigo_mul {a : ℝ} {b : ℝ} (hb : 0 < b) : (fun x => Real.log x ^ 2) =O[Filter.atTop] fun x => a + Real.log (x / b) ^ 2"} +{"name":"Finset.sum_shift_front","declaration":"theorem Finset.sum_shift_front {E : Type u_1} [Ring E] {u : ℕ → E} {n : ℕ} : cumsum u (n + 1) = u 0 + cumsum (shift u) n"} +{"name":"bounded_of_shift","declaration":"theorem bounded_of_shift {u : ℕ → ℝ} (h : Filter.BoundedAtFilter Filter.atTop (shift u)) : Filter.BoundedAtFilter Filter.atTop u"} +{"name":"shift","declaration":"def shift {α : Type u_1} {E : Type u_2} [OfNat α 1] [Add α] (u : α → E) (n : α) : E"} +{"name":"fourier_surjection_on_schwartz","declaration":"theorem fourier_surjection_on_schwartz (f : SchwartzMap ℝ ℂ) : ∃ g, Real.fourierIntegral ⇑g = ⇑f"} +{"name":"hh_integral'","declaration":"theorem hh_integral' : ∫ (t : ℝ) in Set.Ioi 0, hh (1 / (2 * Real.pi)) t = 2 * Real.pi ^ 2"} +{"name":"wiener_ikehara_smooth_sub","declaration":"theorem wiener_ikehara_smooth_sub {A : ℝ} {Ψ : ℝ → ℂ} (h1 : MeasureTheory.Integrable Ψ MeasureTheory.volume) (hplus : closure (Function.support Ψ) ⊆ Set.Ioi 0) : Filter.Tendsto (fun x => (↑A * ∫ (y : ℝ) in Set.Ioi x⁻¹, Ψ y) - ↑A * ∫ (y : ℝ) in Set.Ioi 0, Ψ y) Filter.atTop (nhds 0)"} +{"name":"cancel_main","declaration":"theorem cancel_main {C : ℝ} {f : ℕ → ℝ} {g : ℕ → ℝ} (hf : 0 ≤ f) (hg : 0 ≤ g) (hf' : ∀ (n : ℕ), cumsum f n ≤ C * ↑n) (hg' : Antitone g) (n : ℕ) (hn : 2 ≤ n) : cumsum (f * g) n ≤ C * cumsum g n"} +{"name":"tsum_indicator","declaration":"theorem tsum_indicator {a : ℝ} {b : ℝ} {x : ℝ} {f : ℕ → ℝ} (hx : 0 < x) : ∑' (n : ℕ), f n * Set.indicator (Set.Ico a b) 1 (↑n / x) = Finset.sum (Finset.Ico ⌈a * x⌉₊ ⌈b * x⌉₊) fun n => f n"} +{"name":"sum_range_succ","declaration":"theorem sum_range_succ (a : ℕ → ℝ) (n : ℕ) : (Finset.sum (Finset.range n) fun i => a (i + 1)) = (Finset.sum (Finset.range (n + 1)) fun i => a i) - a 0"} +{"name":"decay_bounds_cor","declaration":"theorem decay_bounds_cor (ψ : W21) : ∃ C, ∀ (u : ℝ), ‖Real.fourierIntegral ψ.toFun u‖ ≤ C / (1 + u ^ 2)"} +{"name":"limiting_fourier_lim3","declaration":"theorem limiting_fourier_lim3 {x : ℝ} {G : ℂ → ℂ} (hG : ContinuousOn G {s | 1 ≤ s.re}) (ψ : CS 2 ℂ) (hx : 1 ≤ x) : Filter.Tendsto (fun σ' => ∫ (t : ℝ), G (↑σ' + ↑t * Complex.I) * ψ.toFun t * ↑x ^ (↑t * Complex.I))\n (nhdsWithin 1 (Set.Ioi 1)) (nhds (∫ (t : ℝ), G (1 + ↑t * Complex.I) * ψ.toFun t * ↑x ^ (↑t * Complex.I)))"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.ZetaBounds.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.ZetaBounds.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..2f6528666018ca7a21e404e6743b3cc92fa11b63 --- /dev/null +++ b/pnt-declarations/PrimeNumberTheoremAnd.ZetaBounds.jsonl @@ -0,0 +1,95 @@ +{"name":"ZetaSum_aux1_5d","declaration":"theorem ZetaSum_aux1_5d {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : IntervalIntegrable (fun u => |↑⌊u⌋ + 1 / 2 - u| / u ^ (s.re + 1)) MeasureTheory.volume a b"} +{"name":"ZetaSum_aux1","declaration":"theorem ZetaSum_aux1 {a : ℕ} {b : ℕ} {s : ℂ} (s_ne_one : s ≠ 1) (s_ne_zero : s ≠ 0) (ha : a ∈ Set.Ioo 0 b) : (Finset.sum (Finset.Ioc ↑a ↑b) fun n => 1 / ↑n ^ s) =\n (↑b ^ (1 - s) - ↑a ^ (1 - s)) / (1 - s) + 1 / 2 * (1 / ↑b ^ s) - 1 / 2 * (1 / ↑a ^ s) +\n s * ∫ (x : ℝ) in ↑a..↑b, (↑⌊x⌋ + 1 / 2 - ↑x) * ↑x ^ (-(s + 1))"} +{"name":"xpos_of_uIcc","declaration":"theorem xpos_of_uIcc {a : ℕ} {b : ℕ} (ha : a ∈ Set.Ioo 0 b) {x : ℝ} (x_in : x ∈ Set.uIcc ↑a ↑b) : 0 < x"} +{"name":"ZetaSum_aux1_5","declaration":"theorem ZetaSum_aux1_5 {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : ∫ (x : ℝ) in a..b, |↑⌊x⌋ + 1 / 2 - x| / x ^ (s.re + 1) ≤ ∫ (x : ℝ) in a..b, 1 / x ^ (s.re + 1)"} +{"name":"riemannzeta","declaration":"/-- We use `ζ` to denote the Rieman zeta function and `ζ₀` to denote the alternative\nRieman zeta function.. -/\ndef riemannzeta : Lean.ParserDescr"} +{"name":"Complex.cpow_inv_tendsto","declaration":"theorem Complex.cpow_inv_tendsto {s : ℂ} (hs : 0 < s.re) : Filter.Tendsto (fun x => (↑x ^ s)⁻¹) Filter.atTop (nhds 0)"} +{"name":"ZetaBnd_aux2","declaration":"theorem ZetaBnd_aux2 {n : ℕ} {t : ℝ} {A : ℝ} {σ : ℝ} (Apos : 0 < A) (σpos : 0 < σ) (n_le_t : ↑n ≤ |t|) (σ_ge : 1 - A / Real.log |t| ≤ σ) : ‖↑n ^ (-(↑σ + ↑t * Complex.I))‖ ≤ (↑n)⁻¹ * Real.exp A"} +{"name":"Ioi_union_Iio_mem_cocompact","declaration":"theorem Ioi_union_Iio_mem_cocompact {a : ℝ} (ha : 0 ≤ a) : Set.Ioi a ∪ Set.Iio (-a) ∈ Filter.cocompact ℝ"} +{"name":"Zeta0EqZeta","declaration":"theorem Zeta0EqZeta {N : ℕ} (N_pos : 0 < N) {s : ℂ} (reS_pos : 0 < s.re) (s_ne_one : s ≠ 1) : riemannZeta0 N s = riemannZeta s"} +{"name":"one_div_cpow_eq_cpow_neg","declaration":"theorem one_div_cpow_eq_cpow_neg (x : ℂ) (s : ℂ) : 1 / x ^ s = x ^ (-s)"} +{"name":"ZetaInvBound1","declaration":"theorem ZetaInvBound1 {σ : ℝ} {t : ℝ} (σ_gt : 1 < σ) : 1 / ‖riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ ‖riemannZeta ↑σ‖ ^ (3 / 4) * ‖riemannZeta (↑σ + 2 * ↑t * Complex.I)‖ ^ (1 / 4)"} +{"name":"integrability_aux₀","declaration":"theorem integrability_aux₀ {a : ℝ} {b : ℝ} : ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.uIcc a b), ‖↑⌊x⌋‖ ≤ max ‖a‖ ‖b‖ + 1"} +{"name":"finsetSum_tendsto_tsum","declaration":"theorem finsetSum_tendsto_tsum {N : ℕ} {f : ℕ → ℂ} (hf : Summable f) : Filter.Tendsto (fun k => Finset.sum (Finset.Ioc N k) fun n => f n) Filter.atTop (nhds (∑' (n : ℕ), f (n + N)))"} +{"name":"Nat.self_div_floor_bound","declaration":"theorem Nat.self_div_floor_bound {t : ℝ} (t_ge : 1 ≤ |t|) : |t| / ↑⌊|t|⌋₊ ∈ Set.Icc 1 2"} +{"name":"Summable_rpow","declaration":"theorem Summable_rpow {s : ℂ} (s_re_gt : 1 < s.re) : Summable fun n => 1 / ↑n ^ s"} +{"name":"ZetaSum_aux1_1'","declaration":"theorem ZetaSum_aux1_1' {a : ℝ} {b : ℝ} {x : ℝ} (apos : 0 < a) (hx : x ∈ Set.Icc a b) : 0 < x"} +{"name":"Tendsto_nhdsWithin_punctured_map_add","declaration":"theorem Tendsto_nhdsWithin_punctured_map_add {f : ℝ → ℝ} (a : ℝ) (x : ℝ) (f_mono : StrictMono f) (f_iso : Isometry f) : Filter.Tendsto (fun y => f y + a) (nhdsWithin x (Set.Ioi x)) (nhdsWithin (f x + a) (Set.Ioi (f x + a)))"} +{"name":"add_le_add_le_add_le_add","declaration":"theorem add_le_add_le_add_le_add {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} {c : α} {d : α} {e : α} {f : α} {g : α} {h : α} (h₁ : a ≤ b) (h₂ : c ≤ d) (h₃ : e ≤ f) (h₄ : g ≤ h) : a + c + e + g ≤ b + d + f + h"} +{"name":"ZetaInvBound2","declaration":"theorem ZetaInvBound2 {σ : ℝ} (hσ : σ ∈ Set.Ioc 1 2) : (fun t => 1 / ‖riemannZeta (↑σ + ↑t * Complex.I)‖) =O[Filter.cocompact ℝ] fun t =>\n (σ - 1) ^ (-3 / 4) * Real.log |t| ^ (1 / 4)"} +{"name":"Tendsto_nhdsWithin_punctured_add","declaration":"theorem Tendsto_nhdsWithin_punctured_add (a : ℝ) (x : ℝ) : Filter.Tendsto (fun y => y + a) (nhdsWithin x (Set.Ioi x)) (nhdsWithin (x + a) (Set.Ioi (x + a)))"} +{"name":"sum_eq_int_deriv_aux","declaration":"theorem sum_eq_int_deriv_aux {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (ha : a ∈ Set.Ico (↑k) b) (b_le_kpOne : b ≤ ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc ⌊a⌋ ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑⌊a⌋ + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑⌊x⌋ + 1 / 2 - ↑x) * deriv φ x"} +{"name":"add_le_add_le_add","declaration":"theorem add_le_add_le_add {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} {c : α} {d : α} {e : α} {f : α} (h₁ : a ≤ b) (h₂ : c ≤ d) (h₃ : e ≤ f) : a + c + e ≤ b + d + f"} +{"name":"ContDiffOn.hasDeriv_deriv","declaration":"theorem ContDiffOn.hasDeriv_deriv {φ : ℝ → ℂ} {s : Set ℝ} (φDiff : ContDiffOn ℝ 1 φ s) {x : ℝ} (x_in_s : s ∈ nhds x) : HasDerivAt φ (deriv φ x) x"} +{"name":"UpperBnd_aux2","declaration":"theorem UpperBnd_aux2 {A : ℝ} {σ : ℝ} {t : ℝ} (A_pos : 0 < A) (A_lt : A < 1) (t_ge : 3 < |t|) (σ_ge : 1 - A / Real.log |t| ≤ σ) : |t| ^ (1 - σ) ≤ Real.exp A"} +{"name":"ZetaDerivUpperBnd","declaration":"theorem ZetaDerivUpperBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n 3 < |t| →\n σ ∈ Set.Icc (1 - A / Real.log |t|) 2 → ‖deriv riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t| ^ 2"} +{"name":"Finset_coe_Nat_Int","declaration":"theorem Finset_coe_Nat_Int (f : ℤ → ℂ) (m : ℕ) (n : ℕ) : (Finset.sum (Finset.Ioc m n) fun x => f ↑x) = Finset.sum (Finset.Ioc ↑m ↑n) fun x => f x"} +{"name":"div_rpow_eq_rpow_div_neg","declaration":"theorem div_rpow_eq_rpow_div_neg {x : ℝ} {y : ℝ} {s : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x ^ s / y ^ s = (y / x) ^ (-s)"} +{"name":"ZetaSum_aux1φderiv","declaration":"theorem ZetaSum_aux1φderiv {s : ℂ} (s_ne_zero : s ≠ 0) {x : ℝ} (xpos : 0 < x) : deriv (fun t => 1 / ↑t ^ s) x = (fun x => -s * ↑x ^ (-(s + 1))) x"} +{"name":"interval_induction","declaration":"theorem interval_induction (P : ℝ → ℝ → Prop) (base : ∀ (a b : ℝ) (k : ℤ), ↑k ≤ a → a < b → b ≤ ↑k + 1 → P a b) (step : ∀ (a : ℝ) (k : ℤ) (b : ℝ), a < ↑k → ↑k < b → P a ↑k → P (↑k) b → P a b) (a : ℝ) (b : ℝ) (hab : a < b) : P a b"} +{"name":"riemannZeta0_zero_aux","declaration":"theorem riemannZeta0_zero_aux (N : ℕ) (Npos : 0 < N) : (Finset.sum (Finset.Ico 0 N) fun x => (↑x)⁻¹) = Finset.sum (Finset.Ico 1 N) fun x => (↑x)⁻¹"} +{"name":"ZetaSum_aux1a","declaration":"theorem ZetaSum_aux1a {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : ‖∫ (x : ℝ) in a..b, (↑⌊x⌋ + 1 / 2 - ↑x) / ↑x ^ (s + 1)‖ ≤ (a ^ (-s.re) - b ^ (-s.re)) / s.re"} +{"name":"ct_aux1","declaration":"def ct_aux1 : ℕ"} +{"name":"tendsto_coe_atTop","declaration":"theorem tendsto_coe_atTop : Filter.Tendsto (fun n => ↑n) Filter.atTop Filter.atTop"} +{"name":"ZetaSum_aux1_4","declaration":"theorem ZetaSum_aux1_4 {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} : ∫ (x : ℝ) in a..b, ‖(↑⌊x⌋ + ↑1 / 2 - ↑x) / ↑x ^ (s + 1)‖ = ∫ (x : ℝ) in a..b, |↑⌊x⌋ + 1 / 2 - x| / x ^ (s + 1).re"} +{"name":"ZetaSum_aux1_5b","declaration":"theorem ZetaSum_aux1_5b {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : IntervalIntegrable (fun u => 1 / u ^ (s.re + 1)) MeasureTheory.volume a b"} +{"name":"riemannzeta0","declaration":"def riemannzeta0 : Lean.ParserDescr"} +{"name":"LinearDerivative_ofReal","declaration":"theorem LinearDerivative_ofReal (x : ℝ) (a : ℂ) (b : ℂ) : HasDerivAt (fun t => a * ↑t + b) a x"} +{"name":"interval_induction_aux_int","declaration":"theorem interval_induction_aux_int (n : ℕ) (P : ℝ → ℝ → Prop) : (∀ (a b : ℝ) (k : ℤ), ↑k ≤ a → a < b → b ≤ ↑k + 1 → P a b) →\n (∀ (a : ℝ) (k : ℤ) (c : ℝ), a < ↑k → ↑k < c → P a ↑k → P (↑k) c → P a c) → ∀ (a b : ℝ), a < b → ↑n = ⌊b⌋ - ⌊a⌋ → P a b"} +{"name":"UpperBnd_aux3","declaration":"theorem UpperBnd_aux3 {A : ℝ} {C : ℝ} {σ : ℝ} {t : ℝ} (Apos : 0 < A) (A_lt_one : A < 1) {N : ℕ} (Npos : 0 < N) (σ_ge : 1 - A / Real.log |t| ≤ σ) (t_ge : 3 < |t|) (N_le_t : ↑N ≤ |t|) (hC : 2 ≤ C) : ‖Finset.sum (Finset.range N) fun n => ↑n ^ (-(↑σ + ↑t * Complex.I))‖ ≤ Real.exp A * C * Real.log |t|"} +{"name":"Finset.Ioc_diff_Ioc","declaration":"/-- ** Partial summation ** (TODO : Add to Mathlib). -/\ntheorem Finset.Ioc_diff_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a : α} {b : α} {c : α} [DecidableEq α] (hb : b ∈ Finset.Icc a c) : Finset.Ioc a b = Finset.Ioc a c \\ Finset.Ioc b c"} +{"name":"ZetaSum_aux1_5c","declaration":"theorem ZetaSum_aux1_5c {a : ℝ} {b : ℝ} {s : ℂ} : let g := fun u => |↑⌊u⌋ + 1 / 2 - u| / u ^ (s.re + 1);\nMeasureTheory.AEStronglyMeasurable g (MeasureTheory.volume.restrict (Ι a b))"} +{"name":"lt_abs_mem_cocompact","declaration":"theorem lt_abs_mem_cocompact {a : ℝ} (ha : 0 ≤ a) : {t | a < |t|} ∈ Filter.cocompact ℝ"} +{"name":"div_rpow_neg_eq_rpow_div","declaration":"theorem div_rpow_neg_eq_rpow_div {x : ℝ} {y : ℝ} {s : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x ^ (-s) / y ^ (-s) = (y / x) ^ s"} +{"name":"C_aux1","declaration":"def C_aux1 : ℕ"} +{"name":"div_cpow_eq_cpow_neg","declaration":"theorem div_cpow_eq_cpow_neg (a : ℂ) (x : ℂ) (s : ℂ) : a / x ^ s = a * x ^ (-s)"} +{"name":"sum_eq_int_deriv","declaration":"theorem sum_eq_int_deriv {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} (a_lt_b : a < b) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc ⌊a⌋ ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑⌊a⌋ + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑⌊x⌋ + 1 / 2 - ↑x) * deriv φ x"} +{"name":"ContDiffOn.continuousOn_deriv","declaration":"theorem ContDiffOn.continuousOn_deriv {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} (φDiff : ContDiffOn ℝ 1 φ (Set.uIoo a b)) : ContinuousOn (deriv φ) (Set.uIoo a b)"} +{"name":"Finset.Ioc_eq_Ico","declaration":"theorem Finset.Ioc_eq_Ico (M : ℕ) (N : ℕ) : Finset.Ioc N M = Finset.Ico (N + 1) (M + 1)"} +{"name":"isPathConnected_aux","declaration":"theorem isPathConnected_aux : IsPathConnected {z | z ≠ 1 ∧ 0 < z.re}"} +{"name":"ZetaBnd_aux1","declaration":"theorem ZetaBnd_aux1 (N : ℕ) (Npos : 1 ≤ N) {σ : ℝ} (hσ : σ ∈ Set.Ioc 0 2) (t : ℝ) (ht : ↑ct_aux1 < |t|) : ‖(↑σ + ↑t * Complex.I) * ∫ (x : ℝ) in Set.Ioi ↑N, (↑⌊x⌋ + 1 / 2 - ↑x) / ↑x ^ (↑σ + ↑t * Complex.I + 1)‖ ≤\n ↑C_aux1 * |t| * ↑N ^ (-σ) / σ"} +{"name":"ZetaSum_aux1_3b","declaration":"theorem ZetaSum_aux1_3b (x : ℝ) : ↑⌊x⌋ + 1 / 2 - x ≤ 1 / 2"} +{"name":"UpperBnd_aux5","declaration":"theorem UpperBnd_aux5 {σ : ℝ} {t : ℝ} (t_ge : 3 < |t|) (σ_le : σ ≤ 2) : (|t| / ↑⌊|t|⌋₊) ^ σ ≤ 4"} +{"name":"tsum_eq_partial_add_tail","declaration":"theorem tsum_eq_partial_add_tail {N : ℕ} (f : ℕ → ℂ) (hf : Summable f) : ∑' (n : ℕ), f n = (Finset.sum (Finset.Ico 0 N) fun n => f n) + ∑' (n : ℕ), f (n + N)"} +{"name":"integrability_aux₁","declaration":"theorem integrability_aux₁ {a : ℝ} {b : ℝ} : IntervalIntegrable (fun x => ↑⌊x⌋) MeasureTheory.volume a b"} +{"name":"Zeta_diff_Bnd","declaration":"theorem Zeta_diff_Bnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ₁ σ₂ t : ℝ),\n 3 < |t| →\n 1 - A / Real.log |t| ≤ σ₁ →\n σ₂ ≤ 2 →\n σ₁ < σ₂ →\n ‖riemannZeta (↑σ₂ + ↑t * Complex.I) - riemannZeta (↑σ₁ + ↑t * Complex.I)‖ ≤\n C * Real.log |t| ^ 2 * (σ₂ - σ₁)"} +{"name":"integral_deriv_mul_eq_sub'","declaration":"theorem integral_deriv_mul_eq_sub' {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {a : ℝ} {b : ℝ} {u : ℝ → A} {v : ℝ → A} {u' : ℝ → A} {v' : ℝ → A} (hu : ∀ x ∈ Set.uIcc a b, HasDerivWithinAt u (u' x) (Set.uIcc a b) x) (hv : ∀ x ∈ Set.uIcc a b, HasDerivWithinAt v (v' x) (Set.uIcc a b) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b) : ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a"} +{"name":"ZetaSum_aux1_1","declaration":"theorem ZetaSum_aux1_1 {a : ℝ} {b : ℝ} {x : ℝ} (apos : 0 < a) (a_lt_b : a < b) (hx : x ∈ Set.uIcc a b) : 0 < x"} +{"name":"ZetaSum_aux1_3","declaration":"theorem ZetaSum_aux1_3 (x : ℝ) : |↑⌊x⌋ + 1 / 2 - x| ≤ 1 / 2"} +{"name":"HolomorphicOn_riemannZeta0","declaration":"theorem HolomorphicOn_riemannZeta0 {N : ℕ} (N_pos : 0 < N) : HolomorphicOn (riemannZeta0 N) {s | s ≠ 1 ∧ 0 < s.re}"} +{"name":"sum_eq_int_deriv_aux1","declaration":"theorem sum_eq_int_deriv_aux1 {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (ha : a ∈ Set.Ico (↑k) b) (b_le_kpOne : b ≤ ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc k ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑k + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑k + 1 / 2 - ↑x) * deriv φ x"} +{"name":"integrability_aux","declaration":"theorem integrability_aux {a : ℝ} {b : ℝ} : IntervalIntegrable (fun x => ↑⌊x⌋ + 1 / 2 - ↑x) MeasureTheory.volume a b"} +{"name":"deriv_fun_re","declaration":"theorem deriv_fun_re {t : ℝ} {f : ℂ → ℂ} (diff : ∀ (σ : ℝ), DifferentiableAt ℂ f (↑σ + ↑t * Complex.I)) : (deriv fun {σ₂} => f (↑σ₂ + ↑t * Complex.I)) = fun σ => deriv f (↑σ + ↑t * Complex.I)"} +{"name":"ZetaSum_aux1₁","declaration":"theorem ZetaSum_aux1₁ {a : ℕ} {b : ℕ} {s : ℂ} (s_ne_one : s ≠ 1) (ha : a ∈ Set.Ioo 0 b) : ∫ (x : ℝ) in ↑a..↑b, 1 / ↑x ^ s = (↑b ^ (1 - s) - ↑a ^ (1 - s)) / (1 - s)"} +{"name":"ZetaUpperBnd","declaration":"theorem ZetaUpperBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n ↑ct_aux1 < |t| → σ ∈ Set.Icc (1 - A / Real.log |t|) 2 → ‖riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t|"} +{"name":"Zeta_eq_int_derivZeta","declaration":"theorem Zeta_eq_int_derivZeta {σ₁ : ℝ} {σ₂ : ℝ} {t : ℝ} (t_ne_zero : t ≠ 0) : ∫ (σ : ℝ) in σ₁..σ₂, deriv riemannZeta (↑σ + ↑t * Complex.I) =\n riemannZeta (↑σ₂ + ↑t * Complex.I) - riemannZeta (↑σ₁ + ↑t * Complex.I)"} +{"name":"ZetaSum_aux2","declaration":"theorem ZetaSum_aux2 {N : ℕ} (N_pos : 0 < N) {s : ℂ} (s_re_gt : 1 < s.re) : ∑' (n : ℕ), 1 / (↑n + ↑N) ^ s =\n -↑N ^ (1 - s) / (1 - s) - ↑N ^ (-s) / 2 + s * ∫ (x : ℝ) in Set.Ioi ↑N, (↑⌊x⌋ + 1 / 2 - ↑x) * ↑x ^ (-(s + 1))"} +{"name":"Complex.cpow_tendsto","declaration":"theorem Complex.cpow_tendsto {s : ℂ} (s_re_gt : 1 < s.re) : Filter.Tendsto (fun x => ↑x ^ (1 - s)) Filter.atTop (nhds 0)"} +{"name":"ZetaSum_aux1_4'","declaration":"theorem ZetaSum_aux1_4' (x : ℝ) (hx : 0 < x) (s : ℂ) : ‖(↑⌊x⌋ + 1 / 2 - ↑x) / ↑x ^ (s + 1)‖ = |↑⌊x⌋ + 1 / 2 - x| / x ^ (s + 1).re"} +{"name":"uIcc_subsets","declaration":"theorem uIcc_subsets {a : ℝ} {b : ℝ} {c : ℝ} (hc : c ∈ Set.Icc a b) : Set.uIcc a c ⊆ Set.uIcc a b ∧ Set.uIcc c b ⊆ Set.uIcc a b"} +{"name":"Real.differentiableAt_cpow_const_of_ne","declaration":"theorem Real.differentiableAt_cpow_const_of_ne (s : ℂ) {x : ℝ} (xpos : 0 < x) : DifferentiableAt ℝ (fun x => ↑x ^ s) x"} +{"name":"neg_s_ne_neg_one","declaration":"theorem neg_s_ne_neg_one {s : ℂ} (s_ne_one : s ≠ 1) : -s ≠ -1"} +{"name":"UpperBnd_aux6","declaration":"theorem UpperBnd_aux6 {σ : ℝ} {t : ℝ} (t_ge : 3 < |t|) (σ_gt : 1 / 2 < σ) (σ_le : σ ≤ 2) (neOne : ↑σ + ↑t * Complex.I ≠ 1) (Npos : 0 < ⌊|t|⌋₊) (N_le_t : ↑⌊|t|⌋₊ ≤ |t|) : ↑⌊|t|⌋₊ ^ (1 - σ) / ‖1 - (↑σ + ↑t * Complex.I)‖ ≤ |t| ^ (1 - σ) * 2 ∧\n ↑⌊|t|⌋₊ ^ (-σ) / 2 ≤ |t| ^ (1 - σ) ∧ ↑⌊|t|⌋₊ ^ (-σ) / σ ≤ 8 * |t| ^ (-σ)"} +{"name":"norm_add₄_le","declaration":"theorem norm_add₄_le {E : Type u_1} [SeminormedAddGroup E] (a : E) (b : E) (c : E) (d : E) : ‖a + b + c + d‖ ≤ ‖a‖ + ‖b‖ + ‖c‖ + ‖d‖"} +{"name":"ZetaSum_aux1_2","declaration":"theorem ZetaSum_aux1_2 {a : ℝ} {b : ℝ} {c : ℝ} (apos : 0 < a) (a_lt_b : a < b) (h : c ≠ 0 ∧ 0 ∉ Set.uIcc a b) : ∫ (x : ℝ) in a..b, 1 / x ^ (c + 1) = (a ^ (-c) - b ^ (-c)) / c"} +{"name":"sum_eq_int_deriv_aux_lt","declaration":"theorem sum_eq_int_deriv_aux_lt {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (ha : a ∈ Set.Ico (↑k) b) (b_lt_kpOne : b < ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc k ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑k + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑k + 1 / 2 - ↑x) * deriv φ x"} +{"name":"ZetaSum_aux1_5a","declaration":"theorem ZetaSum_aux1_5a {a : ℝ} {b : ℝ} (apos : 0 < a) {s : ℂ} (x : ℝ) (h : x ∈ Set.Icc a b) : |↑⌊x⌋ + 1 / 2 - x| / x ^ (s.re + 1) ≤ 1 / x ^ (s.re + 1)"} +{"name":"le_trans₄","declaration":"theorem le_trans₄ {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} {d : α} : a ≤ b → b ≤ c → c ≤ d → a ≤ d"} +{"name":"integrability_aux₂","declaration":"theorem integrability_aux₂ {a : ℝ} {b : ℝ} : IntervalIntegrable (fun x => 1 / 2 - ↑x) MeasureTheory.volume a b"} +{"name":"ZetaNear1BndFilter","declaration":"theorem ZetaNear1BndFilter : (fun σ => riemannZeta ↑σ) =O[nhdsWithin 1 (Set.Ioi 1)] fun σ => 1 / (↑σ - 1)"} +{"name":"ZetaNear1BndExact","declaration":"theorem ZetaNear1BndExact : ∃ c, ∃ (_ : 0 < c), ∀ σ ∈ Set.Ioc 1 2, ‖riemannZeta ↑σ‖ ≤ c / (σ - 1)"} +{"name":"LogDerivZetaBnd","declaration":"theorem LogDerivZetaBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n 3 < |t| →\n σ ∈ Set.Ico (1 - A / Real.log |t| ^ 9) 1 →\n ‖deriv riemannZeta (↑σ + ↑t * Complex.I) / riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t| ^ 9"} +{"name":"ZetaSum_aux1_3a","declaration":"theorem ZetaSum_aux1_3a (x : ℝ) : -(1 / 2) < ↑⌊x⌋ + 1 / 2 - x"} +{"name":"UpperBnd_aux","declaration":"theorem UpperBnd_aux {A : ℝ} {σ : ℝ} {t : ℝ} (A_pos : 0 < A) (A_lt : A < 1) (t_ge : 3 < |t|) (σ_ge : 1 - A / Real.log |t| ≤ σ) : 1 < Real.log |t| ∧ 1 - A < σ ∧ 0 < σ ∧ ↑σ + ↑t * Complex.I ≠ 1"} +{"name":"ZetaSum_aux1φDiff","declaration":"theorem ZetaSum_aux1φDiff {s : ℂ} {x : ℝ} (xpos : 0 < x) : HasDerivAt (fun t => 1 / ↑t ^ s) (deriv (fun t => 1 / ↑t ^ s) x) x"} +{"name":"Complex.one_div_cpow_eq","declaration":"theorem Complex.one_div_cpow_eq {s : ℂ} {x : ℝ} (x_ne : x ≠ 0) : 1 / ↑x ^ s = ↑x ^ (-s)"} +{"name":"ZetaInvBnd","declaration":"theorem ZetaInvBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n 3 < |t| →\n σ ∈ Set.Ico (1 - A / Real.log |t| ^ 9) 1 → 1 / ‖riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t| ^ 7"} +{"name":"riemannZeta0_apply","declaration":"theorem riemannZeta0_apply (N : ℕ) (s : ℂ) : riemannZeta0 N s =\n (Finset.sum (Finset.range N) fun n => 1 / ↑n ^ s) +\n (-↑N ^ (1 - s) / (1 - s) + -↑N ^ (-s) / 2 + s * ∫ (x : ℝ) in Set.Ioi ↑N, (↑⌊x⌋ + 1 / 2 - ↑x) * ↑x ^ (-(s + 1)))"} +{"name":"div_rpow_eq_rpow_neg","declaration":"theorem div_rpow_eq_rpow_neg (a : ℝ) (x : ℝ) (s : ℝ) (hx : 0 ≤ x) : a / x ^ s = a * x ^ (-s)"} +{"name":"HolomophicOn_riemannZeta","declaration":"theorem HolomophicOn_riemannZeta : HolomorphicOn riemannZeta {s | s ≠ 1}"} +{"name":"sum_eq_int_deriv_aux2","declaration":"theorem sum_eq_int_deriv_aux2 {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} (c : ℂ) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : ∫ (x : ℝ) in a..b, (c - ↑x) * deriv φ x = (c - ↑b) * φ b - (c - ↑a) * φ a + ∫ (x : ℝ) in a..b, φ x"} +{"name":"riemannZeta0","declaration":"def riemannZeta0 (N : ℕ) (s : ℂ) : ℂ"} +{"name":"ZetaSum_aux1derivφCont","declaration":"theorem ZetaSum_aux1derivφCont {s : ℂ} (s_ne_zero : s ≠ 0) {a : ℕ} {b : ℕ} (ha : a ∈ Set.Ioo 0 b) : ContinuousOn (deriv fun t => 1 / ↑t ^ s) (Set.uIcc ↑a ↑b)"} +{"name":"sum_eq_int_deriv_aux_eq","declaration":"theorem sum_eq_int_deriv_aux_eq {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (b_eq_kpOne : b = ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc k ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑k + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑k + 1 / 2 - ↑x) * deriv φ x"} +{"name":"Finset.sum_Ioc_add_sum_Ioc","declaration":"theorem Finset.sum_Ioc_add_sum_Ioc {a : ℤ} {b : ℤ} {c : ℤ} (f : ℤ → ℂ) (hb : b ∈ Finset.Icc a c) : ((Finset.sum (Finset.Ioc a b) fun n => f n) + Finset.sum (Finset.Ioc b c) fun n => f n) =\n Finset.sum (Finset.Ioc a c) fun n => f n"} +{"name":"ZetaSum_aux2a","declaration":"theorem ZetaSum_aux2a : ∃ C, ∀ (x : ℝ), |↑⌊x⌋ + 1 / 2 - x| ≤ C"} diff --git a/pnt-declarations/PrimeNumberTheoremAnd.jsonl b/pnt-declarations/PrimeNumberTheoremAnd.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.Approx.ApproxLimit.jsonl b/scilean-declarations/SciLean.Core.Approx.ApproxLimit.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..958d92dff1c202b1dcf0c67bff5d3f20d84df995 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Approx.ApproxLimit.jsonl @@ -0,0 +1,2 @@ +{"name":"SciLean.approx_limit_tactic","declaration":"/-- This tactic eliminates `limit` from an expression that you are approximating\n\nFor example, for goal\n\n```\n ⊢ Approx _ (x + limit n → ∞, (1 + x/n)^n)\n```\n\ncalling `approx_limit n := ` produces goal\n\n```\n n : Nat\n ⊢ Approx _ (x + (1 + x/n)^n)\n```\n\nwhere `` is a proof that this approximation is indeed valid\n\nWarning: The validity proof is not completely correct right now\n-/\ndef SciLean.approx_limit_tactic : Lean.ParserDescr"} +{"name":"SciLean.approxLimitTactic","declaration":"def SciLean.approxLimitTactic : Lean.Elab.Tactic.Tactic"} diff --git a/scilean-declarations/SciLean.Core.Approx.ApproxSolution.jsonl b/scilean-declarations/SciLean.Core.Approx.ApproxSolution.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7fc60da8fc213f182dc6324b7d7eeb4cfbadad9e --- /dev/null +++ b/scilean-declarations/SciLean.Core.Approx.ApproxSolution.jsonl @@ -0,0 +1,7 @@ +{"name":"SciLean.ApproxSolution.val","declaration":"def SciLean.ApproxSolution.val {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} {lN : Filter N} {spec : α → Prop} (a : SciLean.ApproxSolution lN spec) (p : N) : α"} +{"name":"SciLean.ApproxSolution.exact","declaration":"ctor SciLean.ApproxSolution.exact {α : Type u_1} [TopologicalSpace α] [Nonempty α] {spec : α → Prop} (impl : α) (h : spec impl) : SciLean.ApproxSolution ⊤ spec"} +{"name":"SciLean.approx_convergence","declaration":"theorem SciLean.approx_convergence {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} {lN : Filter N} {spec : α → Prop} (approx : SciLean.ApproxSolution lN spec) : ∃ a, Filter.Tendsto (SciLean.ApproxSolution.val approx) lN (nhds a)"} +{"name":"SciLean.ApproxSolution.approx","declaration":"ctor SciLean.ApproxSolution.approx {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} {M : Type} {spec : α → Prop} (specₙ : N → α → Prop) (lN : Filter N) (lM : Filter M) (consistent : ∀ (aₙ : N → α), (∀ (n : N), specₙ n (aₙ n)) → ∀ (a : α), (a = Filter.limit lN fun n => aₙ n) → spec a) (convergence : ∀ (a' : N → M → α),\n (∀ (n : N), specₙ n (Filter.limit lM fun m => a' n m)) →\n ∃ a,\n a =\n Filter.limit (Filter.prod lN lM) fun x =>\n match x with\n | (n, m) => a' n m) (impl : (n : N) → SciLean.ApproxSolution lM (specₙ n)) : SciLean.ApproxSolution (Filter.prod lN lM) spec"} +{"name":"SciLean.ApproxSolution","declaration":"inductive SciLean.ApproxSolution {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} (lN : Filter N) (spec : α → Prop) : Type (max 1 u_1)"} +{"name":"SciLean.approx_consistency","declaration":"theorem SciLean.approx_consistency {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} {lN : Filter N} [T2Space α] {spec : α → Prop} (approx : SciLean.ApproxSolution lN spec) (a : α) : (a = Filter.limit lN fun n => SciLean.ApproxSolution.val approx n) → spec a"} +{"name":"SciLean.ApproxSolution.limit","declaration":"def SciLean.ApproxSolution.limit {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} {lN : Filter N} {spec : α → Prop} (a : SciLean.ApproxSolution lN spec) : α"} diff --git a/scilean-declarations/SciLean.Core.Approx.Basic.jsonl b/scilean-declarations/SciLean.Core.Approx.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..06feb45d508dd07b5ab6eb4605f6d6522fea6e90 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Approx.Basic.jsonl @@ -0,0 +1,5 @@ +{"name":"SciLean.instCoeFunApproxForAll","declaration":"instance SciLean.instCoeFunApproxForAll {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} (lN : Filter N) (a : α) : CoeFun (SciLean.Approx lN a) fun x => N → α"} +{"name":"SciLean.Approx","declaration":"def SciLean.Approx {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : outParam Type} (lN : Filter N) (a : α) : Type (max 1 u_1)"} +{"name":"SciLean.Approx.exact","declaration":"def SciLean.Approx.exact {α : Type u_1} [TopologicalSpace α] [Nonempty α] {a : α} : SciLean.ApproxSolution ⊤ (Eq a)"} +{"name":"SciLean.Approx.limit","declaration":"def SciLean.Approx.limit {α : Type u_1} [TopologicalSpace α] [Nonempty α] {N : Type} {lN : Filter N} (M : Type) (lM : Filter M) {aₙ : N → α} (x : (n : N) → SciLean.Approx lM (aₙ n)) : SciLean.Approx (Filter.prod lN lM) (Filter.limit lN fun n => aₙ n)"} +{"name":"SciLean.«command_Approx__:_:=_By_»","declaration":"def SciLean.«command_Approx__:_:=_By_» : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Distribution.Basic.jsonl b/scilean-declarations/SciLean.Core.Distribution.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e4418b9e349cc220cda093412ec4ab207eaad5fd --- /dev/null +++ b/scilean-declarations/SciLean.Core.Distribution.Basic.jsonl @@ -0,0 +1,85 @@ +{"name":"SciLean.sub_restrict'","declaration":"theorem SciLean.sub_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : T.restrict A - S.restrict A = (T - S).restrict A"} +{"name":"SciLean.Distribution.toMeasure","declaration":"def SciLean.Distribution.toMeasure {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] (f' : 𝒟'(X, R)) : MeasureTheory.Measure X"} +{"name":"SciLean.Distribution.postComp.arg_T.IsSmoothLinarMap_rule","declaration":"theorem SciLean.Distribution.postComp.arg_T.IsSmoothLinarMap_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_5} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (T : W → 𝒟'(X, Y)) (f : Y ⊸[R] Z) (hT : SciLean.IsSmoothLinearMap R T) : SciLean.IsSmoothLinearMap R fun w => SciLean.Distribution.postComp (T w) f"} +{"name":"SciLean.Distribution.smul_extAction","declaration":"theorem SciLean.Distribution.smul_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (r : R) (T : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (r • T).extAction φ L = r • T.extAction φ L"} +{"name":"SciLean.Function.toDistribution_extAction","declaration":"theorem SciLean.Function.toDistribution_extAction {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_1} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (φ : X → R) : (↑f).extAction φ (fun y ⊸[R] fun r ⊸[R] r • y) = ∫' x, φ x • f x ∂MeasureTheory.volume"} +{"name":"SciLean.Distribution.restrict","declaration":"def SciLean.Distribution.restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : 𝒟'(X, Y)"} +{"name":"SciLean.«term𝒟'(_,_)»","declaration":"def SciLean.«term𝒟'(_,_)» : Lean.ParserDescr"} +{"name":"SciLean.restrict_univ","declaration":"theorem SciLean.restrict_univ {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) : T.restrict Set.univ = T"} +{"name":"SciLean.Distribution.extAction_iteD'","declaration":"theorem SciLean.Distribution.extAction_iteD' {R : Type u_2} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_1} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (A : Set X) (B : Set X) (t : 𝒟'(X, U)) (e : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : ((ifD A then\n t\n else\n e).restrict\n B).extAction\n φ L =\n (t.restrict B).extAction (fun x => if x ∈ A then φ x else 0) L +\n (e.restrict B).extAction (fun x => if x ∉ A then φ x else 0) L"} +{"name":"SciLean.Distribution.prod'_extAction","declaration":"theorem SciLean.Distribution.prod'_extAction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_7} [SciLean.Vec R X] {Z : Type u_6} [SciLean.Vec R Z] {X₁ : Type u_2} [SciLean.Vec R X₁] {X₂ : Type u_4} [SciLean.Vec R X₂] {Y₁ : Type u_3} [SciLean.Vec R Y₁] {Y₂ : Type u_5} [SciLean.Vec R Y₂] (p : X₁ → X₂ → X) (T : 𝒟'(X₁, Y₁)) (S : X₁ → 𝒟'(X₂, Y₂)) (L : Y₁ ⊸[R] Y₂ ⊸[R] Z) (K : Z ⊸[R] R ⊸[R] Z) (φ : X → R) : (SciLean.Distribution.prod p T S L).extAction φ K =\n T.extAction (fun x₁ => (S x₁).extAction (fun x₂ => φ (p x₁ x₂)) (fun y₂ ⊸[R] fun r ⊸[R] r • y₂))\n (fun y₁ ⊸[R] fun y₂ ⊸[R] (K ((L y₁) y₂)) 1)"} +{"name":"SciLean.Distribution.ext","declaration":"theorem SciLean.Distribution.ext {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (x : 𝒟'(X, Y)) (y : 𝒟'(X, Y)) : (∀ (φ : 𝒟 X), x φ = y φ) → x = y"} +{"name":"Function.toDistribution","declaration":"def Function.toDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) : 𝒟'(X, Y)"} +{"name":"SciLean.Distribution.extAction.arg_φ.IsSmoothLinearMap","declaration":"theorem SciLean.Distribution.extAction.arg_φ.IsSmoothLinearMap {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (φ : W → X → V) (L : U ⊸[R] V ⊸[R] Z) (hφ : SciLean.IsSmoothLinearMap R φ) : SciLean.IsSmoothLinearMap R fun w => T.extAction (φ w) L"} +{"name":"SciLean.postComp_id","declaration":"theorem SciLean.postComp_id {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (u : 𝒟'(X, Y)) : SciLean.Distribution.postComp u (fun y ⊸[R] y) = u"} +{"name":"SciLean.instInnerDistribution","declaration":"instance SciLean.instInnerDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [MeasureTheory.MeasureSpace X] [SciLean.SemiInnerProductSpace R Y] [Module ℝ Y] : Inner R (𝒟'(X, Y))"} +{"name":"SciLean.HSub.hSub.arg_a0a1.toDistribution_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : (↑fun x => f x - g x) = ↑f - ↑g"} +{"name":"SciLean.iteD.arg_te.IsSmoothLinearMap_rule","declaration":"theorem SciLean.iteD.arg_te.IsSmoothLinearMap_rule {R : Type u_2} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (t : W → 𝒟'(X, Y)) (e : W → 𝒟'(X, Y)) (ht : SciLean.IsSmoothLinearMap R t) (he : SciLean.IsSmoothLinearMap R e) : SciLean.IsSmoothLinearMap R fun w =>\n ifD A then\n t w\n else\n e w"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule'","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (r : R) (f : X → Y) : r • ↑f = ↑fun x => r • f x"} +{"name":"SciLean.Distribution.add_extAction","declaration":"theorem SciLean.Distribution.add_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (T' : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (T + T').extAction φ L = T.extAction φ L + T'.extAction φ L"} +{"name":"SciLean.unexpandIteD","declaration":"def SciLean.unexpandIteD : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.Function.toDistribution_action","declaration":"theorem SciLean.Function.toDistribution_action {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (φ : 𝒟 X) : ↑f φ = ∫' x, φ x • f x ∂MeasureTheory.volume"} +{"name":"SciLean.Distribution.mk_extAction","declaration":"theorem SciLean.Distribution.mk_extAction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : (X → R) → Y) (hT : SciLean.IsSmoothLinearMap R fun φ => T ⇑φ) (φ : X → R) : (fun φ ⊸[R] T ⇑φ).extAction φ (fun y ⊸[R] fun r ⊸[R] r • y) = T φ"} +{"name":"SciLean.Distribution.neg_extAction","declaration":"theorem SciLean.Distribution.neg_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (-T).extAction φ L = -T.extAction φ L"} +{"name":"SciLean.Distribution","declaration":"def SciLean.Distribution (R : Type u_1) [SciLean.RealScalar R] (X : Type u_2) [SciLean.Vec R X] (Y : Type u_3) [SciLean.Vec R Y] : Type (max (max u_2 u_1) u_3)"} +{"name":"SciLean.TestFunction.apply_IsSmoothLinearMap","declaration":"theorem SciLean.TestFunction.apply_IsSmoothLinearMap {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] : SciLean.IsSmoothLinearMap R fun φ => ⇑φ"} +{"name":"SciLean.unexpandDistribution","declaration":"def SciLean.unexpandDistribution : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.instModuleRealDistributionSemiringToAddCommMonoidToAddCommGroupToRCLikeToScalarInstVecSmoothLinearMapTestFunctionSpaceToTopologicalSpaceToUniformSpaceInstTCOrVecDiscreteTopologyInstVecToRCLikeToScalarTestFunctionSpace","declaration":"instance SciLean.instModuleRealDistributionSemiringToAddCommMonoidToAddCommGroupToRCLikeToScalarInstVecSmoothLinearMapTestFunctionSpaceToTopologicalSpaceToUniformSpaceInstTCOrVecDiscreteTopologyInstVecToRCLikeToScalarTestFunctionSpace {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] : Module ℝ (𝒟'(X, Y))"} +{"name":"SciLean.postComp_comp","declaration":"theorem SciLean.postComp_comp {R : Type u_1} [SciLean.RealScalar R] {W : Type u_5} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (x : 𝒟'(X, U)) (g : U ⊸[R] V) (f : V ⊸[R] W) : SciLean.Distribution.postComp (SciLean.Distribution.postComp x g) f =\n SciLean.Distribution.postComp x (fun u ⊸[R] f (g u))"} +{"name":"SciLean.postComp_restrict_extAction","declaration":"theorem SciLean.postComp_restrict_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_5} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (x : 𝒟'(X, U)) (f : U ⊸[R] V) (A : Set X) (φ : X → W) (L : V ⊸[R] W ⊸[R] Z) : ((SciLean.Distribution.postComp x f).restrict A).extAction φ L =\n (x.restrict A).extAction φ (fun u ⊸[R] fun w ⊸[R] (L (f u)) w)"} +{"name":"SciLean.Distribution.extAction'","declaration":"def SciLean.Distribution.extAction' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (φ : X → R) : Y"} +{"name":"SciLean.indextype_sum_restrict'","declaration":"theorem SciLean.indextype_sum_restrict' {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {I : Type u_1} [SciLean.IndexType I] (T : I → 𝒟'(X, R)) (A : Set X) : ∑ i, (T i).restrict A = ( ∑ i, T i).restrict A"} +{"name":"SciLean.instSemiInnerProductSpaceToRCLikeToScalarDistribution","declaration":"instance SciLean.instSemiInnerProductSpaceToRCLikeToScalarDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [MeasureTheory.MeasureSpace X] [SciLean.SemiInnerProductSpace R Y] [Module ℝ Y] : SciLean.SemiInnerProductSpace R (𝒟'(X, Y))"} +{"name":"SciLean.Distribution.IsFunction","declaration":"def SciLean.Distribution.IsFunction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (T : 𝒟'(X, Y)) : Prop"} +{"name":"SciLean.Distribution.extAction","declaration":"def SciLean.Distribution.extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {Z : Type u_5} [SciLean.Vec R Z] (T : 𝒟'(X, Y)) (φ : X → Z) (L : Y ⊸[R] Z ⊸[R] W) : W"} +{"name":"SciLean.add_restrict'","declaration":"theorem SciLean.add_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : T.restrict A + S.restrict A = (T + S).restrict A"} +{"name":"SciLean.instTestFunctionsDistribution","declaration":"instance SciLean.instTestFunctionsDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [MeasureTheory.MeasureSpace X] [SciLean.SemiInnerProductSpace R Y] [Module ℝ Y] : SciLean.TestFunctions (𝒟'(X, Y))"} +{"name":"SciLean.Distribution.postComp","declaration":"def SciLean.Distribution.postComp {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (T : 𝒟'(X, Y)) (f : Y ⊸[R] Z) : 𝒟'(X, Z)"} +{"name":"SciLean.postComp_assoc","declaration":"theorem SciLean.postComp_assoc {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_5} [SciLean.Vec R Y] {U : Type u_3} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (x : 𝒟'(X, U)) (y : U ⊸[R] 𝒟'(Y, V)) (f : V ⊸[R] W) (φ : Y → R) : SciLean.Distribution.postComp (SciLean.Distribution.postComp x y) (fun T ⊸[R] SciLean.Distribution.postComp T f) =\n SciLean.Distribution.postComp x (fun u ⊸[R] SciLean.Distribution.postComp (y u) f)"} +{"name":"SciLean.finset_sum_restrict","declaration":"theorem SciLean.finset_sum_restrict {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {I : Type u_1} [Fintype I] (T : I → 𝒟'(X, Y)) (A : Set X) : (Finset.sum Finset.univ fun i => T i).restrict A = Finset.sum Finset.univ fun i => (T i).restrict A"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule'","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : ↑f + ↑g = ↑fun x => f x + g x"} +{"name":"SciLean.Distribution.zero_extAction","declaration":"theorem SciLean.Distribution.zero_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_3} [SciLean.Vec R W] {X : Type u_5} [SciLean.Vec R X] {U : Type u_2} [SciLean.Vec R U] {V : Type u_4} [SciLean.Vec R V] (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : 0.extAction φ L = 0"} +{"name":"SciLean.action_dirac","declaration":"theorem SciLean.action_dirac {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] (x : X) (φ : 𝒟 X) : (SciLean.dirac x) φ = φ x"} +{"name":"SciLean.Function.toDistribution_zero","declaration":"theorem SciLean.Function.toDistribution_zero {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] : (↑fun x => 0) = 0"} +{"name":"SciLean.term𝒟'_","declaration":"def SciLean.term𝒟'_ : Lean.ParserDescr"} +{"name":"SciLean.Distribution.prod_restrict","declaration":"theorem SciLean.Distribution.prod_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_7} [SciLean.Vec R X] {Z : Type u_6} [SciLean.Vec R Z] {X₁ : Type u_2} [SciLean.Vec R X₁] {X₂ : Type u_4} [SciLean.Vec R X₂] {Y₁ : Type u_3} [SciLean.Vec R Y₁] {Y₂ : Type u_5} [SciLean.Vec R Y₂] (p : X₁ → X₂ → X) (T : 𝒟'(X₁, Y₁)) (S : X₁ → 𝒟'(X₂, Y₂)) (L : Y₁ ⊸[R] Y₂ ⊸[R] Z) (A : Set X) : (SciLean.Distribution.prod p T S L).restrict A =\n SciLean.Distribution.prod p (T.restrict (Set.preimage1 p A)) (fun x₁ => (S x₁).restrict (p x₁ ⁻¹' A)) L"} +{"name":"SciLean.Distribution.IsMeasure","declaration":"def SciLean.Distribution.IsMeasure {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] (f : 𝒟'(X, R)) : Prop"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : (↑fun x => f x + g x) = ↑f + ↑g"} +{"name":"SciLean.HSub.hSub.arg_a0a1.toDistribution_rule'","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : X → Y) (g : X → Y) : ↑f - ↑g = ↑fun x => f x - g x"} +{"name":"SciLean.termIfD_Then_____Else____","declaration":"def SciLean.termIfD_Then_____Else____ : Lean.ParserDescr"} +{"name":"SciLean.finset_sum_restrict'","declaration":"theorem SciLean.finset_sum_restrict' {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {I : Type u_1} [Fintype I] (T : I → 𝒟'(X, Y)) (A : Set X) : (Finset.sum Finset.univ fun i => (T i).restrict A) = (Finset.sum Finset.univ fun i => T i).restrict A"} +{"name":"SciLean.HMul.hMul.arg_a0.toDistribution_rule","declaration":"theorem SciLean.HMul.hMul.arg_a0.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : (↑fun x => f x * r) = r • ↑f"} +{"name":"SciLean.Distribution.toFunction","declaration":"def SciLean.Distribution.toFunction {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (T : 𝒟'(X, Y)) : X → Y"} +{"name":"SciLean.Distribution.extAction.arg_T.IsSmoothLinearMap","declaration":"theorem SciLean.Distribution.extAction.arg_T.IsSmoothLinearMap {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : W → 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] Z) (hT : SciLean.IsSmoothLinearMap R T) : SciLean.IsSmoothLinearMap R fun w => (T w).extAction φ L"} +{"name":"SciLean.iteD_restrict","declaration":"theorem SciLean.iteD_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : (ifD A then\n T\n else\n 0) =\n T.restrict A"} +{"name":"MeasureTheory.Measure.toDistribution","declaration":"def MeasureTheory.Measure.toDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] (μ : MeasureTheory.Measure X) : 𝒟'(X, R)"} +{"name":"SciLean.Distribution.iteD_same","declaration":"theorem SciLean.Distribution.iteD_same {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (u : 𝒟'(X, Y)) : (ifD A then\n u\n else\n u) =\n u"} +{"name":"SciLean.HMul.hMul.arg_a1.toDistribution_rule'","declaration":"theorem SciLean.HMul.hMul.arg_a1.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : r • ↑f = ↑fun x => r • f x"} +{"name":"SciLean.neg_restrict","declaration":"theorem SciLean.neg_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : (-T).restrict A = -T.restrict A"} +{"name":"SciLean.iteD_restrict'","declaration":"theorem SciLean.iteD_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : (ifD A then\n 0\n else\n T) =\n T.restrict Aᶜ"} +{"name":"SciLean.sub_restrict","declaration":"theorem SciLean.sub_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : (T - S).restrict A = T.restrict A - S.restrict A"} +{"name":"SciLean.smul_restrict","declaration":"theorem SciLean.smul_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (r : R) (T : 𝒟'(X, Y)) (A : Set X) : (r • T).restrict A = r • T.restrict A"} +{"name":"SciLean.smul_restrict'","declaration":"theorem SciLean.smul_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (r : R) (T : 𝒟'(X, Y)) (A : Set X) : r • T.restrict A = (r • T).restrict A"} +{"name":"SciLean.Distribution.indextype_sum_extAction","declaration":"theorem SciLean.Distribution.indextype_sum_extAction {R : Type u_2} [SciLean.RealScalar R] {W : Type (max (max u_2 u_3) u_4)} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {U : Type u_4} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] {I : Type u_1} [SciLean.IndexType I] (T : I → 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : ( ∑ i, T i).extAction φ L = ∑ i, (T i).extAction φ L"} +{"name":"SciLean.HMul.hMul.arg_a0.toDistribution_rule'","declaration":"theorem SciLean.HMul.hMul.arg_a0.toDistribution_rule' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : r • ↑f = ↑fun x => f x * r"} +{"name":"SciLean.zero_restrict","declaration":"theorem SciLean.zero_restrict {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) : 0.restrict A = 0"} +{"name":"SciLean.Distribution.prod","declaration":"def SciLean.Distribution.prod {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Z : Type u_3} [SciLean.Vec R Z] {X₁ : Type u_4} [SciLean.Vec R X₁] {X₂ : Type u_5} [SciLean.Vec R X₂] {Y₁ : Type u_6} [SciLean.Vec R Y₁] {Y₂ : Type u_7} [SciLean.Vec R Y₂] (p : X₁ → X₂ → X) (T : 𝒟'(X₁, Y₁)) (S : X₁ → 𝒟'(X₂, Y₂)) (L : Y₁ ⊸[R] Y₂ ⊸[R] Z) : 𝒟'(X, Z)"} +{"name":"SciLean.Distribution.«term⟪_,_⟫[_]»","declaration":"def SciLean.Distribution.«term⟪_,_⟫[_]» : Lean.ParserDescr"} +{"name":"SciLean.Distribution.extAction_iteD","declaration":"theorem SciLean.Distribution.extAction_iteD {R : Type u_2} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_1} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (A : Set X) (t : 𝒟'(X, U)) (e : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (ifD A then\n t\n else\n e).extAction\n φ L =\n t.extAction (fun x => if x ∈ A then φ x else 0) L + e.extAction (fun x => if x ∉ A then φ x else 0) L"} +{"name":"SciLean.add_restrict","declaration":"theorem SciLean.add_restrict {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (S : 𝒟'(X, Y)) (A : Set X) : (T + S).restrict A = T.restrict A + S.restrict A"} +{"name":"SciLean.postComp_extAction","declaration":"theorem SciLean.postComp_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_4} [SciLean.Vec R X] {Z : Type u_5} [SciLean.Vec R Z] {U : Type u_2} [SciLean.Vec R U] {V : Type u_3} [SciLean.Vec R V] {y : U ⊸[R] V} (x : 𝒟'(X, U)) (f : U ⊸[R] V) (φ : X → W) (L : V ⊸[R] W ⊸[R] Z) : (SciLean.Distribution.postComp x y).extAction φ L = x.extAction φ (fun u ⊸[R] fun w ⊸[R] (L (f u)) w)"} +{"name":"SciLean.iteD.arg_cte.toDistribution_rule","declaration":"theorem SciLean.iteD.arg_cte.toDistribution_rule {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (s : Set X) (t : X → Y) (e : X → Y) : (ifD s then\n ↑t\n else\n ↑e) =\n ↑fun x => if x ∈ s then t x else e x"} +{"name":"SciLean.instSMulRealDistribution","declaration":"instance SciLean.instSMulRealDistribution {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] : SMul ℝ (𝒟'(X, Y))"} +{"name":"SciLean.dirac","declaration":"def SciLean.dirac {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] (x : X) : 𝒟'(X, R)"} +{"name":"SciLean.indextype_sum_restrict","declaration":"theorem SciLean.indextype_sum_restrict {R : Type u_2} [SciLean.RealScalar R] {X : Type u_3} [SciLean.Vec R X] {I : Type u_1} [SciLean.IndexType I] (T : I → 𝒟'(X, R)) (A : Set X) : ( ∑ i, T i).restrict A = ∑ i, (T i).restrict A"} +{"name":"SciLean.Distribution.bind","declaration":"def SciLean.Distribution.bind {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {U : Type u_5} [SciLean.Vec R U] {V : Type u_6} [SciLean.Vec R V] (x' : 𝒟'(X, U)) (f : X → 𝒟'(Y, V)) (L : U ⊸[R] V ⊸[R] W) : 𝒟'(Y, W)"} +{"name":"SciLean.Distribution.integrate","declaration":"def SciLean.Distribution.integrate {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) : Y"} +{"name":"SciLean.instCoeMeasureDistributionToVecToRCLikeToScalarInstSemiInnerProductSpace","declaration":"instance SciLean.instCoeMeasureDistributionToVecToRCLikeToScalarInstSemiInnerProductSpace {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasurableSpace X] : Coe (MeasureTheory.Measure X) (𝒟'(X, R))"} +{"name":"SciLean.action_bind","declaration":"theorem SciLean.action_bind {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (x : 𝒟'(X, U)) (f : X → 𝒟'(Y, V)) (L : U ⊸[R] V ⊸[R] W) (φ : 𝒟 Y) : (SciLean.Distribution.bind x f L) φ = x.extAction (fun x' => SciLean.Distribution.extAction' (f x') ⇑φ) L"} +{"name":"SciLean.iteD","declaration":"def SciLean.iteD {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (t : 𝒟'(X, Y)) (e : 𝒟'(X, Y)) : 𝒟'(X, Y)"} +{"name":"SciLean.Distribution.fintype_sum_extAction","declaration":"theorem SciLean.Distribution.fintype_sum_extAction {R : Type u_2} [SciLean.RealScalar R] {W : Type u_5} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {U : Type u_4} [SciLean.Vec R U] {V : Type u_6} [SciLean.Vec R V] {I : Type u_1} [Fintype I] (T : I → 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (Finset.sum Finset.univ fun i => T i).extAction φ L = Finset.sum Finset.univ fun i => (T i).extAction φ L"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (r : R) (f : X → Y) : (↑fun x => r • f x) = r • ↑f"} +{"name":"SciLean.Distribution.sub_extAction","declaration":"theorem SciLean.Distribution.sub_extAction {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {U : Type u_3} [SciLean.Vec R U] {V : Type u_5} [SciLean.Vec R V] (T : 𝒟'(X, U)) (T' : 𝒟'(X, U)) (φ : X → V) (L : U ⊸[R] V ⊸[R] W) : (T - T').extAction φ L = T.extAction φ L - T'.extAction φ L"} +{"name":"SciLean.Distribution.action_iteD","declaration":"theorem SciLean.Distribution.action_iteD {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) (t : 𝒟'(X, Y)) (e : 𝒟'(X, Y)) (φ : 𝒟 X) : (ifD A then\n t\n else\n e)\n φ =\n t.extAction (fun x => if x ∈ A then φ x else 0) (fun y ⊸[R] fun r ⊸[R] r • y) +\n e.extAction (fun x => if x ∉ A then φ x else 0) (fun y ⊸[R] fun r ⊸[R] r • y)"} +{"name":"SciLean.HMul.hMul.arg_a1.toDistribution_rule","declaration":"theorem SciLean.HMul.hMul.arg_a1.toDistribution_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : R) (f : X → R) : (↑fun x => r • f x) = r • ↑f"} +{"name":"SciLean.neg_restrict'","declaration":"theorem SciLean.neg_restrict' {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) (A : Set X) : -T.restrict A = (-T).restrict A"} diff --git a/scilean-declarations/SciLean.Core.Distribution.ParametricDistribDeriv.jsonl b/scilean-declarations/SciLean.Core.Distribution.ParametricDistribDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a014fbecc2d44a50fd4973a64ec79feb4d8ddf5a --- /dev/null +++ b/scilean-declarations/SciLean.Core.Distribution.ParametricDistribDeriv.jsonl @@ -0,0 +1,28 @@ +{"name":"SciLean.Distribution.restrict.arg_T.parDistribDeriv_rule","declaration":"theorem SciLean.Distribution.restrict.arg_T.parDistribDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : W → 𝒟'(X, Y)) (A : Set X) (hT : SciLean.DistribDifferentiable T) : (SciLean.parDistribDeriv fun w => (T w).restrict A) = fun w dw => (SciLean.parDistribDeriv T w dw).restrict A"} +{"name":"SciLean.dirac.arg_xy.DistribDiffrentiable_rule","declaration":"theorem SciLean.dirac.arg_xy.DistribDiffrentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] (x : W → X) (hx : SciLean.CDifferentiable R x) : SciLean.DistribDifferentiable fun w => SciLean.dirac (x w)"} +{"name":"SciLean.Distribution.restrict.arg_T.IsSmoothLinearMap_rule_simple","declaration":"theorem SciLean.Distribution.restrict.arg_T.IsSmoothLinearMap_rule_simple {R : Type u_2} [SciLean.RealScalar R] {X : Type u_1} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (A : Set X) : SciLean.IsSmoothLinearMap R fun T => T.restrict A"} +{"name":"SciLean.Bind.bind.arg_fx.DistribDifferentiable_rule","declaration":"theorem SciLean.Bind.bind.arg_fx.DistribDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_7} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {Z : Type u_2} [SciLean.Vec R Z] {U : Type u_5} [SciLean.Vec R U] {V : Type u_3} [SciLean.Vec R V] (f : X → Y → 𝒟'(Z, V)) (g : X → 𝒟'(Y, U)) (L : U ⊸[R] V ⊸[R] W) (hf : SciLean.DistribDifferentiable fun x =>\n match x with\n | (x, y) => f x y) (hg : SciLean.DistribDifferentiable g) : SciLean.DistribDifferentiable fun x => SciLean.Distribution.bind (g x) (f x) L"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.parDistribDeriv_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.parDistribDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (f : W → 𝒟'(X, Y)) (g : W → 𝒟'(X, Y)) (hf : SciLean.DistribDifferentiable f) (hg : SciLean.DistribDifferentiable g) : (SciLean.parDistribDeriv fun w => f w + g w) = fun w dw =>\n let dy := SciLean.parDistribDeriv f w dw;\n let dz := SciLean.parDistribDeriv g w dw;\n dy + dz"} +{"name":"SciLean.cintegral.arg_f.parDistribDeriv_rule","declaration":"theorem SciLean.cintegral.arg_f.parDistribDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] {Y : Type u_5} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] [Module ℝ Z] [MeasureTheory.MeasureSpace X] [MeasureTheory.MeasureSpace Y] (f : W → X → Y → Z) : (SciLean.parDistribDeriv fun w => ↑fun x => ∫' y, f w x y ∂MeasureTheory.volume) = fun w dw =>\n let Tf := fun w => ↑fun x => ↑fun y => f w x y;\n SciLean.Distribution.postComp (SciLean.parDistribDeriv Tf w dw)\n (fun T ⊸[R] T.extAction (fun x => 1) (fun z ⊸[R] fun r ⊸[R] r • z))"} +{"name":"SciLean.DistribDiffrentiable.const_rule","declaration":"theorem SciLean.DistribDiffrentiable.const_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) : SciLean.DistribDifferentiable fun x => T"} +{"name":"SciLean.distribDifferentiableAt_const_test_fun","declaration":"theorem SciLean.distribDifferentiableAt_const_test_fun {R : Type u_1} [SciLean.RealScalar R] {X : Type u_4} [SciLean.Vec R X] {Y : Type u_2} [SciLean.Vec R Y] {Z : Type u_3} [SciLean.Vec R Z] {f : X → 𝒟'(Y, Z)} {x : X} (hf : SciLean.DistribDifferentiableAt f x) {φ : 𝒟 Y} : SciLean.CDifferentiableAt R (fun x => (f x) φ) x"} +{"name":"SciLean.DistribDifferentiableAt","declaration":"def SciLean.DistribDifferentiableAt {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (f : X → 𝒟'(Y, Z)) (x : X) : Prop"} +{"name":"SciLean.cintegral.arg_f.cderiv_distrib_rule'","declaration":"theorem SciLean.cintegral.arg_f.cderiv_distrib_rule' {R : Type u_2} [SciLean.RealScalar R] {W : Type u_3} [SciLean.Vec R W] {X : Type u_1} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (f : W → X → R) (A : Set X) : ∂ w, ∫' x, f w x ∂MeasureTheory.volume.restrict A = fun w dw =>\n ((SciLean.parDistribDeriv (fun w => ↑fun x => f w x) w dw).restrict A).extAction (fun x => 1)\n (fun y ⊸[R] fun r ⊸[R] r • y)"} +{"name":"SciLean.parDistribDeriv.const_rule","declaration":"theorem SciLean.parDistribDeriv.const_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : 𝒟'(X, Y)) : (SciLean.parDistribDeriv fun x => T) = fun w dw => 0"} +{"name":"SciLean.action_parDistribDeriv","declaration":"theorem SciLean.action_parDistribDeriv {R : Type u_1} [SciLean.RealScalar R] {X : Type u_4} [SciLean.Vec R X] {Y : Type u_2} [SciLean.Vec R Y] {Z : Type u_3} [SciLean.Vec R Z] (f : X → 𝒟'(Y, Z)) (x : X) (dx : X) (φ : 𝒟 Y) : (SciLean.parDistribDeriv f x dx) φ = ∂ (x':=x;dx), (f x') φ"} +{"name":"SciLean.HSub.hSub.arg_a0a1.parDistribDeriv_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.parDistribDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (f : W → 𝒟'(X, Y)) (g : W → 𝒟'(X, Y)) : (SciLean.parDistribDeriv fun w => f w - g w) = fun w dw =>\n let dy := SciLean.parDistribDeriv f w dw;\n let dz := SciLean.parDistribDeriv g w dw;\n dy - dz"} +{"name":"SciLean.DistribDiffrentiable.comp_rule","declaration":"theorem SciLean.DistribDiffrentiable.comp_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_5} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {Z : Type u_2} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] (f : Y → 𝒟'(Z, U)) (g : X → Y) (hf : SciLean.DistribDifferentiable f) (hg : SciLean.CDifferentiable R g) : SciLean.DistribDifferentiable fun x => f (g x)"} +{"name":"SciLean.dirac.arg_x.parDistribDeriv_rule","declaration":"theorem SciLean.dirac.arg_x.parDistribDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_3} [SciLean.Vec R X] (x : W → X) (hx : SciLean.CDifferentiable R x) : (SciLean.parDistribDeriv fun w => SciLean.dirac (x w)) = fun w dw =>\n let xdx := ∂> x w dw;\n SciLean.diracDeriv xdx.1 xdx.2"} +{"name":"SciLean.HSub.hSub.arg_a0a1.DistribDifferentiable_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.DistribDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (f : W → 𝒟'(X, Y)) (g : W → 𝒟'(X, Y)) : SciLean.DistribDifferentiable fun w => f w - g w"} +{"name":"SciLean.Bind.bind.arg_fx.parDistribDiff_rule","declaration":"theorem SciLean.Bind.bind.arg_fx.parDistribDiff_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_4} [SciLean.Vec R X] {Y : Type u_2} [SciLean.Vec R Y] {U : Type u_5} [SciLean.Vec R U] {V : Type u_3} [SciLean.Vec R V] {x : W} (f : W → X → 𝒟'(Y, V)) (g : W → 𝒟'(X, U)) (L : U ⊸[R] V ⊸[R] W) (hf : SciLean.DistribDifferentiable fun x =>\n match x with\n | (w, x) => f w x) (hg : SciLean.DistribDifferentiable g) : (SciLean.parDistribDeriv fun w => SciLean.Distribution.bind (g w) (f w) L) = fun w dw =>\n SciLean.Distribution.bind (SciLean.parDistribDeriv g w dw) (fun x_1 => f x x_1) L +\n SciLean.Distribution.bind (g w) (fun x => SciLean.parDistribDeriv (fun x_1 => f x_1 x) w dw) L"} +{"name":"SciLean.parDistribDeriv.comp_rule","declaration":"theorem SciLean.parDistribDeriv.comp_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_5} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {Z : Type u_2} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] (f : Y → 𝒟'(Z, U)) (g : X → Y) (hf : SciLean.DistribDifferentiable f) (hg : SciLean.CDifferentiable R g) : (SciLean.parDistribDeriv fun x => f (g x)) = fun x dx =>\n let ydy := ∂> g x dx;\n SciLean.parDistribDeriv f ydy.1 ydy.2"} +{"name":"SciLean.diracDeriv","declaration":"def SciLean.diracDeriv {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] (x : X) (dx : X) : 𝒟'(X, R)"} +{"name":"SciLean.parDistribDeriv","declaration":"def SciLean.parDistribDeriv {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (f : X → 𝒟'(Y, Z)) (x : X) (dx : X) : 𝒟'(Y, Z)"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.DistribDifferentiable_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.DistribDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (f : W → 𝒟'(X, Y)) (g : W → 𝒟'(X, Y)) (hf : SciLean.DistribDifferentiable f) (hg : SciLean.DistribDifferentiable g) : SciLean.DistribDifferentiable fun w => f w + g w"} +{"name":"SciLean.cintegral.arg_f.parDistribDeriv_rule'","declaration":"theorem SciLean.cintegral.arg_f.parDistribDeriv_rule' {R : Type u_2} [SciLean.RealScalar R] {W : Type u_3} [SciLean.Vec R W] {X : Type u_4} [SciLean.Vec R X] {Y : Type u_1} [SciLean.Vec R Y] {Z : Type u_5} [SciLean.Vec R Z] [Module ℝ Z] [MeasureTheory.MeasureSpace X] [MeasureTheory.MeasureSpace Y] (f : W → X → Y → Z) (B : X → Set Y) : (SciLean.parDistribDeriv fun w => ↑fun x => ∫' y, f w x y ∂MeasureTheory.volume.restrict (B x)) = fun w dw =>\n let Tf := fun w => ↑fun x => (↑fun y => f w x y).restrict (B x);\n SciLean.Distribution.postComp (SciLean.parDistribDeriv Tf w dw)\n (fun T ⊸[R] T.extAction (fun x => 1) (fun z ⊸[R] fun r ⊸[R] r • z))"} +{"name":"SciLean.cintegral.arg_f.cderiv_distrib_rule","declaration":"theorem SciLean.cintegral.arg_f.cderiv_distrib_rule {R : Type u_3} [SciLean.RealScalar R] {W : Type u_1} [SciLean.Vec R W] {X : Type u_4} [SciLean.Vec R X] {Y : Type u_2} [SciLean.Vec R Y] [Module ℝ Y] [MeasureTheory.MeasureSpace X] (f : W → X → Y) : ∂ w, ∫' x, f w x ∂MeasureTheory.volume = fun w dw =>\n (SciLean.parDistribDeriv (fun w => ↑fun x => f w x) w dw).extAction (fun x => 1) (fun y ⊸[R] fun r ⊸[R] r • y)"} +{"name":"SciLean.Function.toDistribution.arg_f.cderiv_rule","declaration":"theorem SciLean.Function.toDistribution.arg_f.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_4} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] (f : W → X → Y) (hf : ∀ (x : X), SciLean.CDifferentiable R fun x_1 => f x_1 x) : (∂ w, ↑fun x => f w x) = fun w dw =>\n ↑fun x =>\n let dy := ∂ (x_1:=w;dw), f x_1 x;\n dy"} +{"name":"SciLean.Distribution.restrict.arg_T.IsSmoothLinearMap_rule","declaration":"theorem SciLean.Distribution.restrict.arg_T.IsSmoothLinearMap_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] (T : W → 𝒟'(X, Y)) (A : Set X) (hT : SciLean.IsSmoothLinearMap R T) : SciLean.IsSmoothLinearMap R fun w => (T w).restrict A"} +{"name":"SciLean.DistribDifferentiable","declaration":"def SciLean.DistribDifferentiable {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (f : X → 𝒟'(Y, Z)) : Prop"} +{"name":"SciLean.Function.toDistribution.arg_f.CDifferentiable_rule","declaration":"theorem SciLean.Function.toDistribution.arg_f.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type u_4} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] (f : W → X → Y) (hf : ∀ (x : X), SciLean.CDifferentiable R fun x_1 => f x_1 x) : SciLean.CDifferentiable R fun w => ↑fun x => f w x"} +{"name":"SciLean.toDistribution.linear_parDistribDeriv_rule","declaration":"theorem SciLean.toDistribution.linear_parDistribDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_4} [SciLean.Vec R W] {X : Type u_5} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] {Y : Type u_2} [SciLean.Vec R Y] [Module ℝ Y] {Z : Type u_3} [SciLean.Vec R Z] [Module ℝ Z] (f : W → X → Y) (L : Y → Z) (hL : SciLean.IsSmoothLinearMap R L) : (SciLean.parDistribDeriv fun w => ↑fun x => L (f w x)) = fun w dw =>\n SciLean.Distribution.postComp (SciLean.parDistribDeriv (fun w => ↑fun x => f w x) w dw) (fun y ⊸[R] L y)"} diff --git a/scilean-declarations/SciLean.Core.Distribution.ParametricDistribFwdDeriv.jsonl b/scilean-declarations/SciLean.Core.Distribution.ParametricDistribFwdDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e9f931252d6aafa3148348707344e6a16842e178 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Distribution.ParametricDistribFwdDeriv.jsonl @@ -0,0 +1,3 @@ +{"name":"SciLean.parDistribFwdDeriv.bind_rule","declaration":"theorem SciLean.parDistribFwdDeriv.bind_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_6} [SciLean.Vec R W] {X : Type u_7} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {Z : Type u_2} [SciLean.Vec R Z] {U : Type u_5} [SciLean.Vec R U] {V : Type u_3} [SciLean.Vec R V] (f : X → Y → 𝒟'(Z, V)) (g : X → 𝒟'(Y, U)) (L : U ⊸[R] V ⊸[R] W) (hf : SciLean.DistribDifferentiable fun x =>\n match x with\n | (x, y) => f x y) (hg : SciLean.DistribDifferentiable g) : (SciLean.parDistribFwdDeriv fun x => SciLean.Distribution.bind (g x) (f x) L) = fun x dx =>\n let ydy := SciLean.parDistribFwdDeriv g x dx;\n let zdz := fun y => SciLean.parDistribFwdDeriv (fun x => f x y) x dx;\n SciLean.Distribution.bind ydy zdz\n (fun x ⊸[R]\n match x with\n | (r, dr) =>\n fun x ⊸[R]\n match x with\n | (s, ds) => ((L r) s, (L r) ds + (L dr) s))"} +{"name":"SciLean.parDistribFwdDeriv","declaration":"def SciLean.parDistribFwdDeriv {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] {Z : Type u_4} [SciLean.Vec R Z] (f : X → 𝒟'(Y, Z)) (x : X) (dx : X) : 𝒟'(Y, Z × Z)"} +{"name":"SciLean.parDistribFwdDeriv.comp_rule","declaration":"theorem SciLean.parDistribFwdDeriv.comp_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_5} [SciLean.Vec R X] {Y : Type u_4} [SciLean.Vec R Y] {Z : Type u_2} [SciLean.Vec R Z] {U : Type u_3} [SciLean.Vec R U] (f : Y → 𝒟'(Z, U)) (g : X → Y) (hf : SciLean.DistribDifferentiable f) (hg : SciLean.CDifferentiable R g) : (SciLean.parDistribFwdDeriv fun x => f (g x)) = fun x dx =>\n let ydy := ∂> g x dx;\n SciLean.parDistribFwdDeriv f ydy.1 ydy.2"} diff --git a/scilean-declarations/SciLean.Core.Distribution.SimpAttr.jsonl b/scilean-declarations/SciLean.Core.Distribution.SimpAttr.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..9b57921bd0a7fc2bf5b3119c1645e29d41e3caba --- /dev/null +++ b/scilean-declarations/SciLean.Core.Distribution.SimpAttr.jsonl @@ -0,0 +1,12 @@ +{"name":"Parser.Attr.restrict_push","declaration":"def Parser.Attr.restrict_push : Lean.ParserDescr"} +{"name":"Parser.Attr.distrib_eval","declaration":"def Parser.Attr.distrib_eval : Lean.ParserDescr"} +{"name":"Parser.Attr.action_push","declaration":"def Parser.Attr.action_push : Lean.ParserDescr"} +{"name":"Parser.Attr.restrict_pull_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.restrict_pull_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.distrib_eval_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.distrib_eval_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.toDistrib_push","declaration":"def Parser.Attr.toDistrib_push : Lean.ParserDescr"} +{"name":"Parser.Attr.toDistrib_pull","declaration":"def Parser.Attr.toDistrib_pull : Lean.ParserDescr"} +{"name":"Parser.Attr.toDistrib_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.toDistrib_push_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.toDistrib_pull_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.toDistrib_pull_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.restrict_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.restrict_push_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.restrict_pull","declaration":"def Parser.Attr.restrict_pull : Lean.ParserDescr"} +{"name":"Parser.Attr.action_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.action_push_proc : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Distribution.TestFunction.jsonl b/scilean-declarations/SciLean.Core.Distribution.TestFunction.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1c72fbed8ea51b35ee274bebae6178f8d19f4fe3 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Distribution.TestFunction.jsonl @@ -0,0 +1,29 @@ +{"name":"SciLean.instTCOr","declaration":"instance SciLean.instTCOr {A : Sort u_1} {B : Sort u_2} [inst : A] : SciLean.TCOr A B"} +{"name":"SciLean.TCOr.val","declaration":"def SciLean.TCOr.val {A : Sort u_1} {B : Sort u_2} [self : SciLean.TCOr A B] : A ⊕' B"} +{"name":"SciLean.TestFunctionSpace.mk","declaration":"ctor SciLean.TestFunctionSpace.mk {R : Type u} [SciLean.RealScalar R] {X : Type v} [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (toFun : X → R) (is_test_fun : SciLean.IsTestFunction toFun) : 𝒟 X"} +{"name":"SciLean.TestFunctionSpace.eval_IsLinearMap","declaration":"theorem SciLean.TestFunctionSpace.eval_IsLinearMap {X : Type u_2} {R : Type u_1} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (x : X) : IsLinearMap R fun φ => φ x"} +{"name":"SciLean.instAddTestFunctionSpace","declaration":"instance SciLean.instAddTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : Add (𝒟 X)"} +{"name":"SciLean.TestFunctionSpace.eval_CDifferentiable_rule'","declaration":"theorem SciLean.TestFunctionSpace.eval_CDifferentiable_rule' {R : Type u_1} [SciLean.RealScalar R] {W : Type u_3} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] (φ : W → 𝒟 X) (x : W → X) (hφ : SciLean.CDifferentiable R φ) (hx : SciLean.CDifferentiable R x) : SciLean.CDifferentiable R fun w => (φ w) (x w)"} +{"name":"SciLean.instFunLikeTestFunctionSpace","declaration":"instance SciLean.instFunLikeTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : FunLike (𝒟 X) X R"} +{"name":"SciLean.unexpandTestFunctionSpace","declaration":"def SciLean.unexpandTestFunctionSpace : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.instTCOr_1","declaration":"instance SciLean.instTCOr_1 {A : Sort u_1} {B : Sort u_2} [inst : B] : SciLean.TCOr A B"} +{"name":"SciLean.instUniformSpaceTestFunctionSpace","declaration":"instance SciLean.instUniformSpaceTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : UniformSpace (𝒟 X)"} +{"name":"SciLean.term𝒟_","declaration":"def SciLean.term𝒟_ : Lean.ParserDescr"} +{"name":"SciLean.TCOr","declaration":"class SciLean.TCOr (A : Sort u_1) (B : Sort u_2) : Sort (max (max 1 u_1) u_2)"} +{"name":"SciLean.TestFunctionSpace.eval_CDifferentiable","declaration":"theorem SciLean.TestFunctionSpace.eval_CDifferentiable {X : Type u_2} {R : Type u_1} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (x : X) : SciLean.CDifferentiable R fun φ => φ x"} +{"name":"SciLean.TestFunctionSpace.is_test_fun","declaration":"def SciLean.TestFunctionSpace.is_test_fun {R : Type u} [SciLean.RealScalar R] {X : Type v} [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (self : 𝒟 X) : SciLean.IsTestFunction self.toFun"} +{"name":"SciLean.testFunApprox","declaration":"/-- Restrict function `f` by a ball of radius `n` and do smooth by convolution with radius `1/n`.\n\nIf `X` is discrete space then we do not need to do smoothing. -/\nopaque SciLean.testFunApprox {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (n : ℕ) (f : X → R) : 𝒟 X"} +{"name":"SciLean.TestFunctionSpace","declaration":"structure SciLean.TestFunctionSpace (R : Type u) [SciLean.RealScalar R] (X : Type v) [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : Type (max u v)"} +{"name":"SciLean.IsTestFunction.has_compact_support","declaration":"def SciLean.IsTestFunction.has_compact_support {R : Type u} [SciLean.RealScalar R] {X : Type v} [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] {f : X → R} (self : SciLean.IsTestFunction f) : HasCompactSupport f"} +{"name":"SciLean.IsTestFunction.is_smooth","declaration":"def SciLean.IsTestFunction.is_smooth {R : Type u} [SciLean.RealScalar R] {X : Type v} [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] {f : X → R} (self : SciLean.IsTestFunction f) : match SciLean.TCOr.val with\n| PSum.inl val => SciLean.ContCDiff R ⊤ f\n| PSum.inr val => True"} +{"name":"SciLean.instVecToRCLikeToScalarTestFunctionSpace","declaration":"instance SciLean.instVecToRCLikeToScalarTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : SciLean.Vec R (𝒟 X)"} +{"name":"SciLean.TestFunctionSpace.toFun","declaration":"def SciLean.TestFunctionSpace.toFun {R : Type u} [SciLean.RealScalar R] {X : Type v} [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (self : 𝒟 X) : X → R"} +{"name":"SciLean.instSubTestFunctionSpace","declaration":"instance SciLean.instSubTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : Sub (𝒟 X)"} +{"name":"SciLean.TestFunctionSpace.eval_IsSmoothLinearMap","declaration":"theorem SciLean.TestFunctionSpace.eval_IsSmoothLinearMap {X : Type u_2} {R : Type u_1} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (x : X) : SciLean.IsSmoothLinearMap R fun φ => φ x"} +{"name":"SciLean.TestFunctionSpace.eval_CDifferentiableAt_rule","declaration":"theorem SciLean.TestFunctionSpace.eval_CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_3} [SciLean.Vec R W] {X : Type u_2} [SciLean.Vec R X] (w : W) (φ : W → 𝒟 X) (x : W → X) (hφ : SciLean.CDifferentiableAt R φ w) (hx : SciLean.CDifferentiableAt R x w) : SciLean.CDifferentiableAt R (fun w => (φ w) (x w)) w"} +{"name":"SciLean.instNegTestFunctionSpace","declaration":"instance SciLean.instNegTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : Neg (𝒟 X)"} +{"name":"SciLean.IsTestFunction","declaration":"structure SciLean.IsTestFunction {R : Type u} [SciLean.RealScalar R] {X : Type v} [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] (f : X → R) : Prop"} +{"name":"SciLean.instSMulTestFunctionSpace","declaration":"instance SciLean.instSMulTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : SMul R (𝒟 X)"} +{"name":"SciLean.TCOr.mk","declaration":"ctor SciLean.TCOr.mk {A : Sort u_1} {B : Sort u_2} (val : A ⊕' B) : SciLean.TCOr A B"} +{"name":"SciLean.IsTestFunction.mk","declaration":"ctor SciLean.IsTestFunction.mk {R : Type u} [SciLean.RealScalar R] {X : Type v} [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] {f : X → R} (is_smooth : match SciLean.TCOr.val with\n| PSum.inl val => SciLean.ContCDiff R ⊤ f\n| PSum.inr val => True) (has_compact_support : HasCompactSupport f) : SciLean.IsTestFunction f"} +{"name":"SciLean.instZeroTestFunctionSpace","declaration":"instance SciLean.instZeroTestFunctionSpace {X : Type u_1} {R : Type u_2} [SciLean.RealScalar R] [TopologicalSpace X] [space : SciLean.TCOr (SciLean.Vec R X) (DiscreteTopology X)] : Zero (𝒟 X)"} diff --git a/scilean-declarations/SciLean.Core.FloatAsReal.jsonl b/scilean-declarations/SciLean.Core.FloatAsReal.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ae23bcb72f415756b62b649b5e63caf613cc095e --- /dev/null +++ b/scilean-declarations/SciLean.Core.FloatAsReal.jsonl @@ -0,0 +1,21 @@ +{"name":"SciLean.instMeasureSpaceFloat","declaration":"instance SciLean.instMeasureSpaceFloat : MeasureTheory.MeasureSpace Float"} +{"name":"SciLean.instDenselyNormedFieldFloat","declaration":"instance SciLean.instDenselyNormedFieldFloat : DenselyNormedField Float"} +{"name":"SciLean.instRealScalarFloat","declaration":"instance SciLean.instRealScalarFloat : SciLean.RealScalar Float"} +{"name":"SciLean.instSeminormedRingFloat","declaration":"instance SciLean.instSeminormedRingFloat : SeminormedRing Float"} +{"name":"SciLean.instNormedFieldFloat","declaration":"instance SciLean.instNormedFieldFloat : NormedField Float"} +{"name":"SciLean.re_float","declaration":"theorem SciLean.re_float (a : Float) : ↑(RCLike.re a) = a"} +{"name":"SciLean.instLinearOrderedCommRingFloat","declaration":"instance SciLean.instLinearOrderedCommRingFloat : LinearOrderedCommRing Float"} +{"name":"SciLean.instRCLikeFloat","declaration":"instance SciLean.instRCLikeFloat : RCLike Float"} +{"name":"SciLean.instFieldFloat","declaration":"instance SciLean.instFieldFloat : Field Float"} +{"name":"SciLean.instStarRingFloatToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingToNonUnitalSeminormedCommRingToSeminormedCommRingToNormedCommRingToNormedFieldInstDenselyNormedFieldFloat","declaration":"instance SciLean.instStarRingFloatToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingToNonUnitalSeminormedCommRingToSeminormedCommRingToNormedCommRingToNormedFieldInstDenselyNormedFieldFloat : StarRing Float"} +{"name":"SciLean.instIsRealFloat","declaration":"instance SciLean.instIsRealFloat : SciLean.IsReal Float"} +{"name":"SciLean.instDecidableEqFloat","declaration":"instance SciLean.instDecidableEqFloat : DecidableEq Float"} +{"name":"SciLean.instStarRingFloatToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingInstCommRingFloat","declaration":"instance SciLean.instStarRingFloatToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingInstCommRingFloat : StarRing Float"} +{"name":"SciLean.instCompleteSpaceFloatToUniformSpaceToPseudoMetricSpaceInstSeminormedRingFloat","declaration":"instance SciLean.instCompleteSpaceFloatToUniformSpaceToPseudoMetricSpaceInstSeminormedRingFloat : CompleteSpace Float"} +{"name":"SciLean.instLinearOrderedFieldFloat","declaration":"instance SciLean.instLinearOrderedFieldFloat : LinearOrderedField Float"} +{"name":"SciLean.instCommRingFloat","declaration":"instance SciLean.instCommRingFloat : CommRing Float"} +{"name":"SciLean.instNormedAlgebraRealFloatNormedFieldInstSeminormedRingFloat","declaration":"instance SciLean.instNormedAlgebraRealFloatNormedFieldInstSeminormedRingFloat : NormedAlgebra ℝ Float"} +{"name":"SciLean.conj_float","declaration":"theorem SciLean.conj_float (a : Float) : (starRingEnd Float) a = a"} +{"name":"SciLean.instMeasurableSpaceFloat","declaration":"instance SciLean.instMeasurableSpaceFloat : MeasurableSpace Float"} +{"name":"SciLean.im_float","declaration":"theorem SciLean.im_float (a : Float) : ↑(RCLike.im a) = 0"} +{"name":"SciLean.instAlgebraRealFloatInstCommSemiringRealToSemiringToDivisionSemiringToSemifieldToLinearOrderedSemifieldInstLinearOrderedFieldFloat","declaration":"instance SciLean.instAlgebraRealFloatInstCommSemiringRealToSemiringToDivisionSemiringToSemifieldToLinearOrderedSemifieldInstLinearOrderedFieldFloat : Algebra ℝ Float"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.Bijective.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.Bijective.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cfb1b3ee96a6935cbb692bf954e4866d59abce5e --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.Bijective.jsonl @@ -0,0 +1,26 @@ +{"name":"Function.Bijective.HSub.hSub.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.HSub.hSub.arg_a0.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x - y"} +{"name":"Function.Bijective.HSub.hSub.arg_a1.Bijective_rule","declaration":"theorem Function.Bijective.HSub.hSub.arg_a1.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => y - f x"} +{"name":"Function.invFun_comp'","declaration":"theorem Function.invFun_comp' {α : Sort u_1} {β : Sort u_2} [Nonempty α] {f : α → β} (hf : Function.Injective f) {x : α} : Function.invFun f (f x) = x"} +{"name":"Function.Bijective.Equiv.toFun.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Equiv.toFun.arg_a0.Bijective_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y ≃ Z) (g : X → Y) (hf : Function.Bijective g) : Function.Bijective fun x => f (g x)"} +{"name":"Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_group","declaration":"theorem Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => y * f x"} +{"name":"Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_group","declaration":"theorem Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x * y"} +{"name":"Function.Bijective.Neg.neg.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Neg.neg.arg_a0.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => -f x"} +{"name":"Function.Bijective.Function.comp.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Function.comp.arg_a0.Bijective_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y → Z) (hf : Function.Bijective f) (g : X → Y) (hg : Function.Bijective g) : Function.Bijective fun x => (f ∘ g) x"} +{"name":"Function.Bijective.HAdd.hAdd.arg_a1.Bijective_rule","declaration":"theorem Function.Bijective.HAdd.hAdd.arg_a1.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => y + f x"} +{"name":"Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple'","declaration":"theorem Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple' {X : Type u_1} {Y : Type u_2} : Function.Bijective fun xy => (xy.2, xy.1)"} +{"name":"Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_group","declaration":"theorem Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_group {X : Type u_3} {Y : Type u_2} {G : Type u_1} [Group G] [MulAction G Y] (g : G) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => g • f x"} +{"name":"Function.Bijective.comp_rule","declaration":"theorem Function.Bijective.comp_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y → Z) (g : X → Y) (hf : Function.Bijective f) (hg : Function.Bijective g) : Function.Bijective fun x => f (g x)"} +{"name":"Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_field","declaration":"theorem Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) (hy : y ≠ 0) : Function.Bijective fun x => f x / y"} +{"name":"Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_field","declaration":"theorem Function.Bijective.HMul.hMul.arg_a0.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) (hy : y ≠ 0) : Function.Bijective fun x => f x * y"} +{"name":"Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_group","declaration":"theorem Function.Bijective.HDiv.hDiv.arg_a0.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x / y"} +{"name":"Function.Bijective.id_rule","declaration":"theorem Function.Bijective.id_rule {X : Type u_1} : Function.Bijective fun x => x"} +{"name":"Function.Bijective.Equiv.invFun.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.Equiv.invFun.arg_a0.Bijective_rule {X : Type u_3} {Y : Type u_1} {Z : Type u_2} (f : Y ≃ Z) (g : X → Z) (hf : Function.Bijective g) : Function.Bijective fun x => f.invFun (g x)"} +{"name":"Function.Bijective.Inv.inv.arg_a0.Bijective_rule_field","declaration":"theorem Function.Bijective.Inv.inv.arg_a0.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (f : X → Y) (hf : Function.Bijective f) (hf' : ∀ (x : X), f x ≠ 0) : Function.Bijective fun x => (f x)⁻¹"} +{"name":"Function.Bijective.id.arg_a.Bijective_rule","declaration":"theorem Function.Bijective.id.arg_a.Bijective_rule {X : Type u_1} : Function.Bijective fun x => id x"} +{"name":"Function.Bijective.HVAdd.hVAdd.arg_a1.Bijective_rule_group","declaration":"theorem Function.Bijective.HVAdd.hVAdd.arg_a1.Bijective_rule_group {X : Type u_3} {Y : Type u_2} {G : Type u_1} [AddGroup G] [AddAction G Y] (g : G) (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => g +ᵥ f x"} +{"name":"Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple","declaration":"theorem Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule_simple {X : Type u_1} {Y : Type u_2} : Function.Bijective fun xy => (xy.1, xy.2)"} +{"name":"Function.Bijective.Inv.inv.arg_a0.Bijective_rule_group","declaration":"theorem Function.Bijective.Inv.inv.arg_a0.Bijective_rule_group {X : Type u_2} {Y : Type u_1} [Group Y] (f : X → Y) (hf : Function.Bijective f) : Function.Bijective fun x => (f x)⁻¹"} +{"name":"Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_field","declaration":"theorem Function.Bijective.HSMul.hSMul.arg_a1.Bijective_rule_field {X : Type u_3} {Y : Type u_2} {R : Type u_1} [Field R] [MulAction R Y] (r : R) (f : X → Y) (hf : Function.Bijective f) (hr : r ≠ 0) : Function.Bijective fun x => r • f x"} +{"name":"Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule","declaration":"theorem Function.Bijective.Prod.mk.arg_fstsnd.Bijective_rule {X : Type u_5} {X₁ : Type u_1} {X₂ : Type u_3} {Y : Type u_2} {Z : Type u_4} (f : X₁ → Y) (g : X₂ → Z) (p₁ : X → X₁) (p₂ : X → X₂) (hf : Function.Bijective f) (hg : Function.Bijective g) (hp : Function.Bijective fun x => (p₁ x, p₂ x)) : Function.Bijective fun x => (f (p₁ x), g (p₂ x))"} +{"name":"Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_field","declaration":"theorem Function.Bijective.HMul.hMul.arg_a1.Bijective_rule_field {X : Type u_2} {Y : Type u_1} [Field Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) (hy : y ≠ 0) : Function.Bijective fun x => y * f x"} +{"name":"Function.Bijective.HAdd.hAdd.arg_a0.Bijective_rule","declaration":"theorem Function.Bijective.HAdd.hAdd.arg_a0.Bijective_rule {X : Type u_2} {Y : Type u_1} [AddGroup Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : Function.Bijective fun x => f x + y"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.CDifferentiable.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.CDifferentiable.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7f6dd62b439fef07a3076cad44541283f3b089df --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.CDifferentiable.jsonl @@ -0,0 +1,57 @@ +{"name":"SciLean.CDifferentiable.const_rule","declaration":"theorem SciLean.CDifferentiable.const_rule (K : Type u_1) [RCLike K] (X : Type u_2) [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (y : Y) : SciLean.CDifferentiable K fun x => y"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.CDifferentiable_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun x => f x + g x"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.CDifferentiable_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0a1.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) (hx : ∀ (x : X), g x ≠ 0) : SciLean.CDifferentiable K fun x => f x / g x"} +{"name":"SciLean.CDifferentiableAt.apply_rule","declaration":"theorem SciLean.CDifferentiableAt.apply_rule (K : Type u_1) [RCLike K] {ι : Type u_2} (E : ι → Type u_3) [(i : ι) → SciLean.Vec K (E i)] (i : ι) (x : (i : ι) → E i) : SciLean.CDifferentiableAt K (fun x => x i) x"} +{"name":"SciLean.CDifferentiable.apply_rule","declaration":"theorem SciLean.CDifferentiable.apply_rule (K : Type u_1) [RCLike K] {ι : Type u_2} (E : ι → Type u_3) [(i : ι) → SciLean.Vec K (E i)] (i : ι) : SciLean.CDifferentiable K fun x => x i"} +{"name":"SciLean.HSub.hSub.arg_a0a1.CDifferentiableAt_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.CDifferentiableAt K (fun x => f x - g x) x"} +{"name":"SciLean.HDiv.hDiv.arg_a0.CDifferentiableAt_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (r : K) (hf : SciLean.CDifferentiableAt K f x) (hr : r ≠ 0) : SciLean.CDifferentiableAt K (fun x => f x / r) x"} +{"name":"SciLean.CDifferentiableAt","declaration":"/-- `CDifferentiableAt f x` - conveniently differentiable function `f` at point `x`. -/\ndef SciLean.CDifferentiableAt (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (x : X) : Prop"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.CDifferentiableAt_rule","declaration":"def SciLean.HSMul.hSMul.arg_a0a1.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → K) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.CDifferentiableAt K (fun x => f x • g x) x"} +{"name":"SciLean.IndexType.sum.arg_f.CDifferentiableAt_rule","declaration":"theorem SciLean.IndexType.sum.arg_f.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (x : X) (hf : ∀ (i : ι), SciLean.CDifferentiableAt K (fun x => f x i) x) : SciLean.CDifferentiableAt K (fun x => ∑ i, f x i) x"} +{"name":"SciLean.Basis.proj.arg_x.CDifferentiableAt_rule","declaration":"theorem SciLean.Basis.proj.arg_x.CDifferentiableAt_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) (x : X) : SciLean.CDifferentiableAt K (fun x => ℼ i x) x"} +{"name":"SciLean.BasisDuality.toDual.arg_x.CDifferentiableAt_rule","declaration":"theorem SciLean.BasisDuality.toDual.arg_x.CDifferentiableAt_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (x : X) : SciLean.CDifferentiableAt K (fun x => SciLean.BasisDuality.toDual x) x"} +{"name":"SciLean.Inner.inner.arg_a0a1.CDifferentiable_rule","declaration":"theorem SciLean.Inner.inner.arg_a0a1.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable R f) (hg : SciLean.CDifferentiable R g) : SciLean.CDifferentiable R fun x => ⟪f x, g x⟫_R"} +{"name":"SciLean.Neg.neg.arg_a0.CDifferentiable_rule","declaration":"theorem SciLean.Neg.neg.arg_a0.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun x => -f x"} +{"name":"SciLean.HPow.hPow.arg_a0.CDifferentiableAt_rule","declaration":"def SciLean.HPow.hPow.arg_a0.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (n : ℕ) (x : X) (f : X → K) (hf : SciLean.CDifferentiableAt K f x) : SciLean.CDifferentiableAt K (fun x => f x ^ n) x"} +{"name":"SciLean.CDifferentiableAt.pi_rule","declaration":"theorem SciLean.CDifferentiableAt.pi_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (f : X → (i : ι) → E i) (x : X) (hf : ∀ (i : ι), SciLean.CDifferentiableAt K (fun x => f x i) x) : SciLean.CDifferentiableAt K (fun x i => f x i) x"} +{"name":"SciLean.Neg.neg.arg_a0.CDifferentiableAt_rule","declaration":"theorem SciLean.Neg.neg.arg_a0.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (hf : SciLean.CDifferentiableAt K f x) : SciLean.CDifferentiableAt K (fun x => -f x) x"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.CDifferentiableAt_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.CDifferentiableAt K (fun x => f x + g x) x"} +{"name":"SciLean.DualBasis.dualProj.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.DualBasis.dualProj.arg_x.CDifferentiable_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : SciLean.CDifferentiable K fun x => ℼ' i x"} +{"name":"SciLean.Inner.inner.arg_a0a1.CDifferentiableAt_rule","declaration":"theorem SciLean.Inner.inner.arg_a0a1.CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (x : X) (hf : SciLean.CDifferentiableAt R f x) (hg : SciLean.CDifferentiableAt R g x) : SciLean.CDifferentiableAt R (fun x => ⟪f x, g x⟫_R) x"} +{"name":"SciLean.HSMul.hSMul.arg_a1.CDifferentiable_rule_nat","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.CDifferentiable_rule_nat (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℕ) (f : X → Y) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun x => c • f x"} +{"name":"SciLean.BasisDuality.fromDual.arg_x.CDifferentiableAt_rule","declaration":"theorem SciLean.BasisDuality.fromDual.arg_x.CDifferentiableAt_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (x : X) : SciLean.CDifferentiableAt K (fun x => SciLean.BasisDuality.fromDual x) x"} +{"name":"SciLean.HSub.hSub.arg_a0a1.CDifferentiable_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun x => f x - g x"} +{"name":"SciLean.Prod.mk.arg_fstsnd.CDifferentiableAt_rule","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (x : X) (g : X → Y) (hg : SciLean.CDifferentiableAt K g x) (f : X → Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.CDifferentiableAt K (fun x => (g x, f x)) x"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.CDifferentiable_rule","declaration":"def SciLean.HSMul.hSMul.arg_a0a1.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → K) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun x => f x • g x"} +{"name":"SciLean.SciLean.norm₂.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.SciLean.norm₂.arg_x.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] {x : X} (f : X → Y) (hf : SciLean.CDifferentiable R f) (hx : f x ≠ 0) : SciLean.CDifferentiable R fun x => ‖f x‖₂[R]"} +{"name":"SciLean.Prod.fst.arg_self.CDifferentiable_rule","declaration":"theorem SciLean.Prod.fst.arg_self.CDifferentiable_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun x => (f x).1"} +{"name":"SciLean.HSMul.hSMul.arg_a1.CDifferentiable_rule_int","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.CDifferentiable_rule_int (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℤ) (f : X → Y) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun x => c • f x"} +{"name":"SciLean.HMul.hMul.arg_a0a1.CDifferentiableAt_rule","declaration":"def SciLean.HMul.hMul.arg_a0a1.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.CDifferentiableAt K (fun x => f x * g x) x"} +{"name":"SciLean.CDifferentiable","declaration":"/-- `CDifferentiable f` - conveniently differentiable function `f`. -/\ndef SciLean.CDifferentiable (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) : Prop"} +{"name":"SciLean.Inv.inv.arg_a0.CDifferentiableAt_rule","declaration":"def SciLean.Inv.inv.arg_a0.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (hf : SciLean.CDifferentiableAt K f x) (hf' : f x ≠ 0) : SciLean.CDifferentiableAt K (fun x => (f x)⁻¹) x"} +{"name":"SciLean.HDiv.hDiv.arg_a0.CDifferentiable_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (r : K) (hf : SciLean.CDifferentiable K f) (hr : r ≠ 0) : SciLean.CDifferentiable K fun x => f x / r"} +{"name":"SciLean.Basis.proj.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Basis.proj.arg_x.CDifferentiable_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : SciLean.CDifferentiable K fun x => ℼ i x"} +{"name":"SciLean.Prod.mk.arg_fstsnd.CDifferentiable_rule","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (g : X → Y) (hg : SciLean.CDifferentiable K g) (f : X → Z) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun x => (g x, f x)"} +{"name":"SciLean.SciLean.norm₂.arg_x.CDifferentiableAt_rule","declaration":"theorem SciLean.SciLean.norm₂.arg_x.CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.CDifferentiableAt R f x) (hx : f x ≠ 0) : SciLean.CDifferentiableAt R (fun x => ‖f x‖₂[R]) x"} +{"name":"SciLean.DualBasis.dualProj.arg_x.CDifferentiableAt_rule","declaration":"theorem SciLean.DualBasis.dualProj.arg_x.CDifferentiableAt_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) (x : X) : SciLean.CDifferentiableAt K (fun x => ℼ' i x) x"} +{"name":"SciLean.CDifferentiableAt.cdifferentiable_rule","declaration":"theorem SciLean.CDifferentiableAt.cdifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (x : X) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiableAt K f x"} +{"name":"SciLean.HMul.hMul.arg_a0a1.CDifferentiable_rule","declaration":"def SciLean.HMul.hMul.arg_a0a1.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun x => f x * g x"} +{"name":"SciLean.HSMul.hSMul.arg_a1.CDifferentiableAt_rule_nat","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.CDifferentiableAt_rule_nat (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℕ) (f : X → Y) (x : X) (hf : SciLean.CDifferentiableAt K f x) : SciLean.CDifferentiableAt K (fun x => c • f x) x"} +{"name":"SciLean.CDifferentiableAt.const_rule","declaration":"theorem SciLean.CDifferentiableAt.const_rule (K : Type u_1) [RCLike K] (X : Type u_2) [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (y : Y) (x : X) : SciLean.CDifferentiableAt K (fun x => y) x"} +{"name":"SciLean.BasisDuality.fromDual.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.BasisDuality.fromDual.arg_x.CDifferentiable_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : SciLean.CDifferentiable K fun x => SciLean.BasisDuality.fromDual x"} +{"name":"SciLean.IndexType.sum.arg_f.CDifferentiable_rule","declaration":"theorem SciLean.IndexType.sum.arg_f.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.CDifferentiable K fun x => f x i) : SciLean.CDifferentiable K fun x => ∑ i, f x i"} +{"name":"SciLean.Inv.inv.arg_a0.CDifferentiable_rule","declaration":"def SciLean.Inv.inv.arg_a0.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (hf : SciLean.CDifferentiable K f) (hf' : ∀ (x : X), f x ≠ 0) : SciLean.CDifferentiable K fun x => (f x)⁻¹"} +{"name":"SciLean.Prod.snd.arg_self.CDifferentiableAt_rule","declaration":"theorem SciLean.Prod.snd.arg_self.CDifferentiableAt_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.CDifferentiableAt K (fun x => (f x).2) x"} +{"name":"SciLean.HPow.hPow.arg_a0.CDifferentiable_rule","declaration":"def SciLean.HPow.hPow.arg_a0.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {n : ℕ} (f : X → K) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun x => f x ^ n"} +{"name":"SciLean.SciLean.Norm2.norm2.arg_a0.CDifferentiable_rule","declaration":"theorem SciLean.SciLean.Norm2.norm2.arg_a0.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.CDifferentiable R f) : SciLean.CDifferentiable R fun x => ‖f x‖₂²"} +{"name":"SciLean.CDifferentiable.comp_rule","declaration":"theorem SciLean.CDifferentiable.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun x => f (g x)"} +{"name":"SciLean.CDifferentiable.id_rule","declaration":"theorem SciLean.CDifferentiable.id_rule (K : Type u_1) [RCLike K] (X : Type u_2) [SciLean.Vec K X] : SciLean.CDifferentiable K fun x => x"} +{"name":"SciLean.Prod.snd.arg_self.CDifferentiable_rule","declaration":"theorem SciLean.Prod.snd.arg_self.CDifferentiable_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun x => (f x).2"} +{"name":"SciLean.HSMul.hSMul.arg_a1.CDifferentiableAt_rule_int","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.CDifferentiableAt_rule_int (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℤ) (f : X → Y) (x : X) (hf : SciLean.CDifferentiableAt K f x) : SciLean.CDifferentiableAt K (fun x => c • f x) x"} +{"name":"SciLean.Prod.fst.arg_self.CDifferentiableAt_rule","declaration":"theorem SciLean.Prod.fst.arg_self.CDifferentiableAt_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.CDifferentiableAt K (fun x => (f x).1) x"} +{"name":"SciLean.CDifferentiableAt.id_rule","declaration":"theorem SciLean.CDifferentiableAt.id_rule (K : Type u_1) [RCLike K] (X : Type u_2) [SciLean.Vec K X] (x : X) : SciLean.CDifferentiableAt K (fun x => x) x"} +{"name":"SciLean.BasisDuality.toDual.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.BasisDuality.toDual.arg_x.CDifferentiable_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : SciLean.CDifferentiable K fun x => SciLean.BasisDuality.toDual x"} +{"name":"SciLean.SciLean.Norm2.norm2.arg_a0.CDifferentiableAt_rule","declaration":"theorem SciLean.SciLean.Norm2.norm2.arg_a0.CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.CDifferentiableAt R f x) : SciLean.CDifferentiableAt R (fun x => ‖f x‖₂²) x"} +{"name":"SciLean.CDifferentiableAt.comp_rule","declaration":"theorem SciLean.CDifferentiableAt.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (x : X) (hf : SciLean.CDifferentiableAt K f (g x)) (hg : SciLean.CDifferentiableAt K g x) : SciLean.CDifferentiableAt K (fun x => f (g x)) x"} +{"name":"SciLean.CDifferentiable.pi_rule","declaration":"theorem SciLean.CDifferentiable.pi_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.CDifferentiable K fun x => f x i) : SciLean.CDifferentiable K fun x i => f x i"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.CDifferentiableAt_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0a1.CDifferentiableAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) (hx : g x ≠ 0) : SciLean.CDifferentiableAt K (fun x => f x / g x) x"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.ContCDiff.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.ContCDiff.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b3610b34c4d09233098859c84c3d2d86ff6beecb --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.ContCDiff.jsonl @@ -0,0 +1,61 @@ +{"name":"SciLean.Prod.fst.arg_self.ContCDiffAt_rule","declaration":"theorem SciLean.Prod.fst.arg_self.ContCDiffAt_rule (n : ℕ∞) (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.ContCDiffAt K n f x) : SciLean.ContCDiffAt K n (fun x => (f x).1) x"} +{"name":"SciLean.HSMul.hSMul.arg_a1.ContCDiffAt_rule_nat","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.ContCDiffAt_rule_nat (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℕ) (f : X → Y) (x : X) (hf : SciLean.ContCDiffAt K n f x) : SciLean.ContCDiffAt K n (fun x => c • f x) x"} +{"name":"SciLean.HMul.hMul.arg_a0a1.ContCDiffAt_rule","declaration":"def SciLean.HMul.hMul.arg_a0a1.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.ContCDiffAt K n f x) (hg : SciLean.ContCDiffAt K n g x) : SciLean.ContCDiffAt K n (fun x => f x * g x) x"} +{"name":"SciLean.HDiv.hDiv.arg_a0.ContCDiff_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (r : K) (hf : SciLean.ContCDiff K n f) (hr : r ≠ 0) : SciLean.ContCDiff K n fun x => f x / r"} +{"name":"SciLean.BasisDuality.toDual.arg_x.ContCDiffAt_rule","declaration":"theorem SciLean.BasisDuality.toDual.arg_x.ContCDiffAt_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (x : X) : SciLean.ContCDiffAt K n (fun x => SciLean.BasisDuality.toDual x) x"} +{"name":"SciLean.HPow.hPow.arg_a0.ContCDiffAt_rule","declaration":"def SciLean.HPow.hPow.arg_a0.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (m : ℕ) (x : X) (f : X → K) (hf : SciLean.ContCDiffAt K n f x) : SciLean.ContCDiffAt K n (fun x => f x ^ m) x"} +{"name":"SciLean.ContCDiffAt.apply_rule","declaration":"theorem SciLean.ContCDiffAt.apply_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (i : ι) (x : (i : ι) → E i) : SciLean.ContCDiffAt K n (fun x => x i) x"} +{"name":"SciLean.CDifferentaible.ContCDiff_rule","declaration":"theorem SciLean.CDifferentaible.ContCDiff_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.ContCDiff K n f) (h : 0 < n) : SciLean.CDifferentiable K f"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.ContCDiff_rule","declaration":"def SciLean.HSMul.hSMul.arg_a0a1.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → K) (g : X → Y) (hf : SciLean.ContCDiff K n f) (hg : SciLean.ContCDiff K n g) : SciLean.ContCDiff K n fun x => f x • g x"} +{"name":"SciLean.HSub.hSub.arg_a0a1.ContCDiff_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.ContCDiff K n f) (hg : SciLean.ContCDiff K n g) : SciLean.ContCDiff K n fun x => f x - g x"} +{"name":"SciLean.HSub.hSub.arg_a0a1.ContCDiffAt_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.ContCDiffAt K n f x) (hg : SciLean.ContCDiffAt K n g x) : SciLean.ContCDiffAt K n (fun x => f x - g x) x"} +{"name":"SciLean.BasisDuality.fromDual.arg_x.ContCDiff_rule","declaration":"theorem SciLean.BasisDuality.fromDual.arg_x.ContCDiff_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : SciLean.ContCDiff K n fun x => SciLean.BasisDuality.fromDual x"} +{"name":"SciLean.CDifferentaibleAt.ContCDiffAt_rule","declaration":"theorem SciLean.CDifferentaibleAt.ContCDiffAt_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (hf : SciLean.ContCDiffAt K n f x) (h : 0 < n) : SciLean.CDifferentiableAt K f x"} +{"name":"SciLean.ContCDiff.comp_rule","declaration":"theorem SciLean.ContCDiff.comp_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.ContCDiff K n f) (hg : SciLean.ContCDiff K n g) : SciLean.ContCDiff K n fun x => f (g x)"} +{"name":"SciLean.HSMul.hSMul.arg_a1.ContCDiffAt_rule_int","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.ContCDiffAt_rule_int (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℤ) (f : X → Y) (x : X) (hf : SciLean.ContCDiffAt K n f x) : SciLean.ContCDiffAt K n (fun x => c • f x) x"} +{"name":"SciLean.Prod.snd.arg_self.ContCDiff_rule","declaration":"theorem SciLean.Prod.snd.arg_self.ContCDiff_rule (n : ℕ∞) (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiff K n fun x => (f x).2"} +{"name":"SciLean.ContCDiffAt.pi_rule","declaration":"theorem SciLean.ContCDiffAt.pi_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (x : X) (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.ContCDiffAt K n (fun x => f x i) x) : SciLean.ContCDiffAt K n (fun x i => f x i) x"} +{"name":"SciLean.CDifferentaible.ContCDiff_rule'","declaration":"theorem SciLean.CDifferentaible.ContCDiff_rule' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.ContCDiff K ⊤ f) : SciLean.CDifferentiable K f"} +{"name":"SciLean.Prod.mk.arg_fstsnd.ContCDiff_rule","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (g : X → Y) (hg : SciLean.ContCDiff K n g) (f : X → Z) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiff K n fun x => (g x, f x)"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.ContCDiffAt_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.ContCDiffAt K n f x) (hg : SciLean.ContCDiffAt K n g x) : SciLean.ContCDiffAt K n (fun x => f x + g x) x"} +{"name":"SciLean.HSMul.hSMul.arg_a1.ContCDiff_rule_n","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.ContCDiff_rule_n (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℕ) (f : X → Y) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiff K n fun x => c • f x"} +{"name":"SciLean.Norm2.norm2.arg_a0.ContCDiff_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.ContCDiff_rule (n : ℕ∞) {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.ContCDiff R n f) : SciLean.ContCDiff R n fun x => ‖f x‖₂²"} +{"name":"SciLean.Inner.inner.arg_a0a1.ContCDiff_rule","declaration":"theorem SciLean.Inner.inner.arg_a0a1.ContCDiff_rule (n : ℕ∞) {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.ContCDiff R n f) (hg : SciLean.ContCDiff R n g) : SciLean.ContCDiff R n fun x => ⟪f x, g x⟫_R"} +{"name":"SciLean.ContCDiffAt.const_rule","declaration":"theorem SciLean.ContCDiffAt.const_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (y : Y) (x : X) : SciLean.ContCDiffAt K n (fun x => y) x"} +{"name":"SciLean.HMul.hMul.arg_a0a1.ContCDiff_rule","declaration":"def SciLean.HMul.hMul.arg_a0a1.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.ContCDiff K n f) (hg : SciLean.ContCDiff K n g) : SciLean.ContCDiff K n fun x => f x * g x"} +{"name":"SciLean.Neg.neg.arg_a0.ContCDiffAt_rule","declaration":"theorem SciLean.Neg.neg.arg_a0.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (hf : SciLean.ContCDiffAt K n f x) : SciLean.ContCDiffAt K n (fun x => -f x) x"} +{"name":"SciLean.DualBasis.dualProj.arg_x.ContCDiff_rule","declaration":"theorem SciLean.DualBasis.dualProj.arg_x.ContCDiff_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : SciLean.ContCDiff K n fun x => ℼ' i x"} +{"name":"SciLean.Neg.neg.arg_a0.ContCDiff_rule","declaration":"theorem SciLean.Neg.neg.arg_a0.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiff K n fun x => -f x"} +{"name":"SciLean.IndexType.sum.arg_f.ContCDiffAt_rule","declaration":"theorem SciLean.IndexType.sum.arg_f.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (x : X) (hf : ∀ (i : ι), SciLean.ContCDiffAt K n (fun x => f x i) x) : SciLean.ContCDiffAt K n (fun x => ∑ i, f x i) x"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.ContCDiff_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.ContCDiff K n f) (hg : SciLean.ContCDiff K n g) : SciLean.ContCDiff K n fun x => f x + g x"} +{"name":"SciLean.HSMul.hSMul.arg_a1.ContCDiff_rule_int","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.ContCDiff_rule_int (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℤ) (f : X → Y) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiff K n fun x => c • f x"} +{"name":"SciLean.ContCDiff.ContCDiffAt_rule","declaration":"theorem SciLean.ContCDiff.ContCDiffAt_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiffAt K n f x"} +{"name":"SciLean.ContCDiff.const_rule","declaration":"theorem SciLean.ContCDiff.const_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (y : Y) : SciLean.ContCDiff K n fun x => y"} +{"name":"SciLean.Basis.proj.arg_x.ContCDiffAt_rule","declaration":"theorem SciLean.Basis.proj.arg_x.ContCDiffAt_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) (x : X) : SciLean.ContCDiffAt K n (fun x => ℼ i x) x"} +{"name":"SciLean.ContCDiff.id_rule","declaration":"theorem SciLean.ContCDiff.id_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] : SciLean.ContCDiff K n fun x => x"} +{"name":"SciLean.norm₂.arg_x.ContCDiffAt_rule","declaration":"theorem SciLean.norm₂.arg_x.ContCDiffAt_rule (n : ℕ∞) {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.ContCDiffAt R n f x) (hx : f x ≠ 0) : SciLean.ContCDiffAt R n (fun x => ‖f x‖₂[R]) x"} +{"name":"SciLean.Prod.mk.arg_fstsnd.ContCDiffAt_rule","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (x : X) (g : X → Y) (hg : SciLean.ContCDiffAt K n g x) (f : X → Z) (hf : SciLean.ContCDiffAt K n f x) : SciLean.ContCDiffAt K n (fun x => (g x, f x)) x"} +{"name":"SciLean.ContCDiff.le_rule","declaration":"theorem SciLean.ContCDiff.le_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {m : ℕ∞} (f : X → Y) (hf : SciLean.ContCDiff K m f) (h : n ≤ m) : SciLean.ContCDiff K n f"} +{"name":"SciLean.HPow.hPow.arg_a0.ContCDiff_rule","declaration":"def SciLean.HPow.hPow.arg_a0.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (m : ℕ) (f : X → K) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiff K n fun x => f x ^ m"} +{"name":"SciLean.Inner.inner.arg_a0a1.ContCDiffAt_rule","declaration":"theorem SciLean.Inner.inner.arg_a0a1.ContCDiffAt_rule (n : ℕ∞) {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (x : X) (hf : SciLean.ContCDiffAt R n f x) (hg : SciLean.ContCDiffAt R n g x) : SciLean.ContCDiffAt R n (fun x => ⟪f x, g x⟫_R) x"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.ContCDiffAt_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0a1.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.ContCDiffAt K n f x) (hg : SciLean.ContCDiffAt K n g x) (hx : g x ≠ 0) : SciLean.ContCDiffAt K n (fun x => f x / g x) x"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.ContCDiffAt_rule","declaration":"def SciLean.HSMul.hSMul.arg_a0a1.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → K) (g : X → Y) (hf : SciLean.ContCDiffAt K n f x) (hg : SciLean.ContCDiffAt K n g x) : SciLean.ContCDiffAt K n (fun x => f x • g x) x"} +{"name":"SciLean.Prod.fst.arg_self.ContCDiff_rule","declaration":"theorem SciLean.Prod.fst.arg_self.ContCDiff_rule (n : ℕ∞) (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.ContCDiff K n f) : SciLean.ContCDiff K n fun x => (f x).1"} +{"name":"SciLean.ContCDiff.pi_rule","declaration":"theorem SciLean.ContCDiff.pi_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (x : X) (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.ContCDiff K n fun x => f x i) : SciLean.ContCDiff K n fun x i => f x i"} +{"name":"SciLean.norm₂.arg_x.ContCDiff_rule","declaration":"theorem SciLean.norm₂.arg_x.ContCDiff_rule (n : ℕ∞) {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.ContCDiff R n f) (hx : ∀ (x : X), f x ≠ 0) : SciLean.ContCDiff R n fun x => ‖f x‖₂[R]"} +{"name":"SciLean.Prod.snd.arg_self.ContCDiffAt_rule","declaration":"theorem SciLean.Prod.snd.arg_self.ContCDiffAt_rule (n : ℕ∞) (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.ContCDiffAt K n f x) : SciLean.ContCDiffAt K n (fun x => (f x).2) x"} +{"name":"SciLean.ContCDiffAt.id_rule","declaration":"theorem SciLean.ContCDiffAt.id_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] (x : X) : SciLean.ContCDiffAt K n (fun x => x) x"} +{"name":"SciLean.ContCDiffAt.le_rule","declaration":"theorem SciLean.ContCDiffAt.le_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {m : ℕ∞} (x : X) (f : X → Y) (hf : SciLean.ContCDiffAt K m f x) (h : n ≤ m) : SciLean.ContCDiffAt K n f x"} +{"name":"SciLean.Basis.proj.arg_x.ContCDiff_rule","declaration":"theorem SciLean.Basis.proj.arg_x.ContCDiff_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : SciLean.ContCDiff K n fun x => ℼ i x"} +{"name":"SciLean.ContCDiffAt.comp_rule","declaration":"theorem SciLean.ContCDiffAt.comp_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (x : X) (f : Y → Z) (g : X → Y) (hf : SciLean.ContCDiffAt K n f (g x)) (hg : SciLean.ContCDiffAt K n g x) : SciLean.ContCDiffAt K n (fun x => f (g x)) x"} +{"name":"SciLean.ContCDiff","declaration":"/-- `ContCDiff f` - conveniently differentiable function `f`. -/\ndef SciLean.ContCDiff (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) : Prop"} +{"name":"SciLean.DualBasis.dualProj.arg_x.ContCDiffAt_rule","declaration":"theorem SciLean.DualBasis.dualProj.arg_x.ContCDiffAt_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) (x : X) : SciLean.ContCDiffAt K n (fun x => ℼ' i x) x"} +{"name":"SciLean.Norm2.norm2.arg_a0.ContCDiffAt_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.ContCDiffAt_rule (n : ℕ∞) {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.ContCDiffAt R n f x) : SciLean.ContCDiffAt R n (fun x => ‖f x‖₂²) x"} +{"name":"SciLean.ContCDiffAt","declaration":"/-- `CDifferentiableAt f x` - conveniently differentiable function `f` at point `x`. -/\ndef SciLean.ContCDiffAt (K : Type u_1) [RCLike K] (n : ℕ∞) {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (x : X) : Prop"} +{"name":"SciLean.IndexType.sum.arg_f.ContCDiff_rule","declaration":"theorem SciLean.IndexType.sum.arg_f.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.ContCDiff K n fun x => f x i) : SciLean.ContCDiff K n fun x => ∑ i, f x i"} +{"name":"SciLean.CDifferentaibleAt.ContCDiffAt_rule'","declaration":"theorem SciLean.CDifferentaibleAt.ContCDiffAt_rule' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (hf : SciLean.ContCDiffAt K ⊤ f x) : SciLean.CDifferentiableAt K f x"} +{"name":"SciLean.BasisDuality.toDual.arg_x.ContCDiff_rule","declaration":"theorem SciLean.BasisDuality.toDual.arg_x.ContCDiff_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : SciLean.ContCDiff K n fun x => SciLean.BasisDuality.toDual x"} +{"name":"SciLean.HDiv.hDiv.arg_a0.ContCDiffAt_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0.ContCDiffAt_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (r : K) (hf : SciLean.ContCDiffAt K n f x) (hr : r ≠ 0) : SciLean.ContCDiffAt K n (fun x => f x / r) x"} +{"name":"SciLean.BasisDuality.fromDual.arg_x.ContCDiffAt_rule","declaration":"theorem SciLean.BasisDuality.fromDual.arg_x.ContCDiffAt_rule (n : ℕ∞) {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (x : X) : SciLean.ContCDiffAt K n (fun x => SciLean.BasisDuality.fromDual x) x"} +{"name":"SciLean.ContCDiff.apply_rule","declaration":"theorem SciLean.ContCDiff.apply_rule (K : Type u_1) [RCLike K] (n : ℕ∞) {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (i : ι) : SciLean.ContCDiff K n fun x => x i"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.ContCDiff_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0a1.ContCDiff_rule (n : ℕ∞) (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.ContCDiff K n f) (hg : SciLean.ContCDiff K n g) (hx : ∀ (x : X), g x ≠ 0) : SciLean.ContCDiff K n fun x => f x / g x"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.Diffeomorphism.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.Diffeomorphism.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..00c1901e84f323df3ebb113b1a0fbf75223fc8be --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.Diffeomorphism.jsonl @@ -0,0 +1,26 @@ +{"name":"SciLean.Diffeomorphism.bijective","declaration":"def SciLean.Diffeomorphism.bijective {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {f : X → Y} (self : SciLean.Diffeomorphism K f) : Function.Bijective f"} +{"name":"Function.invFun.arg_a1.cderiv_rule","declaration":"theorem Function.invFun.arg_a1.cderiv_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Z) (hf : SciLean.Diffeomorphism K f) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun x => Function.invFun f (g x)) = fun x dx =>\n let z := g x;\n let dz := SciLean.cderiv K g x dx;\n let y := Function.invFun f z;\n let dy := Function.invFun (SciLean.cderiv K f y) dz;\n dy"} +{"name":"Neg.neg.arg_a0.Diffeomorphism_rule","declaration":"theorem Neg.neg.arg_a0.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => -f x"} +{"name":"SciLean.Diffeomorphism.mk","declaration":"ctor SciLean.Diffeomorphism.mk {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {f : X → Y} (bijective : Function.Bijective f) (differentiable : SciLean.CDifferentiable K f) (locally_diffeo : ∀ (x : X), Function.Bijective (SciLean.cderiv K f x)) : SciLean.Diffeomorphism K f"} +{"name":"HSub.hSub.arg_a0.Diffeomorphism_rule","declaration":"theorem HSub.hSub.arg_a0.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (y : Y) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => f x - y"} +{"name":"HSMul.hSMul.arg_a1.Diffeomorphism_rule","declaration":"def HSMul.hSMul.arg_a1.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (r : K) (f : X → Y) (hr : r ≠ 0) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => r • f x"} +{"name":"HMul.hMul.arg_a0.Diffeomorphism_rule","declaration":"def HMul.hMul.arg_a0.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (y : K) (hf : SciLean.Diffeomorphism K f) (hy : y ≠ 0) : SciLean.Diffeomorphism K fun x => f x * y"} +{"name":"SciLean.Diffeomorphism.id_rule","declaration":"theorem SciLean.Diffeomorphism.id_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] : SciLean.Diffeomorphism K fun x => x"} +{"name":"Function.invFun.arg_f.cderiv_rule'","declaration":"theorem Function.invFun.arg_f.cderiv_rule' (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : X → Y → Z) (z : Z) (hf : ∀ (x : X), SciLean.Diffeomorphism K (f x)) (hf' : SciLean.CDifferentiable K fun xy => f xy.1 xy.2) : (SciLean.cderiv K fun x => Function.invFun (f x) z) = fun x dx =>\n let y := Function.invFun (f x) z;\n let dfdx_y := SciLean.cderiv K f x dx y;\n let df'dy := SciLean.cderiv K (Function.invFun (f x)) (f x y) dfdx_y;\n -df'dy"} +{"name":"Function.invFun.arg_f_a1.cderiv_rule","declaration":"theorem Function.invFun.arg_f_a1.cderiv_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : X → Y → Z) (hf : ∀ (x : X), SciLean.Diffeomorphism K (f x)) (hf' : SciLean.CDifferentiable K fun xy => f xy.1 xy.2) : (SciLean.cderiv K fun x z => Function.invFun (f x) z) = fun x dx z =>\n let y := Function.invFun (f x) z;\n let dfdx_y := SciLean.cderiv K f x dx y;\n let df'dy := SciLean.cderiv K (Function.invFun (f x)) (f x y) dfdx_y;\n -df'dy"} +{"name":"SciLean.Diffeomorphism.comp_rule","declaration":"theorem SciLean.Diffeomorphism.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.Diffeomorphism K f) (hg : SciLean.Diffeomorphism K g) : SciLean.Diffeomorphism K fun x => f (g x)"} +{"name":"id.arg_a.Diffeomorphism_rule","declaration":"theorem id.arg_a.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] : SciLean.Diffeomorphism K fun x => id x"} +{"name":"SciLean.Diffeomorphism","declaration":"structure SciLean.Diffeomorphism (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) : Prop"} +{"name":"Nat.iterate.arg_a1.Diffeomorphism_rule","declaration":"def Nat.iterate.arg_a1.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (n : ℕ) (f : X → X) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => f^[n] x"} +{"name":"Function.invFun.arg_fa1.CDifferentiable_rule","declaration":"theorem Function.invFun.arg_fa1.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] {W : Type u_5} [SciLean.Vec K W] (f : X → Y → Z) (g : W → X) (h : W → Z) (hf : ∀ (x : X), SciLean.Diffeomorphism K (f x)) (hf' : SciLean.CDifferentiable K fun xy => f xy.1 xy.2) (hg : SciLean.CDifferentiable K g) (hh : SciLean.CDifferentiable K h) : SciLean.CDifferentiable K fun w => Function.invFun (f (g w)) (h w)"} +{"name":"Function.comp.arg_a0.Diffeomorphism_rule","declaration":"theorem Function.comp.arg_a0.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.Diffeomorphism K f) (hg : SciLean.Diffeomorphism K g) : SciLean.Diffeomorphism K fun x => (f ∘ g) x"} +{"name":"HAdd.hAdd.arg_a0.Diffeomorphism_rule","declaration":"theorem HAdd.hAdd.arg_a0.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (y : Y) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => f x + y"} +{"name":"SciLean.Diffeomorphism.differentiable","declaration":"def SciLean.Diffeomorphism.differentiable {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {f : X → Y} (self : SciLean.Diffeomorphism K f) : SciLean.CDifferentiable K f"} +{"name":"SciLean.Diffeomorphism.locally_diffeo","declaration":"def SciLean.Diffeomorphism.locally_diffeo {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {f : X → Y} (self : SciLean.Diffeomorphism K f) (x : X) : Function.Bijective (SciLean.cderiv K f x)"} +{"name":"HMul.hMul.arg_a1.Diffeomorphism_rule","declaration":"def HMul.hMul.arg_a1.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (y : K) (f : X → K) (hy : y ≠ 0) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => y * f x"} +{"name":"SciLean.diffeomorphism_to_differentiables","declaration":"theorem SciLean.diffeomorphism_to_differentiables (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.Diffeomorphism K f) : SciLean.CDifferentiable K f"} +{"name":"HSub.hSub.arg_a1.Diffeomorphism_rule","declaration":"theorem HSub.hSub.arg_a1.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (y : Y) (f : X → Y) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => y - f x"} +{"name":"HAdd.hAdd.arg_a1.Diffeomorphism_rule","declaration":"theorem HAdd.hAdd.arg_a1.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (y : Y) (f : X → Y) (hf : SciLean.Diffeomorphism K f) : SciLean.Diffeomorphism K fun x => y + f x"} +{"name":"SciLean.SciLean.cderiv.arg_dx.Bijective_rule","declaration":"theorem SciLean.SciLean.cderiv.arg_dx.Bijective_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (x : X) (hf : SciLean.Diffeomorphism K f) : Function.Bijective fun dx => SciLean.cderiv K f x dx"} +{"name":"HDiv.hDiv.arg_a0.Diffeomorphism_rule","declaration":"def HDiv.hDiv.arg_a0.Diffeomorphism_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (r : K) (hf : SciLean.Diffeomorphism K f) (hr : r ≠ 0) : SciLean.Diffeomorphism K fun x => f x / r"} +{"name":"SciLean.diffeomorphism_to_bijective","declaration":"theorem SciLean.diffeomorphism_to_bijective (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.Diffeomorphism K f) : Function.Bijective f"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.Differentiable.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.Differentiable.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.HasAdjDiff.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.HasAdjDiff.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7400a1f3155d5a82d5d23460ac4b13e014fc3883 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.HasAdjDiff.jsonl @@ -0,0 +1,60 @@ +{"name":"SciLean.SciLean.norm₂.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.SciLean.norm₂.arg_x.HasAdjDiff_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hfz : ∀ (x : X), f x ≠ 0) : SciLean.HasAdjDiff R fun x => ‖f x‖₂[R]"} +{"name":"SciLean.HasAdjDiffAt.comp_rule","declaration":"theorem SciLean.HasAdjDiffAt.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt K f (g x)) (hg : SciLean.HasAdjDiffAt K g x) : SciLean.HasAdjDiffAt K (fun x => f (g x)) x"} +{"name":"SciLean.SciLean.semiAdjoint.arg_y.HasAdjDiff_rule","declaration":"theorem SciLean.SciLean.semiAdjoint.arg_y.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {W : Type u_4} [SciLean.SemiInnerProductSpace K W] (f : X → Y) (a0 : W → Y) (hf : SciLean.CDifferentiable K f) (ha0 : SciLean.HasAdjDiff K a0) : SciLean.HasAdjDiff K fun w => SciLean.semiAdjoint K f (a0 w)"} +{"name":"SciLean.Prod.fst.arg_self.HasAdjDiffAt_rule","declaration":"theorem SciLean.Prod.fst.arg_self.HasAdjDiffAt_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (x : X) (f : X → Y × Z) (hf : SciLean.HasAdjDiffAt K f x) : SciLean.HasAdjDiffAt K (fun x => (f x).1) x"} +{"name":"SciLean.HasSemiAdjoint.hasAdjDiff","declaration":"theorem SciLean.HasSemiAdjoint.hasAdjDiff (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasAdjDiff K f"} +{"name":"SciLean.HasAdjDiffAt.apply_rule","declaration":"theorem SciLean.HasAdjDiffAt.apply_rule (K : Type u_1) [RCLike K] {ι : Type u_2} [SciLean.IndexType ι] [DecidableEq ι] {E : ι → Type u_3} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] (x : (i : ι) → E i) (i : ι) : SciLean.HasAdjDiffAt K (fun x => x i) x"} +{"name":"SciLean.IndexType.sum.arg_f.HasAdjDiffAt_rule","declaration":"theorem SciLean.IndexType.sum.arg_f.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_4} [SciLean.IndexType ι] (x : X) (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiffAt K (fun x => f x i) x) : SciLean.HasAdjDiffAt K (fun x => ∑ i, f x i) x"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.HasAdjDiff_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0a1.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : ∀ (x : X), g x ≠ 0) : SciLean.HasAdjDiff K fun x => f x / g x"} +{"name":"SciLean.Inner.inner.arg_a0a1.HasAdjDiff_rule","declaration":"theorem SciLean.Inner.inner.arg_a0a1.HasAdjDiff_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : SciLean.HasAdjDiff R fun x => ⟪f x, g x⟫_R"} +{"name":"SciLean.HasAdjDiff.cdifferentiable","declaration":"theorem SciLean.HasAdjDiff.cdifferentiable (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasAdjDiff K f) : SciLean.CDifferentiable K f"} +{"name":"SciLean.HSMul.hSMul.arg_a1.HasAdjDiffAt_rule","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : K) (g : X → Y) (x : X) (hg : SciLean.HasAdjDiffAt K g x) : SciLean.HasAdjDiffAt K (fun x => c • g x) x"} +{"name":"SciLean.HasAdjDiff.apply_rule","declaration":"theorem SciLean.HasAdjDiff.apply_rule (K : Type u_1) [RCLike K] {ι : Type u_2} [SciLean.IndexType ι] [DecidableEq ι] {E : ι → Type u_3} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] (i : ι) : SciLean.HasAdjDiff K fun x => x i"} +{"name":"SciLean.HasAjdDiff.const_rule","declaration":"theorem SciLean.HasAjdDiff.const_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (y : Y) : SciLean.HasAdjDiff K fun x => y"} +{"name":"SciLean.SciLean.semiAdjoint.arg_y.CDifferentiable_rule","declaration":"theorem SciLean.SciLean.semiAdjoint.arg_y.CDifferentiable_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_4} [SciLean.SemiInnerProductSpace K Y] {W : Type u_1} [SciLean.Vec K W] (f : X → Y) (a0 : W → Y) (hf : SciLean.CDifferentiable K f) (ha0 : SciLean.CDifferentiable K a0) : SciLean.CDifferentiable K fun w => SciLean.semiAdjoint K f (a0 w)"} +{"name":"SciLean.HasAdjDiff.pi_rule","declaration":"theorem SciLean.HasAdjDiff.pi_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {ι : Type u_4} [SciLean.IndexType ι] {E : ι → Type u_3} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : SciLean.HasAdjDiff K fun x i => f x i"} +{"name":"SciLean.HasAdjDiff.hasAdjDiffAt","declaration":"theorem SciLean.HasAdjDiff.hasAdjDiffAt (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiff K f) : SciLean.HasAdjDiffAt K f x"} +{"name":"SciLean.SciLean.norm₂.arg_x.HasAdjDiffAt_rule","declaration":"theorem SciLean.SciLean.norm₂.arg_x.HasAdjDiffAt_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (x : X) (f : X → Y) (hf : SciLean.HasAdjDiffAt R f x) (hfz : f x ≠ 0) : SciLean.HasAdjDiffAt R (fun x => ‖f x‖₂[R]) x"} +{"name":"SciLean.Prod.mk.arg_fstsnd.HasAdjDiffAt_rule","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (x : X) (g : X → Y) (hg : SciLean.HasAdjDiffAt K g x) (f : X → Z) (hf : SciLean.HasAdjDiffAt K f x) : SciLean.HasAdjDiffAt K (fun x => (g x, f x)) x"} +{"name":"SciLean.ite.arg_te.HasAdjDiff_rule","declaration":"theorem SciLean.ite.arg_te.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) (ht : SciLean.HasAdjDiff K t) (he : SciLean.HasAdjDiff K e) : SciLean.HasAdjDiff K fun x => if c then t x else e x"} +{"name":"SciLean.HasAdjDiffAt.pi_rule","declaration":"theorem SciLean.HasAdjDiffAt.pi_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {ι : Type u_4} [SciLean.IndexType ι] {E : ι → Type u_3} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] (f : X → (i : ι) → E i) (x : X) (hf : ∀ (i : ι), SciLean.HasAdjDiffAt K (fun x => f x i) x) : SciLean.HasAdjDiffAt K (fun x i => f x i) x"} +{"name":"SciLean.HMul.hMul.arg_a0a1.HasAdjDiff_rule","declaration":"def SciLean.HMul.hMul.arg_a0a1.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiff K fun x => f x * g x"} +{"name":"SciLean.SciLean.semiAdjoint.arg_y.CDifferentiableAt_rule","declaration":"theorem SciLean.SciLean.semiAdjoint.arg_y.CDifferentiableAt_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_4} [SciLean.SemiInnerProductSpace K Y] {W : Type u_1} [SciLean.Vec K W] (w : W) (f : X → Y) (a0 : W → Y) (hf : SciLean.CDifferentiable K f) (ha0 : SciLean.CDifferentiableAt K a0 w) : SciLean.CDifferentiableAt K (fun w => SciLean.semiAdjoint K f (a0 w)) w"} +{"name":"SciLean.Prod.snd.arg_self.HasAdjDiffAt_rule","declaration":"theorem SciLean.Prod.snd.arg_self.HasAdjDiffAt_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (x : X) (f : X → Y × Z) (hf : SciLean.HasAdjDiffAt K f x) : SciLean.HasAdjDiffAt K (fun x => (f x).2) x"} +{"name":"SciLean.cderiv.arg_dx.hasSemiAdjointAt","declaration":"theorem SciLean.cderiv.arg_dx.hasSemiAdjointAt (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt K f x) : SciLean.HasSemiAdjoint K (SciLean.cderiv K f x)"} +{"name":"SciLean.cderiv.arg_dx.hasSemiAdjoint","declaration":"theorem SciLean.cderiv.arg_dx.hasSemiAdjoint (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiff K f) : SciLean.HasSemiAdjoint K (SciLean.cderiv K f x)"} +{"name":"SciLean.HasAdjDiffAt.const_rule","declaration":"theorem SciLean.HasAdjDiffAt.const_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (x : X) (y : Y) : SciLean.HasAdjDiffAt K (fun x => y) x"} +{"name":"SciLean.dite.arg_te.HasAdjDiffAt_rule","declaration":"theorem SciLean.dite.arg_te.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (x : X) (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) (ht : ∀ (h : c), SciLean.HasAdjDiffAt K (t h) x) (he : ∀ (h : ¬c), SciLean.HasAdjDiffAt K (e h) x) : SciLean.HasAdjDiffAt K (fun x => if x_1 : c then t x_1 x else e x_1 x) x"} +{"name":"SciLean.HasAdjDiffAt.cdifferentiableAt","declaration":"theorem SciLean.HasAdjDiffAt.cdifferentiableAt (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt K f x) : SciLean.CDifferentiableAt K f x"} +{"name":"SciLean.HasAdjDiff.let_rule","declaration":"theorem SciLean.HasAdjDiff.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K ↿f) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiff K fun x =>\n let y := g x;\n f x y"} +{"name":"SciLean.HMul.hMul.arg_a0a1.HasAdjDiffAt_rule","declaration":"def SciLean.HMul.hMul.arg_a0a1.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiffAt K f x) (hg : SciLean.HasAdjDiffAt K g x) : SciLean.HasAdjDiffAt K (fun x => f x * g x) x"} +{"name":"SciLean.HSMul.hSMul.arg_a1.HasAdjDiff_rule","declaration":"theorem SciLean.HSMul.hSMul.arg_a1.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : K) (g : X → Y) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiff K fun x => c • g x"} +{"name":"SciLean.ite.arg_te.HasAdjDiffAt_rule","declaration":"theorem SciLean.ite.arg_te.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (x : X) (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) (ht : SciLean.HasAdjDiffAt K t x) (he : SciLean.HasAdjDiffAt K e x) : SciLean.HasAdjDiffAt K (fun x => if c then t x else e x) x"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.HasAdjDiffAt_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiffAt K f x) (hg : SciLean.HasAdjDiffAt K g x) : SciLean.HasAdjDiffAt K (fun x => f x + g x) x"} +{"name":"SciLean.dite.arg_te.HasAdjDiff_rule","declaration":"theorem SciLean.dite.arg_te.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) (ht : ∀ (h : c), SciLean.HasAdjDiff K (t h)) (he : ∀ (h : ¬c), SciLean.HasAdjDiff K (e h)) : SciLean.HasAdjDiff K fun x => if x_1 : c then t x_1 x else e x_1 x"} +{"name":"SciLean.IndexType.sum.arg_f.HasAdjDiff_rule","declaration":"theorem SciLean.IndexType.sum.arg_f.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : SciLean.HasAdjDiff K fun x => ∑ i, f x i"} +{"name":"SciLean.Prod.fst.arg_self.HasAdjDiff_rule","declaration":"theorem SciLean.Prod.fst.arg_self.HasAdjDiff_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : SciLean.HasAdjDiff K fun x => (f x).1"} +{"name":"SciLean.Prod.mk.arg_fstsnd.HasAdjDiff_rule","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (g : X → Y) (hg : SciLean.HasAdjDiff K g) (f : X → Z) (hf : SciLean.HasAdjDiff K f) : SciLean.HasAdjDiff K fun x => (g x, f x)"} +{"name":"SciLean.HasAdjDiffAt","declaration":"def SciLean.HasAdjDiffAt (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Prop"} +{"name":"SciLean.HasAdjDiffAt.id_rule","declaration":"theorem SciLean.HasAdjDiffAt.id_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (x : X) : SciLean.HasAdjDiffAt K (fun x => x) x"} +{"name":"SciLean.HSub.hSub.arg_a0a1.HasAdjDiffAt_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiffAt K f x) (hg : SciLean.HasAdjDiffAt K g x) : SciLean.HasAdjDiffAt K (fun x => f x - g x) x"} +{"name":"SciLean.SciLean.Norm2.norm2.arg_a0.HasAdjDiffAt_rule","declaration":"theorem SciLean.SciLean.Norm2.norm2.arg_a0.HasAdjDiffAt_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (x : X) (f : X → Y) (hf : SciLean.HasAdjDiffAt R f x) : SciLean.HasAdjDiffAt R (fun x => ‖f x‖₂²) x"} +{"name":"SciLean.HasAdjDiffAt.let_rule","declaration":"theorem SciLean.HasAdjDiffAt.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → Z) (g : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt K (↿f) (x, g x)) (hg : SciLean.HasAdjDiffAt K g x) : SciLean.HasAdjDiffAt K\n (fun x =>\n let y := g x;\n f x y)\n x"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.HasAdjDiffAt_rule","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.HasAdjDiffAt_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] (x : X) {Y : Type u_1} [SciLean.SemiHilbert K Y] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiffAt K f x) (hg : SciLean.HasAdjDiffAt K g x) : SciLean.HasAdjDiffAt K (fun x => f x • g x) x"} +{"name":"SciLean.SciLean.Norm2.norm2.arg_a0.HasAdjDiff_rule","declaration":"theorem SciLean.SciLean.Norm2.norm2.arg_a0.HasAdjDiff_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : SciLean.HasAdjDiff R fun x => ‖f x‖₂²"} +{"name":"SciLean.HasAdjDiff.id_rule","declaration":"theorem SciLean.HasAdjDiff.id_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] : SciLean.HasAdjDiff K fun x => x"} +{"name":"SciLean.HPow.hPow.arg_a0.HasAdjDiffAt_rule","declaration":"def SciLean.HPow.hPow.arg_a0.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (x : X) (n : ℕ) (f : X → K) (hf : SciLean.HasAdjDiffAt K f x) : SciLean.HasAdjDiffAt K (fun x => f x ^ n) x"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.HasAdjDiff_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiff K fun x => f x + g x"} +{"name":"SciLean.Prod.snd.arg_self.HasAdjDiff_rule","declaration":"theorem SciLean.Prod.snd.arg_self.HasAdjDiff_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : SciLean.HasAdjDiff K fun x => (f x).2"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.HasAdjDiff_rule","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.HasAdjDiff_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiHilbert K Y] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiff K fun x => f x • g x"} +{"name":"SciLean.Neg.neg.arg_a0.HasAdjDiff_rule","declaration":"theorem SciLean.Neg.neg.arg_a0.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasAdjDiff K f) : SciLean.HasAdjDiff K fun x => -f x"} +{"name":"SciLean.HasAdjDiff","declaration":"def SciLean.HasAdjDiff (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : Prop"} +{"name":"SciLean.SciLean.cderiv.arg_dx.HasSemiAdjoint_rule","declaration":"theorem SciLean.SciLean.cderiv.arg_dx.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (y : Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasSemiAdjoint K g) : SciLean.HasSemiAdjoint K fun dx => SciLean.cderiv K f y (g dx)"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.HasAdjDiffAt_rule","declaration":"def SciLean.HDiv.hDiv.arg_a0a1.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiffAt K f x) (hg : SciLean.HasAdjDiffAt K g x) (hx : g x ≠ 0) : SciLean.HasAdjDiffAt K (fun x => f x / g x) x"} +{"name":"SciLean.Neg.neg.arg_a0.HasAdjDiffAt_rule","declaration":"theorem SciLean.Neg.neg.arg_a0.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (x : X) (f : X → Y) (hf : SciLean.HasAdjDiffAt K f x) : SciLean.HasAdjDiffAt K (fun x => -f x) x"} +{"name":"SciLean.HasAdjDiff.comp_rule","declaration":"theorem SciLean.HasAdjDiff.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiff K fun x => f (g x)"} +{"name":"SciLean.HPow.hPow.arg_a0.HasAdjDiff_rule","declaration":"def SciLean.HPow.hPow.arg_a0.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (n : ℕ) (f : X → K) (hf : SciLean.HasAdjDiff K f) : SciLean.HasAdjDiff K fun x => f x ^ n"} +{"name":"SciLean.SciLean.cderiv.arg_dx.HasSemiAdjoint_rule_at","declaration":"theorem SciLean.SciLean.cderiv.arg_dx.HasSemiAdjoint_rule_at (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (y : Y) (hf : SciLean.HasAdjDiffAt K f y) (hg : SciLean.HasSemiAdjoint K g) : SciLean.HasSemiAdjoint K fun dx => SciLean.cderiv K f y (g dx)"} +{"name":"SciLean.HSub.hSub.arg_a0a1.HasAdjDiff_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.HasAdjDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiff K fun x => f x - g x"} +{"name":"SciLean.Inner.inner.arg_a0a1.HasAdjDiffAt_rule","declaration":"theorem SciLean.Inner.inner.arg_a0a1.HasAdjDiffAt_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiffAt R f x) (hg : SciLean.HasAdjDiffAt R g x) : SciLean.HasAdjDiffAt R (fun x => ⟪f x, g x⟫_R) x"} +{"name":"SciLean.SciLean.semiAdjoint.arg_y.HasAdjDiffAt_rule","declaration":"theorem SciLean.SciLean.semiAdjoint.arg_y.HasAdjDiffAt_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {W : Type u_4} [SciLean.SemiInnerProductSpace K W] (w : W) (f : X → Y) (a0 : W → Y) (hf : SciLean.CDifferentiable K f) (ha0 : SciLean.HasAdjDiffAt K a0 w) : SciLean.HasAdjDiffAt K (fun w => SciLean.semiAdjoint K f (a0 w)) w"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.HasSemiAdjoint.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.HasSemiAdjoint.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..5180689308686c0876165c7ca8a5ebfbe286ae95 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.HasSemiAdjoint.jsonl @@ -0,0 +1,41 @@ +{"name":"SciLean.semiAdjoint_choose","declaration":"theorem SciLean.semiAdjoint_choose (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {f : X → Y} (hf : SciLean.HasSemiAdjoint K f) : SciLean.semiAdjoint K f = Classical.choose ⋯"} +{"name":"SciLean.HasSemiAdjoint.SciLean.semiAdjoint.arg_y.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.SciLean.semiAdjoint.arg_y.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {W : Type u_2} [SciLean.SemiInnerProductSpace K W] (f : X → Y) (a0 : W → Y) (ha0 : SciLean.HasSemiAdjoint K a0) : SciLean.HasSemiAdjoint K fun w => SciLean.semiAdjoint K f (a0 w)"} +{"name":"SciLean.HasSemiAdjoint.IsLinearMap","declaration":"theorem SciLean.HasSemiAdjoint.IsLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) : IsLinearMap K f"} +{"name":"SciLean.HasSemiAdjoint.const_rule","declaration":"theorem SciLean.HasSemiAdjoint.const_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] : SciLean.HasSemiAdjoint K fun x => 0"} +{"name":"SciLean.HasSemiAdjoint.Neg.neg.arg_a0.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.Neg.neg.arg_a0.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : SciLean.HasSemiAdjoint K fun x => -f x"} +{"name":"SciLean.HasSemiAdjoint.id_rule","declaration":"theorem SciLean.HasSemiAdjoint.id_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] : SciLean.HasSemiAdjoint K fun x => x"} +{"name":"SciLean.HasSemiAdjoint.pi_rule","declaration":"theorem SciLean.HasSemiAdjoint.pi_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {ι : Type u_4} [SciLean.IndexType ι] {E : ι → Type u_3} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.HasSemiAdjoint K fun x => f x i) : SciLean.HasSemiAdjoint K fun x i => f x i"} +{"name":"SciLean.HasSemiAdjoint.HAdd.hAdd.arg_a0a1.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.HAdd.hAdd.arg_a0a1.HasSemiAdjoint_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] [ContinuousAdd Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasSemiAdjoint K f) (hg : SciLean.HasSemiAdjoint K g) : SciLean.HasSemiAdjoint K fun x => f x + g x"} +{"name":"SciLean.HasSemiAdjoint.HDiv.hDiv.arg_a0.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.HDiv.hDiv.arg_a0.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) (hf : SciLean.HasSemiAdjoint K f) (y : K) : SciLean.HasSemiAdjoint K fun x => f x / y"} +{"name":"SciLean.semiAdjoint.arg_y.CDifferentiable_rule","declaration":"theorem SciLean.semiAdjoint.arg_y.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : SciLean.CDifferentiable K fun y => SciLean.semiAdjoint K f y"} +{"name":"SciLean.HasSemiAdjoint.HSMul.hSMul.arg_a0.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.HSMul.hSMul.arg_a0.HasSemiAdjoint_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiHilbert K Y] (f : X → K) (y : Y) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasSemiAdjoint K fun x => f x • y"} +{"name":"SciLean.HasSemiAdjoint.HSMul.hSMul.arg_a1.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.HSMul.hSMul.arg_a1.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : K) (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasSemiAdjoint K fun x => c • f x"} +{"name":"SciLean.HasSemiAdjoint.mk","declaration":"ctor SciLean.HasSemiAdjoint.mk {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {f : X → Y} (semiAdjoint_exists : ∃ f', ∀ (x : X) (y : Y), SciLean.TestFunction x → ⟪y, f x⟫_K = ⟪f' y, x⟫_K) (is_differentiable : SciLean.CDifferentiable K f) : SciLean.HasSemiAdjoint K f"} +{"name":"SciLean.HasSemiAdjoint.HSub.hSub.arg_a0a1.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.HSub.hSub.arg_a0a1.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasSemiAdjoint K f) (hg : SciLean.HasSemiAdjoint K g) : SciLean.HasSemiAdjoint K fun x => f x - g x"} +{"name":"SciLean.HasSemiAdjoint.Prod.snd.arg_self.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.Prod.snd.arg_self.HasSemiAdjoint_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasSemiAdjoint K fun x => (f x).2"} +{"name":"SciLean.semiAdjoint","declaration":"/-- Generalization of adjoint of linear map `f : X → Y`.\n\nIf `f : X → Y` is linear map between Hilbert spaces then `semiAdjoint K f = adjoint K f`.\n\n`semiAdjoint` is a generalization of adjoint to spaces that are not necessarily complete\nand might have inner product defined only on a dense subset, see `SemiInnerProductSpace`\nfor more information.\n -/\ndef SciLean.semiAdjoint (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (y : Y) : X"} +{"name":"SciLean.HasSemiAdjoint.semiAdjoint_exists","declaration":"def SciLean.HasSemiAdjoint.semiAdjoint_exists {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {f : X → Y} (self : SciLean.HasSemiAdjoint K f) : ∃ f', ∀ (x : X) (y : Y), SciLean.TestFunction x → ⟪y, f x⟫_K = ⟪f' y, x⟫_K"} +{"name":"SciLean.HasSemiAdjoint","declaration":"structure SciLean.HasSemiAdjoint (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : Prop"} +{"name":"SciLean.HasSemiAdjoint.SciLean.IndexType.sum.arg_f.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.SciLean.IndexType.sum.arg_f.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasSemiAdjoint K fun x => f x i) : SciLean.HasSemiAdjoint K fun x => ∑ i, f x i"} +{"name":"SciLean.HasSemiAdjoint.dite.arg_te.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.dite.arg_te.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (ht : ∀ (p : c), SciLean.HasSemiAdjoint K (t p)) (e : ¬c → X → Y) (he : ∀ (p : ¬c), SciLean.HasSemiAdjoint K (e p)) : SciLean.HasSemiAdjoint K fun x => if x_1 : c then t x_1 x else e x_1 x"} +{"name":"SciLean.HasSemiAdjoint.HMul.hMul.arg_a1.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.HMul.hMul.arg_a1.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (y' : K) (f : X → K) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasSemiAdjoint K fun x => y' * f x"} +{"name":"SciLean.HasSemiAdjoint.Prod.mk.arg_fstsnd.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.Prod.mk.arg_fstsnd.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (g : X → Y) (hg : SciLean.HasSemiAdjoint K g) (f : X → Z) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasSemiAdjoint K fun x => (g x, f x)"} +{"name":"SciLean.semiAdjoint_move","declaration":"def SciLean.semiAdjoint_move (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (x : X) (y : Y) (hx : SciLean.TestFunction x) (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) : ⟪SciLean.semiAdjoint K f y, x⟫_K = ⟪y, f x⟫_K"} +{"name":"SciLean.semiAdjoint_unique","declaration":"def SciLean.semiAdjoint_unique (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) (f' : Y → X) (hf' : ∀ (x : X) (y : Y), SciLean.TestFunction x → ⟪y, f x⟫_K = ⟪f' y, x⟫_K) : f' = SciLean.semiAdjoint K f"} +{"name":"SciLean.HasSemiAdjoint.ite.arg_te.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.ite.arg_te.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) (ht : SciLean.HasSemiAdjoint K t) (he : SciLean.HasSemiAdjoint K e) : SciLean.HasSemiAdjoint K fun x => if c then t x else e x"} +{"name":"SciLean.HasSemiAdjoint.Inner.inner.arg_a0.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.Inner.inner.arg_a0.HasSemiAdjoint_rule {K : Type u_1} [SciLean.RealScalar K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiHilbert K Y] (f : X → Y) : SciLean.HasSemiAdjoint K f → ∀ (y : Y), SciLean.HasSemiAdjoint K fun x => ⟪f x, y⟫_K"} +{"name":"SciLean.HasSemiAdjoint.CDifferentiable","declaration":"theorem SciLean.HasSemiAdjoint.CDifferentiable (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) : SciLean.CDifferentiable K f"} +{"name":"SciLean.HasSemiAdjoint.is_differentiable","declaration":"def SciLean.HasSemiAdjoint.is_differentiable {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {f : X → Y} (self : SciLean.HasSemiAdjoint K f) : SciLean.CDifferentiable K f"} +{"name":"SciLean.semiAdjoint.arg_fy.CDifferentiable_rule","declaration":"theorem SciLean.semiAdjoint.arg_fy.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_4} [SciLean.SemiInnerProductSpace K Y] {W : Type u_3} [SciLean.SemiInnerProductSpace K W] (f : W → X → Y) (y : W → Y) (hf₁ : SciLean.CDifferentiable K fun x =>\n match x with\n | (w, x) => f w x) (hf₂ : ∀ (w : W), SciLean.HasSemiAdjoint K fun x => f w x) (hy : SciLean.CDifferentiable K y) : SciLean.CDifferentiable K fun w => SciLean.semiAdjoint K (f w) (y w)"} +{"name":"SciLean.HasSemiAdjoint.Prod.fst.arg_self.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.Prod.fst.arg_self.HasSemiAdjoint_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasSemiAdjoint K fun x => (f x).1"} +{"name":"SciLean.semiAdjoint.arg_y.IsSmoothLinearMap_rule","declaration":"theorem SciLean.semiAdjoint.arg_y.IsSmoothLinearMap_rule (K : Type u_1) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : SciLean.IsSmoothLinearMap K fun y => SciLean.semiAdjoint K f y"} +{"name":"SciLean.semiAdjoint.arg_y.IsLinearMap_rule","declaration":"/-- `semiAdjoint K f ·` is always linear because either `f` has adjoint and is linear or\n`semiAdjoint K f ·` is zero function and thus linear too. -/\ntheorem SciLean.semiAdjoint.arg_y.IsLinearMap_rule (K : Type u_1) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : IsLinearMap K fun y => SciLean.semiAdjoint K f y"} +{"name":"SciLean.HasSemiAdjoint.starRingEnd.arg_a.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.starRingEnd.arg_a.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) : SciLean.HasSemiAdjoint K f → SciLean.HasSemiAdjoint K fun x => (starRingEnd K) (f x)"} +{"name":"SciLean.HasSemiAdjoint.Inner.inner.arg_a1.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.Inner.inner.arg_a1.HasSemiAdjoint_rule {K : Type u_1} [SciLean.RealScalar K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiHilbert K Y] (f : X → Y) : SciLean.HasSemiAdjoint K f → ∀ (y : Y), SciLean.HasSemiAdjoint K fun x => ⟪y, f x⟫_K"} +{"name":"SciLean.HasSemiAdjoint.apply_rule","declaration":"theorem SciLean.HasSemiAdjoint.apply_rule (K : Type u_1) [RCLike K] {ι : Type u_2} [SciLean.IndexType ι] [DecidableEq ι] {E : ι → Type u_3} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] (i : ι) : SciLean.HasSemiAdjoint K fun x => x i"} +{"name":"SciLean.HasSemiAdjoint.comp_rule","declaration":"theorem SciLean.HasSemiAdjoint.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.HasSemiAdjoint K f) (hg : SciLean.HasSemiAdjoint K g) : SciLean.HasSemiAdjoint K fun x => f (g x)"} +{"name":"SciLean.semiAdjoint.arg_f.IsSmoothLinearMap_rule","declaration":"theorem SciLean.semiAdjoint.arg_f.IsSmoothLinearMap_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {W : Type u_2} [SciLean.SemiInnerProductSpace K W] (f : W → X → Y) (hf₁ : ∀ (x : X), SciLean.IsSmoothLinearMap K fun x_1 => f x_1 x) (hf₂ : ∀ (w : W), SciLean.HasSemiAdjoint K fun x => f w x) : SciLean.IsSmoothLinearMap K fun w => SciLean.semiAdjoint K (f w)"} +{"name":"SciLean.semi_inner_ext","declaration":"def SciLean.semi_inner_ext (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (x : X) (x' : X) : (∀ (φ : X), SciLean.TestFunction φ → ⟪x, φ⟫_K = ⟪x', φ⟫_K) → x = x'"} +{"name":"SciLean.hasSemiAdjoint.IsSmoothLinearMap","declaration":"theorem SciLean.hasSemiAdjoint.IsSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) : SciLean.IsSmoothLinearMap K f"} +{"name":"SciLean.HasSemiAdjoint.Finset.sum.arg_f.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.Finset.sum.arg_f.HasSemiAdjoint_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_4} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_1} [Fintype ι] (f : X → ι → Y) : (∀ (i : ι), SciLean.HasSemiAdjoint K fun x => f x i) →\n ∀ (A : Finset ι), SciLean.HasSemiAdjoint K fun x => Finset.sum A fun i => f x i"} +{"name":"SciLean.HasSemiAdjoint.HMul.hMul.arg_a0.HasSemiAdjoint_rule","declaration":"theorem SciLean.HasSemiAdjoint.HMul.hMul.arg_a0.HasSemiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) (y' : K) (hf : SciLean.HasSemiAdjoint K f) : SciLean.HasSemiAdjoint K fun x => f x * y'"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.IsAffineMap.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.IsAffineMap.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a07b64e68e9ff2bab0d9a0becb0b4667bdfd6e64 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.IsAffineMap.jsonl @@ -0,0 +1,28 @@ +{"name":"Prod.snd.arg_self.IsAffineMap_rule","declaration":"theorem Prod.snd.arg_self.IsAffineMap_rule {R : Type u_3} {X : Type u_4} {Y : Type u_1} {Z : Type u_2} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Y × Z) (hf : IsAffineMap R f) : IsAffineMap R fun x => (f x).2"} +{"name":"HSMul.hSMul.arg_a1.IsAffineMap_rule'","declaration":"theorem HSMul.hSMul.arg_a1.IsAffineMap_rule' {R : Type u_4} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] {S : Type u_1} [CommRing S] [Module S X] [Module S Y] [SMul S R] [IsScalarTower S R Y] (c : R) (f : X → Y) (hf : IsAffineMap R f) : IsAffineMap S fun x => c • f x"} +{"name":"IsAffineMap.IsAffineMap_const","declaration":"theorem IsAffineMap.IsAffineMap_const {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (y : Y) : IsAffineMap R fun x => y"} +{"name":"Prod.fst.arg_self.IsAffineMap_rule","declaration":"theorem Prod.fst.arg_self.IsAffineMap_rule {R : Type u_3} {X : Type u_4} {Y : Type u_1} {Z : Type u_2} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Y × Z) (hf : IsAffineMap R f) : IsAffineMap R fun x => (f x).1"} +{"name":"HSub.hSub.arg_a0a1.IsAffineMap_rule","declaration":"theorem HSub.hSub.arg_a0a1.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → Y) (g : X → Y) (hf : IsAffineMap R f) (hg : IsAffineMap R g) : IsAffineMap R fun x => f x - g x"} +{"name":"SciLean.BasisDuality.toDual.arg_x.IsAffineMap_rule","declaration":"theorem SciLean.BasisDuality.toDual.arg_x.IsAffineMap_rule {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : IsAffineMap K fun x => SciLean.BasisDuality.toDual x"} +{"name":"HAdd.hAdd.arg_a0a1.IsAffineMap_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → Y) (g : X → Y) (hf : IsAffineMap R f) (hg : IsAffineMap R g) : IsAffineMap R fun x => f x + g x"} +{"name":"HSMul.hSMul.arg_a0.IsAffineMap_rule","declaration":"theorem HSMul.hSMul.arg_a0.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → R) (y : Y) (hf : IsAffineMap R f) : IsAffineMap R fun x => f x • y"} +{"name":"HSMul.hSMul.arg_a1.IsAffineMap_rule_int","declaration":"theorem HSMul.hSMul.arg_a1.IsAffineMap_rule_int {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : ℤ) (f : X → Y) (hf : IsAffineMap R f) : IsAffineMap R fun x => c • f x"} +{"name":"Prod.mk.arg_fstsnd.IsAffineMap_rule","declaration":"theorem Prod.mk.arg_fstsnd.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_4} {Z : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Z) (g : X → Y) (hf : IsAffineMap R f) (hg : IsAffineMap R g) : IsAffineMap R fun x => (g x, f x)"} +{"name":"HSMul.hSMul.arg_a1.IsAffineMap_rule","declaration":"theorem HSMul.hSMul.arg_a1.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : R) (f : X → Y) (hf : IsAffineMap R f) : IsAffineMap R fun x => c • f x"} +{"name":"HSMul.hSMul.arg_a1.IsAffineMap_rule_nat","declaration":"theorem HSMul.hSMul.arg_a1.IsAffineMap_rule_nat {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : ℕ) (f : X → Y) (hf : IsAffineMap R f) : IsAffineMap R fun x => c • f x"} +{"name":"IsAffineMap.IsAffineMap_comp","declaration":"theorem IsAffineMap.IsAffineMap_comp {R : Type u_1} {X : Type u_4} {Y : Type u_2} {Z : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] {f : Y → Z} {g : X → Y} (hf : IsAffineMap R f) (hg : IsAffineMap R g) : IsAffineMap R fun x => f (g x)"} +{"name":"IndexType.sum.arg_f.IsAffineMap_rule","declaration":"theorem IndexType.sum.arg_f.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} {ι : Type u_4} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), IsAffineMap R fun x => f x i) : IsAffineMap R fun x => ∑ i, f x i"} +{"name":"SciLean.Basis.proj.arg_x.IsAffineMap_rule","declaration":"theorem SciLean.Basis.proj.arg_x.IsAffineMap_rule {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : IsAffineMap K fun x => ℼ i x"} +{"name":"HMul.hMul.arg_a1.IsAffineMap_rule","declaration":"theorem HMul.hMul.arg_a1.IsAffineMap_rule {R : Type u_1} {X : Type u_2} [CommRing R] [AddCommGroup X] [Module R X] (f : X → R) (hf : IsAffineMap R f) (y' : R) : IsAffineMap R fun x => y' * f x"} +{"name":"IsAffineMap","declaration":"structure IsAffineMap (R : Type u_1) {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → Y) : Prop"} +{"name":"Neg.neg.arg_a0.IsAffineMap_rule","declaration":"theorem Neg.neg.arg_a0.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → Y) (hf : IsAffineMap R f) : IsAffineMap R fun x => -f x"} +{"name":"IsAffineMap.IsAffineMap_pi","declaration":"theorem IsAffineMap.IsAffineMap_pi {R : Type u_1} {X : Type u_2} {ι : Type u_4} {E : ι → Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module R (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), IsAffineMap R fun x => f x i) : IsAffineMap R fun x i => f x i"} +{"name":"SciLean.DualBasis.dualProj.arg_x.IsAffineMap_rule","declaration":"theorem SciLean.DualBasis.dualProj.arg_x.IsAffineMap_rule {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : IsAffineMap K fun x => ℼ' i x"} +{"name":"IsAffineMap.IsAffineMap_apply","declaration":"theorem IsAffineMap.IsAffineMap_apply {R : Type u_1} {ι : Type u_2} {E : ι → Type u_3} [CommRing R] [(i : ι) → AddCommGroup (E i)] [(i : ι) → Module R (E i)] (i : ι) : IsAffineMap R fun f => f i"} +{"name":"dite.arg_te.IsAffineMap_rule","declaration":"theorem dite.arg_te.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (ht : ∀ (p : c), IsAffineMap R (t p)) (e : ¬c → X → Y) (he : ∀ (p : ¬c), IsAffineMap R (e p)) : IsAffineMap R fun x => if x_1 : c then t x_1 x else e x_1 x"} +{"name":"IsAffineMap.IsAffineMap_id","declaration":"theorem IsAffineMap.IsAffineMap_id {R : Type u_1} {X : Type u_2} [CommRing R] [AddCommGroup X] [Module R X] : IsAffineMap R fun x => x"} +{"name":"HMul.hMul.arg_a0.IsAffineMap_rule","declaration":"theorem HMul.hMul.arg_a0.IsAffineMap_rule {R : Type u_1} {X : Type u_2} [CommRing R] [AddCommGroup X] [Module R X] (f : X → R) (hf : IsAffineMap R f) (y' : R) : IsAffineMap R fun x => f x * y'"} +{"name":"ite.arg_te.IsAffineMap_rule","declaration":"theorem ite.arg_te.IsAffineMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) (ht : IsAffineMap R t) (he : IsAffineMap R e) : IsAffineMap R fun x => if c then t x else e x"} +{"name":"IsAffineMap.mk","declaration":"ctor IsAffineMap.mk {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] {f : X → Y} (is_affine : IsLinearMap R fun x => f x - f 0) : IsAffineMap R f"} +{"name":"IsAffineMap.is_affine","declaration":"def IsAffineMap.is_affine {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommRing R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] {f : X → Y} (self : IsAffineMap R f) : IsLinearMap R fun x => f x - f 0"} +{"name":"SciLean.BasisDuality.fromDual.arg_x.IsAffineMap_rule","declaration":"theorem SciLean.BasisDuality.fromDual.arg_x.IsAffineMap_rule {K : Type u_1} [RCLike K] {IX : Type} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : IsAffineMap K fun x => SciLean.BasisDuality.fromDual x"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.IsContinuousLinearMap.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.IsContinuousLinearMap.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..8f465c98c33fb0d3c447cf505c3aabba6d3cd0f1 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.IsContinuousLinearMap.jsonl @@ -0,0 +1,43 @@ +{"name":"SciLean.IsContinuousLinearMap.Prod.snd.arg_self.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.Prod.snd.arg_self.IsContinuousLinearMap_rule {R : Type u_3} [Semiring R] {X : Type u_4} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_1} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {Z : Type u_2} [TopologicalSpace Z] [AddCommMonoid Z] [Module R Z] (f : X → Y × Z) (hf : SciLean.IsContinuousLinearMap R f) : SciLean.IsContinuousLinearMap R fun x => (f x).2"} +{"name":"SciLean.IsContinuousLinearMap.FunLike.coe.arg_a.IsContinuousLinearMap_rule'","declaration":"theorem SciLean.IsContinuousLinearMap.FunLike.coe.arg_a.IsContinuousLinearMap_rule' {R : Type u_3} [Semiring R] {X : Type u_4} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_2} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {Z : Type u_1} [TopologicalSpace Z] [AddCommMonoid Z] [Module R Z] (f : Y →L[R] Z) (g : X → Y) (hg : SciLean.IsContinuousLinearMap R g) : SciLean.IsContinuousLinearMap R fun x => f (g x)"} +{"name":"SciLean.ContinuousLinearMap.eta_reduce","declaration":"theorem SciLean.ContinuousLinearMap.eta_reduce (R : Type u_3) [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_1} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] (f : X →L[R] Y) : SciLean.ContinuousLinearMap.mk' R ⇑f ⋯ = f"} +{"name":"SciLean.IsContinuousLinearMap.id_rule","declaration":"theorem SciLean.IsContinuousLinearMap.id_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] : SciLean.IsContinuousLinearMap R fun x => x"} +{"name":"SciLean.IsContinuousLinearMap.FunLike.coe.arg_a.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.FunLike.coe.arg_a.IsContinuousLinearMap_rule {R : Type u_3} [Semiring R] {Y : Type u_2} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {Z : Type u_1} [TopologicalSpace Z] [AddCommMonoid Z] [Module R Z] (f : Y →L[R] Z) : SciLean.IsContinuousLinearMap R fun y => f y"} +{"name":"SciLean.«termFun_:_=>L[_]_»","declaration":"def SciLean.«termFun_:_=>L[_]_» : Lean.ParserDescr"} +{"name":"SciLean.IsContinuousLinearMap.cont","declaration":"def SciLean.IsContinuousLinearMap.cont {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {f : X → Y} (self : SciLean.IsContinuousLinearMap R f) : Continuous f"} +{"name":"SciLean.IsContinuousLinearMap.ite.arg_te.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.ite.arg_te.IsContinuousLinearMap_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) (ht : SciLean.IsContinuousLinearMap R t) (he : SciLean.IsContinuousLinearMap R e) : SciLean.IsContinuousLinearMap R fun x => if c then t x else e x"} +{"name":"SciLean.«termFun(_:_)=>L[_]_»","declaration":"def SciLean.«termFun(_:_)=>L[_]_» : Lean.ParserDescr"} +{"name":"SciLean.IsContinuousLinearMap.HSMul.hSMul.arg_a0.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.HSMul.hSMul.arg_a0.IsContinuousLinearMap_rule {R : Type u_1} [Semiring R] {X : Type u_3} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_2} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] [TopologicalSpace R] [ContinuousSMul R Y] (f : X → R) (hf : SciLean.IsContinuousLinearMap R f) (y : Y) : SciLean.IsContinuousLinearMap R fun x => f x • y"} +{"name":"SciLean.IsContinuousLinearMap.Finset.sum.arg_f.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.Finset.sum.arg_f.IsContinuousLinearMap_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {ι : Type u_4} (f : X → ι → Y) : (∀ (i : ι), SciLean.IsContinuousLinearMap R fun x => f x i) →\n ∀ (A : Finset ι), SciLean.IsContinuousLinearMap R fun x => Finset.sum A fun i => f x i"} +{"name":"SciLean.«termFun_=>L[_]_»","declaration":"def SciLean.«termFun_=>L[_]_» : Lean.ParserDescr"} +{"name":"SciLean.IsContinuousLinearMap.ContinuousLinearMap.divRight","declaration":"/-- Creates `fun x =>L[R] g x / y` -/\ndef SciLean.IsContinuousLinearMap.ContinuousLinearMap.divRight {R : Type u_1} [NontriviallyNormedField R] {K : Type u_2} [NontriviallyNormedField K] [NormedAlgebra R K] {X : Type u_3} [TopologicalSpace X] [AddCommMonoid X] [Module R X] (g : X →L[R] K) (y : K) : X →L[R] K"} +{"name":"SciLean.IsContinuousLinearMap.dite.arg_te.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.dite.arg_te.IsContinuousLinearMap_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (ht : ∀ (p : c), SciLean.IsContinuousLinearMap R (t p)) (e : ¬c → X → Y) (he : ∀ (p : ¬c), SciLean.IsContinuousLinearMap R (e p)) : SciLean.IsContinuousLinearMap R fun x => if x_1 : c then t x_1 x else e x_1 x"} +{"name":"SciLean.IsContinuousLinearMap.Prod.mk.arg_fstsnd.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.Prod.mk.arg_fstsnd.IsContinuousLinearMap_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {Z : Type u_4} [TopologicalSpace Z] [AddCommMonoid Z] [Module R Z] (g : X → Y) (hg : SciLean.IsContinuousLinearMap R g) (f : X → Z) (hf : SciLean.IsContinuousLinearMap R f) : SciLean.IsContinuousLinearMap R fun x => (g x, f x)"} +{"name":"SciLean.IsContinuousLinearMap.linear","declaration":"def SciLean.IsContinuousLinearMap.linear {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {f : X → Y} (self : SciLean.IsContinuousLinearMap R f) : IsLinearMap R f"} +{"name":"SciLean.IsContinuousLinearMap.ContinuousLinearMap.mul_right","declaration":"def SciLean.IsContinuousLinearMap.ContinuousLinearMap.mul_right {R : Type u_1} [CommSemiring R] {X : Type u_2} [TopologicalSpace X] [Semiring X] [Algebra R X] [TopologicalSemiring X] (x' : X) : X →L[R] X"} +{"name":"SciLean.IsContinuousLinearMap.const_rule","declaration":"theorem SciLean.IsContinuousLinearMap.const_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] : SciLean.IsContinuousLinearMap R fun x => 0"} +{"name":"SciLean.IsContinuousLinearMap.by_morphism","declaration":"theorem SciLean.IsContinuousLinearMap.by_morphism {R : Type u_3} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_1} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {f : X → Y} (g : X →L[R] Y) (h : ∀ (x : X), f x = g x) : SciLean.IsContinuousLinearMap R f"} +{"name":"SciLean.IsContinuousLinearMap.ContinuousLinearMap.mul_left","declaration":"def SciLean.IsContinuousLinearMap.ContinuousLinearMap.mul_left {R : Type u_1} [CommSemiring R] {X : Type u_2} [TopologicalSpace X] [Semiring X] [Algebra R X] [TopologicalSemiring X] (x' : X) : X →L[R] X"} +{"name":"SciLean.IsContinuousLinearMap.pi_rule","declaration":"theorem SciLean.IsContinuousLinearMap.pi_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommMonoid (E i)] [(i : ι) → Module R (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.IsContinuousLinearMap R fun x => f x i) : SciLean.IsContinuousLinearMap R fun x i => f x i"} +{"name":"SciLean.IsContinuousLinearMap.HSMul.hSMul.arg_a1.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.HSMul.hSMul.arg_a1.IsContinuousLinearMap_rule {R : Type u_1} [CommSemiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] [ContinuousConstSMul R Y] (c : R) (f : X → Y) (hf : SciLean.IsContinuousLinearMap R f) : SciLean.IsContinuousLinearMap R fun x => c • f x"} +{"name":"SciLean.IsContinuousLinearMap.HDiv.hDiv.arg_a0.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.HDiv.hDiv.arg_a0.IsContinuousLinearMap_rule {R : Type u_1} [NontriviallyNormedField R] {K : Type u_2} [NontriviallyNormedField K] [NormedAlgebra R K] {X : Type u_3} [TopologicalSpace X] [AddCommMonoid X] [Module R X] (f : X → K) (hf : SciLean.IsContinuousLinearMap R f) (y : K) : SciLean.IsContinuousLinearMap R fun x => f x / y"} +{"name":"SciLean.IsContinuousLinearMap.isContinuousLinearMap_differentiable","declaration":"theorem SciLean.IsContinuousLinearMap.isContinuousLinearMap_differentiable {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_3} [NormedAddCommGroup Y] [NormedSpace K Y] (f : X → Y) (hf : SciLean.IsContinuousLinearMap K f) : Differentiable K f"} +{"name":"SciLean.IsContinuousLinearMap.Neg.neg.arg_a0.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.Neg.neg.arg_a0.IsContinuousLinearMap_rule {R : Type u_1} [Ring R] {X : Type u_2} [TopologicalSpace X] [AddCommGroup X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommGroup Y] [Module R Y] [TopologicalAddGroup Y] (f : X → Y) (hf : SciLean.IsContinuousLinearMap R f) : SciLean.IsContinuousLinearMap R fun x => -f x"} +{"name":"SciLean.IsContinuousLinearMap.apply_rule","declaration":"theorem SciLean.IsContinuousLinearMap.apply_rule {R : Type u_1} [Semiring R] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → TopologicalSpace (E i)] [(i : ι) → AddCommMonoid (E i)] [(i : ι) → Module R (E i)] (i : ι) : SciLean.IsContinuousLinearMap R fun f => f i"} +{"name":"SciLean.IsContinuousLinearMap","declaration":"structure SciLean.IsContinuousLinearMap (R : Type u_1) [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] (f : X → Y) : Prop"} +{"name":"SciLean.IsContinuousLinearMap.starRingEnd.arg_a.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.starRingEnd.arg_a.IsContinuousLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module K X] (f : X → K) : SciLean.IsContinuousLinearMap K f → SciLean.IsContinuousLinearMap K fun x => (starRingEnd K) (f x)"} +{"name":"SciLean.IsContinuousLinearMap.HMul.hMul.arg_a1.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.HMul.hMul.arg_a1.IsContinuousLinearMap_rule {R : Type u_1} [CommSemiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [Semiring Y] [Algebra R Y] [TopologicalSemiring Y] (f : X → Y) (hf : SciLean.IsContinuousLinearMap R f) (y' : Y) : SciLean.IsContinuousLinearMap R fun x => y' * f x"} +{"name":"SciLean.IsContinuousLinearMap.HAdd.hAdd.arg_a0a1.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.HAdd.hAdd.arg_a0a1.IsContinuousLinearMap_rule {R : Type u_2} [Semiring R] {X : Type u_3} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_1} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] [ContinuousAdd Y] (f : X → Y) (g : X → Y) (hf : SciLean.IsContinuousLinearMap R f) (hg : SciLean.IsContinuousLinearMap R g) : SciLean.IsContinuousLinearMap R fun x => f x + g x"} +{"name":"SciLean.IsContinuousLinearMap.Inner.inner.arg_a0.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.Inner.inner.arg_a0.IsContinuousLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module K X] {Y : Type u_3} [NormedAddCommGroup Y] [InnerProductSpace K Y] [CompleteSpace Y] (f : X → Y) : SciLean.IsContinuousLinearMap K f → ∀ (y : Y), SciLean.IsContinuousLinearMap K fun x => ⟪f x, y⟫_K"} +{"name":"SciLean.IsContinuousLinearMap.Prod.fst.arg_self.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.Prod.fst.arg_self.IsContinuousLinearMap_rule {R : Type u_3} [Semiring R] {X : Type u_4} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_1} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {Z : Type u_2} [TopologicalSpace Z] [AddCommMonoid Z] [Module R Z] (f : X → Y × Z) (hf : SciLean.IsContinuousLinearMap R f) : SciLean.IsContinuousLinearMap R fun x => (f x).1"} +{"name":"SciLean.IsContinuousLinearMap.ContinuousLinearMap.smulLeft","declaration":"/-- Creates `fun x =>L[R] r • g x` -/\ndef SciLean.IsContinuousLinearMap.ContinuousLinearMap.smulLeft {R : Type u_1} [CommSemiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] [ContinuousConstSMul R Y] (g : X →L[R] Y) (r : R) : X →L[R] Y"} +{"name":"SciLean.IsContinuousLinearMap.ContinuousLinearMap.div_right","declaration":"def SciLean.IsContinuousLinearMap.ContinuousLinearMap.div_right {R : Type u_1} [NontriviallyNormedField R] {K : Type u_2} [NontriviallyNormedField K] [NormedAlgebra R K] {X : Type u_3} [TopologicalSpace X] [AddCommMonoid X] [Module R X] (g : X →L[R] K) (y : K) (x : X) : (SciLean.IsContinuousLinearMap.ContinuousLinearMap.divRight g y) x = g x / y"} +{"name":"SciLean.unexpandContinuousLinearMapMk","declaration":"def SciLean.unexpandContinuousLinearMapMk : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.IsContinuousLinearMap.HMul.hMul.arg_a0.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.HMul.hMul.arg_a0.IsContinuousLinearMap_rule {R : Type u_1} [CommSemiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [Semiring Y] [Algebra R Y] [TopologicalSemiring Y] (f : X → Y) (hf : SciLean.IsContinuousLinearMap R f) (y' : Y) : SciLean.IsContinuousLinearMap R fun x => f x * y'"} +{"name":"SciLean.IsContinuousLinearMap.mk","declaration":"ctor SciLean.IsContinuousLinearMap.mk {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {f : X → Y} (linear : IsLinearMap R f) (cont : autoParam (Continuous f) _auto✝) : SciLean.IsContinuousLinearMap R f"} +{"name":"SciLean.IsContinuousLinearMap.Inner.inner.arg_a1.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.Inner.inner.arg_a1.IsContinuousLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module K X] {Y : Type u_3} [NormedAddCommGroup Y] [InnerProductSpace K Y] [CompleteSpace Y] (f : X → Y) : SciLean.IsContinuousLinearMap K f → ∀ (y : Y), SciLean.IsContinuousLinearMap K fun x => ⟪y, f x⟫_K"} +{"name":"SciLean.ContinuousLinearMap.mk'_eval","declaration":"theorem SciLean.ContinuousLinearMap.mk'_eval (R : Type u_1) [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] (x : X) (f : X → Y) (hf : SciLean.IsContinuousLinearMap R f) : (SciLean.ContinuousLinearMap.mk' R f hf) x = f x"} +{"name":"SciLean.ContinuousLinearMap.mk'","declaration":"def SciLean.ContinuousLinearMap.mk' (R : Type u_1) [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] (f : X → Y) (hf : SciLean.IsContinuousLinearMap R f) : X →L[R] Y"} +{"name":"SciLean.IsContinuousLinearMap.comp_rule","declaration":"theorem SciLean.IsContinuousLinearMap.comp_rule {R : Type u_1} [Semiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] {Z : Type u_4} [TopologicalSpace Z] [AddCommMonoid Z] [Module R Z] (g : X → Y) (hg : SciLean.IsContinuousLinearMap R g) (f : Y → Z) (hf : SciLean.IsContinuousLinearMap R f) : SciLean.IsContinuousLinearMap R fun x => f (g x)"} +{"name":"SciLean.IsContinuousLinearMap.ContinuousLinearMap.smul_left","declaration":"def SciLean.IsContinuousLinearMap.ContinuousLinearMap.smul_left {R : Type u_1} [CommSemiring R] {X : Type u_2} [TopologicalSpace X] [AddCommMonoid X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommMonoid Y] [Module R Y] [ContinuousConstSMul R Y] (g : X →L[R] Y) (r : R) (x : X) : (SciLean.IsContinuousLinearMap.ContinuousLinearMap.smulLeft g r) x = r • g x"} +{"name":"SciLean.IsContinuousLinearMap.HSub.hSub.arg_a0a1.IsContinuousLinearMap_rule","declaration":"theorem SciLean.IsContinuousLinearMap.HSub.hSub.arg_a0a1.IsContinuousLinearMap_rule {R : Type u_1} [Ring R] {X : Type u_2} [TopologicalSpace X] [AddCommGroup X] [Module R X] {Y : Type u_3} [TopologicalSpace Y] [AddCommGroup Y] [Module R Y] [TopologicalAddGroup Y] (f : X → Y) (g : X → Y) (hf : SciLean.IsContinuousLinearMap R f) (hg : SciLean.IsContinuousLinearMap R g) : SciLean.IsContinuousLinearMap R fun x => f x - g x"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.IsLinearMap.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.IsLinearMap.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e1519fdf11b4489d8e9b8bb8c955e5abbf3a73ba --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.IsLinearMap.jsonl @@ -0,0 +1,33 @@ +{"name":"SciLean.DualBasis.dualProj.arg_x.IsLinearMap_rule","declaration":"theorem SciLean.DualBasis.dualProj.arg_x.IsLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : IsLinearMap K fun x => ℼ' i x"} +{"name":"IsLinearMap.isLinearMap_const_zero","declaration":"theorem IsLinearMap.isLinearMap_const_zero {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] : IsLinearMap R fun x => 0"} +{"name":"IsLinearMap.LinearMap_coe.apply_right","declaration":"theorem IsLinearMap.LinearMap_coe.apply_right {R : Type u_3} {X : Type u_4} {Y : Type u_2} {Z : Type u_1} [CommSemiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Y →ₗ[R] Z) (y : Y) (hf : IsLinearMap R f) : IsLinearMap R fun x => (f x) y"} +{"name":"Prod.mk.arg_fstsnd.IsLinearMap_rule","declaration":"theorem Prod.mk.arg_fstsnd.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_4} {Z : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Z) (g : X → Y) (hf : IsLinearMap R f) (hg : IsLinearMap R g) : IsLinearMap R fun x => (g x, f x)"} +{"name":"HAdd.hAdd.arg_a0a1.IsLinearMap_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → Y) (g : X → Y) (hf : IsLinearMap R f) (hg : IsLinearMap R g) : IsLinearMap R fun x => f x + g x"} +{"name":"Inner.inner.arg_a0.IsLinearMap_rule","declaration":"theorem Inner.inner.arg_a0.IsLinearMap_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiHilbert R X] (y : X) : IsLinearMap R fun x => ⟪x, y⟫_R"} +{"name":"IndexType.sum.arg_f.IsLinearMap_rule","declaration":"theorem IndexType.sum.arg_f.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} {ι : Type u_4} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), IsLinearMap R fun x => f x i) : IsLinearMap R fun x => ∑ i, f x i"} +{"name":"IsLinearMap.isLinearMap_apply","declaration":"theorem IsLinearMap.isLinearMap_apply {R : Type u_1} {ι : Type u_2} {E : ι → Type u_3} [Semiring R] [(i : ι) → AddCommMonoid (E i)] [(i : ι) → Module R (E i)] (i : ι) : IsLinearMap R fun f => f i"} +{"name":"dite.arg_te.IsLinearMap_rule","declaration":"theorem dite.arg_te.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (ht : ∀ (p : c), IsLinearMap R (t p)) (e : ¬c → X → Y) (he : ∀ (p : ¬c), IsLinearMap R (e p)) : IsLinearMap R fun x => if x_1 : c then t x_1 x else e x_1 x"} +{"name":"SciLean.BasisDuality.toDual.arg_x.IsLinearMap_rule","declaration":"theorem SciLean.BasisDuality.toDual.arg_x.IsLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : IsLinearMap K fun x => SciLean.BasisDuality.toDual x"} +{"name":"HSMul.hSMul.arg_a1.IsLinearMap_rule","declaration":"theorem HSMul.hSMul.arg_a1.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [CommSemiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : R) (f : X → Y) (hf : IsLinearMap R f) : IsLinearMap R fun x => c • f x"} +{"name":"HSMul.hSMul.arg_a1.IsLinearMap_rule_nat","declaration":"theorem HSMul.hSMul.arg_a1.IsLinearMap_rule_nat {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : ℕ) (f : X → Y) (hf : IsLinearMap R f) : IsLinearMap R fun x => c • f x"} +{"name":"HMul.hMul.arg_a1.IsLinearMap_rule","declaration":"theorem HMul.hMul.arg_a1.IsLinearMap_rule {R : Type u_1} {X : Type u_2} [CommSemiring R] [AddCommGroup X] [Module R X] (f : X → R) (hf : IsLinearMap R f) (y' : R) : IsLinearMap R fun x => y' * f x"} +{"name":"HSMul.hSMul.arg_a0.IsLinearMap_rule","declaration":"theorem HSMul.hSMul.arg_a0.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → R) (y : Y) (hf : IsLinearMap R f) : IsLinearMap R fun x => f x • y"} +{"name":"ite.arg_te.IsLinearMap_rule","declaration":"theorem ite.arg_te.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) (ht : IsLinearMap R t) (he : IsLinearMap R e) : IsLinearMap R fun x => if c then t x else e x"} +{"name":"Prod.fst.arg_self.IsLinearMap_rule","declaration":"theorem Prod.fst.arg_self.IsLinearMap_rule {R : Type u_3} {X : Type u_4} {Y : Type u_1} {Z : Type u_2} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Y × Z) (hf : IsLinearMap R f) : IsLinearMap R fun x => (f x).1"} +{"name":"Prod.snd.arg_self.IsLinearMap_rule","declaration":"theorem Prod.snd.arg_self.IsLinearMap_rule {R : Type u_3} {X : Type u_4} {Y : Type u_1} {Z : Type u_2} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Y × Z) (hf : IsLinearMap R f) : IsLinearMap R fun x => (f x).2"} +{"name":"SciLean.Basis.proj.arg_x.IsLinearMap_rule","declaration":"theorem SciLean.Basis.proj.arg_x.IsLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : IsLinearMap K fun x => ℼ i x"} +{"name":"Prod.snd.arg_self.IsLinearMap_rule_simple","declaration":"theorem Prod.snd.arg_self.IsLinearMap_rule_simple {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] : IsLinearMap R fun xy => xy.2"} +{"name":"IsLinearMap.isLinearMap_id","declaration":"theorem IsLinearMap.isLinearMap_id {R : Type u_1} {X : Type u_2} [Semiring R] [AddCommGroup X] [Module R X] : IsLinearMap R fun x => x"} +{"name":"HMul.hMul.arg_a0.IsLinearMap_rule","declaration":"theorem HMul.hMul.arg_a0.IsLinearMap_rule {R : Type u_1} {X : Type u_2} [CommSemiring R] [AddCommGroup X] [Module R X] (f : X → R) (hf : IsLinearMap R f) (y' : R) : IsLinearMap R fun x => f x * y'"} +{"name":"IsLinearMap.LinearMap_coe.apply_left","declaration":"theorem IsLinearMap.LinearMap_coe.apply_left {R : Type u_3} {X : Type u_4} {Y : Type u_2} {Z : Type u_1} [CommSemiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : Y →ₗ[R] Z) (g : X → Y) (hg : IsLinearMap R g) : IsLinearMap R fun x => f (g x)"} +{"name":"Inner.inner.arg_a1.IsLinearMap_rule","declaration":"theorem Inner.inner.arg_a1.IsLinearMap_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiHilbert R X] (y : X) : IsLinearMap R fun x => ⟪y, x⟫_R"} +{"name":"IsLinearMap.isLinearMap_pi","declaration":"theorem IsLinearMap.isLinearMap_pi {R : Type u_1} {X : Type u_2} {ι : Type u_4} {E : ι → Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [(i : ι) → AddCommMonoid (E i)] [(i : ι) → Module R (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), IsLinearMap R fun x => f x i) : IsLinearMap R fun x i => f x i"} +{"name":"by_linear_map","declaration":"theorem by_linear_map {R : Type u_3} {X : Type u_2} {Y : Type u_1} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] {f : X → Y} (g : X →ₗ[R] Y) (h : ∀ (x : X), f x = g x) : IsLinearMap R f"} +{"name":"IsLinearMap.mk'.arg_f.IsLinearMap_rule","declaration":"theorem IsLinearMap.mk'.arg_f.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_4} {Z : Type u_3} [CommSemiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] (f : X → Y → Z) (hf₁ : ∀ (y : Y), IsLinearMap R fun x => f x y) (hf₂ : ∀ (x : X), IsLinearMap R fun x_1 => f x x_1) : IsLinearMap R fun x => IsLinearMap.mk' (f x) ⋯"} +{"name":"Prod.fst.arg_self.IsLinearMap_rule_simple","declaration":"theorem Prod.fst.arg_self.IsLinearMap_rule_simple {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] : IsLinearMap R fun xy => xy.1"} +{"name":"Prod.mk.arg_fstsnd.IsLinearMap_rule_simple","declaration":"theorem Prod.mk.arg_fstsnd.IsLinearMap_rule_simple {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] : IsLinearMap R fun xy => (xy.1, xy.2)"} +{"name":"IsLinearMap.isLinearMap_comp","declaration":"theorem IsLinearMap.isLinearMap_comp {R : Type u_1} {X : Type u_4} {Y : Type u_2} {Z : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] {f : Y → Z} {g : X → Y} (hf : IsLinearMap R f) (hg : IsLinearMap R g) : IsLinearMap R fun x => f (g x)"} +{"name":"HSub.hSub.arg_a0a1.IsLinearMap_rule","declaration":"theorem HSub.hSub.arg_a0a1.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → Y) (g : X → Y) (hf : IsLinearMap R f) (hg : IsLinearMap R g) : IsLinearMap R fun x => f x - g x"} +{"name":"HSMul.hSMul.arg_a1.IsLinearMap_rule_int","declaration":"theorem HSMul.hSMul.arg_a1.IsLinearMap_rule_int {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (c : ℤ) (f : X → Y) (hf : IsLinearMap R f) : IsLinearMap R fun x => c • f x"} +{"name":"SciLean.BasisDuality.fromDual.arg_x.IsLinearMap_rule","declaration":"theorem SciLean.BasisDuality.fromDual.arg_x.IsLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : IsLinearMap K fun x => SciLean.BasisDuality.fromDual x"} +{"name":"Neg.neg.arg_a0.IsLinearMap_rule","declaration":"theorem Neg.neg.arg_a0.IsLinearMap_rule {R : Type u_1} {X : Type u_2} {Y : Type u_3} [Semiring R] [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] (f : X → Y) (hf : IsLinearMap R f) : IsLinearMap R fun x => -f x"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.IsSmoothLinearMap.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.IsSmoothLinearMap.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a3d59dcec5d13be772d9f0ea547519a64188c3fb --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionPropositions.IsSmoothLinearMap.jsonl @@ -0,0 +1,27 @@ +{"name":"SciLean.IsSmoothLinearMap","declaration":"def SciLean.IsSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) : Prop"} +{"name":"SciLean.IsSmoothLinearMap.HMul.hMul.arg_a0.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.HMul.hMul.arg_a0.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (hf : SciLean.IsSmoothLinearMap K f) (y' : K) : SciLean.IsSmoothLinearMap K fun x => f x * y'"} +{"name":"SciLean.IsSmoothLinearMap.pi_rule","declaration":"theorem SciLean.IsSmoothLinearMap.pi_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.IsSmoothLinearMap K fun x => f x i) : SciLean.IsSmoothLinearMap K fun x i => f x i"} +{"name":"SciLean.IsSmoothLinearMap.ite.arg_te.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.ite.arg_te.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) (ht : SciLean.IsSmoothLinearMap K t) (he : SciLean.IsSmoothLinearMap K e) : SciLean.IsSmoothLinearMap K fun x => if c then t x else e x"} +{"name":"SciLean.IsSmoothLinearMap.const_zero_rule","declaration":"theorem SciLean.IsSmoothLinearMap.const_zero_rule (K : Type u_1) [RCLike K] (X : Type u_2) [SciLean.Vec K X] (Y : Type u_3) [SciLean.Vec K Y] : SciLean.IsSmoothLinearMap K fun x => 0"} +{"name":"SciLean.IsSmoothLinearMap.isLinearMap","declaration":"theorem SciLean.IsSmoothLinearMap.isLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {f : X → Y} (hf : SciLean.IsSmoothLinearMap K f) : IsLinearMap K f"} +{"name":"SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a1.IsSmoothLinearMap_rule_nat","declaration":"theorem SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a1.IsSmoothLinearMap_rule_nat {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℕ) (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun x => c • f x"} +{"name":"SciLean.IsSmoothLinearMap.cdifferentiable","declaration":"theorem SciLean.IsSmoothLinearMap.cdifferentiable (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {f : X → Y} (hf : SciLean.IsSmoothLinearMap K f) : SciLean.CDifferentiable K f"} +{"name":"SciLean.IsSmoothLinearMap.Prod.fst.arg_self.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.Prod.fst.arg_self.IsSmoothLinearMap_rule {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun x => (f x).1"} +{"name":"SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a1.IsSmoothLinearMap_rule_int","declaration":"theorem SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a1.IsSmoothLinearMap_rule_int {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : ℤ) (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun x => c • f x"} +{"name":"SciLean.IsSmoothLinearMap.dite.arg_te.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.dite.arg_te.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (ht : ∀ (p : c), SciLean.IsSmoothLinearMap K (t p)) (e : ¬c → X → Y) (he : ∀ (p : ¬c), SciLean.IsSmoothLinearMap K (e p)) : SciLean.IsSmoothLinearMap K fun x => if x_1 : c then t x_1 x else e x_1 x"} +{"name":"SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a1.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a1.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : K) (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun x => c • f x"} +{"name":"SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a0.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.HSMul.hSMul.arg_a0.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → K) (y : Y) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun x => f x • y"} +{"name":"SciLean.IsSmoothLinearMap.HAdd.hAdd.arg_a0a1.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.HAdd.hAdd.arg_a0a1.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.IsSmoothLinearMap K f) (hg : SciLean.IsSmoothLinearMap K g) : SciLean.IsSmoothLinearMap K fun x => f x + g x"} +{"name":"SciLean.IsSmoothLinearMap.SciLean.BasisDuality.fromDual.arg_x.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.SciLean.BasisDuality.fromDual.arg_x.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : SciLean.IsSmoothLinearMap K fun x => SciLean.BasisDuality.fromDual x"} +{"name":"SciLean.IsSmoothLinearMap.SciLean.Basis.proj.arg_x.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.SciLean.Basis.proj.arg_x.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : SciLean.IsSmoothLinearMap K fun x => ℼ i x"} +{"name":"SciLean.IsSmoothLinearMap.id_rule","declaration":"theorem SciLean.IsSmoothLinearMap.id_rule (K : Type u_1) [RCLike K] (X : Type u_2) [SciLean.Vec K X] : SciLean.IsSmoothLinearMap K fun x => x"} +{"name":"SciLean.IsSmoothLinearMap.Prod.mk.arg_fstsnd.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.Prod.mk.arg_fstsnd.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_4} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : X → Z) (g : X → Y) (hf : SciLean.IsSmoothLinearMap K f) (hg : SciLean.IsSmoothLinearMap K g) : SciLean.IsSmoothLinearMap K fun x => (g x, f x)"} +{"name":"SciLean.IsSmoothLinearMap.HMul.hMul.arg_a1.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.HMul.hMul.arg_a1.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (hf : SciLean.IsSmoothLinearMap K f) (y' : K) : SciLean.IsSmoothLinearMap K fun x => y' * f x"} +{"name":"SciLean.IsSmoothLinearMap.HSub.hSub.arg_a0a1.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.HSub.hSub.arg_a0a1.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.IsSmoothLinearMap K f) (hg : SciLean.IsSmoothLinearMap K g) : SciLean.IsSmoothLinearMap K fun x => f x - g x"} +{"name":"SciLean.IsSmoothLinearMap.SciLean.BasisDuality.toDual.arg_x.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.SciLean.BasisDuality.toDual.arg_x.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] : SciLean.IsSmoothLinearMap K fun x => SciLean.BasisDuality.toDual x"} +{"name":"SciLean.IsSmoothLinearMap.apply_rule","declaration":"theorem SciLean.IsSmoothLinearMap.apply_rule (K : Type u_1) [RCLike K] {ι : Type u_2} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (i : ι) : SciLean.IsSmoothLinearMap K fun x => x i"} +{"name":"SciLean.IsSmoothLinearMap.IndexType.sum.arg_f.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.IndexType.sum.arg_f.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.IsSmoothLinearMap K fun x => f x i) : SciLean.IsSmoothLinearMap K fun x => ∑ i, f x i"} +{"name":"SciLean.IsSmoothLinearMap.Neg.neg.arg_a0.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.Neg.neg.arg_a0.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun x => -f x"} +{"name":"SciLean.IsSmoothLinearMap.comp_rule","declaration":"theorem SciLean.IsSmoothLinearMap.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.IsSmoothLinearMap K f) (hg : SciLean.IsSmoothLinearMap K g) : SciLean.IsSmoothLinearMap K fun x => f (g x)"} +{"name":"SciLean.IsSmoothLinearMap.SciLean.DualBasis.dualProj.arg_x.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.SciLean.DualBasis.dualProj.arg_x.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {IX : Type u_3} [SciLean.IndexType IX] [SciLean.LawfulIndexType IX] [DecidableEq IX] {X : Type u_2} [SciLean.FinVec IX K X] (i : IX) : SciLean.IsSmoothLinearMap K fun x => ℼ' i x"} +{"name":"SciLean.IsSmoothLinearMap.Prod.snd.arg_self.IsSmoothLinearMap_rule","declaration":"theorem SciLean.IsSmoothLinearMap.Prod.snd.arg_self.IsSmoothLinearMap_rule {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun x => (f x).2"} diff --git a/scilean-declarations/SciLean.Core.FunctionPropositions.jsonl b/scilean-declarations/SciLean.Core.FunctionPropositions.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.FunctionSpaces.ContCDiffMap.jsonl b/scilean-declarations/SciLean.Core.FunctionSpaces.ContCDiffMap.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..6d224f4be93f90d948a190ef77c9ff377490f474 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionSpaces.ContCDiffMap.jsonl @@ -0,0 +1,40 @@ +{"name":"SciLean.ContCDiffMap_apply_right","declaration":"theorem SciLean.ContCDiffMap_apply_right (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿[K,n] Y) : SciLean.ContCDiff K n fun x => f x"} +{"name":"SciLean.instVecContCDiffMap","declaration":"instance SciLean.instVecContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : SciLean.Vec K (X ⟿[K,n] Y)"} +{"name":"SciLean.ContCDiffMap.mk.arg_f.ContCDiff_rule","declaration":"theorem SciLean.ContCDiffMap.mk.arg_f.ContCDiff_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (f : X → Y → Z) (hf : SciLean.ContCDiff K ⊤ fun xy => f xy.1 xy.2) : SciLean.ContCDiff K ⊤ fun x => fun y ⟿[K,⊤] f x y"} +{"name":"SciLean.ContCDiffMap_eval_CDifferentiable'","declaration":"theorem SciLean.ContCDiffMap_eval_CDifferentiable' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : SciLean.CDifferentiable K fun fx => fx.1 fx.2"} +{"name":"SciLean.«term_⟿[_]_»","declaration":"def SciLean.«term_⟿[_]_» : Lean.TrailingParserDescr"} +{"name":"SciLean.ContCDiffMap.smul_apply","declaration":"theorem SciLean.ContCDiffMap.smul_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿[K,n] Y) (x : X) (r : K) : (r • f) x = r • f x"} +{"name":"SciLean.instUniformSpaceContCDiffMap","declaration":"instance SciLean.instUniformSpaceContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : UniformSpace (X ⟿[K,n] Y)"} +{"name":"SciLean.instAddContCDiffMap","declaration":"instance SciLean.instAddContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : Add (X ⟿[K,n] Y)"} +{"name":"SciLean.instNegContCDiffMap","declaration":"instance SciLean.instNegContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : Neg (X ⟿[K,n] Y)"} +{"name":"SciLean.ContCDiffMap.mk'","declaration":"def SciLean.ContCDiffMap.mk' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (f : X → Y) (hf : SciLean.ContCDiff K n f) : X ⟿[K,n] Y"} +{"name":"SciLean.ContCDiffMap.toFun","declaration":"def SciLean.ContCDiffMap.toFun {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⟿[K,n] Y) : X → Y"} +{"name":"SciLean.ContCDiffMap_partial","declaration":"theorem SciLean.ContCDiffMap_partial (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (n : ℕ) (f : W → X ⟿[K,↑n] Y) (g : W → X) (hf : SciLean.ContCDiff K (↑n) f) (hg : SciLean.ContCDiff K (↑n) g) : SciLean.ContCDiff K ↑n fun w => (f w) (g w)"} +{"name":"SciLean.«term_⟿[_,_]_»","declaration":"def SciLean.«term_⟿[_,_]_» : Lean.TrailingParserDescr"} +{"name":"SciLean.ContCDiffMap_apply_CDifferentiable","declaration":"theorem SciLean.ContCDiffMap_apply_CDifferentiable (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⟿[K,⊤] Y) (g : W → X) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun w => (f w) (g w)"} +{"name":"SciLean.ContCDiffMap.sub_apply","declaration":"theorem SciLean.ContCDiffMap.sub_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿[K,n] Y) (g : X ⟿[K,n] Y) (x : X) : (f - g) x = f x - g x"} +{"name":"SciLean.«term_⟿_»","declaration":"def SciLean.«term_⟿_» : Lean.TrailingParserDescr"} +{"name":"SciLean.ContCDiffMap","declaration":"structure SciLean.ContCDiffMap (K : Type u_1) [RCLike K] (n : ℕ∞) (X : Type u_2) (Y : Type u_3) [SciLean.Vec K X] [SciLean.Vec K Y] : Type (max u_2 u_3)"} +{"name":"SciLean.instSubContCDiffMap","declaration":"instance SciLean.instSubContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : Sub (X ⟿[K,n] Y)"} +{"name":"SciLean.ContCDiffMap.mk'_eval","declaration":"theorem SciLean.ContCDiffMap.mk'_eval {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (x : X) (f : X → Y) (hf : SciLean.ContCDiff K n f) : { toFun := f, is_cont_cdiff_map := hf } x = f x"} +{"name":"SciLean.ContCDiffMap.mk.arg_f.IsSmoothLinearMap_rule","declaration":"theorem SciLean.ContCDiffMap.mk.arg_f.IsSmoothLinearMap_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_4} [SciLean.Vec K Y] {W : Type u_3} [SciLean.Vec K W] (n : ℕ∞) (f : W → X → Y) (hf₁ : SciLean.CDifferentiable K fun x =>\n match x with\n | (w, x) => f w x) (hf₂ : IsLinearMap K f) (hf₃ : ∀ (w : W), SciLean.ContCDiff K n (f w)) : SciLean.IsSmoothLinearMap K fun w => fun x ⟿[K,n] f w x"} +{"name":"SciLean.unexpandContCDiffMap","declaration":"def SciLean.unexpandContCDiffMap : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.instFunLikeContCDiffMap","declaration":"instance SciLean.instFunLikeContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : FunLike (X ⟿[K,n] Y) X Y"} +{"name":"SciLean.ContCDiffMap_apply_linearSmoothMap","declaration":"theorem SciLean.ContCDiffMap_apply_linearSmoothMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (n : ℕ∞) (f : W → X ⟿[K,n] Y) (hf : SciLean.IsSmoothLinearMap K f) (x : X) : SciLean.IsSmoothLinearMap K fun w => (f w) x"} +{"name":"SciLean.instZeroContCDiffMap","declaration":"instance SciLean.instZeroContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : Zero (X ⟿[K,n] Y)"} +{"name":"SciLean.ContCDiffMap.is_cont_cdiff_map","declaration":"def SciLean.ContCDiffMap.is_cont_cdiff_map {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⟿[K,n] Y) : SciLean.ContCDiff K n self.toFun"} +{"name":"SciLean.ContCDiffMap_eta","declaration":"theorem SciLean.ContCDiffMap_eta (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿[K,n] Y) : (fun x ⟿[K,n] f x) = f"} +{"name":"SciLean.unexpandContCDiffMapMk","declaration":"def SciLean.unexpandContCDiffMapMk : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.ContCDiffMap_apply_CDifferentiableAt","declaration":"theorem SciLean.ContCDiffMap_apply_CDifferentiableAt (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⟿[K,⊤] Y) (g : W → X) (w : W) (hf : SciLean.CDifferentiableAt K f w) (hg : SciLean.CDifferentiableAt K g w) : SciLean.CDifferentiableAt K (fun w => (f w) (g w)) w"} +{"name":"SciLean.ContCDiffMap_eval_CDifferentiable","declaration":"theorem SciLean.ContCDiffMap_eval_CDifferentiable (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (h : 0 < n) : SciLean.CDifferentiable K fun fx => fx.1 fx.2"} +{"name":"SciLean.instSMulContCDiffMap","declaration":"instance SciLean.instSMulContCDiffMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : SMul K (X ⟿[K,n] Y)"} +{"name":"SciLean.ContCDiffMap.mk","declaration":"ctor SciLean.ContCDiffMap.mk {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (toFun : X → Y) (is_cont_cdiff_map : SciLean.ContCDiff K n toFun) : X ⟿[K,n] Y"} +{"name":"SciLean.«termFun_⟿_»","declaration":"def SciLean.«termFun_⟿_» : Lean.ParserDescr"} +{"name":"SciLean.ContCDiffMap.zero_apply","declaration":"theorem SciLean.ContCDiffMap.zero_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (n : ℕ∞) (x : X) : 0 x = 0"} +{"name":"SciLean.ContCDiffMap.neg_apply","declaration":"theorem SciLean.ContCDiffMap.neg_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿[K,n] Y) (x : X) : (-f) x = -f x"} +{"name":"SciLean.ContCDiffMap.mk.arg_f.ContCDiff_rule_partial","declaration":"theorem SciLean.ContCDiffMap.mk.arg_f.ContCDiff_rule_partial (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (n : ℕ) (l : ℕ) (k : ℕ) (f : X → Y → Z) (hf : SciLean.ContCDiff K ↑n fun xy => f xy.1 xy.2) (h : l + k ≤ n) : SciLean.ContCDiff K ↑k fun x => fun y ⟿[K,↑l] f x y"} +{"name":"SciLean.ContCDiffMap_apply","declaration":"theorem SciLean.ContCDiffMap_apply (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⟿[K,⊤] Y) (g : W → X) (hf : SciLean.ContCDiff K ⊤ f) (hg : SciLean.ContCDiff K ⊤ g) : SciLean.ContCDiff K ⊤ fun w => (f w) (g w)"} +{"name":"SciLean.«termFun_⟿[_,_]_»","declaration":"def SciLean.«termFun_⟿[_,_]_» : Lean.ParserDescr"} +{"name":"SciLean.ContCDiffMap.eta_reduce","declaration":"theorem SciLean.ContCDiffMap.eta_reduce {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿[K,n] Y) : { toFun := f.toFun, is_cont_cdiff_map := ⋯ } = f"} +{"name":"SciLean.ContCDiffMap.add_apply","declaration":"theorem SciLean.ContCDiffMap.add_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿[K,n] Y) (g : X ⟿[K,n] Y) (x : X) : (f + g) x = f x + g x"} +{"name":"SciLean.«termFun_⟿[_]_»","declaration":"def SciLean.«termFun_⟿[_]_» : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.FunctionSpaces.ContCDiffMapFD.jsonl b/scilean-declarations/SciLean.Core.FunctionSpaces.ContCDiffMapFD.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1a9350f78f51e7c8c5c030f1723f75228d653b0a --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionSpaces.ContCDiffMapFD.jsonl @@ -0,0 +1,47 @@ +{"name":"SciLean.ContCDiffMapFD.dir_independence","declaration":"def SciLean.ContCDiffMapFD.dir_independence {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⟿FD[K,n] Y) (x : X) (dx : X) : self.toFun x dx = self.toFun x 0"} +{"name":"SciLean.ContCDiffMapFD_apply_linearSmoothMap","declaration":"theorem SciLean.ContCDiffMapFD_apply_linearSmoothMap {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] {n : ℕ∞} (f : W → X ⟿FD[K,n] Y) (hf : SciLean.IsSmoothLinearMap K f) (x : X) : SciLean.IsSmoothLinearMap K fun w => (f w) x"} +{"name":"SciLean.ContCDiffMapFD_fwdDeriv_rule","declaration":"theorem SciLean.ContCDiffMapFD_fwdDeriv_rule {K : Type u_4} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] {W : Type u_5} [SciLean.Vec K W] (f : W → (X ⟿FD[K,⊤] Y) × Z) (g : W → X) : (SciLean.fwdDeriv K fun w => (f w).1 (g w)) = fun w dw =>\n let fdfz := SciLean.fwdDeriv K f w dw;\n let xdx := SciLean.fwdDeriv K g w dw;\n let fw := fdfz.1.1;\n let df := fdfz.2.1;\n let zdz := fw.FD xdx.1 xdx.2;\n (zdz.1, df xdx.1 + zdz.2)"} +{"name":"SciLean.ContCDiffMapFD.is_cont_cdiff_map","declaration":"def SciLean.ContCDiffMapFD.is_cont_cdiff_map {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⟿FD[K,n] Y) : SciLean.ContCDiff K n fun x => (self.toFun x 0).1"} +{"name":"SciLean.«termFun_⟿FD[_]_»","declaration":"def SciLean.«termFun_⟿FD[_]_» : Lean.ParserDescr"} +{"name":"SciLean.instSubContCDiffMapFD","declaration":"instance SciLean.instSubContCDiffMapFD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} : Sub (X ⟿FD[K,n] Y)"} +{"name":"SciLean.ContCDiffMapFD_eval_fwdDeriv","declaration":"theorem SciLean.ContCDiffMapFD_eval_fwdDeriv {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) : (SciLean.fwdDeriv K fun x => f x) = f.FD"} +{"name":"SciLean.ContCDiffMapFD.sub_apply","declaration":"theorem SciLean.ContCDiffMapFD.sub_apply {K : Type u_3} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) (g : X ⟿FD[K,n] Y) (x : X) : (f - g) x = f x - g x"} +{"name":"SciLean.ContCDiffMapFD_apply_right","declaration":"theorem SciLean.ContCDiffMapFD_apply_right (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (f : X ⟿FD[K,n] Y) : SciLean.ContCDiff K n fun x => f x"} +{"name":"SciLean.ContCDiffMapFD.mk.arg_f.ContCDiff_rule","declaration":"theorem SciLean.ContCDiffMapFD.mk.arg_f.ContCDiff_rule {K : Type u_2} [RCLike K] {X : Type u_3} [SciLean.Vec K X] {Y : Type u_4} [SciLean.Vec K Y] {Z : Type u_1} [SciLean.Vec K Z] (f : X → Y → Z) (f' : X → Y → Y → Z × Z) (hf : SciLean.ContCDiff K ⊤ fun xy => f xy.1 xy.2) (h : ∀ (x : X), (SciLean.fwdDeriv K fun y => f x y) = f' x) (h' : ∀ (x : X), SciLean.ContCDiff K ⊤ (f x)) : SciLean.ContCDiff K ⊤ fun x => fun y ⟿FD[K,⊤] f x y"} +{"name":"SciLean.«term_⟿FD[_]_»","declaration":"def SciLean.«term_⟿FD[_]_» : Lean.TrailingParserDescr"} +{"name":"SciLean.«term_⟿FD_»","declaration":"def SciLean.«term_⟿FD_» : Lean.TrailingParserDescr"} +{"name":"SciLean.ContCDiffMapFD_apply_CDifferentiableAt","declaration":"theorem SciLean.ContCDiffMapFD_apply_CDifferentiableAt {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⟿FD[K,⊤] Y) (g : W → X) (w : W) (hf : SciLean.CDifferentiableAt K f w) (hg : SciLean.CDifferentiableAt K g w) : SciLean.CDifferentiableAt K (fun w => (f w) (g w)) w"} +{"name":"SciLean.ContCDiffMapFD_eval_cdifferentiable'","declaration":"theorem SciLean.ContCDiffMapFD_eval_cdifferentiable' {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X ⟿FD[K,⊤] Y) : SciLean.CDifferentiable K fun x => f x"} +{"name":"SciLean.instUniformSpaceContCDiffMapFD","declaration":"instance SciLean.instUniformSpaceContCDiffMapFD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} : UniformSpace (X ⟿FD[K,n] Y)"} +{"name":"SciLean.ContCDiffMapFD.neg_apply","declaration":"theorem SciLean.ContCDiffMapFD.neg_apply {K : Type u_3} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) (x : X) : (-f) x = -f x"} +{"name":"SciLean.ContCDiffMapFD_eval_CDifferentiable","declaration":"theorem SciLean.ContCDiffMapFD_eval_CDifferentiable {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} (h : 0 < n) : SciLean.CDifferentiable K fun fx => fx.1 fx.2"} +{"name":"SciLean.ContCDiffMapFD.FD_fst","declaration":"theorem SciLean.ContCDiffMapFD.FD_fst {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) (x : X) (dx : X) : (f.FD x dx).1 = f x"} +{"name":"SciLean.instZeroContCDiffMapFD","declaration":"instance SciLean.instZeroContCDiffMapFD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} : Zero (X ⟿FD[K,n] Y)"} +{"name":"SciLean.ContCDiffMapFD.mk.arg_f.ContCDiff_rule_partial","declaration":"theorem SciLean.ContCDiffMapFD.mk.arg_f.ContCDiff_rule_partial {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (n : ℕ) (l : ℕ) (k : ℕ) (f : X → Y → Z) (hf : SciLean.ContCDiff K ↑n fun xy => f xy.1 xy.2) (h : l + k ≤ n) : SciLean.ContCDiff K ↑k fun x => fun y ⟿FD[K,↑l] f x y"} +{"name":"SciLean.ContCDiffMapFD_partial","declaration":"theorem SciLean.ContCDiffMapFD_partial {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (n : ℕ) (f : W → X ⟿FD[K,↑n] Y) (g : W → X) (hf : SciLean.ContCDiff K (↑n) f) (hg : SciLean.ContCDiff K (↑n) g) : SciLean.ContCDiff K ↑n fun w => (f w) (g w)"} +{"name":"SciLean.ContCDiffMapFD.mk","declaration":"ctor SciLean.ContCDiffMapFD.mk {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (toFun : X → X → Y × Y) (is_cont_cdiff_map : SciLean.ContCDiff K n fun x => (toFun x 0).1) (cderiv_snd : ∂ x, (toFun x 0).1 = fun x dx => (toFun x dx).2) (dir_independence : ∀ (x dx : X), toFun x dx = toFun x 0) : X ⟿FD[K,n] Y"} +{"name":"SciLean.ContCDiffMapFD_cderiv_rule","declaration":"theorem SciLean.ContCDiffMapFD_cderiv_rule {K : Type u_4} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] {W : Type u_5} [SciLean.Vec K W] (f : W → (X ⟿FD[K,⊤] Y) × Z) (g : W → X) : ∂ w, (f w).1 (g w) = fun w dw =>\n let dfz := ∂ f w dw;\n let fwz := f w;\n let x := g w;\n let dx := ∂ g w dw;\n dfz.1 x + (fwz.1.FD x dx).2"} +{"name":"SciLean.instNegContCDiffMapFD","declaration":"instance SciLean.instNegContCDiffMapFD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} : Neg (X ⟿FD[K,n] Y)"} +{"name":"SciLean.ContCDiffMapFD_apply","declaration":"theorem SciLean.ContCDiffMapFD_apply {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⟿FD[K,⊤] Y) (g : W → X) (hf : SciLean.ContCDiff K ⊤ f) (hg : SciLean.ContCDiff K ⊤ g) : SciLean.ContCDiff K ⊤ fun w => (f w) (g w)"} +{"name":"SciLean.instSMulContCDiffMapFD","declaration":"instance SciLean.instSMulContCDiffMapFD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} : SMul K (X ⟿FD[K,n] Y)"} +{"name":"SciLean.ContCDiffMapFD.add_apply","declaration":"theorem SciLean.ContCDiffMapFD.add_apply {K : Type u_3} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) (g : X ⟿FD[K,n] Y) (x : X) : (f + g) x = f x + g x"} +{"name":"SciLean.instFunLikeContCDiffMapFD","declaration":"instance SciLean.instFunLikeContCDiffMapFD (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) : FunLike (X ⟿FD[K,n] Y) X Y"} +{"name":"SciLean.ContCDiffMapFD.toFun","declaration":"def SciLean.ContCDiffMapFD.toFun {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⟿FD[K,n] Y) : X → X → Y × Y"} +{"name":"SciLean.«termFun_⟿FD[_,_]_»","declaration":"def SciLean.«termFun_⟿FD[_,_]_» : Lean.ParserDescr"} +{"name":"SciLean.ContCDiffMapFD_eval_cdifferentiable","declaration":"theorem SciLean.ContCDiffMapFD_eval_cdifferentiable {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) (h : 0 < n) : SciLean.CDifferentiable K fun x => f x"} +{"name":"SciLean.ContCDiffMapFD.mk'","declaration":"def SciLean.ContCDiffMapFD.mk' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (n : ℕ∞) (f : X → Y) (f' : X → X → Y × Y) (h : SciLean.fwdDeriv K f = f') (hf : SciLean.ContCDiff K n f) : X ⟿FD[K,n] Y"} +{"name":"SciLean.unexpandContCDiffMapFDMk","declaration":"def SciLean.unexpandContCDiffMapFDMk : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.instVecContCDiffMapFD","declaration":"instance SciLean.instVecContCDiffMapFD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} : SciLean.Vec K (X ⟿FD[K,n] Y)"} +{"name":"SciLean.ContCDiffMapFD","declaration":"structure SciLean.ContCDiffMapFD (K : Type u_1) [RCLike K] (n : ℕ∞) (X : Type u_2) (Y : Type u_3) [SciLean.Vec K X] [SciLean.Vec K Y] : Type (max u_2 u_3)"} +{"name":"SciLean.ContCDiffMapFD.smul_apply","declaration":"theorem SciLean.ContCDiffMapFD.smul_apply {K : Type u_3} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) (x : X) (r : K) : (r • f) x = r • f x"} +{"name":"SciLean.ContCDiffMapFD_eta","declaration":"theorem SciLean.ContCDiffMapFD_eta {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) : (fun x ⟿FD[K,n] f x) = f"} +{"name":"SciLean.ContCDiffMapFD.zero_apply","declaration":"theorem SciLean.ContCDiffMapFD.zero_apply {K : Type u_3} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {n : ℕ∞} (x : X) : 0 x = 0"} +{"name":"SciLean.unexpandContCDiffMapFD","declaration":"def SciLean.unexpandContCDiffMapFD : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.«term_⟿FD[_,_]_»","declaration":"def SciLean.«term_⟿FD[_,_]_» : Lean.TrailingParserDescr"} +{"name":"SciLean.ContCDiffMapFD_apply_CDifferentiable","declaration":"theorem SciLean.ContCDiffMapFD_apply_CDifferentiable {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⟿FD[K,⊤] Y) (g : W → X) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun w => (f w) (g w)"} +{"name":"SciLean.ContCDiffMapFD.FD","declaration":"def SciLean.ContCDiffMapFD.FD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} (f : X ⟿FD[K,n] Y) (x : X) (dx : X) : Y × Y"} +{"name":"SciLean.«termFun_⟿FD_»","declaration":"def SciLean.«termFun_⟿FD_» : Lean.ParserDescr"} +{"name":"SciLean.instAddContCDiffMapFD","declaration":"instance SciLean.instAddContCDiffMapFD {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {n : ℕ∞} : Add (X ⟿FD[K,n] Y)"} +{"name":"SciLean.ContCDiffMapFD_eval_CDifferentiable'","declaration":"theorem SciLean.ContCDiffMapFD_eval_CDifferentiable' {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : SciLean.CDifferentiable K fun fx => fx.1 fx.2"} +{"name":"SciLean.ContCDiffMapFD.cderiv_snd","declaration":"def SciLean.ContCDiffMapFD.cderiv_snd {K : Type u_1} [RCLike K] {n : ℕ∞} {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⟿FD[K,n] Y) : ∂ x, (self.toFun x 0).1 = fun x dx => (self.toFun x dx).2"} +{"name":"SciLean.ContCDiffMapFD.mk.arg_f.IsSmoothLinearMap_rule","declaration":"theorem SciLean.ContCDiffMapFD.mk.arg_f.IsSmoothLinearMap_rule {K : Type u_2} [RCLike K] {X : Type u_3} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] {n : ℕ∞} (f : W → X → Y) (f' : W → X → X → Y × Y) (hf₁ : SciLean.CDifferentiable K fun x =>\n match x with\n | (w, x) => f w x) (hf₂ : IsLinearMap K f) (hf₃ : ∀ (w : W), SciLean.ContCDiff K n (f w)) (h : ∀ (w : W), (SciLean.fwdDeriv K fun x => f w x) = f' w) (h' : ∀ (w : W), SciLean.ContCDiff K n (f w)) : SciLean.IsSmoothLinearMap K fun w => fun x ⟿FD[K,n] f w x"} diff --git a/scilean-declarations/SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl b/scilean-declarations/SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f6b27885de7b7ea9d6816c06e745804ff78c5b22 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionSpaces.SmoothLinearMap.jsonl @@ -0,0 +1,36 @@ +{"name":"SciLean.SmoothLinearMap.neg_apply","declaration":"theorem SciLean.SmoothLinearMap.neg_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (f : X ⊸[K] Y) (x : X) : (-f) x = -f x"} +{"name":"SciLean.SmoothLinearMap.mk'.arg_f.CDifferentiable_rule","declaration":"theorem SciLean.SmoothLinearMap.mk'.arg_f.CDifferentiable_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⊸[K] Y) (g : W → X) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun w => (f w) (g w)"} +{"name":"SciLean.«term_⊸_»","declaration":"def SciLean.«term_⊸_» : Lean.TrailingParserDescr"} +{"name":"SciLean.SmoothLinearMap.is_smooth_linear_map","declaration":"def SciLean.SmoothLinearMap.is_smooth_linear_map {K : Type u_1} [RCLike K] {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⊸[K] Y) : SciLean.IsSmoothLinearMap K self.toFun"} +{"name":"SciLean.SmoothLinearMap_apply_right","declaration":"theorem SciLean.SmoothLinearMap_apply_right (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X ⊸[K] Y) : SciLean.IsSmoothLinearMap K fun x => f x"} +{"name":"SciLean.SmoothLinearMap.add_apply","declaration":"theorem SciLean.SmoothLinearMap.add_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (f : X ⊸[K] Y) (g : X ⊸[K] Y) (x : X) : (f + g) x = f x + g x"} +{"name":"SciLean.SmoothLinearMap.mk'.arg_f.IsSmoothLinearMap_rule","declaration":"theorem SciLean.SmoothLinearMap.mk'.arg_f.IsSmoothLinearMap_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_4} [SciLean.Vec K Y] {W : Type u_3} [SciLean.Vec K W] (f : W → X → Y) (hf : SciLean.CDifferentiable K fun x =>\n match x with\n | (w, x) => f w x) (hf₁ : ∀ (x : X), SciLean.IsSmoothLinearMap K fun x_1 => f x_1 x) (hf₂ : ∀ (w : W), SciLean.IsSmoothLinearMap K fun x => f w x) : SciLean.IsSmoothLinearMap K fun w => fun x ⊸[K] f w x"} +{"name":"SciLean.«term_⊸[_]_»","declaration":"def SciLean.«term_⊸[_]_» : Lean.TrailingParserDescr"} +{"name":"SciLean.instZeroSmoothLinearMap","declaration":"instance SciLean.instZeroSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : Zero (X ⊸[K] Y)"} +{"name":"SciLean.SmoothLinearMap.indextype_sum_apply","declaration":"theorem SciLean.SmoothLinearMap.indextype_sum_apply (K : Type u_4) [RCLike K] {X : Type u_3} [SciLean.Vec K X] {Y : Type (max u_2 u_3)} [SciLean.Vec K Y] {I : Type u_1} [SciLean.IndexType I] (f : I → X ⊸[K] Y) (x : X) : ( ∑ i, f i) x = ∑ i, (f i) x"} +{"name":"SciLean.unexpandSmoothLinearMapMk'","declaration":"def SciLean.unexpandSmoothLinearMapMk' : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.instVecSmoothLinearMap","declaration":"instance SciLean.instVecSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : SciLean.Vec K (X ⊸[K] Y)"} +{"name":"SciLean.SmoothLinearMap.smul_apply","declaration":"theorem SciLean.SmoothLinearMap.smul_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (f : X ⊸[K] Y) (x : X) (r : K) : (r • f) x = r • f x"} +{"name":"SciLean.instSMulSmoothLinearMap","declaration":"instance SciLean.instSMulSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : SMul K (X ⊸[K] Y)"} +{"name":"SciLean.SmoothLinearMap.eta_reduce","declaration":"theorem SciLean.SmoothLinearMap.eta_reduce {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X ⊸[K] Y) : { toFun := f.toFun, is_smooth_linear_map := ⋯ } = f"} +{"name":"SciLean.instFunLikeSmoothLinearMap","declaration":"instance SciLean.instFunLikeSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : FunLike (X ⊸[K] Y) X Y"} +{"name":"SciLean.«termFun_⊸_»","declaration":"def SciLean.«termFun_⊸_» : Lean.ParserDescr"} +{"name":"SciLean.SmoothLinearMap.ext","declaration":"theorem SciLean.SmoothLinearMap.ext {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X ⊸[K] Y) (g : X ⊸[K] Y) : (∀ (x : X), f x = g x) → f = g"} +{"name":"SciLean.SmoothLinearMap.fintype_sum_apply","declaration":"theorem SciLean.SmoothLinearMap.fintype_sum_apply (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.Vec K X] {Y : Type u_4} [SciLean.Vec K Y] {I : Type u_1} [Fintype I] (f : I → X ⊸[K] Y) (x : X) : (Finset.sum Finset.univ fun i => f i) x = Finset.sum Finset.univ fun i => (f i) x"} +{"name":"SciLean.instUniformSpaceSmoothLinearMap","declaration":"instance SciLean.instUniformSpaceSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : UniformSpace (X ⊸[K] Y)"} +{"name":"SciLean.instSubSmoothLinearMap","declaration":"instance SciLean.instSubSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : Sub (X ⊸[K] Y)"} +{"name":"SciLean.SmoothLinearMap.mk'_eval","declaration":"theorem SciLean.SmoothLinearMap.mk'_eval (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : (SciLean.SmoothLinearMap.mk' K f hf) x = f x"} +{"name":"SciLean.unexpandSmoothLinearMap","declaration":"def SciLean.unexpandSmoothLinearMap : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.unexpandSmoothLinearMapMk","declaration":"def SciLean.unexpandSmoothLinearMapMk : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.instNegSmoothLinearMap","declaration":"instance SciLean.instNegSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : Neg (X ⊸[K] Y)"} +{"name":"SciLean.«termFun_⊸[_]_»","declaration":"def SciLean.«termFun_⊸[_]_» : Lean.ParserDescr"} +{"name":"SciLean.SmoothLinearMap.mk","declaration":"ctor SciLean.SmoothLinearMap.mk {K : Type u_1} [RCLike K] {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (toFun : X → Y) (is_smooth_linear_map : SciLean.IsSmoothLinearMap K toFun) : X ⊸[K] Y"} +{"name":"SciLean.SmoothLinearMap.sub_apply","declaration":"theorem SciLean.SmoothLinearMap.sub_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (f : X ⊸[K] Y) (g : X ⊸[K] Y) (x : X) : (f - g) x = f x - g x"} +{"name":"SciLean.SmoothLinearMap.mk_eval","declaration":"theorem SciLean.SmoothLinearMap.mk_eval {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : { toFun := f, is_smooth_linear_map := hf } x = f x"} +{"name":"SciLean.SmoothLinearMap_apply_left","declaration":"theorem SciLean.SmoothLinearMap_apply_left (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : W → X ⊸[K] Y) (x : X) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.IsSmoothLinearMap K fun w => (f w) x"} +{"name":"SciLean.SmoothLinearMap.toFun","declaration":"def SciLean.SmoothLinearMap.toFun {K : Type u_1} [RCLike K] {X : Type u_2} {Y : Type u_3} [SciLean.Vec K X] [SciLean.Vec K Y] (self : X ⊸[K] Y) : X → Y"} +{"name":"SciLean.instAddSmoothLinearMap","declaration":"instance SciLean.instAddSmoothLinearMap (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] : Add (X ⊸[K] Y)"} +{"name":"SciLean.SmoothLinearMap","declaration":"structure SciLean.SmoothLinearMap (K : Type u_1) [RCLike K] (X : Type u_2) (Y : Type u_3) [SciLean.Vec K X] [SciLean.Vec K Y] : Type (max u_2 u_3)"} +{"name":"SciLean.SmoothLinearMap.mk'","declaration":"def SciLean.SmoothLinearMap.mk' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : X ⊸[K] Y"} +{"name":"SciLean.SmoothLinearMap.apply_zero","declaration":"theorem SciLean.SmoothLinearMap.apply_zero (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X ⊸[K] Y) : f 0 = 0"} +{"name":"SciLean.SmoothLinearMap.zero_apply","declaration":"theorem SciLean.SmoothLinearMap.zero_apply (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (x : X) : 0 x = 0"} diff --git a/scilean-declarations/SciLean.Core.FunctionSpaces.jsonl b/scilean-declarations/SciLean.Core.FunctionSpaces.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.Adjoint.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.Adjoint.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.CDeriv.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.CDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..146602496a2601ab6272f9443a0a0fb3a5fc7f4a --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.CDeriv.jsonl @@ -0,0 +1,57 @@ +{"name":"SciLean.SciLean.cderiv.arg_dx.CDifferentiable_rule","declaration":"theorem SciLean.SciLean.cderiv.arg_dx.CDifferentiable_rule {K : Type u_1} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (y : Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiable K fun dx' => SciLean.cderiv K f y (g dx')"} +{"name":"SciLean.Prod.fst.arg_self.cderiv_rule_at","declaration":"theorem SciLean.Prod.fst.arg_self.cderiv_rule_at {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.cderiv K (fun x => (f x).1) x = fun dx => (SciLean.cderiv K f x dx).1"} +{"name":"SciLean.cderiv.arg_dx.IsLinearMap_rule_simple","declaration":"theorem SciLean.cderiv.arg_dx.IsLinearMap_rule_simple {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (x : X) (hf : SciLean.CDifferentiableAt K f x) : IsLinearMap K fun dx => SciLean.cderiv K f x dx"} +{"name":"SciLean.dite.arg_te.cderiv_rule","declaration":"theorem SciLean.dite.arg_te.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.cderiv K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.cderiv K (t p) y else SciLean.cderiv K (e p) y"} +{"name":"SciLean.cderiv.arg_dx.IsLinearMap_rule","declaration":"theorem SciLean.cderiv.arg_dx.IsLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {W : Type u_4} [SciLean.Vec K W] (f : X → Y) (x : X) (dx : W → X) (hf : SciLean.CDifferentiableAt K f x) (hdx : IsLinearMap K dx) : IsLinearMap K fun w => SciLean.cderiv K f x (dx w)"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.cderiv_rule","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → K) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun x => f x • g x) = fun x =>\n let k := f x;\n let y := g x;\n fun dx => k • SciLean.cderiv K g x dx + SciLean.cderiv K f x dx • y"} +{"name":"SciLean.cderiv_apply","declaration":"theorem SciLean.cderiv_apply {K : Type u_2} [RCLike K] {X : Type u_3} [SciLean.Vec K X] {Y : Type u_4} {Z : Type u_1} [SciLean.Vec K Z] (f : X → Y → Z) (x : X) (dx : X) (y : Y) : SciLean.cderiv K f x dx y = SciLean.cderiv K (fun x' => f x' y) x dx"} +{"name":"SciLean.cderiv.arg_f.IsSmoothLinearMap_rule","declaration":"theorem SciLean.cderiv.arg_f.IsSmoothLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_3} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (f : X → Y → Z) (hf : SciLean.CDifferentiable K fun x =>\n match x with\n | (x, y) => f x y) (hf' : ∀ (y : Y), IsLinearMap K fun x => f x y) : SciLean.IsSmoothLinearMap K fun x => SciLean.cderiv K fun x_1 => f x x_1"} +{"name":"SciLean.IndexType.sum.arg_f.cderiv_rule_at","declaration":"theorem SciLean.IndexType.sum.arg_f.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (x : X) (hf : ∀ (i : ι), SciLean.CDifferentiableAt K (fun x => f x i) x) : SciLean.cderiv K (fun x => ∑ i, f x i) x = fun dx => ∑ i, SciLean.cderiv K (fun x => f x i) x dx"} +{"name":"SciLean.HPow.hPow.arg_a0.cderiv_rule_at","declaration":"def SciLean.HPow.hPow.arg_a0.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (n : ℕ) (x : X) (f : X → K) (hf : SciLean.CDifferentiableAt K f x) : SciLean.cderiv K (fun x => f x ^ n) x = fun dx => ↑n * SciLean.cderiv K f x dx * f x ^ (n - 1)"} +{"name":"SciLean.HSub.hSub.arg_a0.cderiv_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (f : X → Y) (y : Y) : (SciLean.cderiv K fun x => f x - y) = fun x dx => SciLean.cderiv K f x dx"} +{"name":"SciLean.Prod.snd.arg_self.cderiv_rule","declaration":"theorem SciLean.Prod.snd.arg_self.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.CDifferentiable K f) : (SciLean.cderiv K fun x => (f x).2) = fun x dx => (SciLean.cderiv K f x dx).2"} +{"name":"SciLean.cderiv_apply_zero","declaration":"theorem SciLean.cderiv_apply_zero {K : Type u_2} [RCLike K] {X : Type u_3} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] (f : X → Y) (x : X) : SciLean.cderiv K f x 0 = 0"} +{"name":"SciLean.Prod.mk.arg_fstsnd.cderiv_rule","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (g : X → Y) (hg : SciLean.CDifferentiable K g) (f : X → Z) (hf : SciLean.CDifferentiable K f) : (SciLean.cderiv K fun x => (g x, f x)) = fun x dx => (SciLean.cderiv K g x dx, SciLean.cderiv K f x dx)"} +{"name":"SciLean.HSub.hSub.arg_a1.cderiv_rule","declaration":"theorem SciLean.HSub.hSub.arg_a1.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (f : X → Y) (y : Y) : (SciLean.cderiv K fun x => y - f x) = fun x dx => -SciLean.cderiv K f x dx"} +{"name":"SciLean.HAdd.hAdd.arg_a1.cderiv_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a1.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (f : X → Y) (y : Y) : (SciLean.cderiv K fun x => y + f x) = fun x dx => SciLean.cderiv K f x dx"} +{"name":"SciLean.SciLean.norm₂.arg_x.cderiv_rule","declaration":"theorem SciLean.SciLean.norm₂.arg_x.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.CDifferentiable R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.cderiv R fun x => ‖f x‖₂[R]) = fun x dx =>\n let y := f x;\n let dy := SciLean.cderiv R f x dx;\n ‖y‖₂[R]⁻¹ * ⟪dy, y⟫_R"} +{"name":"SciLean.cderiv.comp_rule_at","declaration":"theorem SciLean.cderiv.comp_rule_at (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (x : X) (hf : SciLean.CDifferentiableAt K f (g x)) (hg : SciLean.CDifferentiableAt K g x) : SciLean.cderiv K (fun x => f (g x)) x =\n let y := g x;\n fun dx =>\n let dy := SciLean.cderiv K g x dx;\n let dz := SciLean.cderiv K f y dy;\n dz"} +{"name":"SciLean.Neg.neg.arg_a0.cderiv_rule'","declaration":"theorem SciLean.Neg.neg.arg_a0.cderiv_rule' {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (x : X) (f : X → Y) : SciLean.cderiv K (fun x => -f x) x = fun dx => -SciLean.cderiv K f x dx"} +{"name":"SciLean.IndexType.sum.arg_f.cderiv_rule","declaration":"theorem SciLean.IndexType.sum.arg_f.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.CDifferentiable K fun x => f x i) : (SciLean.cderiv K fun x => ∑ i, f x i) = fun x dx => ∑ i, SciLean.cderiv K (fun x => f x i) x dx"} +{"name":"SciLean.SciLean.cderiv.arg_dx.cderiv_rule_at","declaration":"theorem SciLean.SciLean.cderiv.arg_dx.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (y : Y) (dx : X) (hf : SciLean.CDifferentiableAt K f y) (hg : SciLean.CDifferentiableAt K g dx) : SciLean.cderiv K (fun dx' => SciLean.cderiv K f y (g dx')) dx = fun ddx =>\n let ddy := SciLean.cderiv K g dx ddx;\n SciLean.cderiv K f y ddy"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.cderiv_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun x => f x + g x) = fun x dx => SciLean.cderiv K f x dx + SciLean.cderiv K g x dx"} +{"name":"SciLean.cderiv_of_linear","declaration":"theorem SciLean.cderiv_of_linear {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.IsSmoothLinearMap K f) : SciLean.cderiv K f = fun x dx => f dx"} +{"name":"SciLean.cderiv.pi_rule","declaration":"theorem SciLean.cderiv.pi_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.CDifferentiable K fun x => f x i) : (SciLean.cderiv K fun x i => f x i) = fun x dx i => SciLean.cderiv K (fun x => f x i) x dx"} +{"name":"SciLean.ite.arg_chte.cderiv_rule","declaration":"theorem SciLean.ite.arg_chte.cderiv_rule {K : Type u_2} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (c : X → Prop) [dec : (x : X) → Decidable (c x)] (t : X → Y) (e : X → Y) (ht : ∀ x ∈ closure c, SciLean.CDifferentiableAt K t x) (he : ∀ x ∈ (interior c)ᶜ, SciLean.CDifferentiableAt K e x) (hc : ∀ x ∈ frontier c, SciLean.cderiv K t x = SciLean.cderiv K e x) : (SciLean.cderiv K fun x => if c x then t x else e x) = fun y =>\n if c y then SciLean.cderiv K t y else SciLean.cderiv K e y"} +{"name":"SciLean.Inner.inner.arg_a0a1.cderiv_rule_at","declaration":"theorem SciLean.Inner.inner.arg_a0a1.cderiv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (x : X) (hf : SciLean.CDifferentiableAt R f x) (hg : SciLean.CDifferentiableAt R g x) : SciLean.cderiv R (fun x => ⟪f x, g x⟫_R) x = fun dx =>\n let y₁ := f x;\n let dy₁ := SciLean.cderiv R f x dx;\n let y₂ := g x;\n let dy₂ := SciLean.cderiv R g x dx;\n ⟪dy₁, y₂⟫_R + ⟪y₁, dy₂⟫_R"} +{"name":"SciLean.Inner.inner.arg_a0a1.cderiv_rule","declaration":"theorem SciLean.Inner.inner.arg_a0a1.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable R f) (hg : SciLean.CDifferentiable R g) : (SciLean.cderiv R fun x => ⟪f x, g x⟫_R) = fun x dx =>\n let y₁ := f x;\n let dy₁ := SciLean.cderiv R f x dx;\n let y₂ := g x;\n let dy₂ := SciLean.cderiv R g x dx;\n ⟪dy₁, y₂⟫_R + ⟪y₁, dy₂⟫_R"} +{"name":"SciLean.Neg.neg.arg_a0.cderiv_rule","declaration":"theorem SciLean.Neg.neg.arg_a0.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (f : X → Y) : (SciLean.cderiv K fun x => -f x) = fun x dx => -SciLean.cderiv K f x dx"} +{"name":"SciLean.cderiv.apply_rule","declaration":"theorem SciLean.cderiv.apply_rule (K : Type u_3) [RCLike K] {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → SciLean.Vec K (E i)] (i : ι) : (SciLean.cderiv K fun x => x i) = fun x dx => dx i"} +{"name":"SciLean.HSub.hSub.arg_a0a1.cderiv_rule_at","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.cderiv K (fun x => f x - g x) x = fun dx => SciLean.cderiv K f x dx - SciLean.cderiv K g x dx"} +{"name":"SciLean.SciLean.cderiv.arg_dx.CDifferentiableAt_rule","declaration":"theorem SciLean.SciLean.cderiv.arg_dx.CDifferentiableAt_rule {K : Type u_1} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (y : Y) (dx : X) (hf : SciLean.CDifferentiableAt K f y) (hg : SciLean.CDifferentiableAt K g dx) : SciLean.CDifferentiableAt K (fun dx' => SciLean.cderiv K f y (g dx')) dx"} +{"name":"SciLean.HSMul.hSMul.arg_a0a1.cderiv_rule_at","declaration":"theorem SciLean.HSMul.hSMul.arg_a0a1.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → K) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.cderiv K (fun x => f x • g x) x =\n let k := f x;\n let y := g x;\n fun dx => k • SciLean.cderiv K g x dx + SciLean.cderiv K f x dx • y"} +{"name":"SciLean.cderiv.id_rule","declaration":"theorem SciLean.cderiv.id_rule (K : Type u_2) [RCLike K] {X : Type u_1} [SciLean.Vec K X] : (SciLean.cderiv K fun x => x) = fun x dx => dx"} +{"name":"SciLean.SciLean.norm₂.arg_x.cderiv_rule_at","declaration":"theorem SciLean.SciLean.norm₂.arg_x.cderiv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.CDifferentiableAt R f x) (hx : f x ≠ 0) : SciLean.cderiv R (fun x => ‖f x‖₂[R]) x = fun dx =>\n let y := f x;\n let dy := SciLean.cderiv R f x dx;\n ‖y‖₂[R]⁻¹ * ⟪dy, y⟫_R"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.cderiv_rule_at","declaration":"theorem SciLean.HDiv.hDiv.arg_a0a1.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) (hx : g x ≠ 0) : SciLean.cderiv K (fun x => f x / g x) x =\n let k := f x;\n let k' := g x;\n fun dx => (SciLean.cderiv K f x dx * k' - k * SciLean.cderiv K g x dx) / k' ^ 2"} +{"name":"SciLean.cderiv.pi_rule_at","declaration":"theorem SciLean.cderiv.pi_rule_at (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (f : X → (i : ι) → E i) (x : X) (hf : ∀ (i : ι), SciLean.CDifferentiableAt K (fun x => f x i) x) : SciLean.cderiv K (fun x i => f x i) x = fun dx i => SciLean.cderiv K (fun x => f x i) x dx"} +{"name":"SciLean.HAdd.hAdd.arg_a0a1.cderiv_rule_at","declaration":"theorem SciLean.HAdd.hAdd.arg_a0a1.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.cderiv K (fun x => f x + g x) x = fun dx => SciLean.cderiv K f x dx + SciLean.cderiv K g x dx"} +{"name":"SciLean.HMul.hMul.arg_a0a1.cderiv_rule","declaration":"theorem SciLean.HMul.hMul.arg_a0a1.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun x => f x * g x) = fun x =>\n let fx := f x;\n let gx := g x;\n fun dx => SciLean.cderiv K g x dx * fx + SciLean.cderiv K f x dx * gx"} +{"name":"SciLean.ite.arg_te.cderiv_rule","declaration":"theorem SciLean.ite.arg_te.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.cderiv K fun x => if c then t x else e x) = fun y => if c then SciLean.cderiv K t y else SciLean.cderiv K e y"} +{"name":"SciLean.cderiv.let_rule_at","declaration":"theorem SciLean.cderiv.let_rule_at (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (f : X → Y → Z) (g : X → Y) (x : X) (hf : SciLean.CDifferentiableAt K (↿f) (x, g x)) (hg : SciLean.CDifferentiableAt K g x) : SciLean.cderiv K\n (fun x =>\n let y := g x;\n f x y)\n x =\n let y := g x;\n fun dx =>\n let dy := SciLean.cderiv K g x dx;\n let dz := SciLean.cderiv K (fun xy => f xy.1 xy.2) (x, y) (dx, dy);\n dz"} +{"name":"SciLean.HSub.hSub.arg_a0a1.cderiv_rule","declaration":"theorem SciLean.HSub.hSub.arg_a0a1.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun x => f x - g x) = fun x dx => SciLean.cderiv K f x dx - SciLean.cderiv K g x dx"} +{"name":"SciLean.HPow.hPow.arg_a0.cderiv_rule","declaration":"def SciLean.HPow.hPow.arg_a0.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (n : ℕ) (f : X → K) (hf : SciLean.CDifferentiable K f) : (SciLean.cderiv K fun x => f x ^ n) = fun x dx => ↑n * SciLean.cderiv K f x dx * f x ^ (n - 1)"} +{"name":"SciLean.HMul.hMul.arg_a0a1.cderiv_rule_at","declaration":"theorem SciLean.HMul.hMul.arg_a0a1.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.cderiv K (fun x => f x * g x) x =\n let fx := f x;\n let gx := g x;\n fun dx => SciLean.cderiv K g x dx * fx + SciLean.cderiv K f x dx * gx"} +{"name":"SciLean.cderiv.let_rule","declaration":"theorem SciLean.cderiv.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.CDifferentiable K fun xy => f xy.1 xy.2) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun x =>\n let y := g x;\n f x y) =\n fun x =>\n let y := g x;\n fun dx =>\n let dy := SciLean.cderiv K g x dx;\n let dz := SciLean.cderiv K (fun xy => f xy.1 xy.2) (x, y) (dx, dy);\n dz"} +{"name":"SciLean.cderiv.const_rule","declaration":"theorem SciLean.cderiv.const_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (x : X) : (SciLean.cderiv K fun x_1 => x) = fun x dx => 0"} +{"name":"SciLean.cderiv.comp_rule","declaration":"theorem SciLean.cderiv.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun x => f (g x)) = fun x =>\n let y := g x;\n fun dx =>\n let dy := SciLean.cderiv K g x dx;\n let dz := SciLean.cderiv K f y dy;\n dz"} +{"name":"SciLean.Prod.snd.arg_self.cderiv_rule_at","declaration":"theorem SciLean.Prod.snd.arg_self.cderiv_rule_at {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.cderiv K (fun x => (f x).2) x = fun dx => (SciLean.cderiv K f x dx).2"} +{"name":"SciLean.SciLean.cderiv.arg_dx.cderiv_rule","declaration":"theorem SciLean.SciLean.cderiv.arg_dx.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (y : Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.cderiv K fun dx => SciLean.cderiv K f y (g dx)) = fun dx ddx =>\n let ddy := SciLean.cderiv K g dx ddx;\n SciLean.cderiv K f y ddy"} +{"name":"SciLean.cderiv.arg_f.IsLinearMap_rule","declaration":"theorem SciLean.cderiv.arg_f.IsLinearMap_rule {K : Type u_1} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : X → Y → Z) (hf : ∀ (x : X), SciLean.CDifferentiable K fun y => f x y) (hf' : ∀ (y : Y), IsLinearMap K fun x => f x y) : IsLinearMap K fun x => SciLean.cderiv K fun x_1 => f x x_1"} +{"name":"SciLean.cderiv","declaration":"def SciLean.cderiv (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (x : X) (dx : X) : Y"} +{"name":"SciLean.Prod.fst.arg_self.cderiv_rule","declaration":"theorem SciLean.Prod.fst.arg_self.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.CDifferentiable K f) : (SciLean.cderiv K fun x => (f x).1) = fun x dx => (SciLean.cderiv K f x dx).1"} +{"name":"SciLean.SciLean.Norm2.norm2.arg_a0.cderiv_rule","declaration":"theorem SciLean.SciLean.Norm2.norm2.arg_a0.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.CDifferentiable R f) : (SciLean.cderiv R fun x => ‖f x‖₂²) = fun x dx =>\n let y := f x;\n let dy := SciLean.cderiv R f x dx;\n 2 * ⟪dy, y⟫_R"} +{"name":"SciLean.HDiv.hDiv.arg_a0a1.cderiv_rule","declaration":"theorem SciLean.HDiv.hDiv.arg_a0a1.cderiv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.cderiv K fun x => f x / g x) = fun x =>\n let k := f x;\n let k' := g x;\n fun dx => (SciLean.cderiv K f x dx * k' - k * SciLean.cderiv K g x dx) / k' ^ 2"} +{"name":"SciLean.HAdd.hAdd.arg_a0.cderiv_rule","declaration":"theorem SciLean.HAdd.hAdd.arg_a0.cderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (f : X → Y) (y : Y) : (SciLean.cderiv K fun x => f x + y) = fun x dx => SciLean.cderiv K f x dx"} +{"name":"SciLean.SciLean.Norm2.norm2.arg_a0.cderiv_rule_at","declaration":"theorem SciLean.SciLean.Norm2.norm2.arg_a0.cderiv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.CDifferentiableAt R f x) : SciLean.cderiv R (fun x => ‖f x‖₂²) x = fun dx =>\n let y := f x;\n let dy := SciLean.cderiv R f x dx;\n 2 * ⟪dy, y⟫_R"} +{"name":"SciLean.Prod.mk.arg_fstsnd.cderiv_rule_at","declaration":"theorem SciLean.Prod.mk.arg_fstsnd.cderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (x : X) (g : X → Y) (hg : SciLean.CDifferentiableAt K g x) (f : X → Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.cderiv K (fun x => (g x, f x)) x = fun dx => (SciLean.cderiv K g x dx, SciLean.cderiv K f x dx)"} +{"name":"SciLean.scalarCDeriv","declaration":"def SciLean.scalarCDeriv (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : K → X) (t : K) : X"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.FDeriv.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.FDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f8b3c10204c4422802f73943fbf1c931c8c8a7ab --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.FDeriv.jsonl @@ -0,0 +1,17 @@ +{"name":"SciLean.fderiv.pi_rule_at","declaration":"theorem SciLean.fderiv.pi_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {ι : Type u_4} [Fintype ι] {E : ι → Type u_1} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace K (E i)] (f : X → (i : ι) → E i) (x : X) (hf : ∀ (i : ι), DifferentiableAt K (fun x => f x i) x) : fderiv K (fun x i => f x i) x = SciLean.ContinuousLinearMap.mk' K (fun dx i => (fderiv K (fun x => f x i) x) dx) ⋯"} +{"name":"SciLean.fderiv.id_rule","declaration":"theorem SciLean.fderiv.id_rule {K : Type u_2} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] : (fderiv K fun x => x) = fun x => fun dx =>L[K] dx"} +{"name":"SciLean.fderiv.apply_rule","declaration":"theorem SciLean.fderiv.apply_rule {K : Type u_3} [RCLike K] {ι : Type u_1} [Fintype ι] {E : ι → Type u_2} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace K (E i)] (i : ι) : (fderiv K fun x => x i) = fun x => fun dx =>L[K] dx i"} +{"name":"HAdd.hAdd.arg_a0a1.fderiv_rule_at","declaration":"theorem HAdd.hAdd.arg_a0a1.fderiv_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) (f : X → Y) (g : X → Y) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : fderiv K (fun x => f x + g x) x = fun dx =>L[K] (fderiv K f x) dx + (fderiv K g x) dx"} +{"name":"HPow.hPow.arg_a0.fderiv_rule_at","declaration":"def HPow.hPow.arg_a0.fderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] (n : ℕ) (x : X) (f : X → K) (hf : DifferentiableAt K f x) : fderiv K (fun x => f x ^ n) x = fun dx =>L[K] ↑n * (fderiv K f x) dx * f x ^ (n - 1)"} +{"name":"Neg.neg.arg_a0.fderiv_rule","declaration":"theorem Neg.neg.arg_a0.fderiv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] (f : X → Y) : (fderiv K fun x => -f x) = fun x => fun dx =>L[K] -(fderiv K f x) dx"} +{"name":"HSub.hSub.arg_a0a1.fderiv_rule_at","declaration":"theorem HSub.hSub.arg_a0a1.fderiv_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) (f : X → Y) (g : X → Y) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : fderiv K (fun x => f x - g x) x = fun dx =>L[K] (fderiv K f x) dx - (fderiv K g x) dx"} +{"name":"Prod.snd.arg_self.fderiv_rule_at","declaration":"theorem Prod.snd.arg_self.fderiv_rule_at {K : Type u_4} [RCLike K] {X : Type u_3} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_2} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (f : X → Y × Z) (hf : DifferentiableAt K f x) : fderiv K (fun x => (f x).2) x = fun dx =>L[K] ((fderiv K f x) dx).2"} +{"name":"Prod.mk.arg_fstsnd.fderiv_rule_at","declaration":"theorem Prod.mk.arg_fstsnd.fderiv_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_4} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (g : X → Y) (hg : DifferentiableAt K g x) (f : X → Z) (hf : DifferentiableAt K f x) : fderiv K (fun x => (g x, f x)) x = fun dx =>L[K] ((fderiv K g x) dx, (fderiv K f x) dx)"} +{"name":"HSMul.hSMul.arg_a0a1.fderiv_rule_at","declaration":"theorem HSMul.hSMul.arg_a0a1.fderiv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_3} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) (f : X → K) (g : X → Y) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : fderiv K (fun x => f x • g x) x =\n let k := f x;\n let y := g x;\n fun dx =>L[K] k • (fderiv K g x) dx + (fderiv K f x) dx • y"} +{"name":"SciLean.fderiv.const_rule","declaration":"theorem SciLean.fderiv.const_rule {K : Type u_3} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) : (fderiv K fun x_1 => x) = fun x => fun x =>L[K] 0"} +{"name":"SciLean.fderiv.let_rule_at","declaration":"theorem SciLean.fderiv.let_rule_at {K : Type u_4} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_3} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_1} [NormedAddCommGroup Z] [NormedSpace K Z] (f : X → Y → Z) (g : X → Y) (x : X) (hf : DifferentiableAt K (fun xy => f xy.1 xy.2) (x, g x)) (hg : DifferentiableAt K g x) : fderiv K\n (fun x =>\n let y := g x;\n f x y)\n x =\n let y := g x;\n fun dx =>L[K]\n let dy := (fderiv K g x) dx;\n let dz := (fderiv K (fun xy => f xy.1 xy.2) (x, y)) (dx, dy);\n dz"} +{"name":"Prod.fst.arg_self.fderiv_rule_at","declaration":"theorem Prod.fst.arg_self.fderiv_rule_at {K : Type u_4} [RCLike K] {X : Type u_3} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_2} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (f : X → Y × Z) (hf : DifferentiableAt K f x) : fderiv K (fun x => (f x).1) x = fun dx =>L[K] ((fderiv K f x) dx).1"} +{"name":"HMul.hMul.arg_a0a1.fderiv_rule_at","declaration":"theorem HMul.hMul.arg_a0a1.fderiv_rule_at {K : Type u_2} [RCLike K] {X : Type u_3} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedCommRing Y] [NormedAlgebra K Y] (x : X) (f : X → Y) (g : X → Y) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : fderiv K (fun x => f x * g x) x =\n let fx := f x;\n let gx := g x;\n fun dx =>L[K] (fderiv K g x) dx * fx + (fderiv K f x) dx * gx"} +{"name":"isContinuousLinearMap_fderiv","declaration":"theorem isContinuousLinearMap_fderiv {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_3} [NormedAddCommGroup Y] [NormedSpace K Y] (f : X → Y) (hf : SciLean.IsContinuousLinearMap K f) : fderiv K f = fun x => fun dx =>L[K] f dx"} +{"name":"HDiv.hDiv.arg_a0a1.fderiv_rule_at","declaration":"theorem HDiv.hDiv.arg_a0a1.fderiv_rule_at {K : Type u_2} [RCLike K] {R : Type u_1} [NontriviallyNormedField R] [NormedAlgebra R K] (x : R) (f : R → K) (g : R → K) (hf : DifferentiableAt R f x) (hg : DifferentiableAt R g x) (hx : g x ≠ 0) : fderiv R (fun x => f x / g x) x =\n let k := f x;\n let k' := g x;\n fun dx =>L[R] ((fderiv R f x) dx * k' - k * (fderiv R g x) dx) / k' ^ 2"} +{"name":"SciLean.fderiv.comp_rule_at","declaration":"theorem SciLean.fderiv.comp_rule_at {K : Type u_3} [RCLike K] {X : Type u_4} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_1} [NormedAddCommGroup Z] [NormedSpace K Z] (f : Y → Z) (g : X → Y) (x : X) (hf : DifferentiableAt K f (g x)) (hg : DifferentiableAt K g x) : fderiv K (fun x => f (g x)) x =\n let y := g x;\n fun dx =>L[K]\n let dy := (fderiv K g x) dx;\n let dz := (fderiv K f y) dy;\n dz"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.FwdDeriv.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.FwdDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..80027eefff9a630b35b3dce5cba1f0406de8e174 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.FwdDeriv.jsonl @@ -0,0 +1,39 @@ +{"name":"SciLean.fwdDeriv.SciLean.norm₂.arg_x.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.SciLean.norm₂.arg_x.fwdDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (x : X) (f : X → Y) (hf : SciLean.CDifferentiableAt R f x) (hx : f x ≠ 0) : SciLean.fwdDeriv R (fun x => ‖f x‖₂[R]) x = fun dx =>\n let ydy := SciLean.fwdDeriv R f x dx;\n let ynorm := ‖ydy.1‖₂[R];\n (ynorm, ynorm⁻¹ * ⟪ydy.2, ydy.1⟫_R)"} +{"name":"SciLean.fwdDeriv.Inner.inner.arg_a0a1.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.Inner.inner.arg_a0a1.fwdDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiableAt R f x) (hg : SciLean.CDifferentiableAt R g x) : SciLean.fwdDeriv R (fun x => ⟪f x, g x⟫_R) x = fun dx =>\n let y₁dy₁ := SciLean.fwdDeriv R f x dx;\n let y₂dy₂ := SciLean.fwdDeriv R g x dx;\n (⟪y₁dy₁.1, y₂dy₂.1⟫_R, ⟪y₁dy₁.2, y₂dy₂.1⟫_R + ⟪y₁dy₁.1, y₂dy₂.2⟫_R)"} +{"name":"SciLean.fwdDeriv.Prod.snd.arg_self.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.Prod.snd.arg_self.fwdDeriv_rule_at {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.fwdDeriv K (fun x => (f x).2) x = fun dx =>\n let yzdyz := SciLean.fwdDeriv K f x dx;\n (yzdyz.1.2, yzdyz.2.2)"} +{"name":"SciLean.fwdDeriv.HMul.hMul.arg_a0a1.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.HMul.hMul.arg_a0a1.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.fwdDeriv K (fun x => f x * g x) x = fun dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n (ydy.1 * zdz.1, zdz.2 * ydy.1 + ydy.2 * zdz.1)"} +{"name":"SciLean.fwdDeriv.HSub.hSub.arg_a0a1.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.HSub.hSub.arg_a0a1.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.fwdDeriv K (fun x => f x - g x) x = fun dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n ydy - zdz"} +{"name":"SciLean.fwdDeriv.Inner.inner.arg_a0a1.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.Inner.inner.arg_a0a1.fwdDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable R f) (hg : SciLean.CDifferentiable R g) : (SciLean.fwdDeriv R fun x => ⟪f x, g x⟫_R) = fun x dx =>\n let y₁dy₁ := SciLean.fwdDeriv R f x dx;\n let y₂dy₂ := SciLean.fwdDeriv R g x dx;\n (⟪y₁dy₁.1, y₂dy₂.1⟫_R, ⟪y₁dy₁.2, y₂dy₂.1⟫_R + ⟪y₁dy₁.1, y₂dy₂.2⟫_R)"} +{"name":"SciLean.fwdDeriv.Prod.mk.arg_fstsnd.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.Prod.mk.arg_fstsnd.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (g : X → Y) (hg : SciLean.CDifferentiable K g) (f : X → Z) (hf : SciLean.CDifferentiable K f) : (SciLean.fwdDeriv K fun x => (g x, f x)) = fun x dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n let zdz := SciLean.fwdDeriv K f x dx;\n ((ydy.1, zdz.1), ydy.2, zdz.2)"} +{"name":"SciLean.fwdDeriv","declaration":"def SciLean.fwdDeriv (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (x : X) (dx : X) : Y × Y"} +{"name":"SciLean.fwdDeriv.comp_rule","declaration":"theorem SciLean.fwdDeriv.comp_rule {K : Type u_1} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDeriv K fun x => f (g x)) = fun x dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n let zdz := SciLean.fwdDeriv K f ydy.1 ydy.2;\n zdz"} +{"name":"SciLean.fwdDeriv.Prod.snd.arg_self.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.Prod.snd.arg_self.fwdDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.CDifferentiable K f) : (SciLean.fwdDeriv K fun x => (f x).2) = fun x dx =>\n let yzdyz := SciLean.fwdDeriv K f x dx;\n (yzdyz.1.2, yzdyz.2.2)"} +{"name":"SciLean.fwdDeriv.pi_rule","declaration":"theorem SciLean.fwdDeriv.pi_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.CDifferentiable K fun x => f x i) : (SciLean.fwdDeriv K fun x i => f x i) = fun x dx =>\n (fun i => f x i, fun i => (SciLean.fwdDeriv K (fun x => f x i) x dx).2)"} +{"name":"SciLean.fwdDeriv.HAdd.hAdd.arg_a0a1.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.HAdd.hAdd.arg_a0a1.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDeriv K fun x => f x + g x) = fun x dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n ydy + zdz"} +{"name":"SciLean.fwdDeriv.SciLean.Norm2.norm2.arg_a0.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.SciLean.Norm2.norm2.arg_a0.fwdDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.CDifferentiable R f) : (SciLean.fwdDeriv R fun x => ‖f x‖₂²) = fun x dx =>\n let ydy := SciLean.fwdDeriv R f x dx;\n (‖ydy.1‖₂², 2 * ⟪ydy.2, ydy.1⟫_R)"} +{"name":"SciLean.fwdDeriv.SciLean.norm₂.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.SciLean.norm₂.arg_x.fwdDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.CDifferentiable R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.fwdDeriv R fun x => ‖f x‖₂[R]) = fun x dx =>\n let ydy := SciLean.fwdDeriv R f x dx;\n let ynorm := ‖ydy.1‖₂[R];\n (ynorm, ynorm⁻¹ * ⟪ydy.2, ydy.1⟫_R)"} +{"name":"SciLean.fwdDeriv.Prod.fst.arg_self.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.Prod.fst.arg_self.fwdDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (f : X → Y × Z) (hf : SciLean.CDifferentiable K f) : (SciLean.fwdDeriv K fun x => (f x).1) = fun x dx =>\n let yzdyz := SciLean.fwdDeriv K f x dx;\n (yzdyz.1.1, yzdyz.2.1)"} +{"name":"SciLean.fwdDeriv.const_rule","declaration":"theorem SciLean.fwdDeriv.const_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (y : Y) : (SciLean.fwdDeriv K fun x => y) = fun x dx => (y, 0)"} +{"name":"SciLean.fwdDeriv.comp_rule_at","declaration":"theorem SciLean.fwdDeriv.comp_rule_at {K : Type u_1} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] {Z : Type u_3} [SciLean.Vec K Z] (x : X) (f : Y → Z) (g : X → Y) (hf : SciLean.CDifferentiableAt K f (g x)) (hg : SciLean.CDifferentiableAt K g x) : SciLean.fwdDeriv K (fun x => f (g x)) x = fun dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n let zdz := SciLean.fwdDeriv K f ydy.1 ydy.2;\n zdz"} +{"name":"SciLean.fwdDeriv.SciLean.Norm2.norm2.arg_a0.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.SciLean.Norm2.norm2.arg_a0.fwdDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.Vec R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (x : X) (f : X → Y) (hf : SciLean.CDifferentiableAt R f x) : SciLean.fwdDeriv R (fun x => ‖f x‖₂²) x = fun dx =>\n let ydy := SciLean.fwdDeriv R f x dx;\n (‖ydy.1‖₂², 2 * ⟪ydy.2, ydy.1⟫_R)"} +{"name":"SciLean.fwdDeriv.HMul.hMul.arg_a0a1.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.HMul.hMul.arg_a0a1.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDeriv K fun x => f x * g x) = fun x dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n (ydy.1 * zdz.1, zdz.2 * ydy.1 + ydy.2 * zdz.1)"} +{"name":"SciLean.fwdDeriv.HPow.hPow.arg_a0.fwdDeriv_rule_at","declaration":"def SciLean.fwdDeriv.HPow.hPow.arg_a0.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (n : ℕ) (x : X) (f : X → K) (hf : SciLean.CDifferentiableAt K f x) : SciLean.fwdDeriv K (fun x => f x ^ n) x = fun dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n (ydy.1 ^ n, ↑n * ydy.2 * ydy.1 ^ (n - 1))"} +{"name":"SciLean.fwdDeriv.Prod.mk.arg_fstsnd.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.Prod.mk.arg_fstsnd.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (x : X) (g : X → Y) (hg : SciLean.CDifferentiableAt K g x) (f : X → Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.fwdDeriv K (fun x => (g x, f x)) x = fun dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n let zdz := SciLean.fwdDeriv K f x dx;\n ((ydy.1, zdz.1), ydy.2, zdz.2)"} +{"name":"SciLean.fwdDeriv.ite.arg_te.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.ite.arg_te.fwdDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.fwdDeriv K fun x => if c then t x else e x) = fun y =>\n if c then SciLean.fwdDeriv K t y else SciLean.fwdDeriv K e y"} +{"name":"SciLean.fwdDeriv.let_rule_at","declaration":"theorem SciLean.fwdDeriv.let_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (x : X) (f : X → Y → Z) (g : X → Y) (hf : SciLean.CDifferentiableAt K (fun xy => f xy.1 xy.2) (x, g x)) (hg : SciLean.CDifferentiableAt K g x) : SciLean.fwdDeriv K\n (fun x =>\n let y := g x;\n f x y)\n x =\n fun dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n let zdz := SciLean.fwdDeriv K (fun xy => f xy.1 xy.2) (x, ydy.1) (dx, ydy.2);\n zdz"} +{"name":"SciLean.fwdDeriv.HSMul.hSMul.arg_a0a1.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.HSMul.hSMul.arg_a0a1.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → K) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.fwdDeriv K (fun x => f x • g x) x = fun dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n (ydy.1 • zdz.1, ydy.1 • zdz.2 + ydy.2 • zdz.1)"} +{"name":"SciLean.fwdDeriv.HDiv.hDiv.arg_a0a1.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.HDiv.hDiv.arg_a0a1.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (f : X → K) (g : X → K) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.fwdDeriv K fun x => f x / g x) = fun x dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n (ydy.1 / zdz.1, (ydy.2 * zdz.1 - ydy.1 * zdz.2) / zdz.1 ^ 2)"} +{"name":"SciLean.fwdDeriv.HDiv.hDiv.arg_a0a1.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.HDiv.hDiv.arg_a0a1.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (x : X) (f : X → K) (g : X → K) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) (hx : g x ≠ 0) : SciLean.fwdDeriv K (fun x => f x / g x) x = fun dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n (ydy.1 / zdz.1, (ydy.2 * zdz.1 - ydy.1 * zdz.2) / zdz.1 ^ 2)"} +{"name":"SciLean.fwdDeriv.HAdd.hAdd.arg_a0a1.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.HAdd.hAdd.arg_a0a1.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (x : X) (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiableAt K f x) (hg : SciLean.CDifferentiableAt K g x) : SciLean.fwdDeriv K (fun x => f x + g x) x = fun dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n ydy + zdz"} +{"name":"SciLean.fwdDeriv.Prod.fst.arg_self.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.Prod.fst.arg_self.fwdDeriv_rule_at {K : Type u_3} [RCLike K] {X : Type u_4} [SciLean.Vec K X] {Y : Type u_1} [SciLean.Vec K Y] {Z : Type u_2} [SciLean.Vec K Z] (x : X) (f : X → Y × Z) (hf : SciLean.CDifferentiableAt K f x) : SciLean.fwdDeriv K (fun x => (f x).1) x = fun dx =>\n let yzdyz := SciLean.fwdDeriv K f x dx;\n (yzdyz.1.1, yzdyz.2.1)"} +{"name":"SciLean.fwdDeriv.apply_rule","declaration":"theorem SciLean.fwdDeriv.apply_rule {K : Type u_3} [RCLike K] {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → SciLean.Vec K (E i)] (i : ι) : (SciLean.fwdDeriv K fun x => x i) = fun x dx => (x i, dx i)"} +{"name":"SciLean.fwdDeriv.let_rule","declaration":"theorem SciLean.fwdDeriv.let_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {Z : Type u_4} [SciLean.Vec K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.CDifferentiable K fun xy => f xy.1 xy.2) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDeriv K fun x =>\n let y := g x;\n f x y) =\n fun x dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n let zdz := SciLean.fwdDeriv K (fun xy => f xy.1 xy.2) (x, ydy.1) (dx, ydy.2);\n zdz"} +{"name":"SciLean.fwdDeriv.IndexType.sum.arg_f.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.IndexType.sum.arg_f.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.CDifferentiable K fun x => f x i) : (SciLean.fwdDeriv K fun x => ∑ i, f x i) = fun x dx =>\n let ydy := fun i => SciLean.fwdDeriv K (fun x => f x i) x dx;\n ∑ i, ydy i"} +{"name":"SciLean.fwdDeriv.dite.arg_te.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.dite.arg_te.fwdDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.fwdDeriv K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.fwdDeriv K (t p) y else SciLean.fwdDeriv K (e p) y"} +{"name":"SciLean.fwdDeriv.IndexType.sum.arg_f.fwdDeriv_rule_at","declaration":"theorem SciLean.fwdDeriv.IndexType.sum.arg_f.fwdDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] {ι : Type u_4} [SciLean.IndexType ι] (x : X) (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.CDifferentiableAt K (fun x => f x i) x) : SciLean.fwdDeriv K (fun x => ∑ i, f x i) x = fun dx =>\n let ydy := fun i => SciLean.fwdDeriv K (fun x => f x i) x dx;\n ∑ i, ydy i"} +{"name":"SciLean.fwdDeriv.pi_rule_at","declaration":"theorem SciLean.fwdDeriv.pi_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {ι : Type u_4} {E : ι → Type u_3} [(i : ι) → SciLean.Vec K (E i)] (x : X) (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.CDifferentiableAt K (fun x => f x i) x) : SciLean.fwdDeriv K (fun x i => f x i) x = fun dx =>\n (fun i => f x i, fun i => (SciLean.fwdDeriv K (fun x => f x i) x dx).2)"} +{"name":"SciLean.fwdDeriv.HPow.hPow.arg_a0.fwdDeriv_rule","declaration":"def SciLean.fwdDeriv.HPow.hPow.arg_a0.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] (n : ℕ) (f : X → K) (hf : SciLean.CDifferentiable K f) : (SciLean.fwdDeriv K fun x => f x ^ n) = fun x dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n (ydy.1 ^ n, ↑n * ydy.2 * ydy.1 ^ (n - 1))"} +{"name":"SciLean.fwdDeriv.HSMul.hSMul.arg_a0a1.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.HSMul.hSMul.arg_a0a1.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → K) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDeriv K fun x => f x • g x) = fun x dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n (ydy.1 • zdz.1, ydy.1 • zdz.2 + ydy.2 • zdz.1)"} +{"name":"SciLean.fwdDeriv.Neg.neg.arg_a0.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.Neg.neg.arg_a0.fwdDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [SciLean.Vec K X] {Y : Type u_2} [SciLean.Vec K Y] (f : X → Y) : (SciLean.fwdDeriv K fun x => -f x) = fun x dx => -SciLean.fwdDeriv K f x dx"} +{"name":"SciLean.fwdDeriv.HSub.hSub.arg_a0a1.fwdDeriv_rule","declaration":"theorem SciLean.fwdDeriv.HSub.hSub.arg_a0a1.fwdDeriv_rule {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {Y : Type u_3} [SciLean.Vec K Y] (f : X → Y) (g : X → Y) (hf : SciLean.CDifferentiable K f) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDeriv K fun x => f x - g x) = fun x dx =>\n let ydy := SciLean.fwdDeriv K f x dx;\n let zdz := SciLean.fwdDeriv K g x dx;\n ydy - zdz"} +{"name":"SciLean.fwdDeriv.id_rule","declaration":"theorem SciLean.fwdDeriv.id_rule {K : Type u_2} [RCLike K] {X : Type u_1} [SciLean.Vec K X] : (SciLean.fwdDeriv K fun x => x) = fun x dx => (x, dx)"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.FwdFDeriv.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.FwdFDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b16d298d2c636b7c4199bc21ce2cc5b763d7627e --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.FwdFDeriv.jsonl @@ -0,0 +1,20 @@ +{"name":"SciLean.FwdFDeriv.HDiv.hDiv.arg_a0a1.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.HDiv.hDiv.arg_a0a1.fwdFDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] (x : X) (f : X → K) (g : X → K) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) (hx : g x ≠ 0) : SciLean.fwdFDeriv K (fun x => f x / g x) x = fun dx =>\n let ydy := SciLean.fwdFDeriv K f x dx;\n let zdz := SciLean.fwdFDeriv K g x dx;\n (ydy.1 / zdz.1, (ydy.2 * zdz.1 - ydy.1 * zdz.2) / zdz.1 ^ 2)"} +{"name":"SciLean.FwdFDeriv.ite.arg_te.fwdFDeriv_rule","declaration":"theorem SciLean.FwdFDeriv.ite.arg_te.fwdFDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.fwdFDeriv K fun x => if c then t x else e x) = fun y =>\n if c then SciLean.fwdFDeriv K t y else SciLean.fwdFDeriv K e y"} +{"name":"SciLean.FwdFDeriv.HSub.hSub.arg_a0a1.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.HSub.hSub.arg_a0a1.fwdFDeriv_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) (f : X → Y) (g : X → Y) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : SciLean.fwdFDeriv K (fun x => f x - g x) x = fun dx => SciLean.fwdFDeriv K f x dx - SciLean.fwdFDeriv K g x dx"} +{"name":"SciLean.FwdFDeriv.id_rule","declaration":"theorem SciLean.FwdFDeriv.id_rule {K : Type u_2} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] : (SciLean.fwdFDeriv K fun x => x) = fun x dx => (x, dx)"} +{"name":"SciLean.FwdFDeriv.HMul.hMul.arg_a0a1.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.HMul.hMul.arg_a0a1.fwdFDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] (x : X) (f : X → K) (g : X → K) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : SciLean.fwdFDeriv K (fun x => f x * g x) x = fun dx =>\n let ydy := SciLean.fwdFDeriv K f x dx;\n let zdz := SciLean.fwdFDeriv K g x dx;\n (ydy.1 * zdz.1, zdz.2 * ydy.1 + ydy.2 * zdz.1)"} +{"name":"SciLean.FwdFDeriv.Prod.fst.arg_self.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.Prod.fst.arg_self.fwdFDeriv_rule_at {K : Type u_4} [RCLike K] {X : Type u_3} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_2} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (f : X → Y × Z) (hf : DifferentiableAt K f x) : SciLean.fwdFDeriv K (fun x => (f x).1) x = fun dx =>\n let yzdyz := SciLean.fwdFDeriv K f x dx;\n (yzdyz.1.1, yzdyz.2.1)"} +{"name":"SciLean.FwdFDeriv.HPow.hPow.arg_a0.fwdFDeriv_rule_at","declaration":"def SciLean.FwdFDeriv.HPow.hPow.arg_a0.fwdFDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] (n : ℕ) (x : X) (f : X → K) (hf : DifferentiableAt K f x) : SciLean.fwdFDeriv K (fun x => f x ^ n) x = fun dx =>\n let ydy := SciLean.fwdFDeriv K f x dx;\n (ydy.1 ^ n, ↑n * ydy.2 * ydy.1 ^ (n - 1))"} +{"name":"SciLean.FwdFDeriv.const_rule","declaration":"theorem SciLean.FwdFDeriv.const_rule {K : Type u_3} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] (y : Y) : (SciLean.fwdFDeriv K fun x => y) = fun x dx => (y, 0)"} +{"name":"SciLean.FwdFDeriv.pi_rule_at","declaration":"theorem SciLean.FwdFDeriv.pi_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {ι : Type u_4} [Fintype ι] {E : ι → Type u_1} [(j : ι) → NormedAddCommGroup (E j)] [(j : ι) → NormedSpace K (E j)] (x : X) (f : X → (i : ι) → E i) (hf : ∀ (i : ι), DifferentiableAt K (fun x => f x i) x) : SciLean.fwdFDeriv K (fun x i => f x i) x = fun dx =>\n (fun i => f x i, fun i => (SciLean.fwdFDeriv K (fun x => f x i) x dx).2)"} +{"name":"SciLean.FwdFDeriv.Prod.mk.arg_fstsnd.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.Prod.mk.arg_fstsnd.fwdFDeriv_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_4} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (g : X → Y) (hg : DifferentiableAt K g x) (f : X → Z) (hf : DifferentiableAt K f x) : SciLean.fwdFDeriv K (fun x => (g x, f x)) x = fun dx =>\n let ydy := SciLean.fwdFDeriv K g x dx;\n let zdz := SciLean.fwdFDeriv K f x dx;\n ((ydy.1, zdz.1), ydy.2, zdz.2)"} +{"name":"SciLean.FwdFDeriv.HAdd.hAdd.arg_a0a1.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.HAdd.hAdd.arg_a0a1.fwdFDeriv_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) (f : X → Y) (g : X → Y) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : SciLean.fwdFDeriv K (fun x => f x + g x) x = fun dx => SciLean.fwdFDeriv K f x dx + SciLean.fwdFDeriv K g x dx"} +{"name":"SciLean.FwdFDeriv.let_rule_at","declaration":"theorem SciLean.FwdFDeriv.let_rule_at {K : Type u_4} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_3} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_1} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (f : X → Y → Z) (g : X → Y) (hf : DifferentiableAt K (fun xy => f xy.1 xy.2) (x, g x)) (hg : DifferentiableAt K g x) : SciLean.fwdFDeriv K\n (fun x =>\n let y := g x;\n f x y)\n x =\n fun dx =>\n let ydy := SciLean.fwdFDeriv K g x dx;\n let zdz := SciLean.fwdFDeriv K (fun xy => f xy.1 xy.2) (x, ydy.1) (dx, ydy.2);\n zdz"} +{"name":"SciLean.FwdFDeriv.comp_rule_at","declaration":"theorem SciLean.FwdFDeriv.comp_rule_at {K : Type u_3} [RCLike K] {X : Type u_4} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_1} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (f : Y → Z) (g : X → Y) (hf : DifferentiableAt K f (g x)) (hg : DifferentiableAt K g x) : SciLean.fwdFDeriv K (fun x => f (g x)) x = fun dx =>\n let ydy := SciLean.fwdFDeriv K g x dx;\n let zdz := SciLean.fwdFDeriv K f ydy.1 ydy.2;\n zdz"} +{"name":"SciLean.FwdFDeriv.Neg.neg.arg_a0.fwdFDeriv_rule","declaration":"theorem SciLean.FwdFDeriv.Neg.neg.arg_a0.fwdFDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) (f : X → Y) : SciLean.fwdFDeriv K (fun x => -f x) x = fun dx => -SciLean.fwdFDeriv K f x dx"} +{"name":"SciLean.FwdFDeriv.dite.arg_te.fwdFDeriv_rule","declaration":"theorem SciLean.FwdFDeriv.dite.arg_te.fwdFDeriv_rule {K : Type u_3} [RCLike K] {X : Type u_1} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_2} [NormedAddCommGroup Y] [NormedSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.fwdFDeriv K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.fwdFDeriv K (t p) y else SciLean.fwdFDeriv K (e p) y"} +{"name":"SciLean.FwdFDeriv.apply_rule","declaration":"theorem SciLean.FwdFDeriv.apply_rule {K : Type u_3} [RCLike K] {ι : Type u_1} [Fintype ι] {E : ι → Type u_2} [(j : ι) → NormedAddCommGroup (E j)] [(j : ι) → NormedSpace K (E j)] (i : ι) : (SciLean.fwdFDeriv K fun x => x i) = fun x dx => (x i, dx i)"} +{"name":"SciLean.FwdFDeriv.Prod.snd.arg_self.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.Prod.snd.arg_self.fwdFDeriv_rule_at {K : Type u_4} [RCLike K] {X : Type u_3} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] {Z : Type u_2} [NormedAddCommGroup Z] [NormedSpace K Z] (x : X) (f : X → Y × Z) (hf : DifferentiableAt K f x) : SciLean.fwdFDeriv K (fun x => (f x).2) x = fun dx =>\n let yzdyz := SciLean.fwdFDeriv K f x dx;\n (yzdyz.1.2, yzdyz.2.2)"} +{"name":"SciLean.fwdFDeriv","declaration":"def SciLean.fwdFDeriv (K : Type u_1) [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_3} [NormedAddCommGroup Y] [NormedSpace K Y] (f : X → Y) (x : X) (dx : X) : Y × Y"} +{"name":"SciLean.FwdFDeriv.FinType.sum.arg_f.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.FinType.sum.arg_f.fwdFDeriv_rule_at {K : Type u_3} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_1} [NormedAddCommGroup Y] [NormedSpace K Y] {ι : Type u_4} [Fintype ι] (x : X) (f : X → ι → Y) (hf : ∀ (i : ι), DifferentiableAt K (fun x => f x i) x) : SciLean.fwdFDeriv K (fun x => Finset.sum Finset.univ fun i => f x i) x = fun dx =>\n let ydy := fun i => SciLean.fwdFDeriv K (fun x => f x i) x dx;\n Finset.sum Finset.univ fun i => ydy i"} +{"name":"SciLean.FwdFDeriv.HSMul.hSMul.arg_a0a1.fwdFDeriv_rule_at","declaration":"theorem SciLean.FwdFDeriv.HSMul.hSMul.arg_a0a1.fwdFDeriv_rule_at {K : Type u_1} [RCLike K] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace K X] {Y : Type u_3} [NormedAddCommGroup Y] [NormedSpace K Y] (x : X) (f : X → K) (g : X → Y) (hf : DifferentiableAt K f x) (hg : DifferentiableAt K g x) : SciLean.fwdFDeriv K (fun x => f x • g x) x = fun dx =>\n let ydy := SciLean.fwdFDeriv K f x dx;\n let zdz := SciLean.fwdFDeriv K g x dx;\n (ydy.1 • zdz.1, ydy.1 • zdz.2 + ydy.2 • zdz.1)"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.InvFun.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.InvFun.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1749df4e614c41469f4a71ae87fdf657ec67488d --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.InvFun.jsonl @@ -0,0 +1,20 @@ +{"name":"Function.invFun.HSMul.hSMul.arg_a1.invFun_rule_group","declaration":"def Function.invFun.HSMul.hSMul.arg_a1.invFun_rule_group {X : Type u_1} [Nonempty X] {Y : Type u_2} {G : Type u_3} [Group G] [MulAction G Y] (g : G) (f : X → Y) (hf : Function.Bijective f) : (Function.invFun fun x => g • f x) = fun y => Function.invFun f (g⁻¹ • y)"} +{"name":"Function.invFun.HSMul.hSMul.arg_a1.invFun_rule_field","declaration":"def Function.invFun.HSMul.hSMul.arg_a1.invFun_rule_field {X : Type u_1} [Nonempty X] {Y : Type u_2} {R : Type u_3} [Field R] [MulAction R Y] (r : R) (f : X → Y) (hf : Function.Bijective f) (hr : r ≠ 0) : (Function.invFun fun x => r • f x) = fun y => Function.invFun f (r⁻¹ • y)"} +{"name":"Function.invFun.HMul.hMul.arg_a0.invFun_rule_field","declaration":"def Function.invFun.HMul.hMul.arg_a0.invFun_rule_field {X : Type u_1} [Nonempty X] {Y : Type u_2} [Field Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) (hy : y ≠ 0) : (Function.invFun fun x => f x * y) = fun y' => Function.invFun f (y' / y)"} +{"name":"Function.invFun.comp_rule","declaration":"theorem Function.invFun.comp_rule {X : Type u_3} [Nonempty X] {Y : Type u_1} [Nonempty Y] {Z : Type u_2} (f : Y → Z) (g : X → Y) (hf : Function.Bijective f) (hg : Function.Bijective g) : (Function.invFun fun x => f (g x)) = fun z =>\n let y := Function.invFun f z;\n let x := Function.invFun g y;\n x"} +{"name":"Function.invFun.HAdd.hAdd.arg_a0.invFun_rule","declaration":"theorem Function.invFun.HAdd.hAdd.arg_a0.invFun_rule {X : Type u_2} [Nonempty X] {Y : Type u_1} [AddGroup Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : (Function.invFun fun x => f x + y) = fun y' => Function.invFun f (y' - y)"} +{"name":"Function.invFun.HSub.hSub.arg_a0.invFun_rule","declaration":"theorem Function.invFun.HSub.hSub.arg_a0.invFun_rule {X : Type u_2} [Nonempty X] {Y : Type u_1} [AddGroup Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : (Function.invFun fun x => f x - y) = fun y' => Function.invFun f (y' + y)"} +{"name":"Function.invFun.let_rule","declaration":"theorem Function.invFun.let_rule {X : Type u_5} [Nonempty X] {Y : Type u_1} [Nonempty Y] {Z : Type u_3} {X₁ : Type u_4} [Nonempty X₁] {X₂ : Type u_2} [Nonempty X₂] (f : X₂ → Y → Z) (g : X₁ → Y) (p₁ : X → X₁) (p₂ : X → X₂) (hf : Function.Bijective fun xy => f xy.1 xy.2) (hg : Function.Bijective g) (hp : Function.Bijective fun x => (p₁ x, p₂ x)) : (Function.invFun fun x =>\n let y := g (p₁ x);\n f (p₂ x) y) =\n fun z =>\n let x₂y := Function.invFun (fun xy => f xy.1 xy.2) z;\n let x₁ := Function.invFun g x₂y.2;\n let x := Function.invFun (fun x => (p₁ x, p₂ x)) (x₁, x₂y.1);\n x"} +{"name":"Function.invFun.Inv.inv.arg_a0.invFun_rule_group","declaration":"theorem Function.invFun.Inv.inv.arg_a0.invFun_rule_group {X : Type u_2} [Nonempty X] {Y : Type u_1} [Group Y] (f : X → Y) (hf : Function.Bijective f) : (Function.invFun fun x => (f x)⁻¹) = fun y => Function.invFun f y⁻¹"} +{"name":"Function.invFun.HAdd.hAdd.arg_a1.invFun_rule","declaration":"theorem Function.invFun.HAdd.hAdd.arg_a1.invFun_rule {X : Type u_2} [Nonempty X] {Y : Type u_1} [AddGroup Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : (Function.invFun fun x => y + f x) = fun y' => Function.invFun f (-y + y')"} +{"name":"Function.invFun.Prod.mk.arg_fstsnd.invFun_rule","declaration":"theorem Function.invFun.Prod.mk.arg_fstsnd.invFun_rule {X₁ : Type u_1} [Nonempty X₁] {X₂ : Type u_3} [Nonempty X₂] {X : Type u_5} [Nonempty X] {Y : Type u_2} {Z : Type u_4} (f : X₁ → Y) (g : X₂ → Z) (p₁ : X → X₁) (p₂ : X → X₂) (hf : Function.Bijective f) (hg : Function.Bijective g) (hp : Function.Bijective fun x => (p₁ x, p₂ x)) : (Function.invFun fun x => (f (p₁ x), g (p₂ x))) = fun yz =>\n let x₁ := Function.invFun f yz.1;\n let x₂ := Function.invFun g yz.2;\n let x := Function.invFun (fun x => (p₁ x, p₂ x)) (x₁, x₂);\n x"} +{"name":"Function.invFun.Neg.neg.arg_a0.invFun_rule","declaration":"theorem Function.invFun.Neg.neg.arg_a0.invFun_rule {X : Type u_2} [Nonempty X] {Y : Type u_1} [AddGroup Y] (f : X → Y) (hf : Function.Bijective f) : (Function.invFun fun x => -f x) = fun y => Function.invFun f (-y)"} +{"name":"Function.invFun.Equiv.toFun.arg_a0.invFun_rule","declaration":"theorem Function.invFun.Equiv.toFun.arg_a0.invFun_rule {X : Type u_3} [Nonempty X] {Y : Type u_1} {Z : Type u_2} (f : Y ≃ Z) (g : X → Y) (hf : Function.Bijective g) : (Function.invFun fun x => f (g x)) = fun z => Function.invFun g (f.invFun z)"} +{"name":"Function.invFun.Equiv.invFun.arg_a0.invFun_rule","declaration":"theorem Function.invFun.Equiv.invFun.arg_a0.invFun_rule {X : Type u_3} [Nonempty X] {Y : Type u_1} {Z : Type u_2} (f : Y ≃ Z) (g : X → Z) (hf : Function.Bijective g) : (Function.invFun fun x => f.invFun (g x)) = fun z => Function.invFun g (f z)"} +{"name":"Function.invFun.HMul.hMul.arg_a1.invFun_rule_group","declaration":"def Function.invFun.HMul.hMul.arg_a1.invFun_rule_group {X : Type u_1} [Nonempty X] {Y : Type u_2} [Group Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : (Function.invFun fun x => y * f x) = fun y' => Function.invFun f (y⁻¹ * y')"} +{"name":"Function.invFun.HSub.hSub.arg_a1.invFun_rule","declaration":"theorem Function.invFun.HSub.hSub.arg_a1.invFun_rule {X : Type u_2} [Nonempty X] {Y : Type u_1} [AddGroup Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) : (Function.invFun fun x => y - f x) = fun y' => Function.invFun f (y - y')"} +{"name":"Function.invFun.Inv.inv.arg_a0.invFun_rule_field","declaration":"theorem Function.invFun.Inv.inv.arg_a0.invFun_rule_field {X : Type u_2} [Nonempty X] {Y : Type u_1} [Field Y] (f : X → Y) (hf : Function.Bijective f) (hf' : ∀ (x : X), f x ≠ 0) : (Function.invFun fun x => (f x)⁻¹) = fun y => Function.invFun f y⁻¹"} +{"name":"Function.invFun.HMul.hMul.arg_a0.invFun_rule_group","declaration":"def Function.invFun.HMul.hMul.arg_a0.invFun_rule_group {X : Type u_1} [Nonempty X] {Y : Type u_2} [Group Y] (f : X → Y) (y : Y) (hf : Function.Bijective f) : (Function.invFun fun x => f x * y) = fun y' => Function.invFun f (y' / y)"} +{"name":"Function.invFun.HVAdd.hVAdd.arg_a1.invFun_rule_group","declaration":"def Function.invFun.HVAdd.hVAdd.arg_a1.invFun_rule_group {X : Type u_1} [Nonempty X] {Y : Type u_2} {G : Type u_3} [AddGroup G] [AddAction G Y] (g : G) (f : X → Y) (hf : Function.Bijective f) : (Function.invFun fun x => g +ᵥ f x) = fun y => Function.invFun f (-g +ᵥ y)"} +{"name":"Function.invFun.id_rule","declaration":"theorem Function.invFun.id_rule {X : Type u_1} [Nonempty X] : (Function.invFun fun x => x) = fun x => x"} +{"name":"Function.invFun.HMul.hMul.arg_a1.invFun_rule_field","declaration":"def Function.invFun.HMul.hMul.arg_a1.invFun_rule_field {X : Type u_1} [Nonempty X] {Y : Type u_2} [Field Y] (y : Y) (f : X → Y) (hf : Function.Bijective f) (hy : y ≠ 0) : (Function.invFun fun x => y * f x) = fun y' => Function.invFun f (y⁻¹ * y')"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.Isomorph.RealToFloat.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.Isomorph.RealToFloat.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e7eb550bab9e17c7a4dd62e15ccc9d7d75d03d1b --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.Isomorph.RealToFloat.jsonl @@ -0,0 +1,26 @@ +{"name":"Float.exp.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Float.exp.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => Float.exp (f x)) = fun x => Real.exp (SciLean.isomorph `FloatToReal f x)"} +{"name":"Float.acos.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Float.acos.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => Float.acos (f x)) = fun x => Real.arccos (SciLean.isomorph `FloatToReal f x)"} +{"name":"HAdd.hAdd.arg_a0a1.isomorph_rule_RealToFloat","declaration":"axiom HAdd.hAdd.arg_a0a1.isomorph_rule_RealToFloat {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) (g : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => f x + g x) = fun x =>\n SciLean.isomorph `RealToFloat f x + SciLean.isomorph `RealToFloat g x"} +{"name":"Float.atan.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Float.atan.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => Float.atan (f x)) = fun x => Real.arctan (SciLean.isomorph `FloatToReal f x)"} +{"name":"Neg.neg.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Neg.neg.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => -f x) = fun x => -SciLean.isomorph `FloatToReal f x"} +{"name":"instInvFloat","declaration":"instance instInvFloat : Inv Float"} +{"name":"HDiv.hDiv.arg_a0a1.isomorph_rule_RealToFloat","declaration":"axiom HDiv.hDiv.arg_a0a1.isomorph_rule_RealToFloat {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) (g : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => f x / g x) = fun x =>\n SciLean.isomorph `RealToFloat f x / SciLean.isomorph `RealToFloat g x"} +{"name":"Float.cos.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Float.cos.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => Float.cos (f x)) = fun x => Real.cos (SciLean.isomorph `FloatToReal f x)"} +{"name":"HMul.hMul.arg_a0a1.isomorph_rule_RealToFloat","declaration":"axiom HMul.hMul.arg_a0a1.isomorph_rule_RealToFloat {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) (g : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => f x * g x) = fun x =>\n SciLean.isomorph `RealToFloat f x * SciLean.isomorph `RealToFloat g x"} +{"name":"HDiv.hDiv.arg_a0a1.isomorph_rule_FloatToReal","declaration":"axiom HDiv.hDiv.arg_a0a1.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) (g : α → Float) : (SciLean.isomorph `FloatToReal fun x => f x / g x) = fun x =>\n SciLean.isomorph `FloatToReal f x / SciLean.isomorph `FloatToReal g x"} +{"name":"HAdd.hAdd.arg_a0a1.isomorph_rule_FloatToReal","declaration":"axiom HAdd.hAdd.arg_a0a1.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) (g : α → Float) : (SciLean.isomorph `FloatToReal fun x => f x + g x) = fun x =>\n SciLean.isomorph `FloatToReal f x + SciLean.isomorph `FloatToReal g x"} +{"name":"Neg.neg.arg_a0.isomorph_rule_RealToFloat","declaration":"axiom Neg.neg.arg_a0.isomorph_rule_RealToFloat {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => -f x) = fun x => -SciLean.isomorph `RealToFloat f x"} +{"name":"Float.sin.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Float.sin.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => Float.sin (f x)) = fun x => Real.sin (SciLean.isomorph `FloatToReal f x)"} +{"name":"HSub.hSub.arg_a0a1.isomorph_rule_FloatToReal","declaration":"axiom HSub.hSub.arg_a0a1.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) (g : α → Float) : (SciLean.isomorph `FloatToReal fun x => f x - g x) = fun x =>\n SciLean.isomorph `FloatToReal f x - SciLean.isomorph `FloatToReal g x"} +{"name":"HMul.hMul.arg_a0a1.isomorph_rule_FloatToReal","declaration":"axiom HMul.hMul.arg_a0a1.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) (g : α → Float) : (SciLean.isomorph `FloatToReal fun x => f x * g x) = fun x =>\n SciLean.isomorph `FloatToReal f x * SciLean.isomorph `FloatToReal g x"} +{"name":"Zero.zero.isomorph_rule_FloatToReal","declaration":"axiom Zero.zero.isomorph_rule_FloatToReal : SciLean.floatToReal 0 = 0"} +{"name":"Real.sqrt.arg_x.isomorph_rule_RealToFloat","declaration":"axiom Real.sqrt.arg_x.isomorph_rule_RealToFloat {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => √(f x)) = fun x => Float.sqrt (SciLean.isomorph `RealToFloat f x)"} +{"name":"Prod.snd.arg_self.isomorph_rule","declaration":"theorem Prod.snd.arg_self.isomorph_rule {α : Type u_1} {α' : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {β₁' : Type u_5} {β₂' : Type u_6} [SciLean.IsomorphicType `RealToFloat α α'] [SciLean.IsomorphicType `RealToFloat β₁ β₁'] [SciLean.IsomorphicType `RealToFloat β₂ β₂'] (f : α → β₁ × β₂) : (SciLean.isomorph `RealToFloat fun x => (f x).2) = fun x => (SciLean.isomorph `RealToFloat f x).2"} +{"name":"Real.exp.arg_x.isomorph_rule","declaration":"axiom Real.exp.arg_x.isomorph_rule {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => Real.exp (f x)) = fun x => Float.exp (SciLean.isomorph `RealToFloat f x)"} +{"name":"Inv.inv.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Inv.inv.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => (f x)⁻¹) = fun x => (SciLean.isomorph `FloatToReal f x)⁻¹"} +{"name":"Float.asin.arg_a0.isomorph_rule_FloatToReal","declaration":"axiom Float.asin.arg_a0.isomorph_rule_FloatToReal {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `FloatToReal α α'] (f : α → Float) : (SciLean.isomorph `FloatToReal fun x => Float.asin (f x)) = fun x => Real.arcsin (SciLean.isomorph `FloatToReal f x)"} +{"name":"HSub.hSub.arg_a0a1.isomorph_rule_RealToFloat","declaration":"axiom HSub.hSub.arg_a0a1.isomorph_rule_RealToFloat {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) (g : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => f x - g x) = fun x =>\n SciLean.isomorph `RealToFloat f x - SciLean.isomorph `RealToFloat g x"} +{"name":"One.one.isomorph_rule_FloatToReal","declaration":"axiom One.one.isomorph_rule_FloatToReal : SciLean.floatToReal 1 = 1"} +{"name":"Prod.fst.arg_self.isomorph_rule","declaration":"theorem Prod.fst.arg_self.isomorph_rule {α : Type u_1} {α' : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {β₁' : Type u_5} {β₂' : Type u_6} [SciLean.IsomorphicType `RealToFloat α α'] [SciLean.IsomorphicType `RealToFloat β₁ β₁'] [SciLean.IsomorphicType `RealToFloat β₂ β₂'] (f : α → β₁ × β₂) : (SciLean.isomorph `RealToFloat fun x => (f x).1) = fun x => (SciLean.isomorph `RealToFloat f x).1"} +{"name":"And.arg_ab.isomorph_rule","declaration":"axiom And.arg_ab.isomorph_rule {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → Prop) (g : α → Prop) : (SciLean.isomorph `RealToFloat fun x => f x ∧ g x) = fun x =>\n SciLean.isomorph `RealToFloat f x ∧ SciLean.isomorph `RealToFloat g x"} +{"name":"LE.le.arg_a0a1.isomorph_rule","declaration":"axiom LE.le.arg_a0a1.isomorph_rule {α : Type u_1} {α' : Type u_2} [SciLean.IsomorphicType `RealToFloat α α'] (f : α → ℝ) (g : α → ℝ) : (SciLean.isomorph `RealToFloat fun x => f x ≤ g x) = fun x =>\n SciLean.isomorph `RealToFloat f x ≤ SciLean.isomorph `RealToFloat g x"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.Isomorph.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.Isomorph.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d033a9282fbe873dcdf4dedf77af3e12ebb80ba0 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.Isomorph.jsonl @@ -0,0 +1,9 @@ +{"name":"Prod.mk.isomorph_rule","declaration":"theorem Prod.mk.isomorph_rule {α : Type u_4} {β : Type u_6} {γ : Type u_5} {α' : outParam (Type u_1)} {β' : outParam (Type u_2)} {γ' : outParam (Type u_3)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] [SciLean.IsomorphicType tag γ γ'] (f : α → β) (g : α → γ) : (SciLean.isomorph tag fun x => (f x, g x)) = fun x' =>\n let y' := SciLean.isomorph tag f x';\n let z' := SciLean.isomorph tag g x';\n (y', z')"} +{"name":"SciLean.isomorph.apply_rule","declaration":"theorem SciLean.isomorph.apply_rule {α : Type u_3} {β : Type u_4} {α' : outParam (Type u_1)} {β' : outParam (Type u_2)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] (x : α) : (SciLean.isomorph tag fun f => f x) = fun f => f ((SciLean.IsomorphicType.equiv tag) x)"} +{"name":"SciLean.isomorph.let_rule","declaration":"theorem SciLean.isomorph.let_rule {α : Type u_3} {β : Type u_5} {γ : Type u_4} {α' : outParam (Type u_1)} {β' : outParam (Type u_6)} {γ' : outParam (Type u_2)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] [SciLean.IsomorphicType tag γ γ'] (f : α → β → γ) (g : α → β) : (SciLean.isomorph tag fun x =>\n let y := g x;\n f x y) =\n fun x' =>\n let y' := SciLean.isomorph tag g x';\n SciLean.isomorph tag (fun xy => f xy.1 xy.2) (x', y')"} +{"name":"Prod.fst.isomorph_rule","declaration":"theorem Prod.fst.isomorph_rule {α : Type u_5} {β : Type u_1} {γ : Type u_2} {α' : outParam (Type u_3)} {β' : outParam (Type u_4)} {γ' : outParam (Type u_6)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] [SciLean.IsomorphicType tag γ γ'] (f : α → β × γ) : (SciLean.isomorph tag fun x => (f x).1) = fun x =>\n let f' := SciLean.isomorph tag f;\n (f' x).1"} +{"name":"Prod.snd.isomorph_rule","declaration":"theorem Prod.snd.isomorph_rule {α : Type u_5} {β : Type u_1} {γ : Type u_2} {α' : outParam (Type u_3)} {β' : outParam (Type u_4)} {γ' : outParam (Type u_6)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] [SciLean.IsomorphicType tag γ γ'] (f : α → β × γ) : (SciLean.isomorph tag fun x => (f x).1) = fun x =>\n let f' := SciLean.isomorph tag f;\n (f' x).1"} +{"name":"SciLean.isomorph.pi_rule","declaration":"theorem SciLean.isomorph.pi_rule {α : Type u_4} {β : Type u_5} {γ : Type u_6} {α' : outParam (Type u_1)} {β' : outParam (Type u_2)} {γ' : outParam (Type u_3)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] [SciLean.IsomorphicType tag γ γ'] (f : α → β → γ) : (SciLean.isomorph tag fun x y => f x y) = fun x' =>\n SciLean.isomorph tag (f ((SciLean.IsomorphicType.equiv tag).symm x'))"} +{"name":"SciLean.isomorph.id_rule","declaration":"theorem SciLean.isomorph.id_rule {α : Type u_2} {α' : outParam (Type u_1)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] : (SciLean.isomorph tag fun x => x) = fun x => x"} +{"name":"SciLean.isomorph.const_rule","declaration":"theorem SciLean.isomorph.const_rule {α : Type u_3} {β : Type u_4} {α' : outParam (Type u_1)} {β' : outParam (Type u_2)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] (y : β) : (SciLean.isomorph tag fun x => y) = fun x => (SciLean.IsomorphicType.equiv tag) y"} +{"name":"SciLean.isomorph.comp_rule","declaration":"theorem SciLean.isomorph.comp_rule {α : Type u_3} {β : Type u_5} {γ : Type u_4} {α' : outParam (Type u_1)} {β' : outParam (Type u_6)} {γ' : outParam (Type u_2)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] [SciLean.IsomorphicType tag γ γ'] (f : β → γ) (g : α → β) : (SciLean.isomorph tag fun x => f (g x)) = fun x => SciLean.isomorph tag f (SciLean.isomorph tag g x)"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.Preimage.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.Preimage.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..574e7632c9d9aab622eef49abae3082dc3a43632 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.Preimage.jsonl @@ -0,0 +1,16 @@ +{"name":"Set.preimage1_const","declaration":"theorem Set.preimage1_const {α : Type u_1} {β : Sort u_2} {γ : Type u_3} (c : γ) (C : Set γ) : Set.preimage1 (fun x x => c) C = if c ∈ C then Set.univ else ∅"} +{"name":"Set.Prod.mk.arg_fstsnd.preimage_rule_prod","declaration":"theorem Set.Prod.mk.arg_fstsnd.preimage_rule_prod {α : Type u_3} {β : Type u_1} {γ : Type u_2} (f : α → β) (g : α → γ) (B : Set β) (C : Set γ) : (fun x => (f x, g x)) ⁻¹' Set.prod B C = f ⁻¹' B ∩ g ⁻¹' C"} +{"name":"Set.preimage1","declaration":"def Set.preimage1 {α : Type u_1} {β : Sort u_2} {γ : Type u_3} (f : α → β → γ) (C : Set γ) : Set α"} +{"name":"Set.preimage1_id1","declaration":"theorem Set.preimage1_id1 {α : Type u_1} {β : Sort u_2} (A : Set α) : Set.preimage1 (fun a x => a) A = A"} +{"name":"Set.preimage_const'","declaration":"theorem Set.preimage_const' {α : Type u_2} {β : Type u_1} (b : β) (s : Set β) : (fun x => b) ⁻¹' s = if b ∈ s then Set.univ else ∅"} +{"name":"Set.preimage_comp'","declaration":"theorem Set.preimage_comp' {α : Type u_1} {β : Type u_3} {γ : Type u_2} (f : β → γ) (g : α → β) : (Set.preimage fun x => f (g x)) = fun s => g ⁻¹' (f ⁻¹' s)"} +{"name":"Set.Neg.neg.arg_a1.preimage_rule_Ioo","declaration":"theorem Set.Neg.neg.arg_a1.preimage_rule_Ioo {R : Type u_1} [SciLean.RealScalar R] (a : R) (b : R) : (fun x => -x) ⁻¹' Set.Ioo a b = Set.Ioo (-b) (-a)"} +{"name":"Set.HSub.hSub.arg_a1.preimage_rule_Ioo","declaration":"theorem Set.HSub.hSub.arg_a1.preimage_rule_Ioo {R : Type u_1} [SciLean.RealScalar R] (x' : R) (a : R) (b : R) : (fun x => x' - x) ⁻¹' Set.Ioo a b = Set.Ioo (x' - b) (x' - a)"} +{"name":"Set.preimage1_prod'","declaration":"theorem Set.preimage1_prod' {α : Type u_1} {β : Sort u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ) (g : β → δ) (C : Set γ) (D : Set δ) : Set.preimage1 (fun x y => (g y, f x)) (Set.prod D C) = f ⁻¹' C"} +{"name":"Set.preimage1_prod","declaration":"theorem Set.preimage1_prod {α : Type u_1} {β : Sort u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ) (g : β → δ) (C : Set γ) (D : Set δ) : Set.preimage1 (fun x y => (f x, g y)) (Set.prod C D) = f ⁻¹' C"} +{"name":"Set.Prod.mk.arg_fst.preimage_rule_prod","declaration":"theorem Set.Prod.mk.arg_fst.preimage_rule_prod {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β) (c : γ) : (Set.preimage fun x => (f x, c)) = fun s => f ⁻¹' {y | (y, c) ∈ s}"} +{"name":"Set.preimage1_id2","declaration":"theorem Set.preimage1_id2 {α : Type u_1} {β : Type u_2} (B : Set β) : Set.preimage1 (fun x b => b) B = Set.univ"} +{"name":"Set.HAdd.hAdd.arg_a1.preimage_rule_Ioo","declaration":"theorem Set.HAdd.hAdd.arg_a1.preimage_rule_Ioo {R : Type u_1} [SciLean.RealScalar R] (x' : R) (a : R) (b : R) : (fun x => x' + x) ⁻¹' Set.Ioo a b = Set.Ioo (a - x') (b - x')"} +{"name":"Set.HAdd.hAdd.arg_a0.preimage_rule_Ioo","declaration":"theorem Set.HAdd.hAdd.arg_a0.preimage_rule_Ioo {R : Type u_1} [SciLean.RealScalar R] (x' : R) (a : R) (b : R) : (fun x => x + x') ⁻¹' Set.Ioo a b = Set.Ioo (a - x') (b - x')"} +{"name":"Set.HSub.hSub.arg_a0.preimage_rule_Ioo","declaration":"theorem Set.HSub.hSub.arg_a0.preimage_rule_Ioo {R : Type u_1} [SciLean.RealScalar R] (x' : R) (a : R) (b : R) : (fun x => x - x') ⁻¹' Set.Ioo a b = Set.Ioo (a + x') (b + x')"} +{"name":"Set.Prod.mk.arg_snd.preimage_rule_prod","declaration":"theorem Set.Prod.mk.arg_snd.preimage_rule_prod {α : Type u_1} {β : Type u_2} {γ : Type u_3} (b : β) (g : α → γ) : (Set.preimage fun x => (b, g x)) = fun s => g ⁻¹' {z | (b, z) ∈ s}"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.RevCDeriv.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.RevCDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.RevDeriv.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.RevDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1e47137184098dfb309773f5f0a5f74865153e84 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.RevDeriv.jsonl @@ -0,0 +1,109 @@ +{"name":"Inner.inner.arg_a0a1.revDerivUpdate_rule","declaration":"theorem Inner.inner.arg_a0a1.revDerivUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDerivUpdate R fun x => ⟪f x, g x⟫_R) = fun x =>\n let y₁df := SciLean.revDerivUpdate R f x;\n let y₂dg := SciLean.revDerivUpdate R g x;\n (⟪y₁df.1, y₂dg.1⟫_R, fun dr dx => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1) dx))"} +{"name":"Prod.fst.arg_self.revDerivProjUpdate_rule","declaration":"theorem Prod.fst.arg_self.revDerivProjUpdate_rule {K : Type} [RCLike K] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {X' : Type} {Xi : Type} {XI : Xi → Type} [SciLean.StructType X' Xi XI] [DecidableEq Xi] [SciLean.SemiInnerProductSpace K X'] [(i : Xi) → SciLean.SemiInnerProductSpace K (XI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X' × Y) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K Xi fun x => (f x).1) = fun w =>\n let xydf := SciLean.revDerivProjUpdate K (Xi ⊕ Unit) f w;\n (xydf.1.1, fun i dxy dw => xydf.2 (Sum.inl i) dxy dw)"} +{"name":"HSMul.hSMul.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y Yi YI] [SciLean.IndexType Yi] [SciLean.LawfulIndexType Yi] [DecidableEq Yi] [SciLean.SemiHilbert K Y] [(i : Yi) → SciLean.SemiHilbert K (YI i)] [SciLean.SemiInnerProductSpaceStruct K Y Yi YI] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Yi fun x => f x • g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDerivProjUpdate K Yi g x;\n (ydf.1 • zdg.1, fun i dy dx =>\n let dk := ⟪SciLean.structProj zdg.1 i, dy⟫_K;\n let dy' := (starRingEnd K) ydf.1 • dy;\n let dx := zdg.2 i dy' dx;\n ydf.2 dk dx)"} +{"name":"SciLean.revDeriv'","declaration":"def SciLean.revDeriv' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y × Y ⊸[K] X"} +{"name":"SciLean.revDerivUpdate","declaration":"def SciLean.revDerivUpdate (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y × (Y → X → X)"} +{"name":"HSub.hSub.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Yi fun x => f x - g x) = fun x =>\n let ydf := SciLean.revDerivProjUpdate K Yi f x;\n let ydg := SciLean.revDerivProjUpdate K Yi g x;\n (ydf.1 - ydg.1, fun i dy dx =>\n let dx := ydf.2 i dy dx;\n let dy' := -dy;\n ydg.2 i dy' dx)"} +{"name":"SciLean.norm₂.arg_x.revDerivProj_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDerivProj_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDerivProj R Unit (fun x => ‖f x‖₂[R]) x =\n let ydf := SciLean.revDeriv R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun x dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"} +{"name":"Prod.mk.arg_fstsnd.revDerivUpdate_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (g : X → Y) (f : X → Z) (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => (g x, f x)) = fun x =>\n let ydg := SciLean.revDerivUpdate K g x;\n let zdf := SciLean.revDerivUpdate K f x;\n ((ydg.1, zdf.1), fun dyz dx =>\n let dx := ydg.2 dyz.1 dx;\n zdf.2 dyz.2 dx)"} +{"name":"SciLean.gradient","declaration":"def SciLean.gradient (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y → X"} +{"name":"HSub.hSub.arg_a0a1.revDeriv_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x - g x) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n let ydg := SciLean.revDerivUpdate K g x;\n (ydf.1 - ydg.1, fun dy =>\n let dx := ydf.2 dy;\n let dy' := -dy;\n ydg.2 dy' dx)"} +{"name":"HPow.hPow.arg_a0.revDerivProjUpdate_rule","declaration":"def HPow.hPow.arg_a0.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K Unit fun x => f x ^ n) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n (ydf.1 ^ n, fun x dy dx => ydf.2 (y' * dy) dx)"} +{"name":"Prod.snd.arg_self.revDerivUpdate_rule","declaration":"theorem Prod.snd.arg_self.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => (f x).2) = fun x =>\n let yzdf := SciLean.revDerivProjUpdate K (Unit ⊕ Unit) f x;\n (yzdf.1.2, fun dy dx => yzdf.2 (Sum.inr ()) dy dx)"} +{"name":"SciLean.revDerivProj.apply_rule","declaration":"theorem SciLean.revDerivProj.apply_rule (K : Type u_5) (I : Type u_1) [RCLike K] {ι : Type u_2} [SciLean.IndexType ι] [DecidableEq ι] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (i : ι) : (SciLean.revDerivProj K I fun f => f i) = fun f => (f i, fun j dxj => SciLean.oneHot (i, j) dxj)"} +{"name":"dite.arg_te.revDerivProj_rule","declaration":"theorem dite.arg_te.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : c → X → Y') (e : ¬c → X → Y') : (SciLean.revDerivProj K Yi fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.revDerivProj K Yi (t p) y else SciLean.revDerivProj K Yi (e p) y"} +{"name":"SciLean.revDeriv.comp_rule","declaration":"theorem SciLean.revDeriv.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f (g x)) = fun x =>\n let ydg := SciLean.revDeriv K g x;\n let zdf := SciLean.revDeriv K f ydg.1;\n (zdf.1, fun dz =>\n let dy := zdf.2 dz;\n ydg.2 dy)"} +{"name":"dite.arg_te.revDerivProjUpdate_rule","declaration":"theorem dite.arg_te.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : c → X → Y') (e : ¬c → X → Y') : (SciLean.revDerivProjUpdate K Yi fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.revDerivProjUpdate K Yi (t p) y else SciLean.revDerivProjUpdate K Yi (e p) y"} +{"name":"SciLean.revDerivProjUpdate.let_rule","declaration":"theorem SciLean.revDerivProjUpdate.let_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_4} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K fun x =>\n match x with\n | (x, y) => f x y) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K I fun x =>\n let y := g x;\n f x y) =\n fun x =>\n let ydg' := SciLean.revDerivUpdate K g x;\n let zdf' :=\n SciLean.revDerivProjUpdate K I\n (fun x =>\n match x with\n | (x, y) => f x y)\n (x, ydg'.1);\n (zdf'.1, fun i dei dx =>\n let dxy := zdf'.2 i dei (dx, 0);\n ydg'.2 dxy.2 dxy.1)"} +{"name":"dite.arg_te.revDeriv_rule","declaration":"theorem dite.arg_te.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.revDeriv K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.revDeriv K (t p) y else SciLean.revDeriv K (e p) y"} +{"name":"SciLean.revDerivProjUpdate.apply_rule","declaration":"theorem SciLean.revDerivProjUpdate.apply_rule (K : Type u_5) (I : Type u_1) [RCLike K] {ι : Type u_2} [SciLean.IndexType ι] [DecidableEq ι] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (i : ι) : (SciLean.revDerivProjUpdate K I fun f => f i) = fun f =>\n (f i, fun j dxj df i' => if i = i' then SciLean.structModify j (fun xj => xj + dxj) (df i') else df i')"} +{"name":"SciLean.revDerivProj","declaration":"def SciLean.revDerivProj (K : Type u_1) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_4} {EI : I → Type u_5} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (f : X → E) (x : X) : E × ((i : I) → EI i → X)"} +{"name":"Neg.neg.arg_a0.revDerivProj_rule","declaration":"theorem Neg.neg.arg_a0.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') : (SciLean.revDerivProj K Yi fun x => -f x) = fun x =>\n let ydf := SciLean.revDerivProj K Yi f x;\n (-ydf.1, fun i dy =>\n let dy' := -dy;\n ydf.2 i dy')"} +{"name":"ite.arg_te.revDerivProjUpdate_rule","declaration":"theorem ite.arg_te.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : X → Y') (e : X → Y') : (SciLean.revDerivProjUpdate K Yi fun x => if c then t x else e x) = fun y =>\n if c then SciLean.revDerivProjUpdate K Yi t y else SciLean.revDerivProjUpdate K Yi e y"} +{"name":"Prod.fst.arg_self.revDerivUpdate_rule","declaration":"theorem Prod.fst.arg_self.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => (f x).1) = fun x =>\n let yzdf := SciLean.revDerivProjUpdate K (Unit ⊕ Unit) f x;\n (yzdf.1.1, fun dy dx => yzdf.2 (Sum.inl ()) dy dx)"} +{"name":"Prod.snd.arg_self.revDerivProj_rule","declaration":"theorem Prod.snd.arg_self.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X × Y') (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K Yi fun x => (f x).2) = fun w =>\n let xydf := SciLean.revDerivProj K (Unit ⊕ Yi) f w;\n (xydf.1.2, fun i dxy => xydf.2 (Sum.inr i) dxy)"} +{"name":"SciLean.scalarGradient","declaration":"def SciLean.scalarGradient (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) (x : X) : X"} +{"name":"Prod.fst.arg_self.revDeriv_rule","declaration":"theorem Prod.fst.arg_self.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => (f x).1) = fun x =>\n let yzdf := SciLean.revDerivProj K (Unit ⊕ Unit) f x;\n (yzdf.1.1, fun dy => yzdf.2 (Sum.inl ()) dy)"} +{"name":"HDiv.hDiv.arg_a0a1.revDeriv_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : SciLean.fpropParam (∀ (x : X), g x ≠ 0)) : (SciLean.revDeriv K fun x => f x / g x) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n let zdg := SciLean.revDerivUpdate K g x;\n (ydf.1 / zdg.1, fun dx' =>\n (1 / (starRingEnd K) zdg.1 ^ 2) • zdg.2 (-(starRingEnd K) ydf.1 • dx') ((starRingEnd K) zdg.1 • ydf.2 dx'))"} +{"name":"Neg.neg.arg_a0.revDerivProjUpdate_rule","declaration":"theorem Neg.neg.arg_a0.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') : (SciLean.revDerivProjUpdate K Yi fun x => -f x) = fun x =>\n let ydf := SciLean.revDerivProjUpdate K Yi f x;\n (-ydf.1, fun i dy dx =>\n let dy' := -dy;\n ydf.2 i dy' dx)"} +{"name":"ite.arg_te.revDerivProj_rule","declaration":"theorem ite.arg_te.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : X → Y') (e : X → Y') : (SciLean.revDerivProj K Yi fun x => if c then t x else e x) = fun y =>\n if c then SciLean.revDerivProj K Yi t y else SciLean.revDerivProj K Yi e y"} +{"name":"HSMul.hSMul.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiHilbert K Y] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x • g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDerivUpdate K g x;\n (ydf.1 • zdg.1, fun dy dx =>\n let dk := ⟪zdg.1, dy⟫_K;\n let dy' := (starRingEnd K) ydf.1 • dy;\n let dx := zdg.2 dy' dx;\n ydf.2 dk dx)"} +{"name":"HPow.hPow.arg_a0.revDerivProj_rule","declaration":"def HPow.hPow.arg_a0.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K Unit fun x => f x ^ n) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n (ydf.1 ^ n, fun x dx' => ydf.2 (y' * dx'))"} +{"name":"HMul.hMul.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x * g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDerivUpdate K g x;\n (ydf.1 * zdg.1, fun dx' dx =>\n let dx₁ := (starRingEnd K) zdg.1 * dx';\n let dx₂ := (starRingEnd K) ydf.1 * dx';\n let dx := zdg.2 dx₂ dx;\n ydf.2 dx₁ dx)"} +{"name":"SciLean.revDerivUpdate.comp_rule","declaration":"theorem SciLean.revDerivUpdate.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f (g x)) = fun x =>\n let ydg := SciLean.revDerivUpdate K g x;\n let zdf := SciLean.revDeriv K f ydg.1;\n (zdf.1, fun dz dx =>\n let dy := zdf.2 dz;\n ydg.2 dy dx)"} +{"name":"HSMul.hSMul.arg_a0a1.revDerivProj_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y Yi YI] [SciLean.IndexType Yi] [SciLean.LawfulIndexType Yi] [DecidableEq Yi] [SciLean.SemiHilbert K Y] [(i : Yi) → SciLean.SemiHilbert K (YI i)] [SciLean.SemiInnerProductSpaceStruct K Y Yi YI] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Yi fun x => f x • g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDerivProj K Yi g x;\n (ydf.1 • zdg.1, fun i dy =>\n let dk := ⟪SciLean.structProj zdg.1 i, dy⟫_K;\n let dx := zdg.2 i dy;\n let dx := (starRingEnd K) ydf.1 • dx;\n ydf.2 dk dx)"} +{"name":"SciLean.revDerivUpdate.pi_rule","declaration":"theorem SciLean.revDerivUpdate.pi_rule (K : Type u_1) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {EI : I → Type u_2} [SciLean.IndexType I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → (i : I) → EI i) (hf : ∀ (i : I), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivUpdate K fun x i => f x i) = fun x =>\n let xdf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n (fun i => (xdf i).1, fun dy dx => LeanColls.fold (SciLean.IndexType.univ I) (fun dx i => (xdf i).2 (dy i) dx) dx)"} +{"name":"SciLean.revDerivProjUpdate_fst","declaration":"theorem SciLean.revDerivProjUpdate_fst (K : Type u_5) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_1} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) : (SciLean.revDerivProjUpdate K I f x).1 = f x"} +{"name":"SciLean.revDerivProj_fst","declaration":"theorem SciLean.revDerivProj_fst (K : Type u_5) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_1} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) : (SciLean.revDerivProj K I f x).1 = f x"} +{"name":"SciLean.revDeriv.let_rule","declaration":"theorem SciLean.revDeriv.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K fun xy => f xy.1 xy.2) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x =>\n let y := g x;\n f x y) =\n fun x =>\n let ydg := SciLean.revDerivUpdate K g x;\n let zdf := SciLean.revDeriv K (fun xy => f xy.1 xy.2) (x, ydg.1);\n (zdf.1, fun dz =>\n let dxdy := zdf.2 dz;\n let dx := ydg.2 dxdy.2 dxdy.1;\n dx)"} +{"name":"SciLean.revDeriv.id_rule","declaration":"theorem SciLean.revDeriv.id_rule (K : Type u_2) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] : (SciLean.revDeriv K fun x => x) = fun x => (x, fun dx => dx)"} +{"name":"SciLean.revDerivProj.const_rule","declaration":"theorem SciLean.revDerivProj.const_rule (K : Type u_5) (I : Type u_1) [RCLike K] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (x : E) : (SciLean.revDerivProj K I fun x_1 => x) = fun x_1 => (x, fun i de => 0)"} +{"name":"SciLean.revDerivProjUpdate_snd_zero'","declaration":"theorem SciLean.revDerivProjUpdate_snd_zero' (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (dy : Y) : (SciLean.revDerivUpdate K f x).2 dy 0 = (SciLean.revDeriv K f x).2 dy"} +{"name":"Prod.snd.arg_self.revDeriv_rule","declaration":"theorem Prod.snd.arg_self.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => (f x).2) = fun x =>\n let yzdf := SciLean.revDerivProj K (Unit ⊕ Unit) f x;\n (yzdf.1.2, fun dy => yzdf.2 (Sum.inr ()) dy)"} +{"name":"HDiv.hDiv.arg_a0a1.revDerivProj_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.revDerivProj K Unit fun x => f x / g x) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n let zdg := SciLean.revDerivUpdate K g x;\n (ydf.1 / zdg.1, fun x dx' =>\n (1 / (starRingEnd K) zdg.1 ^ 2) • zdg.2 (-(starRingEnd K) ydf.1 • dx') ((starRingEnd K) zdg.1 • ydf.2 dx'))"} +{"name":"SciLean.Norm2.norm2.arg_a0.revDeriv_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDeriv R fun x => ‖f x‖₂²) = fun x =>\n let ydf := SciLean.revDeriv R f x;\n let ynorm2 := ‖ydf.1‖₂²;\n (ynorm2, fun dr => (2 * dr) • ydf.2 ydf.1)"} +{"name":"Neg.neg.arg_a0.revDeriv_rule","declaration":"theorem Neg.neg.arg_a0.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : SciLean.revDeriv K (fun x => -f x) x =\n let ydf := SciLean.revDeriv K f x;\n (-ydf.1, fun dy =>\n let dx := ydf.2 dy;\n -dx)"} +{"name":"HMul.hMul.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Unit fun x => f x * g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDerivUpdate K g x;\n (ydf.1 * zdg.1, fun x dy dx =>\n let dy₁ := (starRingEnd K) zdg.1 * dy;\n let dy₂ := (starRingEnd K) ydf.1 * dy;\n let dx := zdg.2 dy₂ dx;\n ydf.2 dy₁ dx)"} +{"name":"SciLean.revDeriv","declaration":"def SciLean.revDeriv (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y × (Y → X)"} +{"name":"Neg.neg.arg_a0.revDerivUpdate_rule","declaration":"theorem Neg.neg.arg_a0.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : (SciLean.revDerivUpdate K fun x => -f x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n (-ydf.1, fun dy dx =>\n let dy' := -dy;\n ydf.2 dy' dx)"} +{"name":"SciLean.revDeriv.pi_rule","declaration":"theorem SciLean.revDeriv.pi_rule (K : Type u_1) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {EI : I → Type u_2} [SciLean.IndexType I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → (i : I) → EI i) (hf : ∀ (i : I), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDeriv K fun x i => f x i) = fun x =>\n let xdf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n (fun i => (xdf i).1, fun dy => LeanColls.fold (SciLean.IndexType.univ I) (fun dx i => (xdf i).2 (dy i) dx) 0)"} +{"name":"dite.arg_te.revDerivUpdate_rule","declaration":"theorem dite.arg_te.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.revDerivUpdate K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.revDerivUpdate K (t p) y else SciLean.revDerivUpdate K (e p) y"} +{"name":"Prod.snd.arg_self.revDerivProjUpdate_rule","declaration":"theorem Prod.snd.arg_self.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X × Y') (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K Yi fun x => (f x).2) = fun w =>\n let xydf := SciLean.revDerivProjUpdate K (Unit ⊕ Yi) f w;\n (xydf.1.2, fun i dxy dw => xydf.2 (Sum.inr i) dxy dw)"} +{"name":"SciLean.revDerivProj.id_rule","declaration":"theorem SciLean.revDerivProj.id_rule (K : Type u_4) (I : Type u_1) [RCLike K] {E : Type u_2} {EI : I → Type u_3} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] : (SciLean.revDerivProj K I fun x => x) = fun x => (x, fun i de => SciLean.oneHot i de)"} +{"name":"SciLean.revDerivProj_snd_zero","declaration":"theorem SciLean.revDerivProj_snd_zero (K : Type u_5) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {E : Type u_2} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) (i : I) : (SciLean.revDerivProj K I f x).2 i 0 = 0"} +{"name":"SciLean.revDerivProjUpdate.comp_rule","declaration":"theorem SciLean.revDerivProjUpdate.comp_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K I fun x => f (g x)) = fun x =>\n let ydg' := SciLean.revDerivUpdate K g x;\n let zdf' := SciLean.revDerivProj K I f ydg'.1;\n (zdf'.1, fun i de dx => ydg'.2 (zdf'.2 i de) dx)"} +{"name":"SciLean.norm₂.arg_x.revDeriv_rule","declaration":"theorem SciLean.norm₂.arg_x.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDeriv R fun x => ‖f x‖₂[R]) = fun x =>\n let ydf := SciLean.revDeriv R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"} +{"name":"SciLean.revDerivProjUpdate.const_rule","declaration":"theorem SciLean.revDerivProjUpdate.const_rule (K : Type u_5) (I : Type u_1) [RCLike K] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (x : E) : (SciLean.revDerivProjUpdate K I fun x_1 => x) = fun x_1 => (x, fun i de dx => dx)"} +{"name":"IndexType.sum.arg_f.revDerivProj_rule","declaration":"theorem IndexType.sum.arg_f.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {ι : Type} [SciLean.IndexType ι] [DecidableEq ι] (f : X → ι → Y') (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProj K Yi fun x => ∑ i, f x i) = fun x =>\n let ydf := SciLean.revDerivProjUpdate K (ι × Yi) f x;\n ( ∑ i, ydf.1 i, fun j dy => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => ydf.2 (i, j) dy dx) 0)"} +{"name":"SciLean.norm₂.arg_x.revDerivProj_rule","declaration":"theorem SciLean.norm₂.arg_x.revDerivProj_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDerivProj R Unit fun x => ‖f x‖₂[R]) = fun x =>\n let ydf := SciLean.revDeriv R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun x dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"} +{"name":"HAdd.hAdd.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x + g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let ydg := SciLean.revDerivUpdate K g x;\n (ydf.1 + ydg.1, fun dy dx =>\n let dx := ydf.2 dy dx;\n ydg.2 dy dx)"} +{"name":"Prod.mk.arg_fstsnd.revDerivProj_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] {Z' : Type} {Zi : Type} {ZI : Zi → Type} [SciLean.StructType Z' Zi ZI] [DecidableEq Zi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] [SciLean.SemiInnerProductSpace K Z'] [(i : Zi) → SciLean.SemiInnerProductSpace K (ZI i)] (g : X → Y') (f : X → Z') (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K (Yi ⊕ Zi) fun x => (g x, f x)) = fun x =>\n let ydg := SciLean.revDerivProj K Yi g x;\n let zdf := SciLean.revDerivProj K Zi f x;\n ((ydg.1, zdf.1), fun i dyz =>\n match i, dyz with\n | Sum.inl j, dyz => ydg.2 j dyz\n | Sum.inr j, dyz => zdf.2 j dyz)"} +{"name":"ite.arg_te.revDerivUpdate_rule","declaration":"theorem ite.arg_te.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.revDerivUpdate K fun x => if c then t x else e x) = fun y =>\n if c then SciLean.revDerivUpdate K t y else SciLean.revDerivUpdate K e y"} +{"name":"SciLean.revDerivUpdate_snd_zero'","declaration":"theorem SciLean.revDerivUpdate_snd_zero' (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (dy : Y) : (SciLean.revDerivUpdate K f x).2 dy 0 = (SciLean.revDeriv K f x).2 dy"} +{"name":"SciLean.revDerivUpdate_fst","declaration":"theorem SciLean.revDerivUpdate_fst (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : (SciLean.revDerivUpdate K f x).1 = f x"} +{"name":"Prod.mk.arg_fstsnd.revDerivProjUpdate_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] {Z' : Type} {Zi : Type} {ZI : Zi → Type} [SciLean.StructType Z' Zi ZI] [DecidableEq Zi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] [SciLean.SemiInnerProductSpace K Z'] [(i : Zi) → SciLean.SemiInnerProductSpace K (ZI i)] (g : X → Y') (f : X → Z') (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K (Yi ⊕ Zi) fun x => (g x, f x)) = fun x =>\n let ydg := SciLean.revDerivProjUpdate K Yi g x;\n let zdf := SciLean.revDerivProjUpdate K Zi f x;\n ((ydg.1, zdf.1), fun i dyz dx =>\n match i, dyz with\n | Sum.inl j, dyz => ydg.2 j dyz dx\n | Sum.inr j, dyz => zdf.2 j dyz dx)"} +{"name":"HMul.hMul.arg_a0a1.revDeriv_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x * g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDeriv K g x;\n (ydf.1 * zdg.1, fun dx' =>\n let dx₁ := (starRingEnd K) zdg.1 * dx';\n let dx₂ := (starRingEnd K) ydf.1 * dx';\n let dx := zdg.2 dx₂;\n ydf.2 dx₁ dx)"} +{"name":"Inner.inner.arg_a0a1.revDeriv_rule","declaration":"theorem Inner.inner.arg_a0a1.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDeriv R fun x => ⟪f x, g x⟫_R) = fun x =>\n let y₁df := SciLean.revDeriv R f x;\n let y₂dg := SciLean.revDerivUpdate R g x;\n (⟪y₁df.1, y₂dg.1⟫_R, fun dr => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1)))"} +{"name":"SciLean.revDerivUpdate_snd_zero","declaration":"theorem SciLean.revDerivUpdate_snd_zero (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (dx : X) : (SciLean.revDerivUpdate K f x).2 0 dx = dx"} +{"name":"SciLean.norm₂.arg_x.revDerivUpdate_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDerivUpdate_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDerivUpdate R (fun x => ‖f x‖₂[R]) x =\n let ydf := SciLean.revDerivUpdate R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"} +{"name":"SciLean.revDerivProjUpdate_snd_zero","declaration":"theorem SciLean.revDerivProjUpdate_snd_zero (K : Type u_5) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {E : Type u_2} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) (dx : X) (i : I) : (SciLean.revDerivProjUpdate K I f x).2 i 0 dx = dx"} +{"name":"HAdd.hAdd.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Yi fun x => f x + g x) = fun x =>\n let ydf := SciLean.revDerivProjUpdate K Yi f x;\n let ydg := SciLean.revDerivProjUpdate K Yi g x;\n (ydf.1 + ydg.1, fun i dy dx =>\n let dx := ydf.2 i dy dx;\n ydg.2 i dy dx)"} +{"name":"SciLean.revDerivProjUpdate.pi_rule","declaration":"theorem SciLean.revDerivProjUpdate.pi_rule (K : Type u_1) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_3} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProjUpdate K Unit fun x i => f x i) = fun x =>\n let ydf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n (fun i => (ydf i).1, fun x df dx => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => (ydf i).2 (df i) dx) dx)"} +{"name":"HPow.hPow.arg_a0.revDerivUpdate_rule","declaration":"def HPow.hPow.arg_a0.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => f x ^ n) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n (ydf.1 ^ n, fun dy dx => ydf.2 (y' * dy) dx)"} +{"name":"SciLean.revDeriv_fst","declaration":"theorem SciLean.revDeriv_fst (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : (SciLean.revDeriv K f x).1 = f x"} +{"name":"ite.arg_te.revDeriv_rule","declaration":"theorem ite.arg_te.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.revDeriv K fun x => if c then t x else e x) = fun y =>\n if c then SciLean.revDeriv K t y else SciLean.revDeriv K e y"} +{"name":"HSub.hSub.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x - g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let ydg := SciLean.revDerivUpdate K g x;\n (ydf.1 - ydg.1, fun dy dx =>\n let dx := ydf.2 dy dx;\n let dy' := -dy;\n ydg.2 dy' dx)"} +{"name":"SciLean.norm₂.arg_x.revDerivUpdate_rule","declaration":"theorem SciLean.norm₂.arg_x.revDerivUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDerivUpdate R fun x => ‖f x‖₂[R]) = fun x =>\n let ydf := SciLean.revDerivUpdate R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"} +{"name":"HSMul.hSMul.arg_a0a1.revDeriv_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiHilbert K Y] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x • g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDeriv K g x;\n (ydf.1 • zdg.1, fun dy' =>\n let dk := ⟪zdg.1, dy'⟫_K;\n let dx := zdg.2 dy';\n let dx := (starRingEnd K) ydf.1 • dx;\n ydf.2 dk dx)"} +{"name":"HAdd.hAdd.arg_a0a1.revDerivProj_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Yi fun x => f x + g x) = fun x =>\n let ydf := SciLean.revDerivProj K Yi f x;\n let ydg := SciLean.revDerivProjUpdate K Yi g x;\n (ydf.1 + ydg.1, fun i dy =>\n let dx := ydf.2 i dy;\n ydg.2 i dy dx)"} +{"name":"HMul.hMul.arg_a0a1.revDerivProj_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Unit fun x => f x * g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDeriv K g x;\n (ydf.1 * zdg.1, fun x dy =>\n let dy₁ := (starRingEnd K) zdg.1 * dy;\n let dy₂ := (starRingEnd K) ydf.1 * dy;\n let dx := zdg.2 dy₂;\n ydf.2 dy₁ dx)"} +{"name":"SciLean.Norm2.norm2.arg_a0.revDerivProj_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDerivProj_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDerivProj R Unit fun x => ‖f x‖₂²) = fun x =>\n let ydf := SciLean.revDeriv R f x;\n let ynorm2 := ‖ydf.1‖₂²;\n (ynorm2, fun x dr => (2 * dr) • ydf.2 ydf.1)"} +{"name":"SciLean.Norm2.norm2.arg_a0.revDerivUpdate_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDerivUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDerivUpdate R fun x => ‖f x‖₂²) = fun x =>\n let ydf := SciLean.revDerivUpdate R f x;\n let ynorm2 := ‖ydf.1‖₂²;\n (ynorm2, fun dr dx => ydf.2 ((2 * dr) • ydf.1) dx)"} +{"name":"SciLean.norm₂.arg_x.revDeriv_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDeriv R (fun x => ‖f x‖₂[R]) x =\n let ydf := SciLean.revDeriv R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"} +{"name":"SciLean.norm₂.arg_x.revDerivProjUpdate_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDerivProjUpdate_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDerivProjUpdate R Unit (fun x => ‖f x‖₂[R]) x =\n let ydf := SciLean.revDerivUpdate R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun x dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"} +{"name":"SciLean.revDerivProj.pi_rule","declaration":"theorem SciLean.revDerivProj.pi_rule (K : Type u_1) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_3} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProj K Unit fun x i => f x i) = fun x =>\n let ydf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n (fun i => (ydf i).1, fun x df => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => (ydf i).2 (df i) dx) 0)"} +{"name":"SciLean.revDerivProjUpdate.id_rule","declaration":"theorem SciLean.revDerivProjUpdate.id_rule (K : Type u_4) (I : Type u_1) [RCLike K] {E : Type u_2} {EI : I → Type u_3} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] : (SciLean.revDerivProjUpdate K I fun x => x) = fun x => (x, fun i de dx => SciLean.structModify i (fun ei => ei + de) dx)"} +{"name":"SciLean.revDerivUpdate.id_rule","declaration":"theorem SciLean.revDerivUpdate.id_rule (K : Type u_2) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] : (SciLean.revDerivUpdate K fun x => x) = fun x => (x, fun dx' dx => dx + dx')"} +{"name":"Inner.inner.arg_a0a1.revDerivProj_rule","declaration":"theorem Inner.inner.arg_a0a1.revDerivProj_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDerivProj R Unit fun x => ⟪f x, g x⟫_R) = fun x =>\n let y₁df := SciLean.revDeriv R f x;\n let y₂dg := SciLean.revDerivUpdate R g x;\n (⟪y₁df.1, y₂dg.1⟫_R, fun x dr => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1)))"} +{"name":"SciLean.revDeriv_snd_zero","declaration":"theorem SciLean.revDeriv_snd_zero (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : (SciLean.revDeriv K f x).2 0 = 0"} +{"name":"SciLean.revDeriv.const_rule","declaration":"theorem SciLean.revDeriv.const_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (y : Y) : (SciLean.revDeriv K fun x => y) = fun x => (y, fun x => 0)"} +{"name":"SciLean.revDerivProj.comp_rule","declaration":"theorem SciLean.revDerivProj.comp_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K I fun x => f (g x)) = fun x =>\n let ydg' := SciLean.revDeriv K g x;\n let zdf' := SciLean.revDerivProj K I f ydg'.1;\n (zdf'.1, fun i de => ydg'.2 (zdf'.2 i de))"} +{"name":"HPow.hPow.arg_a0.revDeriv_rule","declaration":"def HPow.hPow.arg_a0.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => f x ^ n) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n (ydf.1 ^ n, fun dx' => ydf.2 (y' * dx'))"} +{"name":"HDiv.hDiv.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.revDerivProjUpdate K Unit fun x => f x / g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDerivUpdate K g x;\n (ydf.1 / zdg.1, fun x dx' dx =>\n let c := 1 / (starRingEnd K) zdg.1 ^ 2;\n let a := -(c * (starRingEnd K) ydf.1);\n let b := c * (starRingEnd K) zdg.1;\n zdg.2 (a • dx') (ydf.2 (b • dx') dx))"} +{"name":"SciLean.norm₂.arg_x.revDerivProjUpdate_rule","declaration":"theorem SciLean.norm₂.arg_x.revDerivProjUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDerivProjUpdate R Unit fun x => ‖f x‖₂[R]) = fun x =>\n let ydf := SciLean.revDerivUpdate R f x;\n let ynorm := ‖ydf.1‖₂[R];\n (ynorm, fun x dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"} +{"name":"HAdd.hAdd.arg_a0a1.revDeriv_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x + g x) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n let ydg := SciLean.revDerivUpdate K g x;\n (ydf.1 + ydg.1, fun dy =>\n let dx := ydf.2 dy;\n ydg.2 dy dx)"} +{"name":"IndexType.sum.arg_f.revDeriv_rule","declaration":"theorem IndexType.sum.arg_f.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {ι : Type} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDeriv K fun x => ∑ i, f x i) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n ( ∑ i, ydf.1 i, fun dy => ydf.2 fun x => dy)"} +{"name":"SciLean.revDerivUpdate.const_rule","declaration":"theorem SciLean.revDerivUpdate.const_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (y : Y) : (SciLean.revDerivUpdate K fun x => y) = fun x => (y, fun x dx => dx)"} +{"name":"Prod.fst.arg_self.revDerivProj_rule","declaration":"theorem Prod.fst.arg_self.revDerivProj_rule {K : Type} [RCLike K] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {X' : Type} {Xi : Type} {XI : Xi → Type} [SciLean.StructType X' Xi XI] [DecidableEq Xi] [SciLean.SemiInnerProductSpace K X'] [(i : Xi) → SciLean.SemiInnerProductSpace K (XI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X' × Y) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K Xi fun x => (f x).1) = fun w =>\n let xydf := SciLean.revDerivProj K (Xi ⊕ Unit) f w;\n (xydf.1.1, fun i dxy => xydf.2 (Sum.inl i) dxy)"} +{"name":"Prod.mk.arg_fstsnd.revDeriv_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (g : X → Y) (f : X → Z) (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => (g x, f x)) = fun x =>\n let ydg := SciLean.revDeriv K g x;\n let zdf := SciLean.revDerivUpdate K f x;\n ((ydg.1, zdf.1), fun dyz =>\n let dx := ydg.2 dyz.1;\n zdf.2 dyz.2 dx)"} +{"name":"Inner.inner.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem Inner.inner.arg_a0a1.revDerivProjUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDerivProjUpdate R Unit fun x => ⟪f x, g x⟫_R) = fun x =>\n let y₁df := SciLean.revDerivUpdate R f x;\n let y₂dg := SciLean.revDerivUpdate R g x;\n (⟪y₁df.1, y₂dg.1⟫_R, fun x dr dx => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1) dx))"} +{"name":"SciLean.Norm2.norm2.arg_a0.revDerivProjUpdate_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDerivProjUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDerivProjUpdate R Unit fun x => ‖f x‖₂²) = fun x =>\n let ydf := SciLean.revDerivUpdate R f x;\n let ynorm2 := ‖ydf.1‖₂²;\n (ynorm2, fun x dr dx => ydf.2 ((2 * dr) • ydf.1) dx)"} +{"name":"SciLean.revDerivProj.let_rule","declaration":"theorem SciLean.revDerivProj.let_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_4} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K fun x =>\n match x with\n | (x, y) => f x y) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K I fun x =>\n let y := g x;\n f x y) =\n fun x =>\n let ydg' := SciLean.revDerivUpdate K g x;\n let zdf' :=\n SciLean.revDerivProj K I\n (fun x =>\n match x with\n | (x, y) => f x y)\n (x, ydg'.1);\n (zdf'.1, fun i dei =>\n let dxy := zdf'.2 i dei;\n ydg'.2 dxy.2 dxy.1)"} +{"name":"SciLean.revDerivUpdate.apply_rule","declaration":"theorem SciLean.revDerivUpdate.apply_rule (K : Type u_3) (I : Type u_1) [RCLike K] {EI : I → Type u_2} [SciLean.IndexType I] [DecidableEq I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (i : I) : (SciLean.revDerivUpdate K fun x => x i) = fun x =>\n (x i, fun dxi dx => SciLean.structModify i (fun dxi' => dxi' + dxi) dx)"} +{"name":"SciLean.revDeriv.apply_rule","declaration":"theorem SciLean.revDeriv.apply_rule (K : Type u_3) (I : Type u_1) [RCLike K] {EI : I → Type u_2} [SciLean.IndexType I] [DecidableEq I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (i : I) : (SciLean.revDeriv K fun x => x i) = fun x => (x i, fun dxi => SciLean.oneHot i dxi)"} +{"name":"SciLean.revDerivUpdate.let_rule","declaration":"theorem SciLean.revDerivUpdate.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K fun xy => f xy.1 xy.2) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x =>\n let y := g x;\n f x y) =\n fun x =>\n let ydg := SciLean.revDerivUpdate K g x;\n let zdf := SciLean.revDerivUpdate K (fun xy => f xy.1 xy.2) (x, ydg.1);\n (zdf.1, fun dz dx =>\n let dxdy := zdf.2 dz (dx, 0);\n let dx := ydg.2 dxdy.2 dxdy.1;\n dx)"} +{"name":"HDiv.hDiv.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.revDerivUpdate K fun x => f x / g x) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n let zdg := SciLean.revDerivUpdate K g x;\n (ydf.1 / zdg.1, fun dx' dx =>\n let c := 1 / (starRingEnd K) zdg.1 ^ 2;\n let a := -(c * (starRingEnd K) ydf.1);\n let b := c * (starRingEnd K) zdg.1;\n zdg.2 (a • dx') (ydf.2 (b • dx') dx))"} +{"name":"HSub.hSub.arg_a0a1.revDerivProj_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Yi fun x => f x - g x) = fun x =>\n let ydf := SciLean.revDerivProj K Yi f x;\n let ydg := SciLean.revDerivProjUpdate K Yi g x;\n (ydf.1 - ydg.1, fun i dy =>\n let dx := ydf.2 i dy;\n let dy' := -dy;\n ydg.2 i dy' dx)"} +{"name":"SciLean.revDerivProjUpdate","declaration":"def SciLean.revDerivProjUpdate (K : Type u_1) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_4} {EI : I → Type u_5} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (f : X → E) (x : X) : E × ((i : I) → EI i → X → X)"} +{"name":"IndexType.sum.arg_f.revDerivProjUpdate_rule","declaration":"theorem IndexType.sum.arg_f.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {ι : Type} [SciLean.IndexType ι] [DecidableEq ι] (f : X → ι → Y') (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProjUpdate K Yi fun x => ∑ i, f x i) = fun x =>\n let ydf := SciLean.revDerivProjUpdate K (ι × Yi) f x;\n ( ∑ i, ydf.1 i, fun j dy dx => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => ydf.2 (i, j) dy dx) dx)"} +{"name":"IndexType.sum.arg_f.revDerivUpdate_rule","declaration":"theorem IndexType.sum.arg_f.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {ι : Type} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivUpdate K fun x => ∑ i, f x i) = fun x =>\n let ydf := SciLean.revDerivUpdate K f x;\n ( ∑ i, ydf.1 i, fun dy dx => ydf.2 (fun x => dy) dx)"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.RevFDeriv.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.RevFDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.SemiAdjoint.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.SemiAdjoint.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..898c047cda5ebedc60816e5757aa7bf8c2ba6af0 --- /dev/null +++ b/scilean-declarations/SciLean.Core.FunctionTransformations.SemiAdjoint.jsonl @@ -0,0 +1,25 @@ +{"name":"SciLean.semiAdjoint.HMul.hMul.arg_a1.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.HMul.hMul.arg_a1.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (c : K) (f : X → K) (hf : SciLean.HasSemiAdjoint K f) : (SciLean.semiAdjoint K fun x => c * f x) = fun y => (starRingEnd K) c • SciLean.semiAdjoint K (fun x => f x) y"} +{"name":"SciLean.semiAdjoint.SciLean.semiAdjoint.arg_y.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.SciLean.semiAdjoint.arg_y.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {W : Type u_4} [SciLean.SemiInnerProductSpace K W] (f : X → Y) (a3 : W → Y) (hf : SciLean.HasSemiAdjoint K f) (ha3 : SciLean.HasSemiAdjoint K a3) : (SciLean.semiAdjoint K fun w => SciLean.semiAdjoint K f (a3 w)) = fun x =>\n let y := f x;\n SciLean.semiAdjoint K a3 y"} +{"name":"SciLean.semiAdjoint.HSMul.hSMul.arg_a0.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.HSMul.hSMul.arg_a0.semiAdjoint_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiHilbert K Y] (y' : Y) (f : X → K) (hf : SciLean.HasSemiAdjoint K f) : (SciLean.semiAdjoint K fun x => f x • y') = fun y => SciLean.semiAdjoint K (fun x => f x) ⟪y', y⟫_K"} +{"name":"SciLean.semiAdjoint.Neg.neg.arg_a0.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.Neg.neg.arg_a0.semiAdjoint_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : (SciLean.semiAdjoint K fun x => -f x) = fun y => -SciLean.semiAdjoint K f y"} +{"name":"SciLean.semiAdjoint.let_rule","declaration":"theorem SciLean.semiAdjoint.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.HasSemiAdjoint K ↿f) (hg : SciLean.HasSemiAdjoint K g) : (SciLean.semiAdjoint K fun x =>\n let y := g x;\n f x y) =\n fun z =>\n let xy := SciLean.semiAdjoint K (fun xy => f xy.1 xy.2) z;\n let x' := SciLean.semiAdjoint K g xy.2;\n xy.1 + x'"} +{"name":"SciLean.semiAdjoint.ite.arg_te.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.ite.arg_te.semiAdjoint_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.semiAdjoint K fun x => if c then t x else e x) = fun y =>\n if c then SciLean.semiAdjoint K t y else SciLean.semiAdjoint K e y"} +{"name":"SciLean.semiAdjoint.apply_rule","declaration":"theorem SciLean.semiAdjoint.apply_rule (K : Type u_3) [RCLike K] {ι : Type u_1} [SciLean.IndexType ι] {E : ι → Type u_2} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] [DecidableEq ι] (i : ι) : (SciLean.semiAdjoint K fun f => f i) = fun y j => if h : i = j then h ▸ y else 0"} +{"name":"SciLean.semiAdjoint.starRingEnd.arg_a0.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.starRingEnd.arg_a0.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) : (SciLean.semiAdjoint K fun x => (starRingEnd K) (f x)) = fun z => SciLean.semiAdjoint K f z"} +{"name":"SciLean.semiAdjoint.comp_rule","declaration":"theorem SciLean.semiAdjoint.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.HasSemiAdjoint K f) (hg : SciLean.HasSemiAdjoint K g) : (SciLean.semiAdjoint K fun x => f (g x)) = fun z =>\n let y := SciLean.semiAdjoint K f z;\n let x := SciLean.semiAdjoint K g y;\n x"} +{"name":"SciLean.semiAdjoint.pi_rule","declaration":"theorem SciLean.semiAdjoint.pi_rule (K : Type u_1) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {ι : Type u_3} [SciLean.IndexType ι] {E : ι → Type u_2} [(i : ι) → SciLean.SemiInnerProductSpace K (E i)] (f : X → (i : ι) → E i) (hf : ∀ (i : ι), SciLean.HasSemiAdjoint K fun x => f x i) : (SciLean.semiAdjoint K fun x i => f x i) = fun x' => ∑ i, SciLean.semiAdjoint K (fun x => f x i) (x' i)"} +{"name":"SciLean.semiAdjoint.IndexType.sum.arg_f.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.IndexType.sum.arg_f.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_3} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasSemiAdjoint K fun x => f x i) : (SciLean.semiAdjoint K fun x => ∑ i, f x i) = fun y => ∑ i, SciLean.semiAdjoint K (fun x => f x i) y"} +{"name":"SciLean.semiAdjoint.Finset.sum.arg_f.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.Finset.sum.arg_f.semiAdjoint_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_4} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_1} [Fintype ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasSemiAdjoint K fun x => f x i) : (SciLean.semiAdjoint K fun x => Finset.sum Finset.univ fun i => f x i) = fun y =>\n Finset.sum Finset.univ fun i => SciLean.semiAdjoint K (fun x => f x i) y"} +{"name":"SciLean.semiAdjoint.const_rule","declaration":"theorem SciLean.semiAdjoint.const_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] : (SciLean.semiAdjoint K fun x => 0) = fun x => 0"} +{"name":"SciLean.semiAdjoint.dite.arg_te.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.dite.arg_te.semiAdjoint_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.semiAdjoint K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n if p : c then SciLean.semiAdjoint K (t p) y else SciLean.semiAdjoint K (e p) y"} +{"name":"SciLean.semiAdjoint.HAdd.hAdd.arg_a0a1.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.HAdd.hAdd.arg_a0a1.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasSemiAdjoint K f) (hg : SciLean.HasSemiAdjoint K g) : (SciLean.semiAdjoint K fun x => f x + g x) = fun y =>\n let x₁ := SciLean.semiAdjoint K f y;\n let x₂ := SciLean.semiAdjoint K g y;\n x₁ + x₂"} +{"name":"SciLean.semiAdjoint.HSub.hSub.arg_a0a1.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.HSub.hSub.arg_a0a1.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasSemiAdjoint K f) (hg : SciLean.HasSemiAdjoint K g) : (SciLean.semiAdjoint K fun x => f x - g x) = fun y =>\n let x₁ := SciLean.semiAdjoint K f y;\n let x₂ := SciLean.semiAdjoint K g y;\n x₁ - x₂"} +{"name":"SciLean.semiAdjoint.HMul.hMul.arg_a0.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.HMul.hMul.arg_a0.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (c : K) (f : X → K) (hf : SciLean.HasSemiAdjoint K f) : (SciLean.semiAdjoint K fun x => f x * c) = fun y => (starRingEnd K) c • SciLean.semiAdjoint K (fun x => f x) y"} +{"name":"SciLean.semiAdjoint.HSMul.hSMul.arg_a1.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.HSMul.hSMul.arg_a1.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (c : K) (g : X → Y) (hg : SciLean.HasSemiAdjoint K g) : (SciLean.semiAdjoint K fun x => c • g x) = fun y => (starRingEnd K) c • SciLean.semiAdjoint K g y"} +{"name":"SciLean.semiAdjoint.id_rule","declaration":"theorem SciLean.semiAdjoint.id_rule (K : Type u_2) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] : (SciLean.semiAdjoint K fun x => x) = fun x => x"} +{"name":"SciLean.semiAdjoint.Prod.fst.arg_self.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.Prod.fst.arg_self.semiAdjoint_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasSemiAdjoint K f) : (SciLean.semiAdjoint K fun x => (f x).1) = fun y => SciLean.semiAdjoint K (fun x => f x) (y, 0)"} +{"name":"SciLean.semiAdjoint.Prod.snd.arg_self.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.Prod.snd.arg_self.semiAdjoint_rule (K : Type u_3) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_2} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasSemiAdjoint K f) : (SciLean.semiAdjoint K fun x => (f x).2) = fun z => SciLean.semiAdjoint K f (0, z)"} +{"name":"SciLean.semiAdjoint.Prod.mk.arg_fstsnd.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.Prod.mk.arg_fstsnd.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (g : X → Y) (f : X → Z) (hg : SciLean.HasSemiAdjoint K g) (hf : SciLean.HasSemiAdjoint K f) : (SciLean.semiAdjoint K fun x => (g x, f x)) = fun yz =>\n let x₁ := SciLean.semiAdjoint K g yz.1;\n let x₂ := SciLean.semiAdjoint K f yz.2;\n x₁ + x₂"} +{"name":"SciLean.semiAdjoint.Inner.inner.arg_a0.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.Inner.inner.arg_a0.semiAdjoint_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiHilbert K Y] (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) (y : Y) : (SciLean.semiAdjoint K fun x => ⟪f x, y⟫_K) = fun z => (starRingEnd K) z • SciLean.semiAdjoint K f y"} +{"name":"SciLean.semiAdjoint.Inner.inner.arg_a1.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.Inner.inner.arg_a1.semiAdjoint_rule (K : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiHilbert K Y] (f : X → Y) (hf : SciLean.HasSemiAdjoint K f) (y : Y) : (SciLean.semiAdjoint K fun x => ⟪y, f x⟫_K) = fun z => z • SciLean.semiAdjoint K f y"} +{"name":"SciLean.semiAdjoint.HDiv.hDiv.arg_a0.semiAdjoint_rule","declaration":"theorem SciLean.semiAdjoint.HDiv.hDiv.arg_a0.semiAdjoint_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) (c : K) (hf : SciLean.HasSemiAdjoint K f) : (SciLean.semiAdjoint K fun x => f x / c) = fun y => ((starRingEnd K) c)⁻¹ • SciLean.semiAdjoint K f y"} diff --git a/scilean-declarations/SciLean.Core.FunctionTransformations.jsonl b/scilean-declarations/SciLean.Core.FunctionTransformations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.Functions.Exp.jsonl b/scilean-declarations/SciLean.Core.Functions.Exp.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a57e62c46b17a0eb05ee80d610876cda22977d07 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Functions.Exp.jsonl @@ -0,0 +1,20 @@ +{"name":"SciLean.Scalar.log_inv","declaration":"theorem SciLean.Scalar.log_inv {R : Type u_1} [SciLean.RealScalar R] (x : R) : SciLean.Scalar.log x⁻¹ = -SciLean.Scalar.log x"} +{"name":"SciLean.Scalar.log.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.log.arg_x.fwdDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (x : W → R) (hx : SciLean.CDifferentiable R x) (hw : ∀ (w : W), x w ≠ 0) : (SciLean.fwdDeriv R fun w => SciLean.Scalar.log (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n let l := SciLean.Scalar.log xdx.1;\n (l, xdx.2 / SciLean.Scalar.abs xdx.1)"} +{"name":"SciLean.Scalar.log.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.log.arg_x.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (x : U → R) (hx : SciLean.HasAdjDiff R x) (hu : ∀ (u : U), x u ≠ 0) : (SciLean.revDeriv R fun u => SciLean.Scalar.log (x u)) = fun u =>\n let xdx := SciLean.revDeriv R x u;\n (SciLean.Scalar.log xdx.1, fun dy => xdx.2 ((SciLean.Scalar.abs (x u))⁻¹ * dy))"} +{"name":"SciLean.Scalar.exp.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.exp.arg_x.HasAdjDiff_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : SciLean.HasAdjDiff C fun u => SciLean.Scalar.exp (x u)"} +{"name":"SciLean.Scalar.log_mul","declaration":"theorem SciLean.Scalar.log_mul {R : Type u_1} [SciLean.RealScalar R] (x : R) (y : R) : SciLean.Scalar.log (x * y) = SciLean.Scalar.log x + SciLean.Scalar.log y"} +{"name":"SciLean.Scalar.exp.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.exp.arg_x.CDifferentiable_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : SciLean.CDifferentiable C fun w => SciLean.Scalar.exp (x w)"} +{"name":"SciLean.Scalar.log.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.log.arg_x.HasAdjDiff_rule {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (x : U → R) (hx : SciLean.HasAdjDiff R x) (hu : ∀ (u : U), x u ≠ 0) : SciLean.HasAdjDiff R fun u => SciLean.Scalar.log (x u)"} +{"name":"SciLean.Scalar.log_div","declaration":"theorem SciLean.Scalar.log_div {R : Type u_1} [SciLean.RealScalar R] (x : R) (y : R) : SciLean.Scalar.log (x / y) = SciLean.Scalar.log x - SciLean.Scalar.log y"} +{"name":"SciLean.Scalar.exp.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.exp.arg_x.fwdDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.fwdDeriv C fun w => SciLean.Scalar.exp (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let e := SciLean.Scalar.exp xdx.1;\n (e, xdx.2 * e)"} +{"name":"SciLean.Scalar.log.arg_x.cderiv_rule_at","declaration":"theorem SciLean.Scalar.log.arg_x.cderiv_rule_at {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (w : W) (x : W → R) (hx : SciLean.CDifferentiableAt R x w) (hw : x w ≠ 0) : SciLean.cderiv R (fun w => SciLean.Scalar.log (x w)) w = fun dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n xdx.2 / SciLean.Scalar.abs xdx.1"} +{"name":"SciLean.Scalar.log.arg_x.revDeriv_rule_at","declaration":"theorem SciLean.Scalar.log.arg_x.revDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (u : U) (x : U → R) (hx : SciLean.HasAdjDiffAt R x u) (hu : x u ≠ 0) : SciLean.revDeriv R (fun u => SciLean.Scalar.log (x u)) u =\n let xdx := SciLean.revDeriv R x u;\n (SciLean.Scalar.log xdx.1, fun dy => xdx.2 ((SciLean.Scalar.abs (x u))⁻¹ * dy))"} +{"name":"SciLean.Scalar.exp.arg_x.ceriv_rule","declaration":"theorem SciLean.Scalar.exp.arg_x.ceriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.cderiv C fun w => SciLean.Scalar.exp (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let e := SciLean.Scalar.exp xdx.1;\n xdx.2 * e"} +{"name":"SciLean.Scalar.exp.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.exp.arg_x.revDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : (SciLean.revDeriv C fun u => SciLean.Scalar.exp (x u)) = fun u =>\n let xdx := SciLean.revDeriv C x u;\n (SciLean.Scalar.exp xdx.1, fun dy => xdx.2 ((starRingEnd C) (SciLean.Scalar.exp xdx.1) * dy))"} +{"name":"SciLean.Scalar.log.arg_x.cderiv_rule","declaration":"theorem SciLean.Scalar.log.arg_x.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (x : W → R) (hx : SciLean.CDifferentiable R x) (hw : ∀ (w : W), x w ≠ 0) : (SciLean.cderiv R fun w => SciLean.Scalar.log (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n xdx.2 / SciLean.Scalar.abs xdx.1"} +{"name":"SciLean.Scalar.log.arg_x.HasAdjDiffAt_rule","declaration":"theorem SciLean.Scalar.log.arg_x.HasAdjDiffAt_rule {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (u : U) (x : U → R) (hx : SciLean.HasAdjDiffAt R x u) (hu : x u ≠ 0) : SciLean.HasAdjDiffAt R (fun u => SciLean.Scalar.log (x u)) u"} +{"name":"SciLean.Scalar.log_one","declaration":"theorem SciLean.Scalar.log_one {R : Type u_1} [SciLean.RealScalar R] : SciLean.Scalar.log 1 = 0"} +{"name":"SciLean.Scalar.log.arg_x.CDifferentiableAt_rule","declaration":"theorem SciLean.Scalar.log.arg_x.CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (w : W) (x : W → R) (hx : SciLean.CDifferentiableAt R x w) (hw : x w ≠ 0) : SciLean.CDifferentiableAt R (fun w => SciLean.Scalar.log (x w)) w"} +{"name":"SciLean.Scalar.log.arg_x.fwdDeriv_rule_at","declaration":"theorem SciLean.Scalar.log.arg_x.fwdDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (w : W) (x : W → R) (hx : SciLean.CDifferentiableAt R x w) (hw : x w ≠ 0) : SciLean.fwdDeriv R (fun w => SciLean.Scalar.log (x w)) w = fun dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n let l := SciLean.Scalar.log xdx.1;\n (l, xdx.2 / SciLean.Scalar.abs xdx.1)"} +{"name":"SciLean.Scalar.log_exp","declaration":"theorem SciLean.Scalar.log_exp {R : Type u_1} [SciLean.RealScalar R] (x : R) : SciLean.Scalar.log (SciLean.Scalar.exp x) = x"} +{"name":"SciLean.Scalar.log.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.log.arg_x.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (x : W → R) (hx : SciLean.CDifferentiable R x) (hw : ∀ (w : W), x w ≠ 0) : SciLean.CDifferentiable R fun w => SciLean.Scalar.log (x w)"} diff --git a/scilean-declarations/SciLean.Core.Functions.Gaussian.jsonl b/scilean-declarations/SciLean.Core.Functions.Gaussian.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..5e26d4690260dbd3cddf80eab33136a2d83ef06a --- /dev/null +++ b/scilean-declarations/SciLean.Core.Functions.Gaussian.jsonl @@ -0,0 +1,7 @@ +{"name":"SciLean.gaussian.arg_μx.CDifferentiableAt_rule","declaration":"theorem SciLean.gaussian.arg_μx.CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {U : Type u_3} [SciLean.SemiHilbert R U] (w : W) (μ : W → U) (σ : R) (x : W → U) (hμ : SciLean.CDifferentiableAt R μ w) (hx : SciLean.CDifferentiableAt R x w) : SciLean.CDifferentiableAt R (fun w => SciLean.gaussian (μ w) σ (x w)) w"} +{"name":"SciLean.gaussian","declaration":"def SciLean.gaussian {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiHilbert R U] (μ : U) (σ : R) (x : U) : R"} +{"name":"SciLean.gaussian.arg_μx.CDifferentiable_rule","declaration":"theorem SciLean.gaussian.arg_μx.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {U : Type u_3} [SciLean.SemiHilbert R U] (μ : W → U) (σ : R) (x : W → U) (hμ : SciLean.CDifferentiable R μ) (hx : SciLean.CDifferentiable R x) : SciLean.CDifferentiable R fun w => SciLean.gaussian (μ w) σ (x w)"} +{"name":"SciLean.gaussian.arg_μx.cderiv_rule","declaration":"theorem SciLean.gaussian.arg_μx.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {U : Type u_3} [SciLean.SemiHilbert R U] (μ : W → U) (σ : R) (x : W → U) (hμ : SciLean.CDifferentiable R μ) (hx : SciLean.CDifferentiable R x) : (SciLean.fwdDeriv R fun w => SciLean.gaussian (μ w) σ (x w)) = fun w dw =>\n let μdμ := SciLean.fwdDeriv R μ w dw;\n let xdx := SciLean.fwdDeriv R x w dw;\n let xdx' := σ⁻¹ • (xdx - μdμ);\n let g := SciLean.gaussian μdμ.1 σ xdx.1;\n (g, -⟪xdx'.1, xdx'.2⟫_R * g)"} +{"name":"SciLean.gaussian.arg_μσx.CDifferentiableAt_rule","declaration":"theorem SciLean.gaussian.arg_μσx.CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {U : Type u_3} [SciLean.SemiHilbert R U] (w : W) (μ : W → U) (σ : W → R) (x : W → U) (hμ : SciLean.CDifferentiableAt R μ w) (hσ : SciLean.CDifferentiableAt R σ w) (hx : SciLean.CDifferentiableAt R x w) (hσ' : σ w ≠ 0) : SciLean.CDifferentiableAt R (fun w => SciLean.gaussian (μ w) (σ w) (x w)) w"} +{"name":"SciLean.log_gaussian","declaration":"theorem SciLean.log_gaussian {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiHilbert R U] (μ : U) (σ : R) (x : U) : SciLean.Scalar.log (SciLean.gaussian μ σ x) =\n let x' := σ⁻¹ • (x - μ);\n -‖x'‖₂² / 2 - SciLean.Scalar.log σ - SciLean.Scalar.log (SciLean.Scalar.sqrt (2 * SciLean.RealScalar.pi))"} +{"name":"SciLean.gaussian.arg_μσx.CDifferentiable_rule","declaration":"theorem SciLean.gaussian.arg_μσx.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (μ : W → R) (σ : W → R) (x : W → R) (hμ : SciLean.CDifferentiable R μ) (hσ : SciLean.CDifferentiable R σ) (hx : SciLean.CDifferentiable R x) (hσ' : ∀ (w : W), σ w ≠ 0) : SciLean.CDifferentiable R fun w => SciLean.gaussian (μ w) (σ w) (x w)"} diff --git a/scilean-declarations/SciLean.Core.Functions.Pow.jsonl b/scilean-declarations/SciLean.Core.Functions.Pow.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ddc99e7c428572e2f01c2df6a13c743851915ae0 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Functions.Pow.jsonl @@ -0,0 +1,10 @@ +{"name":"SciLean.Scalar.sqrt.arg_x.cderiv_rule_at","declaration":"theorem SciLean.Scalar.sqrt.arg_x.cderiv_rule_at {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (w : W) (x : W → R) (hx : SciLean.CDifferentiableAt R x w) (hw : 0 < x w) : SciLean.cderiv R (fun w => SciLean.Scalar.sqrt (x w)) w = fun dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n xdx.2 / (2 * SciLean.Scalar.sqrt xdx.1)"} +{"name":"SciLean.Scalar.sqrt.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.sqrt.arg_x.HasAdjDiff_rule {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (x : U → R) (hx : SciLean.HasAdjDiff R x) (hu : ∀ (u : U), 0 < x u) : SciLean.HasAdjDiff R fun u => SciLean.Scalar.sqrt (x u)"} +{"name":"SciLean.Scalar.sqrt.arg_x.revDeriv_rule_at","declaration":"theorem SciLean.Scalar.sqrt.arg_x.revDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (u : U) (x : U → R) (hx : SciLean.HasAdjDiffAt R x u) (hu : 0 < x u) : SciLean.revDeriv R (fun u => SciLean.Scalar.sqrt (x u)) u =\n let xdx := SciLean.revDeriv R x u;\n let x' := SciLean.Scalar.sqrt xdx.1;\n (x', fun dy => xdx.2 (dy / (2 * x')))"} +{"name":"SciLean.Scalar.sqrt.arg_x.HasAdjDiffAt_rule","declaration":"theorem SciLean.Scalar.sqrt.arg_x.HasAdjDiffAt_rule {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (u : U) (x : U → R) (hx : SciLean.HasAdjDiffAt R x u) (hu : 0 < x u) : SciLean.HasAdjDiffAt R (fun u => SciLean.Scalar.sqrt (x u)) u"} +{"name":"SciLean.Scalar.sqrt.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.sqrt.arg_x.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiInnerProductSpace R U] (x : U → R) (hx : SciLean.HasAdjDiff R x) (hu : ∀ (u : U), 0 < x u) : (SciLean.revDeriv R fun u => SciLean.Scalar.sqrt (x u)) = fun u =>\n let xdx := SciLean.revDeriv R x u;\n let x' := SciLean.Scalar.sqrt xdx.1;\n (x', fun dy => xdx.2 (dy / (2 * x')))"} +{"name":"SciLean.Scalar.sqrt.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.sqrt.arg_x.fwdDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (x : W → R) (hx : SciLean.CDifferentiable R x) (hw : ∀ (w : W), 0 < x w) : (SciLean.fwdDeriv R fun w => SciLean.Scalar.sqrt (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n let x' := SciLean.Scalar.sqrt xdx.1;\n (x', xdx.2 / (2 * x'))"} +{"name":"SciLean.Scalar.sqrt.arg_x.fwdDeriv_rule_at","declaration":"theorem SciLean.Scalar.sqrt.arg_x.fwdDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (w : W) (x : W → R) (hx : SciLean.CDifferentiableAt R x w) (hw : 0 < x w) : SciLean.fwdDeriv R (fun w => SciLean.Scalar.sqrt (x w)) w = fun dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n let x' := SciLean.Scalar.sqrt xdx.1;\n (x', xdx.2 / (2 * x'))"} +{"name":"SciLean.Scalar.sqrt.arg_x.cderiv_rule","declaration":"theorem SciLean.Scalar.sqrt.arg_x.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (x : W → R) (hx : SciLean.CDifferentiable R x) (hw : ∀ (w : W), 0 < x w) : (SciLean.cderiv R fun w => SciLean.Scalar.sqrt (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv R x w dw;\n xdx.2 / (2 * SciLean.Scalar.sqrt xdx.1)"} +{"name":"SciLean.Scalar.sqrt.arg_x.CDifferentiableAt_rule","declaration":"theorem SciLean.Scalar.sqrt.arg_x.CDifferentiableAt_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (w : W) (x : W → R) (hx : SciLean.CDifferentiableAt R x w) (hw : x w ≠ 0) : SciLean.CDifferentiableAt R (fun w => SciLean.Scalar.sqrt (x w)) w"} +{"name":"SciLean.Scalar.sqrt.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.sqrt.arg_x.CDifferentiable_rule {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] (x : W → R) (hx : SciLean.CDifferentiable R x) (hw : ∀ (w : W), x w ≠ 0) : SciLean.CDifferentiable R fun w => SciLean.Scalar.sqrt (x w)"} diff --git a/scilean-declarations/SciLean.Core.Functions.Trigonometric.jsonl b/scilean-declarations/SciLean.Core.Functions.Trigonometric.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c44ac0d3411e4d2e8a488e75c966b2f1070cdfb3 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Functions.Trigonometric.jsonl @@ -0,0 +1,15 @@ +{"name":"SciLean.Scalar.sin.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.HasAdjDiff_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : SciLean.HasAdjDiff C fun u => SciLean.Scalar.sin (x u)"} +{"name":"SciLean.Scalar.cos.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.CDifferentiable_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : SciLean.CDifferentiable C fun w => SciLean.Scalar.cos (x w)"} +{"name":"SciLean.Scalar.tanh.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.revDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : (SciLean.revDeriv C fun u => SciLean.Scalar.tanh (x u)) = fun u =>\n let xdx := SciLean.revDeriv C x u;\n (SciLean.Scalar.tanh xdx.1, fun dy => xdx.2 ((starRingEnd C) (1 - SciLean.Scalar.tanh xdx.1 ^ 2) * dy))"} +{"name":"SciLean.Scalar.cos.arg_x.ceriv_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.ceriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.cderiv C fun w => SciLean.Scalar.cos (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let s := -SciLean.Scalar.sin xdx.1;\n xdx.2 * s"} +{"name":"SciLean.Scalar.cos.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.HasAdjDiff_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : SciLean.HasAdjDiff C fun u => SciLean.Scalar.cos (x u)"} +{"name":"SciLean.Scalar.sin.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.revDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : (SciLean.revDeriv C fun u => SciLean.Scalar.sin (x u)) = fun u =>\n let xdx := SciLean.revDeriv C x u;\n (SciLean.Scalar.sin xdx.1, fun dy => xdx.2 ((starRingEnd C) (SciLean.Scalar.cos xdx.1) * dy))"} +{"name":"SciLean.Scalar.sin.arg_x.ceriv_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.ceriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.cderiv C fun w => SciLean.Scalar.sin (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let c := SciLean.Scalar.cos xdx.1;\n xdx.2 * c"} +{"name":"SciLean.Scalar.cos.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.fwdDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.fwdDeriv C fun w => SciLean.Scalar.cos (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let s := SciLean.Scalar.sin xdx.1;\n let c := SciLean.Scalar.cos xdx.1;\n (c, xdx.2 * -s)"} +{"name":"SciLean.Scalar.tanh.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.CDifferentiable_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : SciLean.CDifferentiable C fun w => SciLean.Scalar.tanh (x w)"} +{"name":"SciLean.Scalar.tanh.arg_x.ceriv_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.ceriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.cderiv C fun w => SciLean.Scalar.tanh (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let t := SciLean.Scalar.tanh xdx.1;\n let dt := 1 - t ^ 2;\n xdx.2 * dt"} +{"name":"SciLean.Scalar.tanh.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.fwdDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.fwdDeriv C fun w => SciLean.Scalar.tanh (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let t := SciLean.Scalar.tanh xdx.1;\n let dt := 1 - t ^ 2;\n (t, xdx.2 * dt)"} +{"name":"SciLean.Scalar.tanh.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.HasAdjDiff_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : SciLean.HasAdjDiff C fun u => SciLean.Scalar.tanh (x u)"} +{"name":"SciLean.Scalar.sin.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.fwdDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.fwdDeriv C fun w => SciLean.Scalar.sin (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let s := SciLean.Scalar.sin xdx.1;\n let c := SciLean.Scalar.cos xdx.1;\n (s, xdx.2 * c)"} +{"name":"SciLean.Scalar.cos.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.revDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : (SciLean.revDeriv C fun u => SciLean.Scalar.cos (x u)) = fun u =>\n let xdx := SciLean.revDeriv C x u;\n (SciLean.Scalar.cos xdx.1, fun dy => xdx.2 (-(starRingEnd C) (SciLean.Scalar.sin xdx.1) * dy))"} +{"name":"SciLean.Scalar.sin.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.CDifferentiable_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : SciLean.CDifferentiable C fun w => SciLean.Scalar.sin (x w)"} diff --git a/scilean-declarations/SciLean.Core.Integral.CIntegral.jsonl b/scilean-declarations/SciLean.Core.Integral.CIntegral.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..98aaaee181726099ccb1b1a64867d5fada91e79b --- /dev/null +++ b/scilean-declarations/SciLean.Core.Integral.CIntegral.jsonl @@ -0,0 +1,26 @@ +{"name":"SciLean.cintegral.arg_f.fwdDeriv_rule","declaration":"theorem SciLean.cintegral.arg_f.fwdDeriv_rule {β : Type u_1} [MeasurableSpace β] {R : Type u_2} [RCLike R] {X : Type u_3} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] [Module ℝ Z] (f : X → β → Z) (μ : MeasureTheory.Measure β) (hf : ∀ (x : β), SciLean.CDifferentiable R fun x_1 => f x_1 x) : (SciLean.fwdDeriv R fun x => ∫' y, f x y ∂μ) = fun x dx => ∫' y, SciLean.fwdDeriv R (fun x => f x y) x dx ∂μ"} +{"name":"SciLean.cintegral.arg_f.IsSmoothLinearMap_rule","declaration":"theorem SciLean.cintegral.arg_f.IsSmoothLinearMap_rule {β : Type u_1} [MeasurableSpace β] {R : Type u_2} [RCLike R] {X : Type u_3} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] [Module ℝ Z] (f : X → β → Z) (μ : MeasureTheory.Measure β) (hf : ∀ (y : β), SciLean.IsSmoothLinearMap R fun x => f x y) : SciLean.IsSmoothLinearMap R fun x => ∫' y, f x y ∂μ"} +{"name":"SciLean.cintegral.arg_f.IsLinearMap_rule","declaration":"theorem SciLean.cintegral.arg_f.IsLinearMap_rule {β : Type u_1} [MeasurableSpace β] {R : Type u_2} [RCLike R] {X : Type u_3} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] [Module ℝ Z] (f : X → β → Z) (μ : MeasureTheory.Measure β) (hf : ∀ (y : β), IsLinearMap R fun x => f x y) : IsLinearMap R fun x => ∫' y, f x y ∂μ"} +{"name":"SciLean.cintegral_prod_mk'","declaration":"theorem SciLean.cintegral_prod_mk' {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] {Y : Type u_3} [AddCommGroup Y] [Module ℝ Y] {f : α → X} {g : α → Y} {μ : MeasureTheory.Measure α} : (∫' x, f x ∂μ, ∫' x, g x ∂μ) = ∫' x, (f x, g x) ∂μ"} +{"name":"SciLean.«term∫'_In_,_∂_»","declaration":"def SciLean.«term∫'_In_,_∂_» : Lean.ParserDescr"} +{"name":"SciLean.cintegral.arg_f.cderiv_rule","declaration":"theorem SciLean.cintegral.arg_f.cderiv_rule {β : Type u_1} [MeasurableSpace β] {R : Type u_2} [RCLike R] {X : Type u_3} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] [Module ℝ Z] (f : X → β → Z) (μ : MeasureTheory.Measure β) (hf : ∀ (y : β), SciLean.CDifferentiable R fun x => f x y) : (SciLean.cderiv R fun x => ∫' y, f x y ∂μ) = fun x dx => ∫' y, SciLean.cderiv R (fun x => f x y) x dx ∂μ"} +{"name":"SciLean.cintegral_unit","declaration":"theorem SciLean.cintegral_unit {X : Type u_1} [AddCommGroup X] [Module ℝ X] {f : Unit → X} : ∫' x, f x ∂MeasureTheory.volume = f ()"} +{"name":"SciLean.cintegral_smul","declaration":"theorem SciLean.cintegral_smul {α : Type u_1} [MeasurableSpace α] {X : Type u_3} [AddCommGroup X] [Module ℝ X] {μ : MeasureTheory.Measure α} {R : Type u_2} [Semiring R] [Module R X] {f : α → X} (r : R) : ∫' x, r • f x ∂μ = r • ∫' x, f x ∂μ"} +{"name":"SciLean.cintegral_measure_map","declaration":"theorem SciLean.cintegral_measure_map {α : Type u_1} [MeasurableSpace α] {β : Type u_3} [MeasurableSpace β] {X : Type u_2} [AddCommGroup X] [Module ℝ X] (g : β → X) (f : α → β) (μ : MeasureTheory.Measure α) : ∫' y, g y ∂MeasureTheory.Measure.map f μ = ∫' x, g (f x) ∂μ"} +{"name":"SciLean.cintegral","declaration":"/-- Convenient integral - the integral I need :)\nIt should be Bochner integral but it should integrate function valued function point wise i.e.\n```\n∫ x, fun y => f x y = fun y => ∫ x, f x y\n```\nwhere rhs can be understoods as Bochenr integral and lhs defined thie `cintegral`. -/\nopaque SciLean.cintegral {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] (f : α → X) (μ : MeasureTheory.Measure α) : X"} +{"name":"SciLean.split_integral_over_set_of_ite","declaration":"theorem SciLean.split_integral_over_set_of_ite {R : Type u_3} [SciLean.RealScalar R] [DecidableRel fun x x_1 => x ≤ x_1] {X : Type u_1} [MeasureTheory.MeasureSpace X] {Y : Type u_2} [AddCommGroup Y] [Module ℝ Y] (φ : X → R) (ψ : X → R) (f : X → Y) (g : X → Y) (A : Set X) : ∫' x, if ψ x ≤ φ x then f x else g x ∂MeasureTheory.volume.restrict A =\n ∫' x, f x ∂MeasureTheory.volume.restrict ({x' | 0 ≤ φ x' - ψ x'} ∩ A) +\n ∫' x, g x ∂MeasureTheory.volume.restrict ({x' | 0 ≤ ψ x' - φ x'} ∩ A)"} +{"name":"SciLean.cintegral_add'","declaration":"theorem SciLean.cintegral_add' {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] {f : α → X} {g : α → X} {μ : MeasureTheory.Measure α} (hf : SciLean.CIntegrable f μ) (hg : SciLean.CIntegrable g μ) : ∫' x, f x ∂μ + ∫' x, g x ∂μ = ∫' x, f x + g x ∂μ"} +{"name":"SciLean.cintegral_zero","declaration":"theorem SciLean.cintegral_zero {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] (μ : MeasureTheory.Measure α) : ∫' x, 0 ∂μ = 0"} +{"name":"SciLean.unexpandCIntegral","declaration":"def SciLean.unexpandCIntegral : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.cintegral_smul'","declaration":"theorem SciLean.cintegral_smul' {α : Type u_1} [MeasurableSpace α] {X : Type u_3} [AddCommGroup X] [Module ℝ X] {μ : MeasureTheory.Measure α} {R : Type u_2} [Semiring R] [Module R X] {f : α → X} (r : R) : r • ∫' x, f x ∂μ = ∫' x, r • f x ∂μ"} +{"name":"SciLean.integral_in_set_simp","declaration":"theorem SciLean.integral_in_set_simp {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] (f : α → X) (g : α → X) (A : Set α) (h : ∀ x ∈ A, f x = g x) (μ : MeasureTheory.Measure α) : ∫' x, f x ∂μ.restrict A = ∫' x, g x ∂μ.restrict A"} +{"name":"SciLean.cintegral_add","declaration":"theorem SciLean.cintegral_add {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] {f : α → X} {g : α → X} {μ : MeasureTheory.Measure α} (hf : SciLean.CIntegrable f μ) (hg : SciLean.CIntegrable g μ) : ∫' x, f x + g x ∂μ = ∫' x, f x ∂μ + ∫' x, g x ∂μ"} +{"name":"SciLean.cintegral.arg_f.push_lambda","declaration":"theorem SciLean.cintegral.arg_f.push_lambda {β : Type u_2} [MeasurableSpace β] {X : Type u_3} [AddCommGroup X] [Module ℝ X] {α : Type u_1} (f : α → β → X) (μ : MeasureTheory.Measure β) : (fun x => ∫' y, f x y ∂μ) = ∫' y, (fun x => f x y) ∂μ"} +{"name":"SciLean.split_integral_of_ite","declaration":"theorem SciLean.split_integral_of_ite {R : Type u_3} [SciLean.RealScalar R] [DecidableRel fun x x_1 => x ≤ x_1] {X : Type u_2} [MeasureTheory.MeasureSpace X] {Y : Type u_1} [AddCommGroup Y] [Module ℝ Y] (φ : X → R) (ψ : X → R) (f : X → Y) (g : X → Y) : ∫' x, if ψ x ≤ φ x then f x else g x ∂MeasureTheory.volume =\n ∫' x, f x ∂MeasureTheory.volume.restrict {x' | ψ x' ≤ φ x'} +\n ∫' x, g x ∂MeasureTheory.volume.restrict {x' | φ x' < ψ x'}"} +{"name":"SciLean.integral_simproc","declaration":"def SciLean.integral_simproc : Lean.Meta.Simp.Simproc"} +{"name":"SciLean.cintegral_prod_mk","declaration":"theorem SciLean.cintegral_prod_mk {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] {Y : Type u_3} [AddCommGroup Y] [Module ℝ Y] {f : α → X} {g : α → Y} {μ : MeasureTheory.Measure α} : ∫' x, (f x, g x) ∂μ = (∫' x, f x ∂μ, ∫' x, g x ∂μ)"} +{"name":"SciLean.CIntegrable","declaration":"opaque SciLean.CIntegrable {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] (f : α → X) (μ : MeasureTheory.Measure α) : Prop"} +{"name":"SciLean.cintegral_lambda","declaration":"theorem SciLean.cintegral_lambda {β : Type u_2} [MeasurableSpace β] {X : Type u_3} [AddCommGroup X] [Module ℝ X] {α : Type u_1} (f : α → β → X) (μ : MeasureTheory.Measure β) : (fun x => ∫' y, f x y ∂μ) = ∫' y, (fun x => f x y) ∂μ"} +{"name":"SciLean.cintegral.arg_f.CDifferentiable_rule","declaration":"theorem SciLean.cintegral.arg_f.CDifferentiable_rule {β : Type u_1} [MeasurableSpace β] {R : Type u_2} [RCLike R] {X : Type u_3} [SciLean.Vec R X] {Z : Type u_4} [SciLean.Vec R Z] [Module ℝ Z] (f : X → β → Z) (μ : MeasureTheory.Measure β) (hf : ∀ (x : β), SciLean.CDifferentiable R fun x_1 => f x_1 x) : SciLean.CDifferentiable R fun x => ∫' y, f x y ∂μ"} +{"name":"SciLean.cintegral_add_measures","declaration":"theorem SciLean.cintegral_add_measures {α : Type u_1} [MeasurableSpace α] {X : Type u_2} [AddCommGroup X] [Module ℝ X] {f : α → X} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} (hμ : SciLean.CIntegrable f μ) (hν : SciLean.CIntegrable f ν) : ∫' x, f x ∂(μ + ν) = ∫' x, f x ∂μ + ∫' x, f x ∂ν"} +{"name":"SciLean.cintegral_dirac","declaration":"theorem SciLean.cintegral_dirac {α : Type u_2} [MeasurableSpace α] {X : Type u_1} [AddCommGroup X] [Module ℝ X] {f : α → X} (p : α) : ∫' x, f x ∂MeasureTheory.Measure.dirac p = f p"} diff --git a/scilean-declarations/SciLean.Core.Meta.GenerateLinearMapSimp.jsonl b/scilean-declarations/SciLean.Core.Meta.GenerateLinearMapSimp.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ed4888bb613adb2e96003495ca93c1e51fa30895 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Meta.GenerateLinearMapSimp.jsonl @@ -0,0 +1,23 @@ +{"name":"IsLinearMap.app_zero₂","declaration":"theorem IsLinearMap.app_zero₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) : g 0 0 = 0"} +{"name":"IsLinearMap.neg_pull","declaration":"theorem IsLinearMap.neg_pull {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) : f (-x) = -f x"} +{"name":"IsLinearMap.sum_push","declaration":"theorem IsLinearMap.sum_push {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (ι : Type) [SciLean.IndexType ι] [SciLean.IndexType ι] (x : ι → X) : ∑ i, f (x i) = f ( ∑ i, x i)"} +{"name":"IsLinearMap.app_zero","declaration":"theorem IsLinearMap.app_zero {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) : f 0 = 0"} +{"name":"IsLinearMap.sum_pull","declaration":"theorem IsLinearMap.sum_pull {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (ι : Type) [SciLean.IndexType ι] [SciLean.IndexType ι] (x : ι → X) : f ( ∑ i, x i) = ∑ i, f (x i)"} +{"name":"IsLinearMap.smul_pull","declaration":"theorem IsLinearMap.smul_pull {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) (k : K) : f (k • x) = k • f x"} +{"name":"IsLinearMap.sub_pull₂","declaration":"theorem IsLinearMap.sub_pull₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (x' : X) (y : Y) (y' : Y) : g (x - x') (y - y') = g x y - g x' y'"} +{"name":"IsLinearMap.sub_push","declaration":"theorem IsLinearMap.sub_push {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) (x' : X) : f x - f x' = f (x - x')"} +{"name":"IsLinearMap.add_pull₂","declaration":"theorem IsLinearMap.add_pull₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (x' : X) (y : Y) (y' : Y) : g (x + x') (y + y') = g x y + g x' y'"} +{"name":"SciLean.generateLinearMapSimp","declaration":"def SciLean.generateLinearMapSimp (ctx : Array Lean.Expr) (isLinearMap : Lean.Expr) (thrmName : Lean.Name) (isSimpAttr : optParam Bool true) (makeSimp : optParam Bool false) : Lean.MetaM Unit"} +{"name":"IsLinearMap.smul_pull₂","declaration":"theorem IsLinearMap.smul_pull₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (y : Y) (k : K) : g (k • x) (k • y) = k • g x y"} +{"name":"IsLinearMap.add_push","declaration":"theorem IsLinearMap.add_push {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) (x' : X) : f x + f x' = f (x + x')"} +{"name":"IsLinearMap.neg_push","declaration":"theorem IsLinearMap.neg_push {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) : -f x = f (-x)"} +{"name":"IsLinearMap.add_push₂","declaration":"theorem IsLinearMap.add_push₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (x' : X) (y : Y) (y' : Y) : g x y + g x' y' = g (x + x') (y + y')"} +{"name":"IsLinearMap.smul_push₂","declaration":"theorem IsLinearMap.smul_push₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (y : Y) (k : K) : k • g x y = g (k • x) (k • y)"} +{"name":"IsLinearMap.sub_pull","declaration":"theorem IsLinearMap.sub_pull {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) (x' : X) : f (x - x') = f x - f x'"} +{"name":"IsLinearMap.neg_pull₂","declaration":"theorem IsLinearMap.neg_pull₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (y : Y) : g (-x) (-y) = -g x y"} +{"name":"IsLinearMap.add_pull","declaration":"theorem IsLinearMap.add_pull {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) (x' : X) : f (x + x') = f x + f x'"} +{"name":"IsLinearMap.smul_push","declaration":"theorem IsLinearMap.smul_push {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {f : X → Y} (hf : IsLinearMap K f) (x : X) (k : K) : k • f x = f (k • x)"} +{"name":"IsLinearMap.sub_push₂","declaration":"theorem IsLinearMap.sub_push₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (x' : X) (y : Y) (y' : Y) : g x y - g x' y' = g (x - x') (y - y')"} +{"name":"SciLean.genLinMapSimpsNotation","declaration":"/-- This commands generates simp lemmas for given linear function.\n\nThe commands is used as\n```\n#generate_linear_map_simps thrmName\n```\nwhere `thrmName` is a name of a theorem that states that function `f` is linear i.e. `IsLinearMap K f`.\n\nThe command generates theorems\n```\n@[add_push] theorem add_push (x x' : X) : f x + f x' = f (x + x') := ...\n@[add_pull] theorem add_pull (x x' : X) : f (x + x') = f x + f x' := ...\n@[sub_push] theorem sub_push (x x' : X) : f x - f x' = f (x - x') := ...\n@[sub_pull] theorem sub_pull (x x' : X) : f (x - x') = f x - f x' := ...\n@[neg_push] theorem neg_push (x : X) : - f x = f (- x) := ...\n@[neg_pull] theorem neg_pull (x : X) : f (- x) = - f x := ...\n@[smul_push] theorem smul_push (x : X) (k : K) : k • f x = f (k • x) := ...\n@[smul_pull] theorem smul_pull (x : X) (k : K) : f (k • x) = k • f x := ...\n@[simp] theorem app_zero : f 0 = 0 := ...\n```\nAll the above attributes are simp attributes. The ideas is that you can propagate\narithmetic operations by calling `simp` e.g. `simp only [add_pull]`.\n\n\nThe command also supports functions jointly linear in two arguments. If we have\n`g : X → Y → Z` and `g_is_linear₂ : IsLinear K fun (x,y) => g x y` then\n```\n#generate_linear_map_simps g_is_linear₂\n```\ngenerates theorems like\n```\n@[add_push] theorem add_push (x x' : X) (y y' : Y) : g x y + g x' y' = g (x + x') (y + y') := ...\n...\n```\n\n-/\ndef SciLean.genLinMapSimpsNotation : Lean.ParserDescr"} +{"name":"SciLean.generateLinearMapSimps","declaration":"/-- Generates bunch of simp theorems given a proof that function is linear.\n\nThe provided theorem should be in the simple form `IsLinearMap K (fun x => foo x)`\nNot in the composition form `IsLinearMap K (fun x => foo (f x))`\n-/\ndef SciLean.generateLinearMapSimps (isLinearMapTheorem : Lean.Name) : Lean.MetaM Unit"} +{"name":"IsLinearMap.neg_push₂","declaration":"theorem IsLinearMap.neg_push₂ {K : Type u'} [CommSemiring K] {X : Type u} [AddCommGroup X] [Module K X] {Y : Type v} [AddCommGroup Y] [Module K Y] {Z : Type w} [AddCommGroup Z] [Module K Z] {g : X → Y → Z} (hg : IsLinearMap K fun xy => g xy.1 xy.2) (x : X) (y : Y) : -g x y = g (-x) (-y)"} diff --git a/scilean-declarations/SciLean.Core.Meta.ToAnyPoint.jsonl b/scilean-declarations/SciLean.Core.Meta.ToAnyPoint.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..296496b6aefade52f5ba9835ba1b64c0ddfa8a0c --- /dev/null +++ b/scilean-declarations/SciLean.Core.Meta.ToAnyPoint.jsonl @@ -0,0 +1 @@ +{"name":"toAnyPointAttr","declaration":"opaque toAnyPointAttr : Lean.TagAttribute"} diff --git a/scilean-declarations/SciLean.Core.Monads.FwdDerivMonad.jsonl b/scilean-declarations/SciLean.Core.Monads.FwdDerivMonad.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3a3a4efa22f719b876c48526d0becb9212d36da3 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Monads.FwdDerivMonad.jsonl @@ -0,0 +1,29 @@ +{"name":"SciLean.fwdDerivM.let_rule","declaration":"theorem SciLean.fwdDerivM.let_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] {Z : Type} [SciLean.Vec K Z] (f : X → Y → m Z) (g : X → Y) (hf : SciLean.CDifferentiableM K fun xy => f xy.1 xy.2) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDerivM K fun x =>\n let y := g x;\n f x y) =\n fun x dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n SciLean.fwdDerivM K (fun xy => f xy.1 xy.2) (x, ydy.1) (dx, ydy.2)"} +{"name":"SciLean.fwdDerivM.const_rule","declaration":"theorem SciLean.fwdDerivM.const_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (y : m Y) (hy : SciLean.CDifferentiableValM K y) : (SciLean.fwdDerivM K fun x => y) = fun x x => SciLean.fwdDerivValM K y"} +{"name":"SciLean.fwdDerivM.pure_rule","declaration":"theorem SciLean.fwdDerivM.pure_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] : (SciLean.fwdDerivM K fun x => pure x) = fun x dx => pure (x, dx)"} +{"name":"Bind.bind.arg_a0a1.CDifferentiableM_rule","declaration":"theorem Bind.bind.arg_a0a1.CDifferentiableM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] {Z : Type} [SciLean.Vec K Z] (a0 : X → m Y) (a1 : X → Y → m Z) (ha0 : SciLean.CDifferentiableM K a0) (ha1 : SciLean.CDifferentiableM K fun xy => a1 xy.1 xy.2) : SciLean.CDifferentiableM K fun x => a0 x >>= a1 x"} +{"name":"SciLean.FwdDerivMonad.fwdDerivM_pair","declaration":"def SciLean.FwdDerivMonad.fwdDerivM_pair {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.Vec K X] [SciLean.Vec K Y] (f : X → m Y) (hf : SciLean.CDifferentiableM K f) : (SciLean.fwdDerivM K fun x => do\n let y ← f x\n pure (x, y)) =\n fun x dx => do\n let ydy ← SciLean.fwdDerivM K f x dx\n pure ((x, ydy.1), dx, ydy.2)"} +{"name":"SciLean.CDifferentiableM.pure_rule","declaration":"theorem SciLean.CDifferentiableM.pure_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] : SciLean.CDifferentiableM K fun x => pure x"} +{"name":"Bind.bind.arg_a0a1.fwdDerivM_rule","declaration":"theorem Bind.bind.arg_a0a1.fwdDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] {Z : Type} [SciLean.Vec K Z] (a0 : X → m Y) (a1 : X → Y → m Z) (ha0 : SciLean.CDifferentiableM K a0) (ha1 : SciLean.CDifferentiableM K fun xy => a1 xy.1 xy.2) : (SciLean.fwdDerivM K fun x => a0 x >>= a1 x) = fun x dx => do\n let ydy ← SciLean.fwdDerivM K a0 x dx\n SciLean.fwdDerivM K (fun xy => a1 xy.1 xy.2) (x, ydy.1) (dx, ydy.2)"} +{"name":"SciLean.FwdDerivMonad.mk","declaration":"ctor SciLean.FwdDerivMonad.mk {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] (fwdDerivM : {X Y : Type} → [inst : SciLean.Vec K X] → [inst : SciLean.Vec K Y] → (X → m Y) → X → X → m' (Y × Y)) (CDifferentiableM : {X Y : Type} → [inst : SciLean.Vec K X] → [inst : SciLean.Vec K Y] → (X → m Y) → Prop) (fwdDerivM_pure : ∀ {X Y : Type} [inst : SciLean.Vec K X] [inst_1 : SciLean.Vec K Y] (f : X → Y),\n SciLean.CDifferentiable K f → (fwdDerivM fun x => pure (f x)) = fun x dx => pure (SciLean.fwdDeriv K f x dx)) (fwdDerivM_bind : ∀ {X Y Z : Type} [inst : SciLean.Vec K X] [inst_1 : SciLean.Vec K Y] [inst_2 : SciLean.Vec K Z] (f : Y → m Z)\n (g : X → m Y),\n CDifferentiableM f →\n CDifferentiableM g →\n (fwdDerivM fun x => g x >>= f) = fun x dx => do\n let ydy ← fwdDerivM g x dx\n fwdDerivM f ydy.1 ydy.2) (fwdDerivM_pair : ∀ {X Y : Type} [inst : SciLean.Vec K X] [inst_1 : SciLean.Vec K Y] (f : X → m Y),\n CDifferentiableM f →\n (fwdDerivM fun x => do\n let y ← f x\n pure (x, y)) =\n fun x dx => do\n let ydy ← fwdDerivM f x dx\n pure ((x, ydy.1), dx, ydy.2)) (CDifferentiableM_pure : ∀ {X Y : Type} [inst : SciLean.Vec K X] [inst_1 : SciLean.Vec K Y] (f : X → Y),\n SciLean.CDifferentiable K f → CDifferentiableM fun x => pure (f x)) (CDifferentiableM_bind : ∀ {X Y Z : Type} [inst : SciLean.Vec K X] [inst_1 : SciLean.Vec K Y] [inst_2 : SciLean.Vec K Z] (f : Y → m Z)\n (g : X → m Y), CDifferentiableM f → CDifferentiableM g → CDifferentiableM fun x => g x >>= f) (CDifferentiableM_pair : ∀ {X Y : Type} [inst : SciLean.Vec K X] [inst_1 : SciLean.Vec K Y] (f : X → m Y),\n CDifferentiableM f →\n CDifferentiableM fun x => do\n let y ← f x\n pure (x, y)) : SciLean.FwdDerivMonad K m m'"} +{"name":"SciLean.CDifferentiableM.const_rule","declaration":"theorem SciLean.CDifferentiableM.const_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (y : m Y) (hy : SciLean.CDifferentiableValM K y) : SciLean.CDifferentiableM K fun x => y"} +{"name":"dite.arg_te.fwdDerivM_rule","declaration":"theorem dite.arg_te.fwdDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : c → X → m Y) (e : ¬c → X → m Y) : (SciLean.fwdDerivM K fun x => if h : c then t h x else e h x) = fun y =>\n if h : c then SciLean.fwdDerivM K (t h) y else SciLean.fwdDerivM K (e h) y"} +{"name":"SciLean.CDifferentiableValM","declaration":"def SciLean.CDifferentiableValM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] (x : m X) : Prop"} +{"name":"SciLean.FwdDerivMonad.CDifferentiableM_bind","declaration":"def SciLean.FwdDerivMonad.CDifferentiableM_bind {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} {Z : Type} [SciLean.Vec K X] [SciLean.Vec K Y] [SciLean.Vec K Z] (f : Y → m Z) (g : X → m Y) (hf : SciLean.CDifferentiableM K f) (hg : SciLean.CDifferentiableM K g) : SciLean.CDifferentiableM K fun x => g x >>= f"} +{"name":"SciLean.FwdDerivMonad","declaration":"class SciLean.FwdDerivMonad (K : Type) [RCLike K] (m : Type → Type) (m' : outParam (Type → Type)) [Monad m] [Monad m'] : Type 1"} +{"name":"SciLean.fwdDerivValM","declaration":"def SciLean.fwdDerivValM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] (x : m X) : m' (X × X)"} +{"name":"Pure.pure.arg_a0.CDifferentiableM_rule","declaration":"theorem Pure.pure.arg_a0.CDifferentiableM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (a0 : X → Y) (ha0 : SciLean.CDifferentiable K a0) : SciLean.CDifferentiableM K fun x => pure (a0 x)"} +{"name":"Pure.pure.arg.CDifferentiableValM_rule","declaration":"theorem Pure.pure.arg.CDifferentiableValM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] (x : X) : SciLean.CDifferentiableValM K (pure x)"} +{"name":"ite.arg_te.fwdDerivM_rule","declaration":"theorem ite.arg_te.fwdDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : X → m Y) (e : X → m Y) : (SciLean.fwdDerivM K fun x => if c then t x else e x) = fun y =>\n if c then SciLean.fwdDerivM K t y else SciLean.fwdDerivM K e y"} +{"name":"SciLean.CDifferentiableM.comp_rule","declaration":"theorem SciLean.CDifferentiableM.comp_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] {Z : Type} [SciLean.Vec K Z] (f : Y → m Z) (g : X → Y) (hf : SciLean.CDifferentiableM K f) (hg : SciLean.CDifferentiable K g) : SciLean.CDifferentiableM K fun x => f (g x)"} +{"name":"SciLean.fwdDerivM.comp_rule","declaration":"theorem SciLean.fwdDerivM.comp_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] {Z : Type} [SciLean.Vec K Z] (f : Y → m Z) (g : X → Y) (hf : SciLean.CDifferentiableM K f) (hg : SciLean.CDifferentiable K g) : (SciLean.fwdDerivM K fun x => f (g x)) = fun x dx =>\n let ydy := SciLean.fwdDeriv K g x dx;\n SciLean.fwdDerivM K f ydy.1 ydy.2"} +{"name":"dite.arg_te.CDifferentiableM_rule","declaration":"theorem dite.arg_te.CDifferentiableM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : c → X → m Y) (e : ¬c → X → m Y) (ht : ∀ (h : c), SciLean.CDifferentiableM K (t h)) (he : ∀ (h : ¬c), SciLean.CDifferentiableM K (e h)) : SciLean.CDifferentiableM K fun x => if h : c then t h x else e h x"} +{"name":"Pure.pure.arg.fwdDerivValM_rule","declaration":"theorem Pure.pure.arg.fwdDerivValM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] (x : X) : SciLean.fwdDerivValM K (pure x) = pure (x, 0)"} +{"name":"SciLean.FwdDerivMonad.CDifferentiableM_pure","declaration":"def SciLean.FwdDerivMonad.CDifferentiableM_pure {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.Vec K X] [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiableM K fun x => pure (f x)"} +{"name":"Pure.pure.arg_a0.fwdDerivM_rule","declaration":"theorem Pure.pure.arg_a0.fwdDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (a0 : X → Y) (ha0 : SciLean.CDifferentiable K a0) : (SciLean.fwdDerivM K fun x => pure (a0 x)) = fun x dx => pure (SciLean.fwdDeriv K a0 x dx)"} +{"name":"SciLean.FwdDerivMonad.CDifferentiableM","declaration":"def SciLean.FwdDerivMonad.CDifferentiableM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.Vec K X] [SciLean.Vec K Y] (f : X → m Y) : Prop"} +{"name":"SciLean.FwdDerivMonad.fwdDerivM_bind","declaration":"def SciLean.FwdDerivMonad.fwdDerivM_bind {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} {Z : Type} [SciLean.Vec K X] [SciLean.Vec K Y] [SciLean.Vec K Z] (f : Y → m Z) (g : X → m Y) (hf : SciLean.CDifferentiableM K f) (hg : SciLean.CDifferentiableM K g) : (SciLean.fwdDerivM K fun x => g x >>= f) = fun x dx => do\n let ydy ← SciLean.fwdDerivM K g x dx\n SciLean.fwdDerivM K f ydy.1 ydy.2"} +{"name":"ite.arg_te.CDifferentiableM_rule","declaration":"theorem ite.arg_te.CDifferentiableM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] {X : Type} [SciLean.Vec K X] {Y : Type} [SciLean.Vec K Y] (c : Prop) [dec : Decidable c] (t : X → m Y) (e : X → m Y) (ht : SciLean.CDifferentiableM K t) (he : SciLean.CDifferentiableM K e) : SciLean.CDifferentiableM K fun x => if c then t x else e x"} +{"name":"SciLean.FwdDerivMonad.CDifferentiableM_pair","declaration":"def SciLean.FwdDerivMonad.CDifferentiableM_pair {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.Vec K X] [SciLean.Vec K Y] (f : X → m Y) (hf : SciLean.CDifferentiableM K f) : SciLean.CDifferentiableM K fun x => do\n let y ← f x\n pure (x, y)"} +{"name":"SciLean.FwdDerivMonad.fwdDerivM_pure","declaration":"def SciLean.FwdDerivMonad.fwdDerivM_pure {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.Vec K X] [SciLean.Vec K Y] (f : X → Y) (hf : SciLean.CDifferentiable K f) : (SciLean.fwdDerivM K fun x => pure (f x)) = fun x dx => pure (SciLean.fwdDeriv K f x dx)"} +{"name":"SciLean.FwdDerivMonad.fwdDerivM","declaration":"def SciLean.FwdDerivMonad.fwdDerivM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.FwdDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.Vec K X] [SciLean.Vec K Y] (f : X → m Y) (x : X) (dx : X) : m' (Y × Y)"} diff --git a/scilean-declarations/SciLean.Core.Monads.RevDerivMonad.jsonl b/scilean-declarations/SciLean.Core.Monads.RevDerivMonad.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c8751e6bc1531329ac8ac27073708488bc39cbec --- /dev/null +++ b/scilean-declarations/SciLean.Core.Monads.RevDerivMonad.jsonl @@ -0,0 +1,28 @@ +{"name":"dite.arg_te.revDerivM_rule","declaration":"theorem dite.arg_te.revDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → m Y) (e : ¬c → X → m Y) : (SciLean.revDerivM K fun x => if h : c then t h x else e h x) = fun y =>\n if h : c then SciLean.revDerivM K (t h) y else SciLean.revDerivM K (e h) y"} +{"name":"Pure.pure.arg_a0.revDerivM_rule","declaration":"theorem Pure.pure.arg_a0.revDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (a0 : X → Y) (ha0 : SciLean.HasAdjDiff K a0) : (SciLean.revDerivM K fun x => pure (a0 x)) = fun x =>\n let ydf := SciLean.revDeriv K a0 x;\n pure (ydf.1, fun dy => pure (ydf.2 dy))"} +{"name":"ite.arg_te.revDerivM_rule","declaration":"theorem ite.arg_te.revDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → m Y) (e : X → m Y) : (SciLean.revDerivM K fun x => if c then t x else e x) = fun y =>\n if c then SciLean.revDerivM K t y else SciLean.revDerivM K e y"} +{"name":"dite.arg_te.HasAdjDiffM_rule","declaration":"theorem dite.arg_te.HasAdjDiffM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → m Y) (e : ¬c → X → m Y) (ht : ∀ (h : c), SciLean.HasAdjDiffM K (t h)) (he : ∀ (h : ¬c), SciLean.HasAdjDiffM K (e h)) : SciLean.HasAdjDiffM K fun x => if h : c then t h x else e h x"} +{"name":"SciLean.revDerivM.let_rule","declaration":"theorem SciLean.revDerivM.let_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → m Z) (g : X → Y) (hf : SciLean.HasAdjDiffM K fun xy => f xy.1 xy.2) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivM K fun x =>\n let y := g x;\n f x y) =\n fun x =>\n let ydg := SciLean.revDeriv K g x;\n do\n let zdf ← SciLean.revDerivM K (fun xy => f xy.1 xy.2) (x, ydg.1)\n pure\n (zdf.1, fun dz => do\n let dxy ← zdf.2 dz\n let dx : X := ydg.2 dxy.2\n pure (dxy.1 + dx))"} +{"name":"SciLean.RevDerivMonad.mk","declaration":"ctor SciLean.RevDerivMonad.mk {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] (revDerivM : {X Y : Type} →\n [inst : SciLean.SemiInnerProductSpace K X] →\n [inst : SciLean.SemiInnerProductSpace K Y] → (X → m Y) → X → m (Y × (Y → m' X))) (HasAdjDiffM : {X Y : Type} →\n [inst : SciLean.SemiInnerProductSpace K X] → [inst : SciLean.SemiInnerProductSpace K Y] → (X → m Y) → Prop) (revDerivM_pure : ∀ {X Y : Type} [inst : SciLean.SemiInnerProductSpace K X] [inst_1 : SciLean.SemiInnerProductSpace K Y] (f : X → Y),\n SciLean.HasAdjDiff K f →\n (revDerivM fun x => pure (f x)) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n pure (ydf.1, fun dy => pure (ydf.2 dy))) (revDerivM_bind : ∀ {X Y Z : Type} [inst : SciLean.SemiInnerProductSpace K X] [inst_1 : SciLean.SemiInnerProductSpace K Y]\n [inst_2 : SciLean.SemiInnerProductSpace K Z] (f : Y → m Z) (g : X → m Y),\n HasAdjDiffM f →\n HasAdjDiffM g →\n (revDerivM fun x => g x >>= f) = fun x => do\n let ydg ← revDerivM g x\n let zdf ← revDerivM f ydg.1\n pure (zdf.1, fun dz => zdf.2 dz >>= ydg.2)) (revDerivM_pair : ∀ {X Y : Type} [inst : SciLean.SemiInnerProductSpace K X] [inst_1 : SciLean.SemiInnerProductSpace K Y] (f : X → m Y),\n HasAdjDiffM f →\n (revDerivM fun x => do\n let y ← f x\n pure (x, y)) =\n fun x => do\n let ydf ← revDerivM f x\n pure\n ((x, ydf.1), fun dxy => do\n let dx ← ydf.2 dxy.2\n pure (dxy.1 + dx))) (HasAdjDiffM_pure : ∀ {X Y : Type} [inst : SciLean.SemiInnerProductSpace K X] [inst_1 : SciLean.SemiInnerProductSpace K Y] (f : X → Y),\n SciLean.HasAdjDiff K f → HasAdjDiffM fun x => pure (f x)) (HasAdjDiffM_bind : ∀ {X Y Z : Type} [inst : SciLean.SemiInnerProductSpace K X] [inst_1 : SciLean.SemiInnerProductSpace K Y]\n [inst_2 : SciLean.SemiInnerProductSpace K Z] (f : Y → m Z) (g : X → m Y),\n HasAdjDiffM f → HasAdjDiffM g → HasAdjDiffM fun x => g x >>= f) (HasAdjDiffM_pair : ∀ {X Y : Type} [inst : SciLean.SemiInnerProductSpace K X] [inst_1 : SciLean.SemiInnerProductSpace K Y] (f : X → m Y),\n HasAdjDiffM f →\n HasAdjDiffM fun x => do\n let y ← f x\n pure (x, y)) : SciLean.RevDerivMonad K m m'"} +{"name":"Pure.pure.HasAdjDiffValM_rule","declaration":"theorem Pure.pure.HasAdjDiffValM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] (x : X) : SciLean.HasAdjDiffValM K (pure x)"} +{"name":"SciLean.RevDerivMonad.HasAdjDiffM_pair","declaration":"def SciLean.RevDerivMonad.HasAdjDiffM_pair {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] (f : X → m Y) (hf : SciLean.HasAdjDiffM K f) : SciLean.HasAdjDiffM K fun x => do\n let y ← f x\n pure (x, y)"} +{"name":"SciLean.RevDerivMonad.revDerivM_pair","declaration":"def SciLean.RevDerivMonad.revDerivM_pair {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] (f : X → m Y) (hf : SciLean.HasAdjDiffM K f) : (SciLean.revDerivM K fun x => do\n let y ← f x\n pure (x, y)) =\n fun x => do\n let ydf ← SciLean.revDerivM K f x\n pure\n ((x, ydf.1), fun dxy => do\n let dx ← ydf.2 dxy.2\n pure (dxy.1 + dx))"} +{"name":"SciLean.revDerivM.const_rule","declaration":"theorem SciLean.revDerivM.const_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (y : m Y) (hy : SciLean.HasAdjDiffValM K y) : (SciLean.revDerivM K fun x => y) = fun x => do\n let ydy ← SciLean.revDerivValM K y\n pure\n (ydy.1, fun dy' => do\n let __discr ← ydy.2 dy'\n let x : Unit := __discr\n pure 0)"} +{"name":"SciLean.HasAdjDiffM.let_rule","declaration":"theorem SciLean.HasAdjDiffM.let_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → m Z) (g : X → Y) (hf : SciLean.HasAdjDiffM K fun xy => f xy.1 xy.2) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiffM K fun x =>\n let y := g x;\n f x y"} +{"name":"SciLean.RevDerivMonad.revDerivM_pure","declaration":"def SciLean.RevDerivMonad.revDerivM_pure {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivM K fun x => pure (f x)) = fun x =>\n let ydf := SciLean.revDeriv K f x;\n pure (ydf.1, fun dy => pure (ydf.2 dy))"} +{"name":"SciLean.HasAdjDiffM.comp_rule","declaration":"theorem SciLean.HasAdjDiffM.comp_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : Y → m Z) (g : X → Y) (hf : SciLean.HasAdjDiffM K f) (hg : SciLean.HasAdjDiff K g) : SciLean.HasAdjDiffM K fun x => f (g x)"} +{"name":"SciLean.RevDerivMonad.HasAdjDiffM_pure","declaration":"def SciLean.RevDerivMonad.HasAdjDiffM_pure {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (hf : SciLean.HasAdjDiff K f) : SciLean.HasAdjDiffM K fun x => pure (f x)"} +{"name":"Bind.bind.arg_a0a1.revDerivM_rule","declaration":"theorem Bind.bind.arg_a0a1.revDerivM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (a0 : X → m Y) (a1 : X → Y → m Z) (ha0 : SciLean.HasAdjDiffM K a0) (ha1 : SciLean.HasAdjDiffM K fun xy => a1 xy.1 xy.2) : (SciLean.revDerivM K fun x => a0 x >>= a1 x) = fun x => do\n let ydg ← SciLean.revDerivM K a0 x\n let zdf ← SciLean.revDerivM K (fun xy => a1 xy.1 xy.2) (x, ydg.1)\n pure\n (zdf.1, fun dz => do\n let dxy ← zdf.2 dz\n let dx ← ydg.2 dxy.2\n pure (dxy.1 + dx))"} +{"name":"SciLean.RevDerivMonad","declaration":"class SciLean.RevDerivMonad (K : Type) [RCLike K] (m : Type → Type) (m' : outParam (Type → Type)) [Monad m] [Monad m'] : Type 1"} +{"name":"ite.arg_te.HasAdjDiffM_rule","declaration":"theorem ite.arg_te.HasAdjDiffM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → m Y) (e : X → m Y) (ht : SciLean.HasAdjDiffM K t) (he : SciLean.HasAdjDiffM K e) : SciLean.HasAdjDiffM K fun x => if c then t x else e x"} +{"name":"SciLean.revDerivM.comp_rule","declaration":"theorem SciLean.revDerivM.comp_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : Y → m Z) (g : X → Y) (hf : SciLean.HasAdjDiffM K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivM K fun x => f (g x)) = fun x =>\n let ydg := SciLean.revDeriv K g x;\n do\n let zdf ← SciLean.revDerivM K f ydg.1\n pure\n (zdf.1, fun dz => do\n let dy ← zdf.2 dz\n pure (ydg.2 dy))"} +{"name":"SciLean.HasAdjDiffValM","declaration":"def SciLean.HasAdjDiffValM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] (x : m X) : Prop"} +{"name":"SciLean.RevDerivMonad.HasAdjDiffM_bind","declaration":"def SciLean.RevDerivMonad.HasAdjDiffM_bind {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} {Z : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] [SciLean.SemiInnerProductSpace K Z] (f : Y → m Z) (g : X → m Y) (hf : SciLean.HasAdjDiffM K f) (hg : SciLean.HasAdjDiffM K g) : SciLean.HasAdjDiffM K fun x => g x >>= f"} +{"name":"Pure.pure.arg_a0.HasAdjDiffM_rule","declaration":"theorem Pure.pure.arg_a0.HasAdjDiffM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (a0 : X → Y) (ha0 : SciLean.HasAdjDiff K a0) : SciLean.HasAdjDiffM K fun x => pure (a0 x)"} +{"name":"Pure.pure.arg.revDerivValM_rule","declaration":"theorem Pure.pure.arg.revDerivValM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] (x : X) : SciLean.revDerivValM K (pure x) = pure (x, fun dy => pure 0)"} +{"name":"SciLean.RevDerivMonad.revDerivM","declaration":"def SciLean.RevDerivMonad.revDerivM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] (f : X → m Y) (x : X) : m (Y × (Y → m' X))"} +{"name":"Bind.bind.arg_a0a1.HasAdjDiffM_rule","declaration":"theorem Bind.bind.arg_a0a1.HasAdjDiffM_rule (K : Type) [RCLike K] {m : Type → Type} {m' : Type → Type} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (a0 : X → m Y) (a1 : X → Y → m Z) (ha0 : SciLean.HasAdjDiffM K a0) (ha1 : SciLean.HasAdjDiffM K fun xy => a1 xy.1 xy.2) : SciLean.HasAdjDiffM K fun x => a0 x >>= a1 x"} +{"name":"SciLean.revDerivValM","declaration":"def SciLean.revDerivValM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] {X : Type} [SciLean.SemiInnerProductSpace K X] (x : m X) : m (X × (X → m' Unit))"} +{"name":"SciLean.RevDerivMonad.revDerivM_bind","declaration":"def SciLean.RevDerivMonad.revDerivM_bind {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} {Z : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] [SciLean.SemiInnerProductSpace K Z] (f : Y → m Z) (g : X → m Y) (hf : SciLean.HasAdjDiffM K f) (hg : SciLean.HasAdjDiffM K g) : (SciLean.revDerivM K fun x => g x >>= f) = fun x => do\n let ydg ← SciLean.revDerivM K g x\n let zdf ← SciLean.revDerivM K f ydg.1\n pure (zdf.1, fun dz => zdf.2 dz >>= ydg.2)"} +{"name":"SciLean.RevDerivMonad.HasAdjDiffM","declaration":"def SciLean.RevDerivMonad.HasAdjDiffM (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [self : SciLean.RevDerivMonad K m m'] {X : Type} {Y : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] (f : X → m Y) : Prop"} +{"name":"SciLean.HasAdjDiffM.const_rule","declaration":"theorem SciLean.HasAdjDiffM.const_rule (K : Type) [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (y : m Y) (hy : SciLean.HasAdjDiffValM K y) : SciLean.HasAdjDiffM K fun x => y"} diff --git a/scilean-declarations/SciLean.Core.Monads.StateT.jsonl b/scilean-declarations/SciLean.Core.Monads.StateT.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3d92c0e66e21486d5d6850f1405eb118e2f3fd91 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Monads.StateT.jsonl @@ -0,0 +1,20 @@ +{"name":"MonadState.get.arg.fwdDerivValM_rule","declaration":"theorem MonadState.get.arg.fwdDerivValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] : SciLean.fwdDerivValM K get = get"} +{"name":"modify.arg_f.HasAdjDiffM_rule","declaration":"theorem modify.arg_f.HasAdjDiffM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → S → S) (ha0 : SciLean.HasAdjDiff K fun xs => f xs.1 xs.2) : SciLean.HasAdjDiffM K fun x => modify (f x)"} +{"name":"getThe.arg.revDerivValM_rule","declaration":"theorem getThe.arg.revDerivValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] : SciLean.revDerivValM K (getThe S) = do\n let __do_lift ← getThe S\n pure (__do_lift, fun ds => modifyThe S fun ds' => ds + ds')"} +{"name":"getThe.arg.CDifferentiableValM_rule","declaration":"theorem getThe.arg.CDifferentiableValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] : SciLean.CDifferentiableValM K (getThe S)"} +{"name":"MonadStateOf.set.arg_a0.revDerivM_rule","declaration":"theorem MonadStateOf.set.arg_a0.revDerivM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] {X : Type} [SciLean.SemiInnerProductSpace K X] (s : X → S) (ha0 : SciLean.HasAdjDiff K s) : (SciLean.revDerivM K fun x => set (s x)) = fun x =>\n let sds := SciLean.revDeriv K s x;\n do\n let __do_lift ← set sds.1\n pure\n (__do_lift, fun x => do\n let __do_lift ← get\n let dx : X := sds.2 __do_lift\n set 0\n pure dx)"} +{"name":"MonadStateOf.set.arg_a0.CDifferentiableM_rule","declaration":"theorem MonadStateOf.set.arg_a0.CDifferentiableM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] {X : Type} [SciLean.Vec K X] (s : X → S) (ha0 : SciLean.CDifferentiable K s) : SciLean.CDifferentiableM K fun x => set (s x)"} +{"name":"MonadState.get.arg.revDerivValM_rule","declaration":"theorem MonadState.get.arg.revDerivValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] : SciLean.revDerivValM K get = do\n let __do_lift ← get\n pure (__do_lift, fun ds => modify fun ds' => ds + ds')"} +{"name":"MonadStateOf.set.arg_a0.HasAdjDiffM_rule","declaration":"theorem MonadStateOf.set.arg_a0.HasAdjDiffM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] {X : Type} [SciLean.SemiInnerProductSpace K X] (s : X → S) (ha0 : SciLean.HasAdjDiff K s) : SciLean.HasAdjDiffM K fun x => set (s x)"} +{"name":"getThe.arg.HasAdjDiffValM_rule","declaration":"theorem getThe.arg.HasAdjDiffValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] : SciLean.HasAdjDiffValM K (getThe S)"} +{"name":"MonadState.get.arg.HasAdjDiffValM_rule","declaration":"theorem MonadState.get.arg.HasAdjDiffValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] : SciLean.HasAdjDiffValM K get"} +{"name":"modify.arg_f.revDerivM_rule","declaration":"theorem modify.arg_f.revDerivM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.SemiInnerProductSpace K S] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → S → S) (ha0 : SciLean.HasAdjDiff K fun xs => f xs.1 xs.2) : (SciLean.revDerivM K fun x => modify (f x)) = fun x => do\n let __do_lift ← get\n let sdf : S × (S → X × S) := SciLean.revDeriv K (fun xs => f xs.1 xs.2) (x, __do_lift)\n set sdf.1\n pure\n ((), fun x => do\n let __do_lift ← get\n let dxs : X × S := sdf.2 __do_lift\n set dxs.2\n pure dxs.1)"} +{"name":"MonadState.get.arg.CDifferentiableValM_rule","declaration":"theorem MonadState.get.arg.CDifferentiableValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] : SciLean.CDifferentiableValM K get"} +{"name":"MonadStateOf.set.arg_a0.fwdDerivM_rule","declaration":"theorem MonadStateOf.set.arg_a0.fwdDerivM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] {X : Type} [SciLean.Vec K X] (s : X → S) (ha0 : SciLean.CDifferentiable K s) : (SciLean.fwdDerivM K fun x => set (s x)) = fun x dx =>\n let sds := SciLean.fwdDeriv K s x dx;\n do\n set sds\n pure ((), ())"} +{"name":"getThe.arg.fwdDerivValM_rule","declaration":"theorem getThe.arg.fwdDerivValM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] : SciLean.fwdDerivValM K (getThe S) = getThe (S × S)"} +{"name":"SciLean.instFwdDerivMonadStateTProdInstMonadStateT","declaration":"instance SciLean.instFwdDerivMonadStateTProdInstMonadStateT {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] (S : Type) [SciLean.Vec K S] : SciLean.FwdDerivMonad K (StateT S m) (StateT (S × S) m')"} +{"name":"SciLean.instRevDerivMonadStateTInstMonadStateT","declaration":"instance SciLean.instRevDerivMonadStateTInstMonadStateT {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.RevDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] (S : Type) [SciLean.SemiInnerProductSpace K S] : SciLean.RevDerivMonad K (StateT S m) (StateT S m')"} +{"name":"modifyThe.arg_f.CDifferentiableM_rule","declaration":"theorem modifyThe.arg_f.CDifferentiableM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] {X : Type} [SciLean.Vec K X] (f : X → S → S) (ha0 : SciLean.CDifferentiable K fun xs => f xs.1 xs.2) : SciLean.CDifferentiableM K fun x => modifyThe S (f x)"} +{"name":"modify.arg_f.fwdDerivM_rule","declaration":"theorem modify.arg_f.fwdDerivM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] {X : Type} [SciLean.Vec K X] (f : X → S → S) (ha0 : SciLean.CDifferentiable K fun xs => f xs.1 xs.2) : (SciLean.fwdDerivM K fun x => modify (f x)) = fun x dx => do\n modify fun sds =>\n let sds := SciLean.fwdDeriv K (fun xs => f xs.1 xs.2) (x, sds.1) (dx, sds.2);\n sds\n pure ((), ())"} +{"name":"modify.arg_f.CDifferentiableM_rule","declaration":"theorem modify.arg_f.CDifferentiableM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] {X : Type} [SciLean.Vec K X] (f : X → S → S) (ha0 : SciLean.CDifferentiable K fun xs => f xs.1 xs.2) : SciLean.CDifferentiableM K fun x => modify (f x)"} +{"name":"modifyThe.arg_f.fwdDerivM_rule","declaration":"theorem modifyThe.arg_f.fwdDerivM_rule {K : Type} [RCLike K] {m : Type → Type} {m' : outParam (Type → Type)} [Monad m] [Monad m'] [SciLean.FwdDerivMonad K m m'] [LawfulMonad m] [LawfulMonad m'] {S : Type} [SciLean.Vec K S] {X : Type} [SciLean.Vec K X] (f : X → S → S) (ha0 : SciLean.CDifferentiable K fun xs => f xs.1 xs.2) : (SciLean.fwdDerivM K fun x => modifyThe S (f x)) = fun x dx => do\n modifyThe (S × S) fun sds =>\n let sds := SciLean.fwdDeriv K (fun xs => f xs.1 xs.2) (x, sds.1) (dx, sds.2);\n sds\n pure ((), ())"} diff --git a/scilean-declarations/SciLean.Core.Monads.jsonl b/scilean-declarations/SciLean.Core.Monads.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.Notation.CDeriv.jsonl b/scilean-declarations/SciLean.Core.Notation.CDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..d73efd41d3080ec9c9712496dfa378ead909eba6 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Notation.CDeriv.jsonl @@ -0,0 +1,11 @@ +{"name":"SciLean.Notation.unexpandScalarCDeriv","declaration":"def SciLean.Notation.unexpandScalarCDeriv : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.Notation.diffBinder","declaration":"def SciLean.Notation.diffBinder : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂_»","declaration":"def SciLean.Notation.«term∂_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂!_,_»","declaration":"def SciLean.Notation.«term∂!_,_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂!(_),_»","declaration":"def SciLean.Notation.«term∂!(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.diffBinderType","declaration":"def SciLean.Notation.diffBinderType : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂!_»","declaration":"def SciLean.Notation.«term∂!_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.unexpandCDeriv","declaration":"def SciLean.Notation.unexpandCDeriv : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.Notation.«term∂_,_»","declaration":"def SciLean.Notation.«term∂_,_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂(_),_»","declaration":"def SciLean.Notation.«term∂(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.diffBinderValue","declaration":"def SciLean.Notation.diffBinderValue : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Notation.FwdDeriv.jsonl b/scilean-declarations/SciLean.Core.Notation.FwdDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ca1355f16acc5cf1b9593c900046525180018acf --- /dev/null +++ b/scilean-declarations/SciLean.Core.Notation.FwdDeriv.jsonl @@ -0,0 +1,7 @@ +{"name":"SciLean.Notation.«term∂>_»","declaration":"def SciLean.Notation.«term∂>_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂>!_»","declaration":"def SciLean.Notation.«term∂>!_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.unexpandFwdDeriv","declaration":"def SciLean.Notation.unexpandFwdDeriv : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.Notation.«term∂>(_),_»","declaration":"def SciLean.Notation.«term∂>(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂>_,_»","declaration":"def SciLean.Notation.«term∂>_,_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂>!(_),_»","declaration":"def SciLean.Notation.«term∂>!(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∂>!_,_»","declaration":"def SciLean.Notation.«term∂>!_,_» : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Notation.Gradient.jsonl b/scilean-declarations/SciLean.Core.Notation.Gradient.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a1ebc993bae3a102b2d8222d4bcb2c69ea05d8ce --- /dev/null +++ b/scilean-declarations/SciLean.Core.Notation.Gradient.jsonl @@ -0,0 +1,8 @@ +{"name":"SciLean.Notation.«term∇!_,_»","declaration":"def SciLean.Notation.«term∇!_,_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.gradNotation1","declaration":"def SciLean.Notation.gradNotation1 : Lean.ParserDescr"} +{"name":"SciLean.Notation.unexpandScalarGradient","declaration":"def SciLean.Notation.unexpandScalarGradient : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.Notation.«term∇_,_»","declaration":"def SciLean.Notation.«term∇_,_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.unexpandGradient","declaration":"def SciLean.Notation.unexpandGradient : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.Notation.«term∇!(_),_»","declaration":"def SciLean.Notation.«term∇!(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∇(_),_»","declaration":"def SciLean.Notation.«term∇(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term∇!_»","declaration":"def SciLean.Notation.«term∇!_» : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Notation.RevCDeriv.jsonl b/scilean-declarations/SciLean.Core.Notation.RevCDeriv.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a2be46c190e67a951c7709517ed1d84a7fcd5edf --- /dev/null +++ b/scilean-declarations/SciLean.Core.Notation.RevCDeriv.jsonl @@ -0,0 +1,7 @@ +{"name":"SciLean.Notation.«term<∂_,_»","declaration":"def SciLean.Notation.«term<∂_,_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term<∂!_,_»","declaration":"def SciLean.Notation.«term<∂!_,_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term<∂_»","declaration":"def SciLean.Notation.«term<∂_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term<∂!_»","declaration":"def SciLean.Notation.«term<∂!_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term<∂!(_),_»","declaration":"def SciLean.Notation.«term<∂!(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.«term<∂(_),_»","declaration":"def SciLean.Notation.«term<∂(_),_» : Lean.ParserDescr"} +{"name":"SciLean.Notation.unexpandRevDeriv","declaration":"def SciLean.Notation.unexpandRevDeriv : Lean.PrettyPrinter.Unexpander"} diff --git a/scilean-declarations/SciLean.Core.Notation.jsonl b/scilean-declarations/SciLean.Core.Notation.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.NotationOverField.jsonl b/scilean-declarations/SciLean.Core.NotationOverField.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..4e7dbd03281d11c2ee7d999d332968180ac62f7e --- /dev/null +++ b/scilean-declarations/SciLean.Core.NotationOverField.jsonl @@ -0,0 +1,2 @@ +{"name":"SciLean.NotationOverField.«termDefaultScalar%»","declaration":"def SciLean.NotationOverField.«termDefaultScalar%» : Lean.ParserDescr"} +{"name":"SciLean.NotationOverField.commandSet_default_scalar_","declaration":"def SciLean.NotationOverField.commandSet_default_scalar_ : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Objects.FinVec.jsonl b/scilean-declarations/SciLean.Core.Objects.FinVec.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..254b237b4ef79ace2da8f9871ff3392337d6ce3e --- /dev/null +++ b/scilean-declarations/SciLean.Core.Objects.FinVec.jsonl @@ -0,0 +1,58 @@ +{"name":"SciLean.FinVec.to_dual","declaration":"def SciLean.FinVec.to_dual {ι : outParam (Type u_1)} {K : Type u_2} {X : Type u_3} [outParam (SciLean.IndexType ι)] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [self : SciLean.FinVec ι K X] {x : X} : SciLean.BasisDuality.toDual x = ∑ i, ℼ i x • ⅇ' i"} +{"name":"SciLean.instBasisDuality","declaration":"instance SciLean.instBasisDuality (K : Type u_1) [RCLike K] : SciLean.BasisDuality K"} +{"name":"SciLean.instBasisUnit","declaration":"instance SciLean.instBasisUnit (K : Type u_1) [RCLike K] : SciLean.Basis Unit K K"} +{"name":"SciLean.BasisDuality","declaration":"/-- This should somehow relate to raising and lowering indices but I forgot how.\n\nTODO: add explanation why this is useful\n-/\nclass SciLean.BasisDuality (X : Type u) : Type u"} +{"name":"SciLean.instOrthonormalBasisSumProdToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisSumProdToBasisToBasisToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToAddCommGroupToVecToSemiInnerProductSpaceToSemiHilbertToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToAddCommGroupToVecToSemiInnerProductSpaceToSemiHilbertInstInnerProdToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingToNonUnitalSeminormedCommRingToSeminormedCommRingToNormedCommRingToInnerToInner","declaration":"instance SciLean.instOrthonormalBasisSumProdToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisSumProdToBasisToBasisToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToAddCommGroupToVecToSemiInnerProductSpaceToSemiHilbertToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToAddCommGroupToVecToSemiInnerProductSpaceToSemiHilbertInstInnerProdToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocCommSemiringToNonUnitalNonAssocCommRingToNonUnitalCommRingToNonUnitalSeminormedCommRingToSeminormedCommRingToNormedCommRingToInnerToInner {ι : Type u_1} {K : Type u_2} {X : Type u_3} [RCLike K] {κ : Type u_4} {Y : Type u_5} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [SciLean.IndexType κ] [SciLean.LawfulIndexType κ] [DecidableEq κ] [SciLean.FinVec ι K X] [SciLean.OrthonormalBasis ι K X] [SciLean.FinVec κ K Y] [SciLean.OrthonormalBasis κ K Y] : SciLean.OrthonormalBasis (ι ⊕ κ) K (X × Y)"} +{"name":"SciLean.BasisDuality.mk","declaration":"ctor SciLean.BasisDuality.mk {X : Type u} (toDual : X → X) (fromDual : X → X) : SciLean.BasisDuality X"} +{"name":"SciLean.DualBasis.dualProj","declaration":"def SciLean.DualBasis.dualProj {ι : outParam (Type v)} {K : outParam (Type w)} {X : Type u} [self : SciLean.DualBasis ι K X] (i : ι) (x : X) : K"} +{"name":"SciLean.Basis.proj","declaration":"def SciLean.Basis.proj {ι : outParam (Type v)} {K : outParam (Type w)} {X : Type u} [self : SciLean.Basis ι K X] (i : ι) (x : X) : K"} +{"name":"SciLean.BasisDuality.fromDual","declaration":"def SciLean.BasisDuality.fromDual {X : Type u} [self : SciLean.BasisDuality X] (x : X) : X"} +{"name":"SciLean.instBasisDualityProd","declaration":"instance SciLean.instBasisDualityProd {X : Type u_1} {Y : Type u_2} [SciLean.BasisDuality X] [SciLean.BasisDuality Y] : SciLean.BasisDuality (X × Y)"} +{"name":"SciLean.instDualBasisSumProd","declaration":"instance SciLean.instDualBasisSumProd {X : Type u_1} {Y : Type u_2} {ι : Type u_3} {κ : Type u_4} {K : Type u_5} [SciLean.DualBasis ι K X] [SciLean.DualBasis κ K Y] [Zero X] [Zero Y] : SciLean.DualBasis (ι ⊕ κ) K (X × Y)"} +{"name":"SciLean.instGetElemTrue","declaration":"instance SciLean.instGetElemTrue {ι : Type u_1} {K : Type u_2} {X : Type u_3} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] : GetElem X ι K fun x x => True"} +{"name":"SciLean.DualBasis.mk","declaration":"ctor SciLean.DualBasis.mk {ι : outParam (Type v)} {K : outParam (Type w)} {X : Type u} (dualBasis : ι → X) (dualProj : ι → X → K) : SciLean.DualBasis ι K X"} +{"name":"SciLean.instBasisSumProd","declaration":"instance SciLean.instBasisSumProd {X : Type u_1} {Y : Type u_2} {ι : Type u_3} {κ : Type u_4} {K : Type u_5} [SciLean.Basis ι K X] [SciLean.Basis κ K Y] [Zero X] [Zero Y] : SciLean.Basis (ι ⊕ κ) K (X × Y)"} +{"name":"SciLean.inner_proj_dualProj","declaration":"theorem SciLean.inner_proj_dualProj {ι : Type u_1} {K : Type u_2} {X : Type u_2} : ∀ {x : SciLean.IndexType ι} [inst : SciLean.LawfulIndexType ι] [inst_1 : DecidableEq ι] [inst_2 : RCLike K]\n [inst_3 : SciLean.FinVec ι K X] (x_1 y : X), ⟪x_1, y⟫_K = ∑ i, ℼ i x_1 * ℼ' i y"} +{"name":"SciLean.basis_ext","declaration":"theorem SciLean.basis_ext {ι : Type u_1} {K : Type u_2} {X : Type u_3} : ∀ {x : SciLean.IndexType ι} [inst : SciLean.LawfulIndexType ι] [inst_1 : DecidableEq ι] [inst_2 : RCLike K]\n [inst_3 : SciLean.FinVec ι K X] (x_1 y : X), (∀ (i : ι), ⟪x_1, ⅇ i⟫_K = ⟪y, ⅇ i⟫_K) → x_1 = y"} +{"name":"SciLean.termⅇ_","declaration":"/-- `ⅇ i` is the i-th basis vector -/\ndef SciLean.termⅇ_ : Lean.ParserDescr"} +{"name":"SciLean.termℼ'_","declaration":"/-- `ℼ' i x` is projection of `x` onto i-th dual basis vector `ⅇ' i` -/\ndef SciLean.termℼ'_ : Lean.ParserDescr"} +{"name":"SciLean.FinVec.from_dual","declaration":"def SciLean.FinVec.from_dual {ι : outParam (Type u_1)} {K : Type u_2} {X : Type u_3} [outParam (SciLean.IndexType ι)] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [self : SciLean.FinVec ι K X] {x : X} : SciLean.BasisDuality.fromDual x = ∑ i, ℼ' i x • ⅇ i"} +{"name":"SciLean.instDualBasisForAll","declaration":"instance SciLean.instDualBasisForAll {ι : Type u_1} {K : Type u_2} [DecidableEq ι] [RCLike K] : SciLean.DualBasis ι K (ι → K)"} +{"name":"SciLean.instFinVecProdForAllInstIndexTypeProdInstLawfulIndexTypeProdInstIndexTypeProdInstDecidableEqProd","declaration":"instance SciLean.instFinVecProdForAllInstIndexTypeProdInstLawfulIndexTypeProdInstIndexTypeProdInstDecidableEqProd {ι : Type} {κ : Type} {K : Type u_1} {X : Type u_1} [SciLean.IndexType ι] [SciLean.IndexType κ] [SciLean.LawfulIndexType ι] [SciLean.LawfulIndexType κ] [DecidableEq ι] [DecidableEq κ] [RCLike K] [SciLean.FinVec κ K X] : SciLean.FinVec (ι × κ) K (ι → X)"} +{"name":"SciLean.dualBasis_ext","declaration":"theorem SciLean.dualBasis_ext {ι : Type u_1} {K : Type u_2} {X : Type u_3} : ∀ {x : SciLean.IndexType ι} [inst : SciLean.LawfulIndexType ι] [inst_1 : DecidableEq ι] [inst_2 : RCLike K]\n [inst_3 : SciLean.FinVec ι K X] (x_1 y : X), (∀ (i : ι), ⟪x_1, ⅇ' i⟫_K = ⟪y, ⅇ' i⟫_K) → x_1 = y"} +{"name":"SciLean.dualProj_dualBasis","declaration":"theorem SciLean.dualProj_dualBasis {ι : Type u_2} {K : Type u_1} {X : Type u_3} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] (i : ι) (j : ι) : ℼ' i ⅇ' j = if i = j then 1 else 0"} +{"name":"SciLean.instBasisDualityForAll","declaration":"instance SciLean.instBasisDualityForAll {ι : Type u_1} {X : Type u_2} [SciLean.BasisDuality X] : SciLean.BasisDuality (ι → X)"} +{"name":"SciLean.DualBasis.dualBasis","declaration":"def SciLean.DualBasis.dualBasis {ι : outParam (Type v)} {K : outParam (Type w)} {X : Type u} [self : SciLean.DualBasis ι K X] (i : ι) : X"} +{"name":"SciLean.FinVec.mk","declaration":"ctor SciLean.FinVec.mk {ι : outParam (Type u_1)} {K : Type u_2} {X : Type u_3} [outParam (SciLean.IndexType ι)] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [toSemiHilbert : SciLean.SemiHilbert K X] [toBasis : SciLean.Basis ι K X] [toDualBasis : SciLean.DualBasis ι K X] [toBasisDuality : SciLean.BasisDuality X] (is_basis : ∀ (x : X), x = ∑ i, ℼ i x • ⅇ i) (duality : ∀ (i j : ι), ⟪ⅇ i, ⅇ' j⟫_K = if i = j then 1 else 0) (to_dual : ∀ {x : X}, SciLean.BasisDuality.toDual x = ∑ i, ℼ i x • ⅇ' i) (from_dual : ∀ {x : X}, SciLean.BasisDuality.fromDual x = ∑ i, ℼ' i x • ⅇ i) : SciLean.FinVec ι K X"} +{"name":"SciLean.OrthonormalBasis","declaration":"class SciLean.OrthonormalBasis (ι : Type u_1) (K : Type u_2) (X : Type u_3) [Semiring K] [SciLean.Basis ι K X] [Inner K X] : Prop"} +{"name":"SciLean.instFinVecUnitInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnitInstDecidableEqPUnit","declaration":"instance SciLean.instFinVecUnitInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnitInstDecidableEqPUnit {K : Type u_1} [RCLike K] : SciLean.FinVec Unit K K"} +{"name":"SciLean.OrthonormalBasis.mk","declaration":"ctor SciLean.OrthonormalBasis.mk {ι : Type u_1} {K : Type u_2} {X : Type u_3} [Semiring K] [SciLean.Basis ι K X] [Inner K X] (is_orthogonal : ∀ (i j : ι), i ≠ j → ⟪ⅇ i, ⅇ j⟫_K = 0) (is_orthonormal : ∀ (i : ι), ⟪ⅇ i, ⅇ i⟫_K = 1) : SciLean.OrthonormalBasis ι K X"} +{"name":"SciLean.instBasisProdForAll","declaration":"instance SciLean.instBasisProdForAll {ι : Type u_1} {κ : Type u_2} {K : Type u_3} {X : Type u_4} [DecidableEq ι] [SciLean.Basis κ K X] [Zero X] : SciLean.Basis (ι × κ) K (ι → X)"} +{"name":"SciLean.instFinVecForAll","declaration":"instance SciLean.instFinVecForAll {ι : Type} {K : Type v} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] : SciLean.FinVec ι K (ι → K)"} +{"name":"SciLean.FinVec.duality","declaration":"def SciLean.FinVec.duality {ι : outParam (Type u_1)} {K : Type u_2} {X : Type u_3} [outParam (SciLean.IndexType ι)] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [self : SciLean.FinVec ι K X] (i : ι) (j : ι) : ⟪ⅇ i, ⅇ' j⟫_K = if i = j then 1 else 0"} +{"name":"SciLean.DualBasis","declaration":"/-- Dual basis of the space `X` over the field `K` indexed by `ι`\n\nThe class `FinVec ι K X` guarantees that any element `x : X` can be writtens as:\n```\n∑ i, proj i x • basis i\n```\nand that it is dual to the normal basis\n```\n⟪basis i, dualBasis j⟫[K] = if i=j then 1 else 0\n```\n-/\nclass SciLean.DualBasis (ι : outParam (Type v)) (K : outParam (Type w)) (X : Type u) : Type (max (max u v) w)"} +{"name":"SciLean.instDualBasisProdForAll","declaration":"instance SciLean.instDualBasisProdForAll {ι : Type u_1} {κ : Type u_2} {K : Type u_3} {X : Type u_4} [DecidableEq ι] [SciLean.DualBasis κ K X] [Zero X] : SciLean.DualBasis (ι × κ) K (ι → X)"} +{"name":"SciLean.Basis","declaration":"/-- Basis of the space `X` over the field `K` indexed by `ι`\n\nThe class `FinVec ι K X` guarantees that any element `x : X` can be writtens as:\n```\n∑ i, proj i x • basis i\n```\n-/\nclass SciLean.Basis (ι : outParam (Type v)) (K : outParam (Type w)) (X : Type u) : Type (max (max u v) w)"} +{"name":"SciLean.Basis.basis","declaration":"def SciLean.Basis.basis {ι : outParam (Type v)} {K : outParam (Type w)} {X : Type u} [self : SciLean.Basis ι K X] (i : ι) : X"} +{"name":"SciLean.instBasisForAll","declaration":"instance SciLean.instBasisForAll {ι : Type u_1} {K : Type u_2} [DecidableEq ι] [RCLike K] : SciLean.Basis ι K (ι → K)"} +{"name":"SciLean.termⅇ'_","declaration":"/-- `ⅇ' i` is the i-th dual basis vector -/\ndef SciLean.termⅇ'_ : Lean.ParserDescr"} +{"name":"SciLean.inner_dualBasis_proj","declaration":"theorem SciLean.inner_dualBasis_proj {ι : Type u_3} {K : Type u_1} {X : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] (i : ι) (x : X) : ⟪x, ⅇ' i⟫_K = ℼ i x"} +{"name":"SciLean.Basis.mk","declaration":"ctor SciLean.Basis.mk {ι : outParam (Type v)} {K : outParam (Type w)} {X : Type u} (basis : ι → X) (proj : ι → X → K) : SciLean.Basis ι K X"} +{"name":"SciLean.«termⅇ'[_]_»","declaration":"/-- `ⅇ'[X] i` is the i-th dual basis vector of type `X` -/\ndef SciLean.«termⅇ'[_]_» : Lean.ParserDescr"} +{"name":"SciLean.termℼ_","declaration":"/-- `ℼ i x` is projection of `x` onto i-th basis vector `ⅇ i` -/\ndef SciLean.termℼ_ : Lean.ParserDescr"} +{"name":"SciLean.«termⅇ[_]_»","declaration":"/-- `ⅇ[X] i` is the i-th basis vector of type `X` -/\ndef SciLean.«termⅇ[_]_» : Lean.ParserDescr"} +{"name":"SciLean.FinVec.is_basis","declaration":"def SciLean.FinVec.is_basis {ι : outParam (Type u_1)} {K : Type u_2} {X : Type u_3} [outParam (SciLean.IndexType ι)] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [self : SciLean.FinVec ι K X] (x : X) : x = ∑ i, ℼ i x • ⅇ i"} +{"name":"SciLean.instDualBasisUnit","declaration":"instance SciLean.instDualBasisUnit (K : Type u_1) [RCLike K] : SciLean.DualBasis Unit K K"} +{"name":"SciLean.instOrthonormalBasisProdForAllToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisProdForAllToInnerInstSemiInnerProductSpaceForAllToSemiInnerProductSpaceToSemiHilbert","declaration":"instance SciLean.instOrthonormalBasisProdForAllToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisProdForAllToInnerInstSemiInnerProductSpaceForAllToSemiInnerProductSpaceToSemiHilbert {ι : Type u_1} {K : Type u_2} {X : Type u_3} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] {κ : Type u_4} [SciLean.IndexType ι] [SciLean.IndexType κ] [SciLean.LawfulIndexType ι] [SciLean.LawfulIndexType κ] [Zero X] [SciLean.Basis κ K X] [SciLean.OrthonormalBasis κ K X] : SciLean.OrthonormalBasis (ι × κ) K (ι → X)"} +{"name":"SciLean.instFinVecSumProdInstIndexTypeSumInstLawfulIndexTypeSumInstIndexTypeSumInstDecidableEqSum_1","declaration":"instance SciLean.instFinVecSumProdInstIndexTypeSumInstLawfulIndexTypeSumInstIndexTypeSumInstDecidableEqSum_1 {ι : Type u_1} {κ : Type u_2} {K : Type u_3} {X : Type (max u_4 u_5)} {Y : Type (max u_4 u_5)} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [SciLean.IndexType κ] [SciLean.LawfulIndexType κ] [DecidableEq κ] [RCLike K] [SciLean.FinVec ι K X] [SciLean.FinVec κ K Y] : SciLean.FinVec (ι ⊕ κ) K (X × Y)"} +{"name":"SciLean.proj_zero","declaration":"theorem SciLean.proj_zero {ι : Type u_2} {K : Type u_1} {X : Type u_3} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] (i : ι) : ℼ i 0 = 0"} +{"name":"SciLean.proj_basis","declaration":"theorem SciLean.proj_basis {ι : Type u_2} {K : Type u_1} {X : Type u_3} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] (i : ι) (j : ι) : ℼ i ⅇ j = if i = j then 1 else 0"} +{"name":"SciLean.instOrthonormalBasisUnitToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisUnitInstInner","declaration":"instance SciLean.instOrthonormalBasisUnitToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisUnitInstInner {K : Type u_1} [RCLike K] : SciLean.OrthonormalBasis Unit K K"} +{"name":"SciLean.inner_basis_dualBasis","declaration":"theorem SciLean.inner_basis_dualBasis {ι : Type u_3} {K : Type u_1} {X : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] (i : ι) (j : ι) : ⟪ⅇ i, ⅇ' j⟫_K = if i = j then 1 else 0"} +{"name":"SciLean.FinVec","declaration":"/-- -/\nclass SciLean.FinVec (ι : outParam (Type u_1)) (K : Type u_2) (X : Type u_3) [outParam (SciLean.IndexType ι)] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] : Type (max (max u_1 u_2) u_3)"} +{"name":"SciLean.BasisDuality.toDual","declaration":"def SciLean.BasisDuality.toDual {X : Type u} [self : SciLean.BasisDuality X] (x : X) : X"} +{"name":"SciLean.OrthonormalBasis.is_orthogonal","declaration":"def SciLean.OrthonormalBasis.is_orthogonal {ι : Type u_1} {K : Type u_2} {X : Type u_3} [Semiring K] [SciLean.Basis ι K X] [Inner K X] [self : SciLean.OrthonormalBasis ι K X] (i : ι) (j : ι) : i ≠ j → ⟪ⅇ i, ⅇ j⟫_K = 0"} +{"name":"SciLean.instGetElemNatLtInstLTNatCard","declaration":"instance SciLean.instGetElemNatLtInstLTNatCard {ι : Type u_1} {K : Type u_2} {X : Type u_3} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] : GetElem X ℕ K fun x i => i < SciLean.IndexType.card ι"} +{"name":"SciLean.OrthonormalBasis.is_orthonormal","declaration":"def SciLean.OrthonormalBasis.is_orthonormal {ι : Type u_1} {K : Type u_2} {X : Type u_3} [Semiring K] [SciLean.Basis ι K X] [Inner K X] [self : SciLean.OrthonormalBasis ι K X] (i : ι) : ⟪ⅇ i, ⅇ i⟫_K = 1"} +{"name":"SciLean.inner_dualBasis_basis","declaration":"theorem SciLean.inner_dualBasis_basis {ι : Type u_3} {K : Type u_1} {X : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] (i : ι) (j : ι) : ⟪ⅇ' i, ⅇ j⟫_K = if i = j then 1 else 0"} +{"name":"SciLean.inner_basis_dualProj","declaration":"theorem SciLean.inner_basis_dualProj {ι : Type u_3} {K : Type u_1} {X : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] [RCLike K] [SciLean.FinVec ι K X] (i : ι) (x : X) : ⟪x, ⅇ i⟫_K = ℼ' i x"} diff --git a/scilean-declarations/SciLean.Core.Objects.IsReal.jsonl b/scilean-declarations/SciLean.Core.Objects.IsReal.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..83ea7477dd0009cceed22a731568d1d488b38976 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Objects.IsReal.jsonl @@ -0,0 +1,28 @@ +{"name":"SciLean.ExtendedReal","declaration":"inductive SciLean.ExtendedReal (R : Type u_1) : Type u_1"} +{"name":"SciLean.IsReal.is_real","declaration":"def SciLean.IsReal.is_real {R : semiOutParam (Type u_1)} [self : SciLean.IsReal R] (x : R) : RCLike.im x = 0"} +{"name":"SciLean.ComputableDist","declaration":"class SciLean.ComputableDist (R : Type u_1) (X : Type u_2) [SciLean.IsReal R] [Dist X] : Type (max u_1 u_2)"} +{"name":"SciLean.ComputableNorm.is_norm","declaration":"def SciLean.ComputableNorm.is_norm {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [Norm X] [self : SciLean.ComputableNorm R X] (x : X) : RCLike.re (SciLean.cnorm x) = ‖x‖"} +{"name":"SciLean.NotationOverField.«term‖_‖»","declaration":"/-- Norm, usually `‖x‖ = (∑ i, ‖x i‖^p)^(1/p)` for some `p`\n\nWARRNING: This is override for normal norm notation that provides computable version of norm if available.\n-/\ndef SciLean.NotationOverField.«term‖_‖» : Lean.ParserDescr"} +{"name":"SciLean.instZeroExtendedReal","declaration":"instance SciLean.instZeroExtendedReal (R : Type u_1) [SciLean.IsReal R] : Zero (SciLean.ExtendedReal R)"} +{"name":"SciLean.ComputableNorm.cnorm","declaration":"def SciLean.ComputableNorm.cnorm {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [Norm X] [self : SciLean.ComputableNorm R X] : X → R"} +{"name":"SciLean.instIsRealReal","declaration":"instance SciLean.instIsRealReal : SciLean.IsReal ℝ"} +{"name":"SciLean.IsReal.mk","declaration":"ctor SciLean.IsReal.mk {R : semiOutParam (Type u_1)} [toRCLike : RCLike R] (is_real : ∀ (x : R), RCLike.im x = 0) : SciLean.IsReal R"} +{"name":"SciLean.ComputableNorm.mk","declaration":"ctor SciLean.ComputableNorm.mk {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [Norm X] (cnorm : X → R) (is_norm : ∀ (x : X), RCLike.re (cnorm x) = ‖x‖) : SciLean.ComputableNorm R X"} +{"name":"SciLean.instBotExtendedReal","declaration":"instance SciLean.instBotExtendedReal (R : Type u_1) [SciLean.IsReal R] : Bot (SciLean.ExtendedReal R)"} +{"name":"SciLean.ComputableEDist.is_edist","declaration":"def SciLean.ComputableEDist.is_edist {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [EDist X] [self : SciLean.ComputableEDist R X] (x : X) (y : X) : SciLean.ExtendedReal.toEReal R (SciLean.cedist x y) = ↑(edist x y)"} +{"name":"SciLean.instOrdExtendedReal","declaration":"instance SciLean.instOrdExtendedReal (R : Type u_1) [SciLean.IsReal R] [Ord R] : Ord (SciLean.ExtendedReal R)"} +{"name":"SciLean.ExtendedReal.toEReal","declaration":"def SciLean.ExtendedReal.toEReal (R : Type u_1) [SciLean.IsReal R] (x : SciLean.ExtendedReal R) : EReal"} +{"name":"SciLean.ComputableDist.mk","declaration":"ctor SciLean.ComputableDist.mk {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [Dist X] (cdist : X → X → R) (is_dist : ∀ (x y : X), RCLike.re (cdist x y) = dist x y) : SciLean.ComputableDist R X"} +{"name":"SciLean.ComputableEDist","declaration":"class SciLean.ComputableEDist (R : Type u_1) (X : Type u_2) [SciLean.IsReal R] [EDist X] : Type (max u_1 u_2)"} +{"name":"SciLean.instTopExtendedReal","declaration":"instance SciLean.instTopExtendedReal (R : Type u_1) [SciLean.IsReal R] : Top (SciLean.ExtendedReal R)"} +{"name":"SciLean.ComputableDist.is_dist","declaration":"def SciLean.ComputableDist.is_dist {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [Dist X] [self : SciLean.ComputableDist R X] (x : X) (y : X) : RCLike.re (SciLean.cdist x y) = dist x y"} +{"name":"SciLean.ExtendedReal.negInf","declaration":"ctor SciLean.ExtendedReal.negInf {R : Type u_1} : SciLean.ExtendedReal R"} +{"name":"SciLean.IsReal","declaration":"class SciLean.IsReal (R : semiOutParam (Type u_1)) : Type u_1"} +{"name":"SciLean.instOneExtendedReal","declaration":"instance SciLean.instOneExtendedReal (R : Type u_1) [SciLean.IsReal R] : One (SciLean.ExtendedReal R)"} +{"name":"SciLean.ComputableDist.cdist","declaration":"def SciLean.ComputableDist.cdist {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [Dist X] [self : SciLean.ComputableDist R X] : X → X → R"} +{"name":"SciLean.instNegExtendedReal","declaration":"instance SciLean.instNegExtendedReal (R : Type u_1) [SciLean.IsReal R] : Neg (SciLean.ExtendedReal R)"} +{"name":"SciLean.ExtendedReal.posInf","declaration":"ctor SciLean.ExtendedReal.posInf {R : Type u_1} : SciLean.ExtendedReal R"} +{"name":"SciLean.ExtendedReal.val","declaration":"ctor SciLean.ExtendedReal.val {R : Type u_1} (r : R) : SciLean.ExtendedReal R"} +{"name":"SciLean.ComputableEDist.cedist","declaration":"def SciLean.ComputableEDist.cedist {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [EDist X] [self : SciLean.ComputableEDist R X] : X → X → SciLean.ExtendedReal R"} +{"name":"SciLean.ComputableEDist.mk","declaration":"ctor SciLean.ComputableEDist.mk {R : Type u_1} {X : Type u_2} [SciLean.IsReal R] [EDist X] (cedist : X → X → SciLean.ExtendedReal R) (is_edist : ∀ (x y : X), SciLean.ExtendedReal.toEReal R (cedist x y) = ↑(edist x y)) : SciLean.ComputableEDist R X"} +{"name":"SciLean.ComputableNorm","declaration":"class SciLean.ComputableNorm (R : Type u_1) (X : Type u_2) [SciLean.IsReal R] [Norm X] : Type (max u_1 u_2)"} diff --git a/scilean-declarations/SciLean.Core.Objects.IsomorphicType.RealToFloat.jsonl b/scilean-declarations/SciLean.Core.Objects.IsomorphicType.RealToFloat.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f4b3a707c9442529e96cf395d4e6b8eb0d5eb7a0 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Objects.IsomorphicType.RealToFloat.jsonl @@ -0,0 +1,9 @@ +{"name":"SciLean.isntIsomorphicTypeId","declaration":"instance SciLean.isntIsomorphicTypeId {α : Sort u} : SciLean.IsomorphicType `RealToFloat α α"} +{"name":"SciLean.instIsomorphicTypeMkStr1ForAllForAll","declaration":"instance SciLean.instIsomorphicTypeMkStr1ForAllForAll {α : Type u_1} {α' : Type u_2} {β : Type u_3} {β' : Type u_4} [SciLean.IsomorphicType `RealToFloat α α'] [SciLean.IsomorphicType `RealToFloat β β'] : SciLean.IsomorphicType `RealToFloat (α → β) (α' → β')"} +{"name":"SciLean.instIsomorphicTypeRealToFloatProd","declaration":"instance SciLean.instIsomorphicTypeRealToFloatProd {α : Type u_1} {α' : Type u_2} {β : Type u_3} {β' : Type u_4} [SciLean.IsomorphicType `RealToFloat α α'] [SciLean.IsomorphicType `RealToFloat β β'] : SciLean.IsomorphicType `RealToFloat (α × β) (α' × β')"} +{"name":"SciLean.instIsomorphicTypeMkStr1RealFloat","declaration":"def SciLean.instIsomorphicTypeMkStr1RealFloat : SciLean.IsomorphicType `RealToFloat ℝ Float"} +{"name":"SciLean.floatToReal","declaration":"def SciLean.floatToReal (x : Float) : ℝ"} +{"name":"SciLean.realFloatEquiv","declaration":"/-- This axiom is obviously contradictory. We use it to compile programs that were designed only for reals or proper fields. With this axiom we can plug `Float` to function that would only accept types that have instance of `RCLike`\n\n-/\naxiom SciLean.realFloatEquiv : ℝ ≃ Float"} +{"name":"SciLean.isomorphicType_equiv_id","declaration":"theorem SciLean.isomorphicType_equiv_id {α : Type u_1} (a : α) : (SciLean.IsomorphicType.equiv `RealToFloat) a = a"} +{"name":"SciLean.realToFloat","declaration":"def SciLean.realToFloat (x : ℝ) : Float"} +{"name":"SciLean.instIsomorphicTypeMkStr1FloatReal","declaration":"def SciLean.instIsomorphicTypeMkStr1FloatReal : SciLean.IsomorphicType `FloatToReal Float ℝ"} diff --git a/scilean-declarations/SciLean.Core.Objects.IsomorphicType.jsonl b/scilean-declarations/SciLean.Core.Objects.IsomorphicType.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..8e45427e500aa11e50dc233a362170fd0bcebf25 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Objects.IsomorphicType.jsonl @@ -0,0 +1,11 @@ +{"name":"SciLean.instIsomorphicTypeForAllForAll","declaration":"instance SciLean.instIsomorphicTypeForAllForAll {α : Type u_1} {β : Type u_2} {α' : outParam (Type u_3)} {β' : outParam (Type u_4)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] : SciLean.IsomorphicType tag (α → β) (α' → β')"} +{"name":"SciLean.isomorph.ext","declaration":"theorem SciLean.isomorph.ext {α : Type u_2} {α' : outParam (Type u_1)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] (a : α) (b : α) : (SciLean.IsomorphicType.equiv tag) a = (SciLean.IsomorphicType.equiv tag) b → a = b"} +{"name":"SciLean.invIsomorph","declaration":"def SciLean.invIsomorph {α : Type u_1} {β : Type u_2} {α' : outParam (Type u_3)} {β' : outParam (Type u_4)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] (f : α' → β') (a : α) : β"} +{"name":"SciLean.IsomorphicType","declaration":"/-- Type `α` is isomorphic to `α'`.\n\nThink about this class such that for each `tag` we get partial function `Type → Type`.\n\nFor example we have\n\n-/\nclass SciLean.IsomorphicType (tag : Lean.Name) (α : Sort u_1) (α' : outParam (Sort u_2)) : Sort (max (max 1 u_1) u_2)"} +{"name":"SciLean.isomorph.funext","declaration":"theorem SciLean.isomorph.funext {α : Type u_4} {α' : outParam (Type u_3)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] {β : Type u_1} {β' : Type u_2} [SciLean.IsomorphicType tag β β'] (f : α → β) (g : α → β) : SciLean.isomorph tag f = SciLean.isomorph tag g → f = g"} +{"name":"SciLean.IsomorphicType.mk","declaration":"ctor SciLean.IsomorphicType.mk {tag : Lean.Name} {α : Sort u_1} {α' : outParam (Sort u_2)} (equiv : α ≃ α') : SciLean.IsomorphicType tag α α'"} +{"name":"SciLean.isomorph","declaration":"def SciLean.isomorph {α : Type u_1} {β : Type u_2} {α' : outParam (Type u_3)} {β' : outParam (Type u_4)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] (f : α → β) (a' : α') : β'"} +{"name":"SciLean.instIsomorphicTypeProdProd","declaration":"instance SciLean.instIsomorphicTypeProdProd {α : Type u_1} {β : Type u_2} {α' : outParam (Type u_3)} {β' : outParam (Type u_4)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] : SciLean.IsomorphicType tag (α × β) (α' × β')"} +{"name":"SciLean.isomorph.app","declaration":"theorem SciLean.isomorph.app {α : Type u_3} {β : Type u_2} {α' : outParam (Type u_4)} {β' : outParam (Type u_1)} (tag : Lean.Name) [SciLean.IsomorphicType tag α α'] [SciLean.IsomorphicType tag β β'] (f : α → β) (x : α) : (SciLean.IsomorphicType.equiv tag) (f x) = SciLean.isomorph tag f ((SciLean.IsomorphicType.equiv tag) x)"} +{"name":"SciLean.instIsomorphicType","declaration":"instance SciLean.instIsomorphicType (tag : Lean.Name) (P : Prop) : SciLean.IsomorphicType tag P P"} +{"name":"SciLean.IsomorphicType.equiv","declaration":"def SciLean.IsomorphicType.equiv (tag : Lean.Name) {α : Sort u_1} {α' : outParam (Sort u_2)} [self : SciLean.IsomorphicType tag α α'] : α ≃ α'"} diff --git a/scilean-declarations/SciLean.Core.Objects.Scalar.jsonl b/scilean-declarations/SciLean.Core.Objects.Scalar.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e142ae9216d6e2d985ed8284a3dfef3f5ece08f7 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Objects.Scalar.jsonl @@ -0,0 +1,58 @@ +{"name":"SciLean.Scalar.oftoReal","declaration":"theorem SciLean.Scalar.oftoReal {R : Type u_1} [SciLean.RealScalar R] (x : R) : SciLean.Scalar.ofReal R (SciLean.Scalar.toReal R x) = x"} +{"name":"SciLean.Scalar.real","declaration":"def SciLean.Scalar.real {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : R"} +{"name":"SciLean.Scalar.tan","declaration":"def SciLean.Scalar.tan {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : K"} +{"name":"SciLean.Scalar.make","declaration":"def SciLean.Scalar.make {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] : R → R → K"} +{"name":"SciLean.RealScalar.asin","declaration":"def SciLean.RealScalar.asin {R : semiOutParam (Type u_1)} [self : SciLean.RealScalar R] (x : R) : R"} +{"name":"SciLean.RealScalar","declaration":"/-- `R` behaves as real numbers\n\nThis class allows us to write code independent of particular implementation of real numbers.\n\nSee `Scalar` for motivation for this class.\n-/\nclass SciLean.RealScalar (R : semiOutParam (Type u_1)) : Type u_1"} +{"name":"SciLean.Scalar.sin_def","declaration":"def SciLean.Scalar.sin_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toComplex (SciLean.Scalar.sin x) = Complex.sin (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.Scalar.pow","declaration":"def SciLean.Scalar.pow {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) (y : K) : K"} +{"name":"SciLean.scalar_sqrt_one","declaration":"theorem SciLean.scalar_sqrt_one {R : Type u_1} [SciLean.RealScalar R] : SciLean.Scalar.sqrt 1 = 1"} +{"name":"SciLean.Scalar.real_def","declaration":"def SciLean.Scalar.real_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toReal K (SciLean.Scalar.real x) = RCLike.re (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.Scalar.exp_def","declaration":"def SciLean.Scalar.exp_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toComplex (SciLean.Scalar.exp x) = Complex.exp (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.Scalar.exp","declaration":"def SciLean.Scalar.exp {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : K"} +{"name":"SciLean.Scalar.tanh_def","declaration":"def SciLean.Scalar.tanh_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toComplex (SciLean.Scalar.tanh x) = Complex.tanh (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.Scalar.oftoENNReal","declaration":"theorem SciLean.Scalar.oftoENNReal {R : Type u_1} [SciLean.RealScalar R] (x : R) : ↑↑x = max x 0"} +{"name":"SciLean.Scalar.toENNReal","declaration":"def SciLean.Scalar.toENNReal {R : Type u_1} [SciLean.RealScalar R] (x : R) : ENNReal"} +{"name":"SciLean.Scalar.imag","declaration":"def SciLean.Scalar.imag {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : R"} +{"name":"SciLean.Scalar.abs_def","declaration":"def SciLean.Scalar.abs_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toReal K (SciLean.Scalar.abs x) = Complex.abs (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.scalar_max_one_zero","declaration":"theorem SciLean.scalar_max_one_zero {R : Type u_1} [SciLean.RealScalar R] : max 1 0 = 1"} +{"name":"SciLean.Scalar.make_def","declaration":"def SciLean.Scalar.make_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : R) (y : R) : if ∀ (y : K), RCLike.im y = 0 then\n SciLean.Scalar.toComplex (SciLean.Scalar.make x y) = { re := SciLean.Scalar.toReal K x, im := 0 }\nelse\n SciLean.Scalar.toComplex (SciLean.Scalar.make x y) =\n { re := SciLean.Scalar.toReal K x, im := SciLean.Scalar.toReal K y }"} +{"name":"SciLean.Scalar.ofComplex","declaration":"def SciLean.Scalar.ofComplex {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] : ℂ → K"} +{"name":"SciLean.RealScalar.mk","declaration":"ctor SciLean.RealScalar.mk {R : semiOutParam (Type u_1)} [toScalar : SciLean.Scalar R R] [toMin : Min R] [toMax : Max R] [toOrd : Ord R] (le_total : ∀ (a b : R), a ≤ b ∨ b ≤ a) (decidableLE : DecidableRel fun x x_1 => x ≤ x_1) (decidableEq : DecidableEq R) (decidableLT : DecidableRel fun x x_1 => x < x_1) (min_def : autoParam (∀ (a b : R), min a b = if a ≤ b then a else b) _auto✝) (max_def : autoParam (∀ (a b : R), max a b = if a ≤ b then b else a) _auto✝) (compare_eq_compareOfLessAndEq : autoParam (∀ (a b : R), compare a b = compareOfLessAndEq a b) _auto✝) (is_real : ∀ (x : R), RCLike.im x = 0) (asin : R → R) (asin_def : ∀ (x : R), SciLean.Scalar.toReal R (asin x) = Real.arcsin (SciLean.Scalar.toReal R x)) (acos : R → R) (acos_def : ∀ (x : R), SciLean.Scalar.toReal R (acos x) = Real.arccos (SciLean.Scalar.toReal R x)) (atan : R → R) (atan_def : ∀ (x : R), SciLean.Scalar.toReal R (atan x) = Real.arctan (SciLean.Scalar.toReal R x)) : SciLean.RealScalar R"} +{"name":"SciLean.scalar_abs_one","declaration":"theorem SciLean.scalar_abs_one {R : Type u_1} [SciLean.RealScalar R] : SciLean.Scalar.abs 1 = 1"} +{"name":"SciLean.Scalar.imag_def","declaration":"def SciLean.Scalar.imag_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toReal K (SciLean.Scalar.imag x) = RCLike.im (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.RealScalar.atan","declaration":"def SciLean.RealScalar.atan {R : semiOutParam (Type u_1)} [self : SciLean.RealScalar R] (x : R) : R"} +{"name":"SciLean.Scalar.ofENNReal","declaration":"def SciLean.Scalar.ofENNReal {R : Type u_1} [SciLean.RealScalar R] (x : ENNReal) : R"} +{"name":"SciLean.Scalar.abs","declaration":"def SciLean.Scalar.abs {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : R"} +{"name":"SciLean.Scalar.log_def","declaration":"def SciLean.Scalar.log_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toComplex (SciLean.Scalar.log x) = Complex.log (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.Scalar.pow_def","declaration":"def SciLean.Scalar.pow_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) (y : K) : if ∀ (z : K), RCLike.im z = 0 then\n SciLean.Scalar.toReal K (SciLean.Scalar.real (SciLean.Scalar.pow x y)) =\n (SciLean.Scalar.toComplex x ^ SciLean.Scalar.toComplex y).re\nelse SciLean.Scalar.toComplex (SciLean.Scalar.pow x y) = SciLean.Scalar.toComplex x ^ SciLean.Scalar.toComplex y"} +{"name":"SciLean.conj_for_real_scalar","declaration":"theorem SciLean.conj_for_real_scalar {R : Type u_1} [SciLean.RealScalar R] (r : R) : (starRingEnd R) r = r"} +{"name":"SciLean.scalar_abs_neg","declaration":"theorem SciLean.scalar_abs_neg {R : Type u_1} [SciLean.RealScalar R] (r : R) : SciLean.Scalar.abs (-r) = SciLean.Scalar.abs r"} +{"name":"SciLean.scalar_sqrt_zero","declaration":"theorem SciLean.scalar_sqrt_zero {R : Type u_1} [SciLean.RealScalar R] : SciLean.Scalar.sqrt 0 = 0"} +{"name":"SciLean.Scalar.toComplex","declaration":"def SciLean.Scalar.toComplex {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] : K → ℂ"} +{"name":"SciLean.Scalar.toReal","declaration":"def SciLean.Scalar.toReal {R : outParam (Type u_1)} (K : semiOutParam (Type u_2)) [self : SciLean.Scalar R K] : R → ℝ"} +{"name":"SciLean.scalar_min_zero_one","declaration":"theorem SciLean.scalar_min_zero_one {R : Type u_1} [SciLean.RealScalar R] : min 0 1 = 0"} +{"name":"SciLean.instRealScalarReal","declaration":"instance SciLean.instRealScalarReal : SciLean.RealScalar ℝ"} +{"name":"SciLean.Scalar.log","declaration":"def SciLean.Scalar.log {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : K"} +{"name":"SciLean.scalar_abs_zero","declaration":"theorem SciLean.scalar_abs_zero {R : Type u_1} [SciLean.RealScalar R] : SciLean.Scalar.abs 0 = 0"} +{"name":"SciLean.Scalar.cos","declaration":"def SciLean.Scalar.cos {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : K"} +{"name":"SciLean.scalar_div_one","declaration":"theorem SciLean.scalar_div_one {R : Type u_1} [SciLean.RealScalar R] (x : R) : x / 1 = x"} +{"name":"SciLean.instScalarRealComplex","declaration":"instance SciLean.instScalarRealComplex : SciLean.Scalar ℝ ℂ"} +{"name":"SciLean.Scalar.sqrt_def","declaration":"def SciLean.Scalar.sqrt_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : if ∀ (y : K), RCLike.im y = 0 then\n RCLike.re (SciLean.Scalar.toComplex (SciLean.Scalar.sqrt x)) = √(RCLike.re (SciLean.Scalar.toComplex x))\nelse SciLean.Scalar.toComplex (SciLean.Scalar.sqrt x) = Complex.cpow (SciLean.Scalar.toComplex x) (1 / 2)"} +{"name":"SciLean.Scalar.sin","declaration":"def SciLean.Scalar.sin {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : K"} +{"name":"SciLean.RealScalar.pi","declaration":"def SciLean.RealScalar.pi {R : Type u_1} [SciLean.RealScalar R] : R"} +{"name":"SciLean.Scalar.cos_def","declaration":"def SciLean.Scalar.cos_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toComplex (SciLean.Scalar.cos x) = Complex.cos (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.RealScalar.is_real","declaration":"def SciLean.RealScalar.is_real {R : semiOutParam (Type u_1)} [self : SciLean.RealScalar R] (x : R) : RCLike.im x = 0"} +{"name":"SciLean.Scalar","declaration":"/-- `K` are real or complex numbers over real numbers `R`\n\nThis class allows us to write code independent of particular implementation of real or complex numbers.\n\nThe main motivation for this class is to treat floating point numbers as real numbers but to minimize the impact of such unsoundness. We can write code with valid proofs and only at the last step before compilation provide inconsistent instance `Scalar Float Float`.\n\nAn alternative approach to get executable code would be to add a custom compiler step which would replace every occurance of real or complex numbers with their floating point equivalent. Implementing such compiler step turned out to be quite a non-trivial task thus we are taking this type class approach. -/\nclass SciLean.Scalar (R : outParam (Type u_1)) (K : semiOutParam (Type u_2)) : Type (max u_1 u_2)"} +{"name":"SciLean.Scalar.mk","declaration":"ctor SciLean.Scalar.mk {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [toRCLike : RCLike K] (toComplex : K → ℂ) (toReal : R → ℝ) (ofReal : ℝ → R) (ofComplex : ℂ → K) (make : R → R → K) (make_def : ∀ (x y : R),\n if ∀ (y : K), RCLike.im y = 0 then toComplex (make x y) = { re := toReal x, im := 0 }\n else toComplex (make x y) = { re := toReal x, im := toReal y }) (real : K → R) (real_def : ∀ (x : K), toReal (real x) = RCLike.re (toComplex x)) (imag : K → R) (imag_def : ∀ (x : K), toReal (imag x) = RCLike.im (toComplex x)) (sin : K → K) (sin_def : ∀ (x : K), toComplex (sin x) = Complex.sin (toComplex x)) (cos : K → K) (cos_def : ∀ (x : K), toComplex (cos x) = Complex.cos (toComplex x)) (tan : K → K) (tan_def : ∀ (x : K), toComplex (tan x) = Complex.tan (toComplex x)) (tanh : K → K) (tanh_def : ∀ (x : K), toComplex (tanh x) = Complex.tanh (toComplex x)) (exp : K → K) (exp_def : ∀ (x : K), toComplex (exp x) = Complex.exp (toComplex x)) (log : K → K) (log_def : ∀ (x : K), toComplex (log x) = Complex.log (toComplex x)) (sqrt : K → K) (sqrt_def : ∀ (x : K),\n if ∀ (y : K), RCLike.im y = 0 then RCLike.re (toComplex (sqrt x)) = √(RCLike.re (toComplex x))\n else toComplex (sqrt x) = Complex.cpow (toComplex x) (1 / 2)) (pow : K → K → K) (pow_def : ∀ (x y : K),\n if ∀ (z : K), RCLike.im z = 0 then toReal (real (pow x y)) = (toComplex x ^ toComplex y).re\n else toComplex (pow x y) = toComplex x ^ toComplex y) (abs : K → R) (abs_def : ∀ (x : K), toReal (abs x) = Complex.abs (toComplex x)) : SciLean.Scalar R K"} +{"name":"SciLean.Scalar.tan_def","declaration":"def SciLean.Scalar.tan_def {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : SciLean.Scalar.toComplex (SciLean.Scalar.tan x) = Complex.tan (SciLean.Scalar.toComplex x)"} +{"name":"SciLean.RealScalar.atan_def","declaration":"def SciLean.RealScalar.atan_def {R : semiOutParam (Type u_1)} [self : SciLean.RealScalar R] (x : R) : SciLean.Scalar.toReal R (SciLean.RealScalar.atan x) = Real.arctan (SciLean.Scalar.toReal R x)"} +{"name":"SciLean.Scalar.tanh","declaration":"def SciLean.Scalar.tanh {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : K"} +{"name":"SciLean.Scalar.sqrt","declaration":"def SciLean.Scalar.sqrt {R : outParam (Type u_1)} {K : semiOutParam (Type u_2)} [self : SciLean.Scalar R K] (x : K) : K"} +{"name":"SciLean.RealScalar.acos_def","declaration":"def SciLean.RealScalar.acos_def {R : semiOutParam (Type u_1)} [self : SciLean.RealScalar R] (x : R) : SciLean.Scalar.toReal R (SciLean.RealScalar.acos x) = Real.arccos (SciLean.Scalar.toReal R x)"} +{"name":"SciLean.Scalar.ofReal","declaration":"def SciLean.Scalar.ofReal {R : outParam (Type u_1)} (K : semiOutParam (Type u_2)) [self : SciLean.Scalar R K] : ℝ → R"} +{"name":"SciLean.RealScalar.acos","declaration":"def SciLean.RealScalar.acos {R : semiOutParam (Type u_1)} [self : SciLean.RealScalar R] (x : R) : R"} +{"name":"SciLean.scalar_min_one_zero","declaration":"theorem SciLean.scalar_min_one_zero {R : Type u_1} [SciLean.RealScalar R] : min 1 0 = 0"} +{"name":"SciLean.instHPow","declaration":"instance SciLean.instHPow {R : Type u_1} {K : Type u_2} [SciLean.Scalar R K] : HPow K K K"} +{"name":"SciLean.scalar_max_zero_one","declaration":"theorem SciLean.scalar_max_zero_one {R : Type u_1} [SciLean.RealScalar R] : max 0 1 = 1"} +{"name":"SciLean.RealScalar.asin_def","declaration":"def SciLean.RealScalar.asin_def {R : semiOutParam (Type u_1)} [self : SciLean.RealScalar R] (x : R) : SciLean.Scalar.toReal R (SciLean.RealScalar.asin x) = Real.arcsin (SciLean.Scalar.toReal R x)"} diff --git a/scilean-declarations/SciLean.Core.Objects.SemiInnerProductSpace.jsonl b/scilean-declarations/SciLean.Core.Objects.SemiInnerProductSpace.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3a7e97dce08c8fbf533194d23d1df188710771c6 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Objects.SemiInnerProductSpace.jsonl @@ -0,0 +1,55 @@ +{"name":"SciLean.TestFunctions.TestFunction","declaration":"def SciLean.TestFunctions.TestFunction {X : Type u_1} [self : SciLean.TestFunctions X] : Set X"} +{"name":"SciLean.instSemiInnerProductSpace_1","declaration":"instance SciLean.instSemiInnerProductSpace_1 {K : Type u_1} {X : Type u_2} [RCLike K] [NormedAddCommGroup X] [InnerProductSpace K X] [CompleteSpace X] : SciLean.SemiInnerProductSpace K X"} +{"name":"SciLean.norm₂_squared_nat","declaration":"theorem SciLean.norm₂_squared_nat {R : Type u_1} {K : Type u_2} {X : Type u_3} [SciLean.Scalar R K] [SciLean.Norm2 K X] (x : X) : ‖x‖₂ ^ 2 = ‖x‖₂²"} +{"name":"SciLean.«term‖_‖₂²»","declaration":"def SciLean.«term‖_‖₂²» : Lean.ParserDescr"} +{"name":"SciLean.norm₂_prod","declaration":"theorem SciLean.norm₂_prod {R : Type u_1} {K : Type u_2} {X : Type u_3} {Y : Type u_4} [SciLean.Scalar R K] [AddCommMonoid K] [Inner K X] [Inner K Y] (x : X) (y : Y) : ‖(x, y)‖₂ = SciLean.Scalar.sqrt (‖x‖₂² + ‖y‖₂²)"} +{"name":"SciLean.instInner","declaration":"instance SciLean.instInner {K : Type u_1} [RCLike K] : Inner K K"} +{"name":"SciLean.«term⟪_,_⟫»","declaration":"def SciLean.«term⟪_,_⟫» : Lean.ParserDescr"} +{"name":"SciLean.SemiInnerProductSpace","declaration":"/-- SemiInnerProductSpace is almost InnerProductSpace but `⟪x,y⟫` does not make\nsense for all elements `x y : X`. For example, `C∞(ℝ, ℝ)` or `ℕ → ℝ` are almost\ninner product spaces but `∫ x : ℝ, f x * g x` or `∑ i : ℕ, a i * b i` are not\nmeaningful for all `f, g` or `a, b`. Therefore we introduce notion of test functions\nand `⟪x, φ⟫` has meaning only when `φ` is test function, `x` can be arbitrary.\n\nThe important property is that deciding if an element is zero, `x = 0`, can be\ndetermined by testing `⟪x, ϕ⟫[K] = 0` for all test functions `φ`. This is known\nas fundamental lemma of the calculus of variations.\nhttps://en.wikipedia.org/wiki/Fundamental_lemma_of_the_calculus_of_variations\n\nThis also allows a definition of adjoint between two semi-inner product spaces, see `semiAdjoint`.\n-/\nclass SciLean.SemiInnerProductSpace (K : Type u_1) [RCLike K] (X : Type u_2) : Type (max u_1 u_2)"} +{"name":"SciLean.«term‖_‖₂»","declaration":"def SciLean.«term‖_‖₂» : Lean.ParserDescr"} +{"name":"SciLean.instSemiHilbertUnit","declaration":"instance SciLean.instSemiHilbertUnit {K : Type u_1} [RCLike K] : SciLean.SemiHilbert K Unit"} +{"name":"SciLean.SemiInnerProductSpace.add_left","declaration":"def SciLean.SemiInnerProductSpace.add_left {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] (x : X) (y : X) (z : X) : SciLean.TestFunction x ∧ SciLean.TestFunction y ∨ SciLean.TestFunction z → ⟪x + y, z⟫_K = ⟪x, z⟫_K + ⟪y, z⟫_K"} +{"name":"SciLean.norm₂","declaration":"def SciLean.norm₂ (K : Type u_1) {R : Type u_2} {X : Type u_3} [SciLean.Scalar R K] [SciLean.Norm2 K X] (x : X) : K"} +{"name":"SciLean.SemiInnerProductSpace.conj_sym","declaration":"def SciLean.SemiInnerProductSpace.conj_sym {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] (x : X) (y : X) : (starRingEnd K) ⟪y, x⟫_K = ⟪x, y⟫_K"} +{"name":"SciLean.SemiInnerProductSpace.inner_with_testfun_is_continuous","declaration":"def SciLean.SemiInnerProductSpace.inner_with_testfun_is_continuous {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] (ϕ : X) : SciLean.TestFunction ϕ → Continuous fun x => ⟪x, ϕ⟫_K"} +{"name":"SciLean.instSemiHilbertProd","declaration":"instance SciLean.instSemiHilbertProd {K : Type u_1} [RCLike K] (X : Type u_2) (Y : Type u_3) [SciLean.SemiHilbert K X] [SciLean.SemiHilbert K Y] : SciLean.SemiHilbert K (X × Y)"} +{"name":"SciLean.instTestFunctions","declaration":"instance SciLean.instTestFunctions {K : Type u_1} [RCLike K] : SciLean.TestFunctions K"} +{"name":"SciLean.SemiHilbert","declaration":"/-- Almost Hilbert space but does not have to be complete. It is only c∞-complete.\n\nThe important property is that the norm `‖x‖₂` and inner product `⟪x,y⟫` is meaningful for any `x y : X`. For general semi-inner prodcut space the norm and inner product is well defined only for `x ∈ TestFunction`-/\nclass SciLean.SemiHilbert (K : Type u_1) [RCLike K] (X : Type u_2) : Type (max u_1 u_2)"} +{"name":"SciLean.instSemiHilbertForAll","declaration":"instance SciLean.instSemiHilbertForAll {K : Type u_1} [RCLike K] (ι : Type u_2) (X : ι → Type u_3) [(i : ι) → SciLean.SemiHilbert K (X i)] [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] : SciLean.SemiHilbert K ((i : ι) → X i)"} +{"name":"SciLean.instSemiInnerProductSpaceForAll","declaration":"instance SciLean.instSemiInnerProductSpaceForAll {K : Type u_1} [RCLike K] (ι : Type u_2) (X : ι → Type u_3) [(i : ι) → SciLean.SemiInnerProductSpace K (X i)] [SciLean.IndexType ι] : SciLean.SemiInnerProductSpace K ((i : ι) → X i)"} +{"name":"SciLean.instSemiInnerProductSpaceUnit","declaration":"instance SciLean.instSemiInnerProductSpaceUnit {K : Type u_1} [RCLike K] : SciLean.SemiInnerProductSpace K Unit"} +{"name":"SciLean.Norm2.norm2","declaration":"def SciLean.Norm2.norm2 {K : Type u_1} {X : Type u_2} [self : SciLean.Norm2 K X] : X → K"} +{"name":"SciLean.unexpandNorm₂","declaration":"def SciLean.unexpandNorm₂ : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.vecNormalize","declaration":"def SciLean.vecNormalize {R : Type u_1} (C : Type u_2) [SciLean.Scalar R C] {X : Type u_3} [SciLean.SemiHilbert C X] (x : X) : X"} +{"name":"SciLean.SemiInnerProductSpace.inner_norm2","declaration":"def SciLean.SemiInnerProductSpace.inner_norm2 {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] (x : X) : ⟪x, x⟫_K = ‖x‖₂²"} +{"name":"SciLean.norm₂_scalar","declaration":"theorem SciLean.norm₂_scalar {R : Type u_1} [SciLean.RealScalar R] (x : R) : ‖x‖₂[R] = SciLean.Scalar.abs x"} +{"name":"SciLean.instTestFunctionsForAll","declaration":"instance SciLean.instTestFunctionsForAll (ι : Type u_1) (X : ι → Type u_2) [(i : ι) → SciLean.TestFunctions (X i)] : SciLean.TestFunctions ((i : ι) → X i)"} +{"name":"SciLean.instSemiInnerProductSpace","declaration":"instance SciLean.instSemiInnerProductSpace {K : Type u_1} [RCLike K] : SciLean.SemiInnerProductSpace K K"} +{"name":"SciLean.«term‖_‖₂²[_]»","declaration":"def SciLean.«term‖_‖₂²[_]» : Lean.ParserDescr"} +{"name":"SciLean.SemiInnerProductSpace.inner_pos","declaration":"def SciLean.SemiInnerProductSpace.inner_pos {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] (x : X) : SciLean.TestFunction x → RCLike.re ⟪x, x⟫_K ≥ 0 ∧ RCLike.im ⟪x, x⟫_K = 0"} +{"name":"SciLean.SemiInnerProductSpace.mk","declaration":"ctor SciLean.SemiInnerProductSpace.mk {K : Type u_1} [RCLike K] {X : Type u_2} [toVec : SciLean.Vec K X] [toInner : Inner K X] [toTestFunctions : SciLean.TestFunctions X] [toNorm2 : SciLean.Norm2 K X] (add_left : ∀ (x y z : X),\n SciLean.TestFunction x ∧ SciLean.TestFunction y ∨ SciLean.TestFunction z → ⟪x + y, z⟫_K = ⟪x, z⟫_K + ⟪y, z⟫_K) (smul_left : ∀ (x y : X) (r : K), ⟪r • x, y⟫_K = (starRingEnd K) r * ⟪x, y⟫_K) (conj_sym : ∀ (x y : X), (starRingEnd K) ⟪y, x⟫_K = ⟪x, y⟫_K) (inner_pos : ∀ (x : X), SciLean.TestFunction x → RCLike.re ⟪x, x⟫_K ≥ 0 ∧ RCLike.im ⟪x, x⟫_K = 0) (inner_ext : ∀ (x : X), x = 0 ↔ ∀ (ϕ : X), SciLean.TestFunction ϕ → ⟪x, ϕ⟫_K = 0) (is_lin_subspace : SciLean.VecProp K SciLean.TestFunction) (inner_norm2 : ∀ (x : X), ⟪x, x⟫_K = ‖x‖₂²) (inner_with_testfun_is_continuous : ∀ (ϕ : X), SciLean.TestFunction ϕ → Continuous fun x => ⟪x, ϕ⟫_K) : SciLean.SemiInnerProductSpace K X"} +{"name":"SciLean.SemiHilbert.test_functions_true","declaration":"def SciLean.SemiHilbert.test_functions_true {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiHilbert K X] (x : X) : SciLean.TestFunction x"} +{"name":"SciLean.norm2_scalar","declaration":"theorem SciLean.norm2_scalar {R : Type u_1} [SciLean.RealScalar R] (x : R) : ‖x‖₂² = x ^ 2"} +{"name":"SciLean.TestFunctions.mk","declaration":"ctor SciLean.TestFunctions.mk {X : Type u_1} (TestFunction : Set X) : SciLean.TestFunctions X"} +{"name":"SciLean.TestFunctions","declaration":"/-- TestFunctions defines a subset of well behaved elemets w.r.t. to the inner product.\nFor example:\n 1. test function on `ℕ → ℝ` are sequences with only finitely many non-zero elements\n 2. test function on `C∞(ℝ, ℝ)` are functions with compact support\n\nsee `SemiInnerProductSpace` for more information\n-/\nclass SciLean.TestFunctions (X : Type u_1) : Type u_1"} +{"name":"SciLean.Norm2.mk","declaration":"ctor SciLean.Norm2.mk {K : Type u_1} {X : Type u_2} (norm2 : X → K) : SciLean.Norm2 K X"} +{"name":"SciLean.instTestFunctionsProd","declaration":"instance SciLean.instTestFunctionsProd (X : Type u_1) (Y : Type u_2) [SciLean.TestFunctions X] [SciLean.TestFunctions Y] : SciLean.TestFunctions (X × Y)"} +{"name":"SciLean.unexpandNorm2","declaration":"def SciLean.unexpandNorm2 : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.SemiInnerProductSpace.is_lin_subspace","declaration":"def SciLean.SemiInnerProductSpace.is_lin_subspace {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] : SciLean.VecProp K SciLean.TestFunction"} +{"name":"SciLean.instInnerProd","declaration":"instance SciLean.instInnerProd (K : Type u_1) (X : Type u_2) (Y : Type u_3) [AddCommMonoid K] [Inner K X] [Inner K Y] : Inner K (X × Y)"} +{"name":"SciLean.instSemiInnerProductSpaceProd","declaration":"instance SciLean.instSemiInnerProductSpaceProd {K : Type u_1} [RCLike K] (X : Type u_2) (Y : Type u_3) [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Y] : SciLean.SemiInnerProductSpace K (X × Y)"} +{"name":"SciLean.SemiHilbert.mk","declaration":"ctor SciLean.SemiHilbert.mk {K : Type u_1} [RCLike K] {X : Type u_2} [toSemiInnerProductSpace : SciLean.SemiInnerProductSpace K X] (test_functions_true : ∀ (x : X), SciLean.TestFunction x) : SciLean.SemiHilbert K X"} +{"name":"SciLean.Norm2","declaration":"/-- Square of L₂ norm over the field `K` -/\nclass SciLean.Norm2 (K : Type u_1) (X : Type u_2) : Type (max u_1 u_2)"} +{"name":"SciLean.SemiInnerProductSpace.mkSorryProofs","declaration":"def SciLean.SemiInnerProductSpace.mkSorryProofs {K : Type u_1} [RCLike K] {α : Type u_2} [SciLean.Vec K α] [Inner K α] [SciLean.TestFunctions α] : SciLean.SemiInnerProductSpace K α"} +{"name":"SciLean.scalar_norm","declaration":"theorem SciLean.scalar_norm {R : Type u_1} [SciLean.RealScalar R] (r : R) : ‖r‖₂[R] = SciLean.Scalar.abs r"} +{"name":"SciLean.«term⟪_,_⟫[_]»","declaration":"def SciLean.«term⟪_,_⟫[_]» : Lean.ParserDescr"} +{"name":"SciLean.instInnerUnit","declaration":"instance SciLean.instInnerUnit {K : Type u_1} [RCLike K] : Inner K Unit"} +{"name":"SciLean.unexpandInner","declaration":"def SciLean.unexpandInner : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.norm₂_squared","declaration":"theorem SciLean.norm₂_squared {R : Type u_1} {K : Type u_2} {X : Type u_3} [SciLean.Scalar R K] [SciLean.Norm2 K X] (x : X) : ‖x‖₂ ^ 2 = ‖x‖₂²"} +{"name":"SciLean.instSemiHilbert","declaration":"instance SciLean.instSemiHilbert {K : Type u_1} [RCLike K] : SciLean.SemiHilbert K K"} +{"name":"SciLean.SemiInnerProductSpace.inner_ext","declaration":"def SciLean.SemiInnerProductSpace.inner_ext {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] (x : X) : x = 0 ↔ ∀ (ϕ : X), SciLean.TestFunction ϕ → ⟪x, ϕ⟫_K = 0"} +{"name":"SciLean.«term‖_‖₂[_]»","declaration":"def SciLean.«term‖_‖₂[_]» : Lean.ParserDescr"} +{"name":"SciLean.SemiInnerProductSpace.smul_left","declaration":"def SciLean.SemiInnerProductSpace.smul_left {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.SemiInnerProductSpace K X] (x : X) (y : X) (r : K) : ⟪r • x, y⟫_K = (starRingEnd K) r * ⟪x, y⟫_K"} +{"name":"SciLean.instInnerForAll","declaration":"instance SciLean.instInnerForAll (K : Type u_1) (ι : Type u_2) (X : ι → Type u_3) [AddCommMonoid K] [(i : ι) → Inner K (X i)] [SciLean.IndexType ι] : Inner K ((i : ι) → X i)"} +{"name":"SciLean.instTestFunctionsUnit","declaration":"instance SciLean.instTestFunctionsUnit {K : Type u_1} [RCLike K] : SciLean.TestFunctions Unit"} +{"name":"SciLean.instNorm2","declaration":"instance SciLean.instNorm2 {K : Type u_1} {X : Type u_2} [Inner K X] : SciLean.Norm2 K X"} diff --git a/scilean-declarations/SciLean.Core.Objects.Vec.jsonl b/scilean-declarations/SciLean.Core.Objects.Vec.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..2e8597e8ccf195ee5e808cc919f79ff341926082 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Objects.Vec.jsonl @@ -0,0 +1,45 @@ +{"name":"SciLean.instVecSum","declaration":"instance SciLean.instVecSum {K : Type u_1} [RCLike K] (X : Type) (Y : Type) (TX : X → Type) (TY : Y → Type) (xy : X ⊕ Y) [(x : X) → SciLean.Vec K (TX x)] [(y : Y) → SciLean.Vec K (TY y)] : SciLean.Vec K ((TX⊕TY) xy)"} +{"name":"SciLean.instVecForAll","declaration":"instance SciLean.instVecForAll {K : Type u_1} [RCLike K] (α : Type u_2) (X : α → Type u_3) [inst : (a : α) → SciLean.Vec K (X a)] : SciLean.Vec K ((a : α) → X a)"} +{"name":"SciLean.VecProp.smul","declaration":"def SciLean.VecProp.smul {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [self : SciLean.VecProp K P] (r : K) (x : X) : P x → P (r • x)"} +{"name":"SciLean.instNegSubtype","declaration":"instance SciLean.instNegSubtype {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [inst : SciLean.VecProp K P] : Neg { x // P x }"} +{"name":"SciLean.Vec.mkSorryProofs","declaration":"def SciLean.Vec.mkSorryProofs {K : Type u_1} [RCLike K] {α : Type u_2} [Add α] [Sub α] [Neg α] [Zero α] [SMul K α] [UniformSpace α] : SciLean.Vec K α"} +{"name":"SciLean.Vec.mk","declaration":"ctor SciLean.Vec.mk {K : Type u_1} [RCLike K] {X : Type u_2} [toAddCommGroup : AddCommGroup X] [toUniformSpace : UniformSpace X] [toUniformAddGroup : UniformAddGroup X] [toModule : Module K X] [toContinuousSMul : ContinuousSMul K X] (scalar_wise_smooth : ∀ (c : K → X), SciLean.Curve.Smooth c ↔ ∀ (x' : X →L[K] K), SciLean.Curve.Smooth (⇑x' ∘ c)) : SciLean.Vec K X"} +{"name":"SciLean.Curve.DifferentiableAt","declaration":"def SciLean.Curve.DifferentiableAt {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (x : K) : Prop"} +{"name":"SciLean.Curve.HasDerivAtFilter","declaration":"def SciLean.Curve.HasDerivAtFilter {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (f' : F) (x : K) (L : Filter K) : Prop"} +{"name":"SciLean.AddCommGroup.mkSorryProofs","declaration":"def SciLean.AddCommGroup.mkSorryProofs {α : Type u_1} [Add α] [Sub α] [Neg α] [Zero α] : AddCommGroup α"} +{"name":"SciLean.VecProp.add","declaration":"def SciLean.VecProp.add {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [self : SciLean.VecProp K P] (x : X) (y : X) : P x → P y → P (x + y)"} +{"name":"SciLean.Vec.scalar_wise_smooth","declaration":"def SciLean.Vec.scalar_wise_smooth {K : Type u_1} [RCLike K] {X : Type u_2} [self : SciLean.Vec K X] (c : K → X) : SciLean.Curve.Smooth c ↔ ∀ (x' : X →L[K] K), SciLean.Curve.Smooth (⇑x' ∘ c)"} +{"name":"SciLean.TopologicalAddGroup.mkSorryProofs","declaration":"def SciLean.TopologicalAddGroup.mkSorryProofs {α : Type u_1} [Add α] [Sub α] [Neg α] [Zero α] [TopologicalSpace α] : TopologicalAddGroup α"} +{"name":"SciLean.Curve.ContDiffAt","declaration":"def SciLean.Curve.ContDiffAt {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (x : K) (n : ℕ∞) : Prop"} +{"name":"SciLean.SubNegMonoid.mkSorryProofs","declaration":"def SciLean.SubNegMonoid.mkSorryProofs {α : Type u_1} [Add α] [Sub α] [Neg α] [Zero α] : SubNegMonoid α"} +{"name":"SciLean.Curve.SmoothAt","declaration":"def SciLean.Curve.SmoothAt {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (x : K) : Prop"} +{"name":"SciLean.Curve.HasDerivAt","declaration":"def SciLean.Curve.HasDerivAt {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (f' : F) (x : K) : Prop"} +{"name":"SciLean.instInhabited","declaration":"instance SciLean.instInhabited {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] : Inhabited X"} +{"name":"SciLean.MulAction.mkSorryProofs","declaration":"def SciLean.MulAction.mkSorryProofs {α : Type u_1} {β : Type u_2} [Monoid α] [SMul α β] : MulAction α β"} +{"name":"SciLean.instZeroSubtype","declaration":"instance SciLean.instZeroSubtype {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [inst : SciLean.VecProp K P] : Zero { x // P x }"} +{"name":"SciLean.UniformAddGroup.mkSorryProofs","declaration":"def SciLean.UniformAddGroup.mkSorryProofs {α : Type u_1} [Add α] [Sub α] [Neg α] [Zero α] [UniformSpace α] : UniformAddGroup α"} +{"name":"SciLean.instVecUnit","declaration":"instance SciLean.instVecUnit {K : Type u_1} [RCLike K] : SciLean.Vec K Unit"} +{"name":"SciLean.VecProp","declaration":"class SciLean.VecProp (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] (P : X → Prop) : Prop"} +{"name":"SciLean.Curve.Smooth","declaration":"def SciLean.Curve.Smooth {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) : Prop"} +{"name":"SciLean.Vec","declaration":"/-- Vectors space `X` over field `K`\n\nMore precisely this is Convenient Vector Space which is a special class of vector spaces\nwhich allow very general definition of differentiability. In particular, the space `C∞(ℝ,ℝ)`,\nsmooth functions on real numbers, is Convenient Vector Spaces but fails to be Banach space.\n-/\nclass SciLean.Vec (K : Type u_1) [RCLike K] (X : Type u_2) : Type (max u_1 u_2)"} +{"name":"SciLean.AddSemigroup.mkSorryProofs","declaration":"def SciLean.AddSemigroup.mkSorryProofs {α : Type u_1} [Add α] : AddSemigroup α"} +{"name":"SciLean.ContinuousNeg.mkSorryProofs","declaration":"def SciLean.ContinuousNeg.mkSorryProofs {α : Type u_1} [Neg α] [TopologicalSpace α] : ContinuousNeg α"} +{"name":"SciLean.VecProp.mk","declaration":"ctor SciLean.VecProp.mk {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} (add : ∀ (x y : X), P x → P y → P (x + y)) (neg : ∀ (x : X), P x → P (-x)) (smul : ∀ (r : K) (x : X), P x → P (r • x)) (zero : P 0) : SciLean.VecProp K P"} +{"name":"SciLean.VecProp.zero","declaration":"def SciLean.VecProp.zero {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [self : SciLean.VecProp K P] : P 0"} +{"name":"SciLean.instAddSubtype","declaration":"instance SciLean.instAddSubtype {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [inst : SciLean.VecProp K P] : Add { x // P x }"} +{"name":"SciLean.AddGroup.mkSorryProofs","declaration":"def SciLean.AddGroup.mkSorryProofs {α : Type u_1} [Add α] [Sub α] [Neg α] [Zero α] : AddGroup α"} +{"name":"SciLean.Curve.ContDiffAt'","declaration":"def SciLean.Curve.ContDiffAt' {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (x : K) (n : ℕ) : Prop"} +{"name":"SciLean.Curve.deriv","declaration":"def SciLean.Curve.deriv {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (x : K) : F"} +{"name":"SciLean.«term_⊕_»","declaration":"def SciLean.«term_⊕_» : Lean.TrailingParserDescr"} +{"name":"SciLean.instSubSubtype","declaration":"instance SciLean.instSubSubtype {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [inst : SciLean.VecProp K P] : Sub { x // P x }"} +{"name":"SciLean.VecProp.neg","declaration":"def SciLean.VecProp.neg {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [self : SciLean.VecProp K P] (x : X) : P x → P (-x)"} +{"name":"SciLean.Module.mkSorryProofs","declaration":"def SciLean.Module.mkSorryProofs {α : Type u_1} {β : Type u_2} [Semiring α] [addcommgroup : AddCommGroup β] [SMul α β] : Module α β"} +{"name":"SciLean.instSMulSubtype","declaration":"instance SciLean.instSMulSubtype {K : Type u_1} [RCLike K] {X : Type u_2} [SciLean.Vec K X] {P : X → Prop} [inst : SciLean.VecProp K P] : SMul K { x // P x }"} +{"name":"SciLean.instVecProd","declaration":"instance SciLean.instVecProd {K : Type u_1} [RCLike K] {U : Type u_2} {V : Type u_3} [instU : SciLean.Vec K U] [instV : SciLean.Vec K V] : SciLean.Vec K (U × V)"} +{"name":"SciLean.Curve.ContDiff","declaration":"def SciLean.Curve.ContDiff {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) (n : ℕ∞) : Prop"} +{"name":"SciLean.Curve.Differentiable","declaration":"def SciLean.Curve.Differentiable {K : Type u} [NontriviallyNormedField K] {F : Type v} [AddCommGroup F] [Module K F] [TopologicalSpace F] (f : K → F) : Prop"} +{"name":"SciLean.DistribMulAction.mkSorryProofs","declaration":"def SciLean.DistribMulAction.mkSorryProofs {α : Type u_1} {β : Type u_2} [Monoid α] [AddMonoid β] [SMul α β] : DistribMulAction α β"} +{"name":"SciLean.ContinuousSMul.mkSorryProofs","declaration":"def SciLean.ContinuousSMul.mkSorryProofs {K : Type u_1} [RCLike K] {α : Type u_2} [SMul K α] [TopologicalSpace α] : ContinuousSMul K α"} +{"name":"SciLean.instVec","declaration":"instance SciLean.instVec {K : Type u_1} [RCLike K] : SciLean.Vec K K"} +{"name":"SciLean.ContinuousAdd.mkSorryProofs","declaration":"def SciLean.ContinuousAdd.mkSorryProofs {α : Type u_1} [Add α] [TopologicalSpace α] : ContinuousAdd α"} +{"name":"SciLean.AddMonoid.mkSorryProofs","declaration":"def SciLean.AddMonoid.mkSorryProofs {α : Type u_1} [Add α] [Zero α] : AddMonoid α"} diff --git a/scilean-declarations/SciLean.Core.Rand.Condition.jsonl b/scilean-declarations/SciLean.Core.Rand.Condition.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a342153b24d63d43b5531049113c53b1862a2ad0 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Condition.jsonl @@ -0,0 +1,4 @@ +{"name":"SciLean.Rand.conditionSnd","declaration":"/-- Condition on the second variable of a pair. -/\ndef SciLean.Rand.conditionSnd {X : Type} {Y : Type} (r : SciLean.Rand (X × Y)) (y : Y) : SciLean.Rand X"} +{"name":"SciLean.Rand.HasCondition","declaration":"/-- Given random variable `r : Rand (X×Y)`. Can we condition `r` based on observation of `X`?\n\nCan we come up with conditional rand variable `r₂ : X₁ → Rand X₁` -/\ndef SciLean.Rand.HasCondition {X : Type} {Y : Type} (r : SciLean.Rand (X × Y)) : Prop"} +{"name":"SciLean.Rand.condition","declaration":"def SciLean.Rand.condition {X : Type} {X₁ : Type} {X₂ : Type} (r : SciLean.Rand X) (dec : X₁ × X₂ ≃ X) (x₁ : X₁) : SciLean.Rand X₂"} +{"name":"SciLean.Rand.conditionFst","declaration":"/-- Condition on the first variable of a pair. -/\ndef SciLean.Rand.conditionFst {X : Type} {Y : Type} (r : SciLean.Rand (X × Y)) (x : X) : SciLean.Rand Y"} diff --git a/scilean-declarations/SciLean.Core.Rand.Distributions.Flip.jsonl b/scilean-declarations/SciLean.Core.Rand.Distributions.Flip.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ebc16576227bd040e5057785e6e1c0ae1f8d6927 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Distributions.Flip.jsonl @@ -0,0 +1,8 @@ +{"name":"SciLean.Rand.flip","declaration":"def SciLean.Rand.flip {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (x : R) : SciLean.Rand Bool"} +{"name":"SciLean.Rand.add_as_flip_E","declaration":"theorem SciLean.Rand.add_as_flip_E {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] {X : Type u_1} [AddCommGroup X] [Module R X] [Module ℝ X] {x : X} {y : X} (θ : R) (h : θ ∈ Set.Ioo 0 1) : x + y = (SciLean.Rand.flip θ).𝔼 fun b => if b = true then θ⁻¹ • x else (1 - θ)⁻¹ • y"} +{"name":"SciLean.Rand.flip.integral","declaration":"theorem SciLean.Rand.flip.integral {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] {X : Type u_1} [AddCommGroup X] [Module R X] [Module ℝ X] (θ : R) (f : Bool → X) : ∫' x, f x ∂(SciLean.Rand.flip θ).ℙ = θ • f true + (1 - θ) • f false"} +{"name":"SciLean.Rand.instLawfulRandBoolFlipInstMeasurableSpace","declaration":"instance SciLean.Rand.instLawfulRandBoolFlipInstMeasurableSpace {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (θ : R) : SciLean.Rand.LawfulRand (SciLean.Rand.flip θ)"} +{"name":"SciLean.Rand.flip.pdf","declaration":"theorem SciLean.Rand.flip.pdf {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (x : R) : SciLean.Rand.pdf R (SciLean.Rand.flip x) MeasureTheory.Measure.count = fun b =>\n let x := (x ⊔ 0) ⊓ 1;\n if b = true then x else 1 - x"} +{"name":"SciLean.Rand.flip.measure","declaration":"theorem SciLean.Rand.flip.measure {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (θ : R) : (SciLean.Rand.flip θ).ℙ =\n ENNReal.ofReal (SciLean.Scalar.toReal R θ) • MeasureTheory.Measure.dirac true +\n ENNReal.ofReal (SciLean.Scalar.toReal R (1 - θ)) • MeasureTheory.Measure.dirac false"} +{"name":"SciLean.Rand.flip.pdf_wrt_flip","declaration":"theorem SciLean.Rand.flip.pdf_wrt_flip {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (θ : R) (θ' : R) : SciLean.Rand.pdf R (SciLean.Rand.flip θ) (SciLean.Rand.flip θ').ℙ = fun b =>\n if b = true then θ / θ' else (1 - θ) / (1 - θ')"} +{"name":"SciLean.Rand.flip.E","declaration":"theorem SciLean.Rand.flip.E {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] {X : Type u_1} [AddCommGroup X] [Module R X] [Module ℝ X] (θ : R) (f : Bool → X) : (SciLean.Rand.flip θ).𝔼 f = θ • f true + (1 - θ) • f false"} diff --git a/scilean-declarations/SciLean.Core.Rand.Distributions.Normal.jsonl b/scilean-declarations/SciLean.Core.Rand.Distributions.Normal.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..c92b326e6763e931ae80e34b520238d11bbfeeca --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Distributions.Normal.jsonl @@ -0,0 +1,11 @@ +{"name":"SciLean.Rand.instLawfulRandNormalToMeasurableSpace","declaration":"instance SciLean.Rand.instLawfulRandNormalToMeasurableSpace {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (μ : R) (σ : R) : SciLean.Rand.LawfulRand (SciLean.Rand.normal μ σ)"} +{"name":"SciLean.Rand.normal_reparameterize","declaration":"theorem SciLean.Rand.normal_reparameterize {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (μ : R) (σ : R) : SciLean.Rand.normal μ σ = (SciLean.Rand.normal 0 1).map fun x => σ • x + μ"} +{"name":"SciLean.Rand.normalFDμ","declaration":"def SciLean.Rand.normalFDμ {R : Type u_1} [SciLean.RealScalar R] {U : Type u_2} [SciLean.SemiHilbert R U] [MeasureTheory.MeasureSpace U] (μ : U) (dμ : U) (σ : R) : 𝒟'(U, R × R)"} +{"name":"SciLean.Rand.normal","declaration":"/-- Normal random variable with mean `μ` and standard deviation `σ`. -/\ndef SciLean.Rand.normal {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (μ : R) (σ : R) : SciLean.Rand R"} +{"name":"SciLean.Rand.normalFDμ_score","declaration":"theorem SciLean.Rand.normalFDμ_score {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] {Y : Type u_2} [SciLean.Vec R Y] {Z : Type u_1} [SciLean.Vec R Z] [Module ℝ Z] (μ : R) (dμ : R) (σ : R) (f : R → Y) (L : (R × R) ⊸[R] Y ⊸[R] Z) : (SciLean.Rand.normalFDμ μ dμ σ).extAction f L =\n (SciLean.Rand.normal μ σ).𝔼 fun x => (L (1, -(1 / σ ^ 2) * ⟪x - μ, -dμ⟫_R)) (f x)"} +{"name":"SciLean.Rand.normal.map_add_left","declaration":"theorem SciLean.Rand.normal.map_add_left {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (μ : R) (σ : R) (θ : R) : ((SciLean.Rand.normal μ σ).map fun x => θ + x) = SciLean.Rand.normal (θ + μ) σ"} +{"name":"SciLean.Rand.normal.pdf","declaration":"theorem SciLean.Rand.normal.pdf {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (μ : R) (σ : R) : SciLean.Rand.pdf R (SciLean.Rand.normal μ σ) MeasureTheory.volume = SciLean.gaussian μ σ"} +{"name":"SciLean.Rand.normal.map_sub_right","declaration":"theorem SciLean.Rand.normal.map_sub_right {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (μ : R) (σ : R) (θ : R) : ((SciLean.Rand.normal μ σ).map fun x => x - θ) = SciLean.Rand.normal (μ - θ) σ"} +{"name":"SciLean.Rand.boxMuller","declaration":"def SciLean.Rand.boxMuller {R : Type u_1} [SciLean.RealScalar R] (u : R) (v : R) : R × R"} +{"name":"SciLean.Rand.normal.map_add_right","declaration":"theorem SciLean.Rand.normal.map_add_right {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (μ : R) (σ : R) (θ : R) : ((SciLean.Rand.normal μ σ).map fun x => x + θ) = SciLean.Rand.normal (μ + θ) σ"} +{"name":"SciLean.Rand.normal.arg_μ.parDistribFwdDeriv","declaration":"theorem SciLean.Rand.normal.arg_μ.parDistribFwdDeriv {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] {W : Type u_1} [SciLean.Vec R W] (μ : W → R) (σ : R) (hμ : SciLean.CDifferentiable R μ) : (SciLean.parDistribFwdDeriv fun w => ↑(SciLean.Rand.normal (μ w) σ).ℙ) = fun w dw =>\n let μdμ := ∂> μ w dw;\n SciLean.Rand.normalFDμ μdμ.1 μdμ.2 σ"} diff --git a/scilean-declarations/SciLean.Core.Rand.Distributions.Sphere.jsonl b/scilean-declarations/SciLean.Core.Rand.Distributions.Sphere.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..f7cc5027500bf0d8836629131c977f27d1df758c --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Distributions.Sphere.jsonl @@ -0,0 +1,12 @@ +{"name":"SciLean.Rand.uniform_sphere_density","declaration":"theorem SciLean.Rand.uniform_sphere_density (x : SciLean.Vec3) (r : Float) : MeasureTheory.volume.rnDeriv (SciLean.Rand.uniform (SciLean.Rand.sphere x r)).ℙ = fun y =>\n ENNReal.ofReal (SciLean.Scalar.toReal Float (4 * SciLean.RealScalar.pi * r ^ 2))"} +{"name":"SciLean.Rand.sphere_volume","declaration":"theorem SciLean.Rand.sphere_volume {R : Type u_2} [SciLean.RealScalar R] {ι : Type u_3} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] {X : Type u_1} [SciLean.FinVec ι R X] (x : X) (r : R) : ↑↑MeasureTheory.volume Set.univ = ↑(4 * SciLean.RealScalar.pi * r ^ 2)"} +{"name":"SciLean.Rand.instToStringSphere","declaration":"instance SciLean.Rand.instToStringSphere {R : Type u_1} [SciLean.RealScalar R] {ι : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] {X : Type u_3} [SciLean.FinVec ι R X] (x : X) (r : R) [ToString X] : ToString (SciLean.Rand.sphere x r)"} +{"name":"SciLean.Rand.instMeasureSpaceSphere","declaration":"instance SciLean.Rand.instMeasureSpaceSphere {R : Type u_1} [SciLean.RealScalar R] {ι : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] {X : Type u_3} [SciLean.FinVec ι R X] (x : X) (r : R) : MeasureTheory.MeasureSpace (SciLean.Rand.sphere x r)"} +{"name":"SciLean.Rand.sphereMapDensity","declaration":"def SciLean.Rand.sphereMapDensity (f : SciLean.Vec3 → SciLean.Vec3) (x : SciLean.Vec3) : Float"} +{"name":"SciLean.Rand.mkSphereMap","declaration":"def SciLean.Rand.mkSphereMap (f : SciLean.Vec3 → SciLean.Vec3) (x : SciLean.Vec3) : SciLean.Vec3"} +{"name":"SciLean.Rand.elongate","declaration":"def SciLean.Rand.elongate (s : Float) (v : SciLean.Vec3) (x : SciLean.Vec3) : SciLean.Vec3"} +{"name":"SciLean.Rand.instUniformRandSphereFloatInstRealScalarFloatFinOfNatNatInstOfNatNatInstIndexTypeFinInstLawfulIndexTypeFinInstIndexTypeFinInstDecidableEqFinVec3InstFinVecToRCLikeToScalarInstArrayTypeVec3FinOfNatNatInstOfNatNatFloat","declaration":"instance SciLean.Rand.instUniformRandSphereFloatInstRealScalarFloatFinOfNatNatInstOfNatNatInstIndexTypeFinInstLawfulIndexTypeFinInstIndexTypeFinInstDecidableEqFinVec3InstFinVecToRCLikeToScalarInstArrayTypeVec3FinOfNatNatInstOfNatNatFloat (x : SciLean.Vec3) (r : Float) : SciLean.Rand.UniformRand (SciLean.Rand.sphere x r)"} +{"name":"SciLean.Rand.ball","declaration":"def SciLean.Rand.ball {R : Type u_1} [SciLean.RealScalar R] {ι : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] {X : Type u_3} [SciLean.FinVec ι R X] (x : X) (r : R) : Set X"} +{"name":"SciLean.Rand.sphere","declaration":"def SciLean.Rand.sphere {R : Type u_1} [SciLean.RealScalar R] {ι : Type u_2} [SciLean.IndexType ι] [SciLean.LawfulIndexType ι] [DecidableEq ι] {X : Type u_3} [SciLean.FinVec ι R X] (x : X) (r : R) : Type u_3"} +{"name":"SciLean.Rand.elongateDensity","declaration":"def SciLean.Rand.elongateDensity (s : Float) (v : SciLean.Vec3) (x : SciLean.Vec3) : Float"} +{"name":"SciLean.Rand.mkSphereMap'","declaration":"/-- On sphere this is identical to `mkSphereMap` but the jacobian of this map is equal to the\ntwo dimensional jacobian of `mkSphereMap f : S² → S²` which we need to compute as -/\ndef SciLean.Rand.mkSphereMap' (f : SciLean.Vec3 → SciLean.Vec3) (x : SciLean.Vec3) : SciLean.Vec3"} diff --git a/scilean-declarations/SciLean.Core.Rand.Distributions.Uniform.jsonl b/scilean-declarations/SciLean.Core.Rand.Distributions.Uniform.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..2450b91aacb46bf1aa42eb07aa6652485c4af335 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Distributions.Uniform.jsonl @@ -0,0 +1,6 @@ +{"name":"SciLean.Rand.integral_as_uniform_E_in_set","declaration":"theorem SciLean.Rand.integral_as_uniform_E_in_set {X : Type} [MeasureTheory.MeasureSpace X] (R : Type u_1) [SciLean.RealScalar R] {Y : Type u_2} [AddCommGroup Y] [SMul R Y] [Module ℝ Y] (f : X → Y) (A : Set X) [SciLean.Rand.UniformRand ↑A] : ∫' x, f x ∂MeasureTheory.volume.restrict A =\n (SciLean.Rand.uniform ↑A).𝔼 fun x =>\n let V := ↑(↑↑MeasureTheory.volume Set.univ);\n V • f ↑x"} +{"name":"SciLean.Rand.uniform","declaration":"def SciLean.Rand.uniform (X : Type) [SciLean.Rand.UniformRand X] : SciLean.Rand X"} +{"name":"SciLean.Rand.UniformRand.mk","declaration":"ctor SciLean.Rand.UniformRand.mk {X : Type} (uniform : SciLean.Rand X) : SciLean.Rand.UniformRand X"} +{"name":"SciLean.Rand.integral_as_uniform_E","declaration":"theorem SciLean.Rand.integral_as_uniform_E {X : Type} [MeasureTheory.MeasureSpace X] (R : Type u_1) [SciLean.RealScalar R] {Y : Type u_2} [AddCommGroup Y] [Module R Y] [Module ℝ Y] (f : X → Y) (μ : MeasureTheory.Measure X) [SciLean.Rand.UniformRand X] : ∫' x, f x ∂μ =\n (SciLean.Rand.uniform X).𝔼 fun x =>\n let V := ↑(↑↑MeasureTheory.volume Set.univ);\n V • f x"} +{"name":"SciLean.Rand.UniformRand","declaration":"class SciLean.Rand.UniformRand (X : Type) : Type"} +{"name":"SciLean.Rand.UniformRand.uniform","declaration":"def SciLean.Rand.UniformRand.uniform {X : Type} [self : SciLean.Rand.UniformRand X] : SciLean.Rand X"} diff --git a/scilean-declarations/SciLean.Core.Rand.Distributions.UniformI.jsonl b/scilean-declarations/SciLean.Core.Rand.Distributions.UniformI.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..56e037c133c1289567b51d0901ff6bee222abcb2 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Distributions.UniformI.jsonl @@ -0,0 +1,11 @@ +{"name":"SciLean.Rand.uniformI.measure","declaration":"theorem SciLean.Rand.uniformI.measure {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] : (SciLean.uniformI R).ℙ = MeasureTheory.volume.restrict (Set.Ioo 0 1)"} +{"name":"SciLean.Rand.instUniformRandElemIocToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder","declaration":"instance SciLean.Rand.instUniformRandElemIocToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) : SciLean.Rand.UniformRand ↑(Set.Ioc a b)"} +{"name":"SciLean.Rand.instLawfulRandUniformIToMeasurableSpace","declaration":"instance SciLean.Rand.instLawfulRandUniformIToMeasurableSpace {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] : SciLean.Rand.LawfulRand (SciLean.uniformI R)"} +{"name":"SciLean.Rand.Set.Ioc_volume","declaration":"theorem SciLean.Rand.Set.Ioc_volume {R : Type u_1} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) (h : a ≤ b) : ↑↑MeasureTheory.volume (Set.Ioc a b) = ↑(b - a)"} +{"name":"SciLean.Rand.instUniformRandElemIcoToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder","declaration":"instance SciLean.Rand.instUniformRandElemIcoToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) : SciLean.Rand.UniformRand ↑(Set.Ico a b)"} +{"name":"SciLean.Rand.uniformI.pdf","declaration":"theorem SciLean.Rand.uniformI.pdf {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] : SciLean.Rand.pdf R (SciLean.uniformI R) MeasureTheory.volume = fun x => if 0 < x ∧ x < 1 then 1 else 0"} +{"name":"SciLean.Rand.Set.Ico_volume","declaration":"theorem SciLean.Rand.Set.Ico_volume {R : Type u_1} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) (h : a ≤ b) : ↑↑MeasureTheory.volume (Set.Ico a b) = ↑(b - a)"} +{"name":"SciLean.Rand.Set.Icc_volume","declaration":"theorem SciLean.Rand.Set.Icc_volume {R : Type u_1} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) (h : a ≤ b) : ↑↑MeasureTheory.volume (Set.Icc a b) = ↑(b - a)"} +{"name":"SciLean.Rand.instUniformRandElemIccToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder","declaration":"instance SciLean.Rand.instUniformRandElemIccToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) : SciLean.Rand.UniformRand ↑(Set.Icc a b)"} +{"name":"SciLean.Rand.instUniformRandElemIooToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder","declaration":"instance SciLean.Rand.instUniformRandElemIooToPreorderToPartialOrderToSemilatticeInfToLatticeInstDistribLatticeToLinearOrder {R : Type} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) : SciLean.Rand.UniformRand ↑(Set.Ioo a b)"} +{"name":"SciLean.Rand.Set.Ioo_volume","declaration":"theorem SciLean.Rand.Set.Ioo_volume {R : Type u_1} [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] (a : R) (b : R) (h : a ≤ b) : ↑↑MeasureTheory.volume (Set.Ioo a b) = ↑(b - a)"} diff --git a/scilean-declarations/SciLean.Core.Rand.Distributions.WalkOnSpheres.jsonl b/scilean-declarations/SciLean.Core.Rand.Distributions.WalkOnSpheres.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3b916ff102b8fc93a37f277c2840ba235522834e --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Distributions.WalkOnSpheres.jsonl @@ -0,0 +1,10 @@ +{"name":"SciLean.Rand.walkOnSpheres","declaration":"def SciLean.Rand.walkOnSpheres {Y : Type} [SciLean.SemiHilbert Float Y] (φ : SciLean.Vec3 → Float) (g : SciLean.Vec3 → Y) (n : ℕ) (x : SciLean.Vec3) : SciLean.Rand Y"} +{"name":"SciLean.Rand.harmonicRec'_CDifferentiable","declaration":"theorem SciLean.Rand.harmonicRec'_CDifferentiable {Y : Type} [SciLean.SemiHilbert Float Y] [Module ℝ Y] (n : ℕ) : SciLean.CDifferentiable Float fun w => SciLean.Rand.harmonicRec' n w.1 w.2.1 w.2.2"} +{"name":"SciLean.Rand.harmonicRec'_fwdDeriv_rand","declaration":"def SciLean.Rand.harmonicRec'_fwdDeriv_rand {Y : Type} [SciLean.SemiHilbert Float Y] (n : ℕ) (φ : SciLean.Vec3 ⟿FD[Float,⊤] Float) (dφ : SciLean.Vec3 ⟿FD[Float,⊤] Float) (g : SciLean.Vec3 ⟿FD[Float,⊤] Y) (dg : SciLean.Vec3 ⟿FD[Float,⊤] Y) (x : SciLean.Vec3) (dx : SciLean.Vec3) : SciLean.Rand (Y × Y)"} +{"name":"SciLean.Rand.pi'","declaration":"def SciLean.Rand.pi' : Float"} +{"name":"SciLean.Rand.harmonicRec","declaration":"def SciLean.Rand.harmonicRec {Y : Type} [SciLean.SemiHilbert Float Y] [Module ℝ Y] (n : ℕ) (φ : SciLean.Vec3 → Float) (g : SciLean.Vec3 → Y) (x : SciLean.Vec3) : Y"} +{"name":"SciLean.Rand.harmonicRec.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Rand.harmonicRec.arg_x.CDifferentiable_rule {Y : Type} [SciLean.SemiHilbert Float Y] [Module ℝ Y] (n : ℕ) (φ : SciLean.Vec3 → Float) (g : SciLean.Vec3 → Y) (hφ : SciLean.CDifferentiable Float φ) (hg : SciLean.CDifferentiable Float g) : SciLean.CDifferentiable Float fun x => SciLean.Rand.harmonicRec n φ g x"} +{"name":"SciLean.Rand.harmonicRec.arg_x.fwdDeriv_randApprox","declaration":"def SciLean.Rand.harmonicRec.arg_x.fwdDeriv_randApprox {Y : Type} [SciLean.SemiHilbert Float Y] (n : ℕ) (φ : SciLean.Vec3 → Float) (φ' : SciLean.Vec3 → SciLean.Vec3 → Float × Float) (g : SciLean.Vec3 → Y) (g' : SciLean.Vec3 → SciLean.Vec3 → Y × Y) (x : SciLean.Vec3) (dx : SciLean.Vec3) : SciLean.Rand (Y × Y)"} +{"name":"SciLean.Rand.harmonicRec_fwdDeriv","declaration":"def SciLean.Rand.harmonicRec_fwdDeriv {Y : Type} [SciLean.SemiHilbert Float Y] [Module ℝ Y] (n : ℕ) (φ : SciLean.Vec3 → Float) (φ' : SciLean.Vec3 → SciLean.Vec3 → Float × Float) (g : SciLean.Vec3 → Y) (g' : SciLean.Vec3 → SciLean.Vec3 → Y × Y) : SciLean.Vec3 → SciLean.Vec3 → Y × Y"} +{"name":"SciLean.Rand.harmonicRec'","declaration":"def SciLean.Rand.harmonicRec' {Y : Type} [SciLean.SemiHilbert Float Y] [Module ℝ Y] (n : ℕ) (φ : SciLean.Vec3 ⟿FD[Float,⊤] Float) (g : SciLean.Vec3 ⟿FD[Float,⊤] Y) (x : SciLean.Vec3) : Y"} +{"name":"SciLean.Rand.harmonicRec'_fwdDeriv","declaration":"def SciLean.Rand.harmonicRec'_fwdDeriv {Y : Type} [SciLean.SemiHilbert Float Y] [Module ℝ Y] (n : ℕ) (x : (SciLean.Vec3 ⟿FD[Float,⊤] Float) × (SciLean.Vec3 ⟿FD[Float,⊤] Y) × SciLean.Vec3) (dx : (SciLean.Vec3 ⟿FD[Float,⊤] Float) × (SciLean.Vec3 ⟿FD[Float,⊤] Y) × SciLean.Vec3) : Y × Y"} diff --git a/scilean-declarations/SciLean.Core.Rand.ExpectedValue.jsonl b/scilean-declarations/SciLean.Core.Rand.ExpectedValue.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ea72f00426a0cef74a33bd47d0dbf3b9ef95541d --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.ExpectedValue.jsonl @@ -0,0 +1,3 @@ +{"name":"SciLean.Rand.𝔼_deriv_as_distribDeriv","declaration":"theorem SciLean.Rand.𝔼_deriv_as_distribDeriv {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] {X : Type} [SciLean.Vec R X] [MeasureTheory.MeasureSpace X] (r : W → SciLean.Rand X) (f : W → X → Y) : ∂ w, (r w).𝔼 (f w) = fun w dw =>\n SciLean.Distribution.integrate\n (SciLean.parDistribDeriv (fun w => ↑fun x => SciLean.Rand.pdf R (r w) MeasureTheory.volume x • f w x) w dw)"} +{"name":"SciLean.Rand.𝔼.arg_r.cderiv_rule","declaration":"theorem SciLean.Rand.𝔼.arg_r.cderiv_rule {R : Type u_3} [SciLean.RealScalar R] {W : Type u_1} [SciLean.Vec R W] {X : Type} [MeasurableSpace X] [SciLean.Vec R X] {Y : Type u_2} [SciLean.Vec R Y] [Module ℝ Y] (r : W → SciLean.Rand X) (f : X → Y) : ∂ w, (r w).𝔼 f = fun w dw =>\n let d := SciLean.parDistribDeriv (fun w => ↑(r w).ℙ) w dw;\n d.extAction f (fun r ⊸[R] fun y ⊸[R] r • y)"} +{"name":"SciLean.Rand.𝔼.arg_rf.cderiv_rule'","declaration":"theorem SciLean.Rand.𝔼.arg_rf.cderiv_rule' {R : Type u_1} [SciLean.RealScalar R] {W : Type u_2} [SciLean.Vec R W] {X : Type} [MeasurableSpace X] [SciLean.Vec R X] {Y : Type u_3} [SciLean.Vec R Y] [Module ℝ Y] (r : W → SciLean.Rand X) (f : W → X → Y) (hf : ∀ (x : X), SciLean.CDifferentiable R fun x_1 => f x_1 x) : ∂ w, (r w).𝔼 (f w) = fun w dw =>\n let dr := SciLean.parDistribFwdDeriv (fun w => ↑(r w).ℙ) w dw;\n let df := fun x => ∂> (x_1:=w;dw), f x_1 x;\n dr.extAction df (fun rdr ⊸[R] fun ydy ⊸[R] rdr.1 • ydy.2 + rdr.2 • ydy.1)"} diff --git a/scilean-declarations/SciLean.Core.Rand.Model.jsonl b/scilean-declarations/SciLean.Core.Rand.Model.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0913e7a58902a9f4f2a1c31d3cadc46cceaa2ac3 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Model.jsonl @@ -0,0 +1,6 @@ +{"name":"SciLean.Rand.modelBind","declaration":"/-- Special form of bind for `Rand` for which it is easy to compute conditional probabilities and\nprobability densities. Most likely you want to use this bind when defining probabilistic model. -/\ndef SciLean.Rand.modelBind {X : Type} {Y : Type} (x : SciLean.Rand X) (f : X → SciLean.Rand Y) : SciLean.Rand (X × Y)"} +{"name":"SciLean.«termLet_~____»","declaration":"def SciLean.«termLet_~____» : Lean.ParserDescr"} +{"name":"SciLean.modelBind_condition","declaration":"theorem SciLean.modelBind_condition {X : Type} {Y : Type} (r : SciLean.Rand X) (f : X → SciLean.Rand Y) (x' : X) : SciLean.Rand.conditionFst\n (let x ~ r\n f x)\n x' =\n f x'"} +{"name":"SciLean.modelBind_pdf","declaration":"theorem SciLean.modelBind_pdf {R : Type u_1} [SciLean.RealScalar R] {X : Type} {Y : Type} [MeasureTheory.MeasureSpace X] [MeasureTheory.MeasureSpace Y] (r : SciLean.Rand X) (f : X → SciLean.Rand Y) : SciLean.Rand.pdf R\n (let x ~ r\n f x)\n MeasureTheory.volume =\n fun xy => SciLean.Rand.pdf R r MeasureTheory.volume xy.1 * SciLean.Rand.pdf R (f xy.1) MeasureTheory.volume xy.2"} +{"name":"SciLean.modelBind_pdf_prod","declaration":"theorem SciLean.modelBind_pdf_prod {R : Type u_1} [SciLean.RealScalar R] {X : Type} {Y : Type} [MeasurableSpace X] [MeasurableSpace Y] (r : SciLean.Rand X) (f : X → SciLean.Rand Y) (μ : MeasureTheory.Measure X) (ν : MeasureTheory.Measure Y) : SciLean.Rand.pdf R\n (let x ~ r\n f x)\n (MeasureTheory.Measure.prod μ ν) =\n fun xy => SciLean.Rand.pdf R r μ xy.1 * SciLean.Rand.pdf R (f xy.1) ν xy.2"} +{"name":"SciLean.unexpandRandBind","declaration":"def SciLean.unexpandRandBind : Lean.PrettyPrinter.Unexpander"} diff --git a/scilean-declarations/SciLean.Core.Rand.PushPullExpectation.jsonl b/scilean-declarations/SciLean.Core.Rand.PushPullExpectation.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..b17c4ebea2463b684c7e60b649adce814bec2e29 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.PushPullExpectation.jsonl @@ -0,0 +1,21 @@ +{"name":"SciLean.Rand.ite_pull_mean","declaration":"theorem SciLean.Rand.ite_pull_mean {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] {c : Prop} [Decidable c] (t : SciLean.Rand X) (e : SciLean.Rand X) : (if c then SciLean.Rand.mean t else SciLean.Rand.mean e) = SciLean.Rand.mean (if c then t else e)"} +{"name":"SciLean.Rand.push_E_lambda","declaration":"theorem SciLean.Rand.push_E_lambda {X : Type u_1} {Y : Type} [MeasurableSpace Y] {Z : Type u_2} [AddCommGroup Z] [Module ℝ Z] (r : SciLean.Rand Y) (f : X → Y → Z) : (r.𝔼 fun y x => f x y) = fun x => r.𝔼 fun y => f x y"} +{"name":"SciLean.Rand.bind_pull_mean","declaration":"theorem SciLean.Rand.bind_pull_mean {X : Type} [MeasurableSpace X] {Y : Type} [AddCommGroup Y] [Module ℝ Y] [MeasurableSpace Y] (x : SciLean.Rand X) (f : X → SciLean.Rand Y) : (SciLean.Rand.mean do\n let x' ← x\n pure (SciLean.Rand.mean (f x'))) =\n SciLean.Rand.mean (x >>= f)"} +{"name":"SciLean.Rand.pull_mean_add_2","declaration":"theorem SciLean.Rand.pull_mean_add_2 {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : X) (y : SciLean.Rand X) : x + SciLean.Rand.mean y =\n SciLean.Rand.mean do\n let y' ← y\n pure (x + y')"} +{"name":"SciLean.Rand.pull_E_affine","declaration":"theorem SciLean.Rand.pull_E_affine {X : Type} [MeasurableSpace X] {Y : Type u_1} [AddCommGroup Y] [Module ℝ Y] {Z : Type u_2} [AddCommGroup Z] [Module ℝ Z] (r : SciLean.Rand X) (φ : X → Y) (f : Y → Z) (hf : autoParam (IsAffineMap ℝ f) _auto✝) : f (r.𝔼 φ) = r.𝔼 fun x => f (φ x)"} +{"name":"SciLean.Rand.pull_mean_smul","declaration":"theorem SciLean.Rand.pull_mean_smul {R : Type u_1} [SciLean.RealScalar R] {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] [Module R X] (r : R) (x : SciLean.Rand X) : r • SciLean.Rand.mean x =\n SciLean.Rand.mean do\n let x' ← x\n pure (r • x')"} +{"name":"SciLean.Rand.pull_E_lambda","declaration":"theorem SciLean.Rand.pull_E_lambda {X : Type u_1} {Y : Type} [MeasurableSpace Y] {Z : Type u_2} [AddCommGroup Z] [Module ℝ Z] (r : SciLean.Rand Y) (f : X → Y → Z) : (fun x => r.𝔼 fun y => f x y) = r.𝔼 fun y x => f x y"} +{"name":"SciLean.Rand.pull_E_list_recOn","declaration":"theorem SciLean.Rand.pull_E_list_recOn {α : Type u_1} {C : List α → Type} [(n : List α) → AddCommGroup (C n)] [(n : List α) → Module ℝ (C n)] [(n : List α) → MeasurableSpace (C n)] {D : List α → Type} [(n : List α) → MeasurableSpace (D n)] (l : List α) (x₀ : C []) (r : (head : α) → (tail : List α) → SciLean.Rand (D (head :: tail))) (f : (head : α) → (tail : List α) → C tail → D (head :: tail) → C (head :: tail)) (hf : ∀ (head : α) (tail : List α) (d : D (head :: tail)), IsAffineMap ℝ fun x => f head tail x d) : (List.recOn l x₀ fun head tail x => (r head tail).𝔼 (f head tail x)) =\n SciLean.Rand.mean\n (List.recOn l (pure x₀) fun head tail x => do\n let x' ← x\n let y' ← r head tail\n pure (f head tail x' y'))"} +{"name":"SciLean.Rand.ite_pull_mean_f","declaration":"theorem SciLean.Rand.ite_pull_mean_f {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] {c : Prop} [Decidable c] (t : X) (e : SciLean.Rand X) : (if c then t else SciLean.Rand.mean e) = SciLean.Rand.mean (if c then pure t else e)"} +{"name":"SciLean.Rand.push_E_affine","declaration":"theorem SciLean.Rand.push_E_affine {X : Type} [MeasurableSpace X] {Y : Type u_1} [AddCommGroup Y] [Module ℝ Y] {Z : Type u_2} [AddCommGroup Z] [Module ℝ Z] (r : SciLean.Rand X) (φ : X → Y) (f : Y → Z) (hf : autoParam (IsAffineMap ℝ f) _auto✝) : (r.𝔼 fun x => f (φ x)) = f (r.𝔼 φ)"} +{"name":"SciLean.Rand.pull_mean_add","declaration":"theorem SciLean.Rand.pull_mean_add {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : SciLean.Rand X) (y : SciLean.Rand X) : SciLean.Rand.mean x + SciLean.Rand.mean y =\n SciLean.Rand.mean do\n let x' ← x\n let y' ← y\n pure (x' + y')"} +{"name":"SciLean.Rand.pull_mean_sub","declaration":"theorem SciLean.Rand.pull_mean_sub {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : SciLean.Rand X) (y : SciLean.Rand X) : SciLean.Rand.mean x - SciLean.Rand.mean y =\n SciLean.Rand.mean do\n let x' ← x\n let y' ← y\n pure (x' - y')"} +{"name":"SciLean.Rand.pull_mean_mul","declaration":"theorem SciLean.Rand.pull_mean_mul {R : Type} [SciLean.RealScalar R] (r : R) (x : SciLean.Rand R) : r * SciLean.Rand.mean x =\n SciLean.Rand.mean do\n let x' ← x\n pure (r * x')"} +{"name":"SciLean.Rand.pull_mean_add_1","declaration":"theorem SciLean.Rand.pull_mean_add_1 {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : SciLean.Rand X) (y : X) : SciLean.Rand.mean x + y =\n SciLean.Rand.mean do\n let x' ← x\n pure (x' + y)"} +{"name":"SciLean.Rand.ite_push_mean","declaration":"theorem SciLean.Rand.ite_push_mean {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] {c : Prop} [Decidable c] (t : SciLean.Rand X) (e : SciLean.Rand X) : SciLean.Rand.mean (if c then t else e) = if c then SciLean.Rand.mean t else SciLean.Rand.mean e"} +{"name":"SciLean.Rand.ite_pull_E","declaration":"theorem SciLean.Rand.ite_pull_E {X : Type} [MeasurableSpace X] {Y : Type u_1} [AddCommGroup Y] [Module ℝ Y] {c : Prop} [Decidable c] (t : SciLean.Rand X) (e : SciLean.Rand X) (φ : X → Y) (ψ : X → Y) : (if c then t.𝔼 φ else e.𝔼 ψ) = (if c then t else e).𝔼 (if c then φ else ψ)"} +{"name":"SciLean.Rand.pull_E_nat_recOn","declaration":"theorem SciLean.Rand.pull_E_nat_recOn (C : ℕ → Type) [(n : ℕ) → AddCommGroup (C n)] [(n : ℕ) → Module ℝ (C n)] [(n : ℕ) → MeasurableSpace (C n)] (D : ℕ → Type) [(n : ℕ) → MeasurableSpace (D n)] {n : ℕ} (x₀ : C 0) (r : (n : ℕ) → SciLean.Rand (D n)) (f : (n : ℕ) → C n → D n → C (n + 1)) (hf : ∀ (n : ℕ) (d : D n), IsAffineMap ℝ fun x => f n x d) : (Nat.recOn n x₀ fun n x => (r n).𝔼 (f n x)) =\n SciLean.Rand.mean\n (Nat.recOn n (pure x₀) fun n x => do\n let x' ← x\n let y' ← r n\n pure (f n x' y'))"} +{"name":"SciLean.Rand.ite_push_E","declaration":"theorem SciLean.Rand.ite_push_E {X : Type} [MeasurableSpace X] {Y : Type u_1} [AddCommGroup Y] [Module ℝ Y] {c : Prop} [Decidable c] (t : SciLean.Rand X) (e : SciLean.Rand X) (φ : X → Y) : (if c then t else e).𝔼 φ = if c then t.𝔼 φ else e.𝔼 φ"} +{"name":"SciLean.Rand.pull_mean_neg","declaration":"theorem SciLean.Rand.pull_mean_neg {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : SciLean.Rand X) : -SciLean.Rand.mean x =\n SciLean.Rand.mean do\n let x' ← x\n pure (-x')"} +{"name":"SciLean.Rand.ite_pull_mean_t","declaration":"theorem SciLean.Rand.ite_pull_mean_t {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] {c : Prop} [Decidable c] (t : SciLean.Rand X) (e : X) : (if c then SciLean.Rand.mean t else e) = SciLean.Rand.mean (if c then t else pure e)"} +{"name":"SciLean.Rand.pull_mean_div","declaration":"theorem SciLean.Rand.pull_mean_div {R : Type} [SciLean.RealScalar R] (x : SciLean.Rand R) (y : R) : SciLean.Rand.mean x / y =\n SciLean.Rand.mean do\n let x' ← x\n pure (x' / y)"} diff --git a/scilean-declarations/SciLean.Core.Rand.Rand.jsonl b/scilean-declarations/SciLean.Core.Rand.Rand.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ce9a6af274b2f81aa3229a26511668d322943713 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Rand.jsonl @@ -0,0 +1,47 @@ +{"name":"SciLean.Rand.map","declaration":"def SciLean.Rand.map {X : Type} {Y : Type} (r : SciLean.Rand X) (f : X → Y) : SciLean.Rand Y"} +{"name":"SciLean.Rand.ext","declaration":"/-- Extensionality of random variable.\n\nWARNING: This theorem is inconsistent!!! The random generators `x.rand` and `y.rand` might differ.\n We are not trying to model pseudo-random numbers. We assume that every random number\n generator is a true random number generator. Thus the result of any probabilistic program\n should be independent on the exact generator up to some randomness.\n\nTODO: We might quotient all the random number generators corresponding to the measure `x.ℙ` under\nthe assumption that they are all true random generators. I believe that such type would be\na singleton i.e. all the random number generators are all the same.\n-/\naxiom SciLean.Rand.ext {X : Type} (x : SciLean.Rand X) (y : SciLean.Rand X) : Erased.out x.spec = Erased.out y.spec → x = y"} +{"name":"SciLean.Rand.pdf_wrt_self","declaration":"theorem SciLean.Rand.pdf_wrt_self {X : Type} {R : Type u_1} [SciLean.RealScalar R] [MeasurableSpace X] (x : SciLean.Rand X) [SciLean.Rand.LawfulRand x] : SciLean.Rand.pdf R x x.ℙ = 1"} +{"name":"SciLean.Rand.pdf","declaration":"/-- Probability density function of `x` w.r.t. the measure `ν`. -/\ndef SciLean.Rand.pdf {X : Type} (R : Type u_1) [SciLean.RealScalar R] [MeasurableSpace X] (x : SciLean.Rand X) (ν : autoParam (MeasureTheory.Measure X) _auto✝) : X → R"} +{"name":"SciLean.Rand.LawfulRand.mk","declaration":"ctor SciLean.Rand.LawfulRand.mk {X : Type} {x : SciLean.Rand X} [MeasurableSpace X] (is_measure : Function.IsMeasure (Erased.out x.spec)) (is_prob : MeasureTheory.IsProbabilityMeasure x.ℙ) : SciLean.Rand.LawfulRand x"} +{"name":"SciLean.Rand.ℙ","declaration":"/-- Probability measure of a random variable -/\ndef SciLean.Rand.ℙ {X : Type} [MeasurableSpace X] (r : SciLean.Rand X) : MeasureTheory.Measure X"} +{"name":"SciLean.Rand.map_ℙ","declaration":"theorem SciLean.Rand.map_ℙ {X : Type} {Y : Type} [MeasurableSpace X] [MeasurableSpace Y] (r : SciLean.Rand X) (f : X → Y) : (r.map f).ℙ = MeasureTheory.Measure.map f r.ℙ"} +{"name":"SciLean.Rand.LawfulRand.is_prob","declaration":"def SciLean.Rand.LawfulRand.is_prob {X : Type} {x : SciLean.Rand X} [MeasurableSpace X] [self : SciLean.Rand.LawfulRand x] : MeasureTheory.IsProbabilityMeasure x.ℙ"} +{"name":"SciLean.Rand.ite_pdf","declaration":"theorem SciLean.Rand.ite_pdf {X : Type} {R : Type u_1} [SciLean.RealScalar R] [MeasurableSpace X] (c : Prop) [Decidable c] (t : SciLean.Rand X) (e : SciLean.Rand X) (μ : MeasureTheory.Measure X) : SciLean.Rand.pdf R (if c then t else e) μ = if c then SciLean.Rand.pdf R t μ else SciLean.Rand.pdf R e μ"} +{"name":"SciLean.Rand.mean","declaration":"def SciLean.Rand.mean {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (r : SciLean.Rand X) : X"} +{"name":"SciLean.Rand.instMonadRand","declaration":"instance SciLean.Rand.instMonadRand : Monad SciLean.Rand"} +{"name":"SciLean.Rand.spec","declaration":"/-- `spec` defines a probability measure by computing an expectation. This means if `x : Rand X`\ncorresponds to a probability measure `μ` then for `φ : X → ℝ`\n```\nx.spec.out φ = ∫ x, φ x ∂μ\n```\n\nUsing `(X→ℝ)→ℝ` instead of `Measure X` for the specification of random variables has the\nadvantage that we can reuse Lean's `do` notation.\n-/\ndef SciLean.Rand.spec {X : Type} (self : SciLean.Rand X) : Erased ((X → ℝ) → ℝ)"} +{"name":"SciLean.Rand.E_smul","declaration":"theorem SciLean.Rand.E_smul {X : Type} {Y : Type u_1} [MeasurableSpace X] [AddCommGroup Y] [Module ℝ Y] (r : SciLean.Rand X) (φ : X → ℝ) (y : Y) : (r.𝔼 fun x' => φ x' • y) = r.𝔼 φ • y"} +{"name":"SciLean.Rand.instIsProbabilityMeasureℙ","declaration":"instance SciLean.Rand.instIsProbabilityMeasureℙ {X : Type} [MeasurableSpace X] (x : SciLean.Rand X) [inst : SciLean.Rand.LawfulRand x] : MeasureTheory.IsProbabilityMeasure x.ℙ"} +{"name":"SciLean.Rand.LawfulRand","declaration":"/-- Specification of `x : Rand X` is really saying that it is a probability measure. -/\nclass SciLean.Rand.LawfulRand {X : Type} (x : SciLean.Rand X) [MeasurableSpace X] : Type"} +{"name":"SciLean.Rand.unexpandRandCombine","declaration":"def SciLean.Rand.unexpandRandCombine : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.Rand.bind_mean","declaration":"theorem SciLean.Rand.bind_mean {X : Type} {Y : Type} [MeasurableSpace X] [MeasurableSpace Y] [AddCommGroup Y] [Module ℝ Y] (x : SciLean.Rand X) (f : X → SciLean.Rand Y) : SciLean.Rand.mean (x >>= f) = x.𝔼 fun x' => SciLean.Rand.mean (f x')"} +{"name":"SciLean.Rand.LawfulRand.is_measure","declaration":"def SciLean.Rand.LawfulRand.is_measure {X : Type} {x : SciLean.Rand X} [MeasurableSpace X] [self : SciLean.Rand.LawfulRand x] : Function.IsMeasure (Erased.out x.spec)"} +{"name":"SciLean.Rand.pure_𝔼","declaration":"theorem SciLean.Rand.pure_𝔼 {X : Type} {Y : Type u_1} [MeasurableSpace X] [AddCommGroup Y] [Module ℝ Y] (x : X) (φ : X → Y) : (pure x).𝔼 φ = φ x"} +{"name":"SciLean.erase_out","declaration":"theorem SciLean.erase_out {α : Sort u_1} (a : α) : Erased.out (SciLean.erase a) = a"} +{"name":"SciLean.Rand.instLawfulMonadRandInstMonadRand","declaration":"instance SciLean.Rand.instLawfulMonadRandInstMonadRand : LawfulMonad SciLean.Rand"} +{"name":"SciLean.Rand.E_zero","declaration":"theorem SciLean.Rand.E_zero {X : Type} [MeasurableSpace X] {U : Type u_1} [AddCommGroup U] [Module ℝ U] (r : SciLean.Rand X) : (r.𝔼 fun x => 0) = 0"} +{"name":"SciLean.Rand.combine_pdf","declaration":"theorem SciLean.Rand.combine_pdf {X : Type} {R : Type} [SciLean.RealScalar R] [MeasurableSpace X] [MeasureTheory.MeasureSpace R] (x : SciLean.Rand X) (y : SciLean.Rand X) (μ : MeasureTheory.Measure X) (θ : R) : SciLean.Rand.pdf R (x +[θ] y) μ = fun x' => (1 - θ) * SciLean.Rand.pdf R x μ x' + θ * SciLean.Rand.pdf R y μ x'"} +{"name":"SciLean.Rand.instHAddRand_1","declaration":"instance SciLean.Rand.instHAddRand_1 {X : Type} [Add X] : HAdd (SciLean.Rand X) X (SciLean.Rand X)"} +{"name":"SciLean.Rand.fst","declaration":"/-- Marginal distribution for the first component of a pair. -/\ndef SciLean.Rand.fst {X : Type} {Y : Type} (r : SciLean.Rand (X × Y)) : SciLean.Rand X"} +{"name":"SciLean.uniformI","declaration":"def SciLean.uniformI (R : Type) [SciLean.RealScalar R] [MeasureTheory.MeasureSpace R] : SciLean.Rand R"} +{"name":"SciLean.Rand.mean_add","declaration":"theorem SciLean.Rand.mean_add {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : SciLean.Rand X) (x' : X) : SciLean.Rand.mean x + x' = SciLean.Rand.mean (x + x')"} +{"name":"SciLean.Rand.instHAddRand","declaration":"instance SciLean.Rand.instHAddRand {X : Type} [Add X] : HAdd X (SciLean.Rand X) (SciLean.Rand X)"} +{"name":"SciLean.Rand.rand","declaration":"/-- `rand` is a pseudo randon number generator implemented using the \"Standard\" number generator\n-/\ndef SciLean.Rand.rand {X : Type} (self : SciLean.Rand X) : StateM StdGen X"} +{"name":"SciLean.erase","declaration":"def SciLean.erase {α : Sort u_1} (a : α) : Erased α"} +{"name":"Function.IsMeasure","declaration":"def Function.IsMeasure {X : Type u_1} [MeasurableSpace X] (F : (X → ℝ) → ℝ) : Prop"} +{"name":"SciLean.Rand","declaration":"/-- `x : Rand X` is a random variable of type `X`\n\nYou can:\n - generate sample with `x.get : IO X`\n - get probability measure with `x.ℙ : Measure X`\n\nThe internal fields `spec` and `rand` are just an internal implementation of `Rand` and should not\nbe accessed by normal users.\n\nTODO: Hide implementation using quotients or something like that\n-/\nstructure SciLean.Rand (X : Type) : Type"} +{"name":"SciLean.Rand.swap_bind","declaration":"theorem SciLean.Rand.swap_bind {X : Type} {Y : Type} {Z : Type} (f : X → Y → Z) (x : SciLean.Rand X) (y : SciLean.Rand Y) : (do\n let x' ← x\n let y' ← y\n pure (f x' y')) =\n do\n let y' ← y\n let x' ← x\n pure (f x' y')"} +{"name":"SciLean.Rand.«term_+[_]_»","declaration":"/-- `x +[θ] y` return random variable `(1-θ)*x + θ*y`.\nIn other words\n- `x` is generated with probability `1-θ`\n- `y` is generated with probability `θ` -/\ndef SciLean.Rand.«term_+[_]_» : Lean.TrailingParserDescr"} +{"name":"SciLean.Rand.expectedValue_as_mean","declaration":"theorem SciLean.Rand.expectedValue_as_mean {X : Type} {Y : Type} [MeasurableSpace X] [MeasurableSpace Y] [AddCommGroup Y] [Module ℝ Y] (x : SciLean.Rand X) (φ : X → Y) : x.𝔼 φ = SciLean.Rand.mean (x.map φ)"} +{"name":"SciLean.Rand.snd","declaration":"/-- Marginal distribution for the second component of a pair. -/\ndef SciLean.Rand.snd {X : Type} {Y : Type} (r : SciLean.Rand (X × Y)) : SciLean.Rand Y"} +{"name":"SciLean.Rand.pure_mean","declaration":"theorem SciLean.Rand.pure_mean {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : X) : SciLean.Rand.mean (pure x) = x"} +{"name":"SciLean.Rand.bind_E","declaration":"theorem SciLean.Rand.bind_E {X : Type} {Y : Type} {Z : Type u_1} [MeasurableSpace X] [MeasurableSpace Y] [AddCommGroup Z] [Module ℝ Z] (r : SciLean.Rand X) (f : X → SciLean.Rand Y) (φ : Y → Z) : (r >>= f).𝔼 φ = r.𝔼 fun x' => (f x').𝔼 φ"} +{"name":"SciLean.Rand.mk","declaration":"ctor SciLean.Rand.mk {X : Type} (spec : Erased ((X → ℝ) → ℝ)) (rand : StateM StdGen X) : SciLean.Rand X"} +{"name":"SciLean.Rand.get","declaration":"/-- Generate rundom number using IO randomness -/\ndef SciLean.Rand.get {X : Type} (x : SciLean.Rand X) : IO X"} +{"name":"SciLean.Rand.combine","declaration":"def SciLean.Rand.combine {X : Type} {R : Type} [SciLean.RealScalar R] [MeasurableSpace X] [MeasureTheory.MeasureSpace R] (x : SciLean.Rand X) (y : SciLean.Rand X) (θ : R) : SciLean.Rand X"} +{"name":"SciLean.Rand.pure_ℙ","declaration":"theorem SciLean.Rand.pure_ℙ {X : Type} [MeasurableSpace X] (x : X) : (pure x).ℙ = MeasureTheory.Measure.dirac x"} +{"name":"SciLean.Rand.reparameterize","declaration":"theorem SciLean.Rand.reparameterize {X : Type} {Y : Type} {Z : Type u_1} [MeasurableSpace X] [MeasurableSpace Y] [AddCommGroup Z] [Module ℝ Z] [Nonempty X] (f : X → Y) (hf : Function.Injective f) {r : SciLean.Rand X} {φ : X → Z} : r.𝔼 φ =\n let invf := Function.invFun f;\n (r.map f).𝔼 fun y => φ (invf y)"} +{"name":"SciLean.Rand.bind_pdf","declaration":"theorem SciLean.Rand.bind_pdf {X : Type} {Y : Type} {R : Type u_1} [SciLean.RealScalar R] [MeasurableSpace X] [MeasurableSpace Y] (ν : MeasureTheory.Measure Y) (x : SciLean.Rand X) (f : X → SciLean.Rand Y) : SciLean.Rand.pdf R (x >>= f) ν = fun y => ∫ (x' : X), SciLean.Rand.pdf R (f x') ν y ∂x.ℙ"} +{"name":"SciLean.Rand.mean_add'","declaration":"theorem SciLean.Rand.mean_add' {X : Type} [MeasurableSpace X] [AddCommGroup X] [Module ℝ X] (x : SciLean.Rand X) (x' : X) : x' + SciLean.Rand.mean x = SciLean.Rand.mean (x' + x)"} +{"name":"SciLean.Rand.E_add","declaration":"theorem SciLean.Rand.E_add {X : Type} [MeasurableSpace X] {U : Type u_1} [AddCommGroup U] [Module ℝ U] (r : SciLean.Rand X) (φ : X → U) (ψ : X → U) (hφ : SciLean.CIntegrable φ r.ℙ) (hψ : SciLean.CIntegrable ψ r.ℙ) : (r.𝔼 fun x => φ x + ψ x) = r.𝔼 φ + r.𝔼 ψ"} +{"name":"SciLean.Rand.𝔼","declaration":"def SciLean.Rand.𝔼 {X : Type} {Y : Type u_1} [MeasurableSpace X] [AddCommGroup Y] [Module ℝ Y] (r : SciLean.Rand X) (φ : X → Y) : Y"} diff --git a/scilean-declarations/SciLean.Core.Rand.SimpAttr.jsonl b/scilean-declarations/SciLean.Core.Rand.SimpAttr.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0cb357652279a1e40ede5ff62aec317a40b1ff76 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.SimpAttr.jsonl @@ -0,0 +1,8 @@ +{"name":"Parser.Attr.rand_push_E","declaration":"def Parser.Attr.rand_push_E : Lean.ParserDescr"} +{"name":"Parser.Attr.rand_pull_E_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.rand_pull_E_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.rand_push_E_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.rand_push_E_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.rand_AD_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.rand_AD_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.rand_simp_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.rand_simp_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.rand_AD","declaration":"def Parser.Attr.rand_AD : Lean.ParserDescr"} +{"name":"Parser.Attr.rand_simp","declaration":"def Parser.Attr.rand_simp : Lean.ParserDescr"} +{"name":"Parser.Attr.rand_pull_E","declaration":"def Parser.Attr.rand_pull_E : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Rand.Tactic.jsonl b/scilean-declarations/SciLean.Core.Rand.Tactic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..4844cfa5802cdfc8bda5c8a699d631d7632e9d58 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.Tactic.jsonl @@ -0,0 +1,7 @@ +{"name":"SciLean.Rand.termDerive_random_approx_By_","declaration":"def SciLean.Rand.termDerive_random_approx_By_ : Lean.ParserDescr"} +{"name":"SciLean.Rand.convRand_pull_E","declaration":"def SciLean.Rand.convRand_pull_E : Lean.ParserDescr"} +{"name":"SciLean.Rand.print_mean_variance","declaration":"def SciLean.Rand.print_mean_variance {R : Type} [SciLean.RealScalar R] [ToString R] (r : SciLean.Rand R) (n : ℕ) (msg : String) : IO Unit"} +{"name":"SciLean.Rand.convRand_compute_mean","declaration":"def SciLean.Rand.convRand_compute_mean : Lean.ParserDescr"} +{"name":"SciLean.Rand.convRand_push_E","declaration":"def SciLean.Rand.convRand_push_E : Lean.ParserDescr"} +{"name":"SciLean.Rand.convRand_AD","declaration":"def SciLean.Rand.convRand_AD : Lean.ParserDescr"} +{"name":"SciLean.Rand.«convRand_fdE_as_E_,_»","declaration":"def SciLean.Rand.«convRand_fdE_as_E_,_» : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.Rand.VariationalInference.jsonl b/scilean-declarations/SciLean.Core.Rand.VariationalInference.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..2a84e8fe934747a3b520d331d5910a983d9c670a --- /dev/null +++ b/scilean-declarations/SciLean.Core.Rand.VariationalInference.jsonl @@ -0,0 +1,4 @@ +{"name":"SciLean.KLDiv.arg_P.cderiv_rule","declaration":"theorem SciLean.KLDiv.arg_P.cderiv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type} [MeasurableSpace X] {W : Type u_2} [SciLean.Vec R W] [SciLean.Vec R X] (P : W → SciLean.Rand X) (Q : SciLean.Rand X) : ∂ w, SciLean.KLDiv (P w) Q = fun w dw =>\n let dP := SciLean.parDistribDeriv (fun w => ↑(P w).ℙ) w dw;\n SciLean.Distribution.extAction' dP fun x => SciLean.Scalar.log (SciLean.Rand.pdf R (P w) Q.ℙ x) - 1"} +{"name":"SciLean.KLDiv","declaration":"/-- Kullback–Leibler divergence of `Dₖₗ(P‖Q)` -/\ndef SciLean.KLDiv {R : Type u_1} [SciLean.RealScalar R] {X : Type} [MeasurableSpace X] (P : SciLean.Rand X) (Q : SciLean.Rand X) : R"} +{"name":"SciLean.kldiv_elbo","declaration":"/-- Express `Kullback–Leibler divergence` as log evidence + ELBO -/\ntheorem SciLean.kldiv_elbo {R : Type u_1} [SciLean.RealScalar R] {X : Type} {Z : Type} [MeasureTheory.MeasureSpace Z] [MeasureTheory.MeasureSpace X] (P : SciLean.Rand (Z × X)) (Q : SciLean.Rand Z) (x : X) : SciLean.KLDiv Q (SciLean.Rand.conditionSnd P x) =\n SciLean.Scalar.log (SciLean.Rand.pdf R P.snd MeasureTheory.volume x) - SciLean.ELBO P Q x"} +{"name":"SciLean.ELBO","declaration":"/-- Evidence Lower Bound\n\n```\n𝔼_Q [log Q(Z) - log P(Z,X)]\n```\nreference: https://en.wikipedia.org/wiki/Variational_Bayesian_methods#Evidence_lower_bound\n -/\ndef SciLean.ELBO {R : Type u_1} [SciLean.RealScalar R] {X : Type} {Z : Type} [MeasureTheory.MeasureSpace Z] [MeasureTheory.MeasureSpace X] (P : SciLean.Rand (Z × X)) (Q : SciLean.Rand Z) (x : X) : R"} diff --git a/scilean-declarations/SciLean.Core.Rand.jsonl b/scilean-declarations/SciLean.Core.Rand.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Core.Simp.jsonl b/scilean-declarations/SciLean.Core.Simp.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..3d59e3354287c7301c34a6933a745c6bb476a1d3 --- /dev/null +++ b/scilean-declarations/SciLean.Core.Simp.jsonl @@ -0,0 +1,20 @@ +{"name":"Parser.Attr.add_pull_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.add_pull_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.add_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.add_push_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.smul_push","declaration":"def Parser.Attr.smul_push : Lean.ParserDescr"} +{"name":"Parser.Attr.sum_pull_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.sum_pull_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.neg_push","declaration":"def Parser.Attr.neg_push : Lean.ParserDescr"} +{"name":"Parser.Attr.sum_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.sum_push_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.add_push","declaration":"def Parser.Attr.add_push : Lean.ParserDescr"} +{"name":"Parser.Attr.add_pull","declaration":"def Parser.Attr.add_pull : Lean.ParserDescr"} +{"name":"Parser.Attr.neg_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.neg_push_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.sub_pull","declaration":"def Parser.Attr.sub_pull : Lean.ParserDescr"} +{"name":"Parser.Attr.sum_pull","declaration":"def Parser.Attr.sum_pull : Lean.ParserDescr"} +{"name":"Parser.Attr.neg_pull","declaration":"def Parser.Attr.neg_pull : Lean.ParserDescr"} +{"name":"Parser.Attr.sub_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.sub_push_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.neg_pull_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.neg_pull_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.sub_push","declaration":"def Parser.Attr.sub_push : Lean.ParserDescr"} +{"name":"Parser.Attr.sum_push","declaration":"def Parser.Attr.sum_push : Lean.ParserDescr"} +{"name":"Parser.Attr.sub_pull_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.sub_pull_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.smul_pull_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.smul_pull_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.smul_push_proc","declaration":"/-- Simplification procedure -/\ndef Parser.Attr.smul_push_proc : Lean.ParserDescr"} +{"name":"Parser.Attr.smul_pull","declaration":"def Parser.Attr.smul_pull : Lean.ParserDescr"} diff --git a/scilean-declarations/SciLean.Core.jsonl b/scilean-declarations/SciLean.Core.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Data.ArraySet.jsonl b/scilean-declarations/SciLean.Data.ArraySet.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..9994d31b1ebf1bf1a04a690f6086993cdd9aa9e6 --- /dev/null +++ b/scilean-declarations/SciLean.Data.ArraySet.jsonl @@ -0,0 +1,19 @@ +{"name":"Array.toArraySet","declaration":"def Array.toArraySet {α : Type u_1} [ord : Ord α] (as : Array α) : SciLean.ArraySet α"} +{"name":"SciLean.ArraySet.lexOrd","declaration":"def SciLean.ArraySet.lexOrd {α : Type u_1} [Ord α] (a : SciLean.ArraySet α) (b : SciLean.ArraySet α) : Ordering"} +{"name":"SciLean.instHashableArraySet","declaration":"instance SciLean.instHashableArraySet : {α : Type u_1} → {ord : Ord α} → [inst : Hashable α] → Hashable (SciLean.ArraySet α)"} +{"name":"SciLean.ArraySet.mem","declaration":"def SciLean.ArraySet.mem {α : Type u_1} [ord : Ord α] (as : SciLean.ArraySet α) (a : α) [DecidableEq α] : Bool"} +{"name":"SciLean.ArraySet.instGetElemArraySetNatLtInstLTNatSize","declaration":"instance SciLean.ArraySet.instGetElemArraySetNatLtInstLTNatSize {α : Type u_1} [ord : Ord α] : GetElem (SciLean.ArraySet α) Nat α fun as i => i < SciLean.ArraySet.size as"} +{"name":"SciLean.ArraySet.instToStringArraySet","declaration":"instance SciLean.ArraySet.instToStringArraySet {α : Type u_1} [ord : Ord α] [ToString α] : ToString (SciLean.ArraySet α)"} +{"name":"SciLean.ArraySet.size","declaration":"def SciLean.ArraySet.size {α : Type u_1} [ord : Ord α] (as : SciLean.ArraySet α) : Nat"} +{"name":"SciLean.ArraySet.instDecidableSubsetArraySetInstHasSubsetArraySet","declaration":"instance SciLean.ArraySet.instDecidableSubsetArraySetInstHasSubsetArraySet {α : Type u_1} [ord : Ord α] (a : SciLean.ArraySet α) (b : SciLean.ArraySet α) : Decidable (a ⊆ b)"} +{"name":"SciLean.ArraySet.instMembershipArraySet","declaration":"instance SciLean.ArraySet.instMembershipArraySet {α : Type u_1} [ord : Ord α] [DecidableEq α] : Membership α (SciLean.ArraySet α)"} +{"name":"SciLean.ArraySet.isSet","declaration":"def SciLean.ArraySet.isSet {α : Type u_1} [ord : Ord α] (self : SciLean.ArraySet α) : Array.sortAndDeduplicate self.data = self.data"} +{"name":"SciLean.ArraySet.mk","declaration":"ctor SciLean.ArraySet.mk {α : Type u_1} [ord : Ord α] (data : Array α) (isSet : Array.sortAndDeduplicate data = data) : SciLean.ArraySet α"} +{"name":"SciLean.ArraySet.instHasSubsetArraySet","declaration":"instance SciLean.ArraySet.instHasSubsetArraySet {α : Type u_1} [ord : Ord α] : HasSubset (SciLean.ArraySet α)"} +{"name":"SciLean.ArraySet.instDecidableMemArraySetInstMembershipArraySet","declaration":"instance SciLean.ArraySet.instDecidableMemArraySetInstMembershipArraySet {α : Type u_1} [ord : Ord α] [DecidableEq α] (a : α) (as : SciLean.ArraySet α) : Decidable (a ∈ as)"} +{"name":"SciLean.ArraySet.instDecidableEqArraySet","declaration":"instance SciLean.ArraySet.instDecidableEqArraySet {α : Type u_1} [ord : Ord α] (a : SciLean.ArraySet α) (b : SciLean.ArraySet α) [DecidableEq α] : Decidable (a = b)"} +{"name":"SciLean.ArraySet","declaration":"/-- `Array α` that is guaranteed to be sorted based on `[Ord α]` and has no duplicates.\n\nWARRNING: `Ord α` is assumed to be lawful, currently there is no typeclass for it.\n-/\nstructure SciLean.ArraySet (α : Type u_1) [ord : Ord α] : Type u_1"} +{"name":"SciLean.ArraySet.instCoeArraySetArray","declaration":"instance SciLean.ArraySet.instCoeArraySetArray {α : Type u_1} [ord : Ord α] : Coe (SciLean.ArraySet α) (Array α)"} +{"name":"SciLean.ArraySet.toArray","declaration":"def SciLean.ArraySet.toArray {α : Type u_1} [ord : Ord α] (as : SciLean.ArraySet α) : Array α"} +{"name":"SciLean.ArraySet.data","declaration":"def SciLean.ArraySet.data {α : Type u_1} [ord : Ord α] (self : SciLean.ArraySet α) : Array α"} +{"name":"SciLean.ArraySet.toList","declaration":"def SciLean.ArraySet.toList {α : Type u_1} [ord : Ord α] (as : SciLean.ArraySet α) : List α"} diff --git a/scilean-declarations/SciLean.Data.ArrayType.Algebra.jsonl b/scilean-declarations/SciLean.Data.ArrayType.Algebra.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..1ceaf166b8176fe57706d07237c9f8396294f062 --- /dev/null +++ b/scilean-declarations/SciLean.Data.ArrayType.Algebra.jsonl @@ -0,0 +1,25 @@ +{"name":"SciLean.ArrayType.instVecStructInstStructTypeInstVec","declaration":"instance SciLean.ArrayType.instVecStructInstStructTypeInstVec {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {K : Type u_4} [RCLike K] [SciLean.ArrayType Cont Idx Elem] [SciLean.Vec K Elem] : SciLean.VecStruct K Cont Idx fun x => Elem"} +{"name":"SciLean.ArrayType.instOrthonormalBasisToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisToInnerToSemiInnerProductSpaceInstSemiHilbertToSemiHilbertUnitInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnitInstDecidableEqPUnitInstFinVecUnitInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnitInstDecidableEqPUnit","declaration":"instance SciLean.ArrayType.instOrthonormalBasisToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToDenselyNormedFieldInstBasisToInnerToSemiInnerProductSpaceInstSemiHilbertToSemiHilbertUnitInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnitInstDecidableEqPUnitInstFinVecUnitInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnitInstDecidableEqPUnit {Cont : Type u_1} {Idx : Type u_2} [SciLean.IndexType Idx] [DecidableEq Idx] {K : Type u_1} [RCLike K] [SciLean.ArrayType Cont Idx K] : SciLean.OrthonormalBasis Idx K Cont"} +{"name":"SciLean.ArrayType.instFinVec","declaration":"instance SciLean.ArrayType.instFinVec {Cont : Type u_1} {Idx : Type u_2} [SciLean.IndexType Idx] [SciLean.LawfulIndexType Idx] [DecidableEq Idx] {K : Type u_1} [RCLike K] [SciLean.ArrayType Cont Idx K] : SciLean.FinVec Idx K Cont"} +{"name":"SciLean.ArrayType.instModuleToSemiringToAddCommMonoidInstAddCommGroup","declaration":"instance SciLean.ArrayType.instModuleToSemiringToAddCommMonoidInstAddCommGroup {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {R : Type u_4} [CommSemiring R] [SciLean.ArrayType Cont Idx Elem] [AddCommGroup Elem] [Module R Elem] : Module R Cont"} +{"name":"SciLean.ArrayType.instSemiInnerProductSpace","declaration":"instance SciLean.ArrayType.instSemiInnerProductSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {K : Type u_1} [RCLike K] [SciLean.ArrayType Cont Idx Elem] [SciLean.SemiInnerProductSpace K Elem] : SciLean.SemiInnerProductSpace K Cont"} +{"name":"SciLean.ArrayType.instSMulStructInstStructTypeInstSMul","declaration":"instance SciLean.ArrayType.instSMulStructInstStructTypeInstSMul {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {K : Type u_4} [SciLean.ArrayType Cont Idx Elem] [SMul K Elem] : SciLean.SMulStruct K Cont Idx fun x => Elem"} +{"name":"SciLean.ArrayType.instBasisDuality","declaration":"instance SciLean.ArrayType.instBasisDuality {Cont : Type u_1} {Idx : Type u_2} {K : Type u_3} [SciLean.ArrayType Cont Idx K] : SciLean.BasisDuality Cont"} +{"name":"SciLean.ArrayType.instAddStructInstStructTypeInstAdd","declaration":"instance SciLean.ArrayType.instAddStructInstStructTypeInstAdd {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Add Elem] : SciLean.AddStruct Cont Idx fun x => Elem"} +{"name":"SciLean.ArrayType.instContinuousSMulInstSMulInstTopologicalSpace","declaration":"instance SciLean.ArrayType.instContinuousSMulInstSMulInstTopologicalSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {R : Type u_4} [SciLean.ArrayType Cont Idx Elem] [TopologicalSpace R] [TopologicalSpace Elem] [SMul R Elem] [ContinuousSMul R Elem] : ContinuousSMul R Cont"} +{"name":"SciLean.ArrayType.instAddCommGroup","declaration":"instance SciLean.ArrayType.instAddCommGroup {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [AddCommGroup Elem] : AddCommGroup Cont"} +{"name":"SciLean.ArrayType.instVec","declaration":"instance SciLean.ArrayType.instVec {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {K : Type u_4} [RCLike K] [SciLean.ArrayType Cont Idx Elem] [SciLean.Vec K Elem] : SciLean.Vec K Cont"} +{"name":"SciLean.ArrayType.instUniformSpace","declaration":"instance SciLean.ArrayType.instUniformSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.ArrayType Cont Idx Elem] [UniformSpace Elem] : UniformSpace Cont"} +{"name":"SciLean.ArrayType.instTopologicalSpace","declaration":"instance SciLean.ArrayType.instTopologicalSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.ArrayType Cont Idx Elem] [TopologicalSpace Elem] : TopologicalSpace Cont"} +{"name":"SciLean.ArrayType.instAddGroup","declaration":"instance SciLean.ArrayType.instAddGroup {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [AddGroup Elem] : AddGroup Cont"} +{"name":"SciLean.ArrayType.instInner","declaration":"instance SciLean.ArrayType.instInner {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {K : Type u_4} [RCLike K] [SciLean.ArrayType Cont Idx Elem] [Inner K Elem] : Inner K Cont"} +{"name":"SciLean.ArrayType.instSemiHilbert","declaration":"instance SciLean.ArrayType.instSemiHilbert {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] {K : Type u_1} [RCLike K] [SciLean.ArrayType Cont Idx Elem] [SciLean.SemiHilbert K Elem] : SciLean.SemiHilbert K Cont"} +{"name":"SciLean.ArrayType.instSemiInnerProductSpaceStructInstStructTypeInstSemiInnerProductSpace","declaration":"instance SciLean.ArrayType.instSemiInnerProductSpaceStructInstStructTypeInstSemiInnerProductSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] [SciLean.LawfulIndexType Idx] {K : Type u_1} [RCLike K] [SciLean.ArrayType Cont Idx Elem] [SciLean.SemiInnerProductSpace K Elem] : SciLean.SemiInnerProductSpaceStruct K Cont Idx fun x => Elem"} +{"name":"SciLean.ArrayType.instCompleteSpaceInstUniformSpace","declaration":"instance SciLean.ArrayType.instCompleteSpaceInstUniformSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.ArrayType Cont Idx Elem] [UniformSpace Elem] [CompleteSpace Elem] : CompleteSpace Cont"} +{"name":"SciLean.ArrayType.instTestFunctions","declaration":"instance SciLean.ArrayType.instTestFunctions {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} {K : Type u_4} [RCLike K] [SciLean.ArrayType Cont Idx Elem] [SciLean.Vec K Elem] [SciLean.TestFunctions Elem] : SciLean.TestFunctions Cont"} +{"name":"SciLean.ArrayType.instBasis","declaration":"instance SciLean.ArrayType.instBasis {Cont : Type u_1} {Idx : Type u_2} [DecidableEq Idx] {K : Type u_3} [RCLike K] [SciLean.ArrayType Cont Idx K] : SciLean.Basis Idx K Cont"} +{"name":"SciLean.ArrayType.instMeasurableSpace","declaration":"instance SciLean.ArrayType.instMeasurableSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.ArrayType Cont Idx Elem] [MeasurableSpace Elem] : MeasurableSpace Cont"} +{"name":"SciLean.ArrayType.instUniformAddGroupInstUniformSpaceInstAddGroup","declaration":"instance SciLean.ArrayType.instUniformAddGroupInstUniformSpaceInstAddGroup {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [UniformSpace Elem] [AddGroup Elem] [UniformAddGroup Elem] : UniformAddGroup Cont"} +{"name":"SciLean.ArrayType.instDualBasis","declaration":"instance SciLean.ArrayType.instDualBasis {Cont : Type u_1} {Idx : Type u_2} [DecidableEq Idx] {K : Type u_3} [RCLike K] [SciLean.ArrayType Cont Idx K] : SciLean.DualBasis Idx K Cont"} +{"name":"SciLean.ArrayType.instMeasureSpace","declaration":"instance SciLean.ArrayType.instMeasureSpace {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.ArrayType Cont Idx Elem] [MeasureTheory.MeasureSpace Elem] : MeasureTheory.MeasureSpace Cont"} +{"name":"SciLean.ArrayType.instZeroStructInstStructTypeInstZero","declaration":"instance SciLean.ArrayType.instZeroStructInstStructTypeInstZero {Cont : Type u_1} {Elem : outParam (Type u_2)} {Idx : Type u_3} [SciLean.ArrayType Cont Idx Elem] [Zero Elem] : SciLean.ZeroStruct Cont Idx fun x => Elem"} diff --git a/scilean-declarations/SciLean.Data.ArrayType.Basic.jsonl b/scilean-declarations/SciLean.Data.ArrayType.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..aa9ff258046437fa0b88319fe5d276e32550b96b --- /dev/null +++ b/scilean-declarations/SciLean.Data.ArrayType.Basic.jsonl @@ -0,0 +1,46 @@ +{"name":"SciLean.ArrayType.instMul","declaration":"instance SciLean.ArrayType.instMul {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Mul Elem] : Mul Cont"} +{"name":"SciLean.ArrayType.mapMono","declaration":"def SciLean.ArrayType.mapMono {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] (f : Elem → Elem) (cont : Cont) : Cont"} +{"name":"SciLean.ArrayType.div_get","declaration":"theorem SciLean.ArrayType.div_get {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Div Elem] (x : Cont) (y : Cont) (i : Idx) : (x / y)[i] = x[i] / y[i]"} +{"name":"SciLean.ArrayType.instZero","declaration":"instance SciLean.ArrayType.instZero {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.ArrayType Cont Idx Elem] [Zero Elem] : Zero Cont"} +{"name":"SciLean.ArrayType.instDecidableEq","declaration":"instance SciLean.ArrayType.instDecidableEq {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [DecidableEq Elem] : DecidableEq Cont"} +{"name":"SciLean.ArrayType.getElem_mapIdxMono","declaration":"theorem SciLean.ArrayType.getElem_mapIdxMono {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] (f : Idx → Elem → Elem) (cont : Cont) (i : Idx) : (SciLean.ArrayType.mapIdxMono f cont)[i] = f i cont[i]"} +{"name":"SciLean.ArrayType.instLE","declaration":"instance SciLean.ArrayType.instLE {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.ArrayType Cont Idx Elem] [LE Elem] : LE Cont"} +{"name":"SciLean.ArrayType.eta","declaration":"theorem SciLean.ArrayType.eta {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.ArrayType Cont Idx Elem] (cont : Cont) : (LeanColls.Indexed.ofFn fun i => cont[i]) = cont"} +{"name":"SciLean.ArrayType.argMinCore","declaration":"def SciLean.ArrayType.argMinCore {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LT Elem] [(x y : Elem) → Decidable (x < y)] [Inhabited Idx] (cont : Cont) : Idx × Elem"} +{"name":"SciLean.ArrayType.instDecidableLeInstLE","declaration":"instance SciLean.ArrayType.instDecidableLeInstLE {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LE Elem] [(x y : Elem) → Decidable (x ≤ y)] (f : Cont) (g : Cont) : Decidable (f ≤ g)"} +{"name":"SciLean.ArrayType.mapIdxMono","declaration":"def SciLean.ArrayType.mapIdxMono {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] (f : Idx → Elem → Elem) (cont : Cont) : Cont"} +{"name":"SciLean.ArrayType.idxMax","declaration":"def SciLean.ArrayType.idxMax {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LT Elem] [(x y : Elem) → Decidable (x < y)] [Inhabited Idx] (cont : Cont) : Idx"} +{"name":"SciLean.ArrayType.instSub","declaration":"instance SciLean.ArrayType.instSub {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Sub Elem] : Sub Cont"} +{"name":"SciLean.ArrayType.zero_get","declaration":"theorem SciLean.ArrayType.zero_get {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.ArrayType Cont Idx Elem] [Zero Elem] (i : Idx) : 0[i] = 0"} +{"name":"SciLean.ArrayType.instToString","declaration":"instance SciLean.ArrayType.instToString {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.ArrayType Cont Idx Elem] [ToString Elem] [SciLean.IndexType Idx] : ToString Cont"} +{"name":"SciLean.ArrayType.instDiv","declaration":"instance SciLean.ArrayType.instDiv {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Div Elem] : Div Cont"} +{"name":"SciLean.ArrayType.instLT","declaration":"instance SciLean.ArrayType.instLT {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.ArrayType Cont Idx Elem] [LT Elem] : LT Cont"} +{"name":"SciLean.ArrayType.sum_ofFn","declaration":"theorem SciLean.ArrayType.sum_ofFn {Cont : Type u_1} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [AddCommMonoid Elem] {ι : Type u_2} [SciLean.IndexType ι] (f : ι → Idx → Elem) : ( ∑ j, LeanColls.Indexed.ofFn fun i => f j i) = LeanColls.Indexed.ofFn fun i => ∑ j, f j i"} +{"name":"SciLean.ArrayType.mul_get","declaration":"theorem SciLean.ArrayType.mul_get {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Mul Elem] (x : Cont) (y : Cont) (i : Idx) : (x * y)[i] = x[i] * y[i]"} +{"name":"SciLean.ArrayType.neg_ofFn","declaration":"theorem SciLean.ArrayType.neg_ofFn {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Neg Elem] (f : Idx → Elem) : -LeanColls.Indexed.ofFn f = LeanColls.Indexed.ofFn fun i => -f i"} +{"name":"SciLean.ArrayType.neg_get","declaration":"theorem SciLean.ArrayType.neg_get {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Neg Elem] (x : Cont) (i : Idx) : (-x)[i] = -x[i]"} +{"name":"SciLean.ArrayType.instDecidableLtInstLT","declaration":"instance SciLean.ArrayType.instDecidableLtInstLT {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LT Elem] [(x y : Elem) → Decidable (x < y)] (f : Cont) (g : Cont) : Decidable (f < g)"} +{"name":"SciLean.ArrayType.getElem_mapMono","declaration":"theorem SciLean.ArrayType.getElem_mapMono {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] (f : Elem → Elem) (cont : Cont) (i : Idx) : (SciLean.ArrayType.mapMono f cont)[i] = f cont[i]"} +{"name":"SciLean.ArrayType.add_ofFn","declaration":"theorem SciLean.ArrayType.add_ofFn {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Add Elem] (f : Idx → Elem) (g : Idx → Elem) : LeanColls.Indexed.ofFn f + LeanColls.Indexed.ofFn g = LeanColls.Indexed.ofFn fun i => f i + g i"} +{"name":"SciLean.ArrayType.smul_get","declaration":"theorem SciLean.ArrayType.smul_get {Cont : Type u_3} {Idx : outParam (Type u_4)} {Elem : outParam (Type u_2)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] {R : Type u_1} [SMul R Elem] (r : R) (x : Cont) (i : Idx) : (r • x)[i] = r • x[i]"} +{"name":"SciLean.ArrayType.instInv","declaration":"instance SciLean.ArrayType.instInv {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Inv Elem] : Inv Cont"} +{"name":"SciLean.ArrayType.instSMul","declaration":"instance SciLean.ArrayType.instSMul {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] {R : Type u_4} [SMul R Elem] : SMul R Cont"} +{"name":"SciLean.ArrayType.instOne","declaration":"instance SciLean.ArrayType.instOne {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.ArrayType Cont Idx Elem] [One Elem] : One Cont"} +{"name":"SciLean.ArrayType.one_get","declaration":"theorem SciLean.ArrayType.one_get {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.ArrayType Cont Idx Elem] [One Elem] (i : Idx) : 1[i] = 1"} +{"name":"SciLean.ArrayType.get_injective","declaration":"def SciLean.ArrayType.get_injective {Cont : Type u} {Idx : outParam (Type v)} {Elem : outParam (Type w)} [self : SciLean.ArrayType Cont Idx Elem] : Function.Injective fun c i => c[i]"} +{"name":"SciLean.ArrayType.argMaxCore","declaration":"def SciLean.ArrayType.argMaxCore {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LT Elem] [(x y : Elem) → Decidable (x < y)] [Inhabited Idx] (cont : Cont) : Idx × Elem"} +{"name":"Array.toArrayType","declaration":"/-- Converts array to ArrayType -/\ndef Array.toArrayType {Elem : Type u_1} (Cont : Type u) (Idx : Type v) [SciLean.IndexType Idx] [LeanColls.Indexed Cont Idx Elem] (a : Array Elem) (h : SciLean.IndexType.card Idx = Array.size a) : Cont"} +{"name":"SciLean.ArrayType.add_get","declaration":"theorem SciLean.ArrayType.add_get {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Add Elem] (x : Cont) (y : Cont) (i : Idx) : (x + y)[i] = x[i] + y[i]"} +{"name":"SciLean.ArrayType.sub_ofFn","declaration":"theorem SciLean.ArrayType.sub_ofFn {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Sub Elem] (f : Idx → Elem) (g : Idx → Elem) : LeanColls.Indexed.ofFn f - LeanColls.Indexed.ofFn g = LeanColls.Indexed.ofFn fun i => f i - g i"} +{"name":"SciLean.ArrayType.mk","declaration":"ctor SciLean.ArrayType.mk {Cont : Type u} {Idx : outParam (Type v)} {Elem : outParam (Type w)} [toIndexed : LeanColls.Indexed Cont Idx Elem] [toLawfulIndexed : LeanColls.LawfulIndexed Cont Idx Elem] (get_injective : Function.Injective fun c i => c[i]) : SciLean.ArrayType Cont Idx Elem"} +{"name":"SciLean.ArrayType.smul_ofFn","declaration":"theorem SciLean.ArrayType.smul_ofFn {Cont : Type u_3} {Idx : outParam (Type u_4)} {Elem : outParam (Type u_2)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] {K : Type u_1} [SMul K Elem] (f : Idx → Elem) (a : K) : a • LeanColls.Indexed.ofFn f = LeanColls.Indexed.ofFn fun i => a • f i"} +{"name":"SciLean.instStructType","declaration":"instance SciLean.instStructType {Cont : Type u_1} {Idx : Type u_2} {Elem : Type u_3} [SciLean.ArrayType Cont Idx Elem] : SciLean.StructType Cont Idx fun x => Elem"} +{"name":"SciLean.ArrayType","declaration":"/-- This class says that `Cont` behaves like an array with `Elem` values indexed by `Idx`\n\nExamples for `Idx = Fin n` and `Elem = ℝ` are: `ArrayN ℝ n` or `ℝ^{n}`\n\nFor `array : Cont` you can:\n1. get values: `Indexed.get array i : Elem` for `i : Idx`\n2. set values: `Indexed.set array i x : Cont` for `i : Idx` and `x : Elem`\n3. make new a array: `Indexed.ofFn f : Cont` for `f : Idx → Elem`\n\nAlternative notation:\n1. `array[x]`\n2. in `do` block: `array[x] := y`, `array[x] += y`, ...\n3. `λ [x] => f x` this notation works only if the type `Cont` can be infered from the context\n Common use: `let array : Cont := λ [x] => f x` where the type asscription `: Cont` is important.\n-/\nclass SciLean.ArrayType (Cont : Type u) (Idx : outParam (Type v)) (Elem : outParam (Type w)) : Type (max (max u v) (w + 1))"} +{"name":"SciLean.ArrayType.instNeg","declaration":"instance SciLean.ArrayType.instNeg {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Neg Elem] : Neg Cont"} +{"name":"SciLean.ArrayType.max","declaration":"def SciLean.ArrayType.max {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LT Elem] [(x y : Elem) → Decidable (x < y)] [Inhabited Idx] (cont : Cont) : Elem"} +{"name":"SciLean.ArrayType.min","declaration":"def SciLean.ArrayType.min {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LT Elem] [(x y : Elem) → Decidable (x < y)] [Inhabited Idx] (cont : Cont) : Elem"} +{"name":"SciLean.ArrayType.sub_get","declaration":"theorem SciLean.ArrayType.sub_get {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Sub Elem] (x : Cont) (y : Cont) (i : Idx) : (x - y)[i] = x[i] - y[i]"} +{"name":"SciLean.ArrayType.getElem_map","declaration":"theorem SciLean.ArrayType.getElem_map {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] (f : Elem → Elem) (cont : Cont) (i : Idx) : (SciLean.ArrayType.mapMono f cont)[i] = f cont[i]"} +{"name":"SciLean.ArrayType.idxMin","declaration":"def SciLean.ArrayType.idxMin {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [LT Elem] [(x y : Elem) → Decidable (x < y)] [Inhabited Idx] (cont : Cont) : Idx"} +{"name":"SciLean.ArrayType.ext","declaration":"theorem SciLean.ArrayType.ext {Cont : Type u_2} {Idx : outParam (Type u_3)} {Elem : outParam (Type u_1)} [SciLean.ArrayType Cont Idx Elem] (x : Cont) (y : Cont) : (∀ (i : Idx), x[i] = y[i]) → x = y"} +{"name":"SciLean.ArrayType.instAdd","declaration":"instance SciLean.ArrayType.instAdd {Cont : Type u_1} {Idx : outParam (Type u_2)} {Elem : outParam (Type u_3)} [SciLean.IndexType Idx] [SciLean.ArrayType Cont Idx Elem] [Add Elem] : Add Cont"} diff --git a/scilean-declarations/SciLean.Data.ArrayType.Notation.jsonl b/scilean-declarations/SciLean.Data.ArrayType.Notation.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e98f37b6320fbe1db3ad5c84ac0a53a5c72d474c --- /dev/null +++ b/scilean-declarations/SciLean.Data.ArrayType.Notation.jsonl @@ -0,0 +1,16 @@ +{"name":"SciLean.ArrayType.PowerNotation.dimSpec_","declaration":"def SciLean.ArrayType.PowerNotation.dimSpec_ : Lean.ParserDescr"} +{"name":"SciLean.ArrayType.PowerNotation.typeIntPower","declaration":"/-- `x^[y]` is either array type or iterated function depending on the type of `x`.\n\n**iterated function** `f^[n]` call `f` n-times e.g. `f^[3] = f∘f∘f`\n\n**type product** `Float^[n]` array of n elemts with values in `Float`\n\nThe array notation is quite flexible and allows you to create arrays indexed with various types.\nExamples where `n m k l : USize`, `a b : Int64` and `ι κ : Type` are types with `Index _` instance:\n- `Float^[n]` index type: `Idx n` i.e. numbers `0,...,n-1`\n- `Float^[n,m]` index type: `Idx n × Idx m` i.e. paris `(0,0),(0,1),...,(1,0),(1,1),...,(n-1,m-1)`\n- `Float^[[-a:b]]` index type :`Idx' (-a) b` i.e. closed interval from `-a` to `b` (`={-a, -a+1, ..., b-1, b}`)\n- `Float^[k,l,m]` index type: `Idx k × Idx l × Idx m` - type product is right associated by default\n- `Float^[[k,l],m]` index type: `(Idx k × Idx l) × Idx m` - left associated product requires explicit brackets\n- `Float^[ι]` index type `ι` - generic type with `Index ι` instances\n- `Float^[ι,[10,[-a:b]],κ]` index type: `ι × (Idx 10 × Idx' (-a) b) × κ` - mix of all above\n-/\ndef SciLean.ArrayType.PowerNotation.typeIntPower : Lean.TrailingParserDescr"} +{"name":"SciLean.«term⊞_=>_»","declaration":"def SciLean.«term⊞_=>_» : Lean.ParserDescr"} +{"name":"SciLean.ArrayType.PowerNotation.«dimSpec[_:_]»","declaration":"def SciLean.ArrayType.PowerNotation.«dimSpec[_:_]» : Lean.ParserDescr"} +{"name":"SciLean.introElemNotation","declaration":"def SciLean.introElemNotation {Cont : Type u_1} {Idx : Type u_2} {Elem : Type u_3} [DecidableEq Idx] [SciLean.ArrayType Cont Idx Elem] [SciLean.ArrayTypeNotation Cont Idx Elem] (f : Idx → Elem) : Cont"} +{"name":"SciLean.ArrayType.PowerNotation.«dimSpec[_]»","declaration":"def SciLean.ArrayType.PowerNotation.«dimSpec[_]» : Lean.ParserDescr"} +{"name":"SciLean.unexpandIntroElemNotation","declaration":"def SciLean.unexpandIntroElemNotation : Lean.PrettyPrinter.Unexpander"} +{"name":"Lean.Parser.Category.dimSpec","declaration":"def Lean.Parser.Category.dimSpec : Lean.Parser.Category"} +{"name":"SciLean.ArrayType.PowerNotation.dimSpec.quot","declaration":"def SciLean.ArrayType.PowerNotation.dimSpec.quot : Lean.ParserDescr"} +{"name":"SciLean.unexpandArrayToArrayType","declaration":"def SciLean.unexpandArrayToArrayType : Lean.PrettyPrinter.Unexpander"} +{"name":"SciLean.ArrayTypeNotation","declaration":"class SciLean.ArrayTypeNotation (Cont : outParam (Type u_1)) (Idx : Type u_2) (Elem : Type u_3) : Type"} +{"name":"SciLean.typeOf","declaration":"def SciLean.typeOf {α : Sort u_1} : α → Sort u_1"} +{"name":"SciLean.ArrayTypeNotation.mk","declaration":"ctor SciLean.ArrayTypeNotation.mk {Cont : outParam (Type u_1)} {Idx : Type u_2} {Elem : Type u_3} : SciLean.ArrayTypeNotation Cont Idx Elem"} +{"name":"SciLean.arrayTypeLiteral","declaration":"def SciLean.arrayTypeLiteral : Lean.ParserDescr"} +{"name":"SciLean.arrayTypeCont","declaration":"def SciLean.arrayTypeCont (Idx : Type u_1) (Elem : Type u_2) {Cont : outParam (Type u_3)} [SciLean.ArrayTypeNotation Cont Idx Elem] : outParam (Type u_3)"} +{"name":"SciLean.ArrayType.PowerNotation.expand'","declaration":"opaque SciLean.ArrayType.PowerNotation.expand' (l : List (Lean.TSyntax `dimSpec)) : Lean.Elab.TermElabM Lean.Expr"} diff --git a/scilean-declarations/SciLean.Data.ArrayType.Properties.jsonl b/scilean-declarations/SciLean.Data.ArrayType.Properties.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..02e5e0bd6cefb8815406296289c52950340505f9 --- /dev/null +++ b/scilean-declarations/SciLean.Data.ArrayType.Properties.jsonl @@ -0,0 +1,26 @@ +{"name":"GetElem.getElem.arg_cont.revCDerivProj_rule","declaration":"theorem GetElem.getElem.arg_cont.revCDerivProj_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] {J : Type u_1} {ElemJ : J → Type u_2} [SciLean.StructType Elem J ElemJ] [SciLean.IndexType J] [SciLean.LawfulIndexType J] [DecidableEq J] [(j : J) → SciLean.SemiInnerProductSpace K (ElemJ j)] [SciLean.SemiInnerProductSpaceStruct K Elem J ElemJ] (f : X → Cont) (idx : Idx) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K J fun x => (f x)[idx]) = fun x =>\n let ydf := SciLean.revDerivProj K (Idx × J) f x;\n (ydf.1[idx], fun j delem => ydf.2 (idx, j) delem)"} +{"name":"LeanColls.Indexed.set.arg_contelem.IsSmoothLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.set.arg_contelem.IsSmoothLinearMap_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.Vec K Elem] (idx : Idx) : SciLean.IsSmoothLinearMap K fun x =>\n match x with\n | (cont, elem) => LeanColls.Indexed.set cont idx elem"} +{"name":"LeanColls.Indexed.get.arg_cont.IsContinuousLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.get.arg_cont.IsContinuousLinearMap_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [NormedAddCommGroup Elem] [NormedSpace K Elem] (idx : Idx) : SciLean.IsContinuousLinearMap K fun cont => cont[idx]"} +{"name":"LeanColls.Indexed.ofFn.arg_cont.revCDerivProjUpdate_rule","declaration":"theorem LeanColls.Indexed.ofFn.arg_cont.revCDerivProjUpdate_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (f : X → Idx → Elem) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K Idx fun x => LeanColls.Indexed.ofFn (f x)) = fun x =>\n let fdf := SciLean.revDerivProjUpdate K Idx f x;\n (LeanColls.Indexed.ofFn fdf.1, fdf.2)"} +{"name":"GetElem.getElem.arg_cont.revCDeriv_rule","declaration":"theorem GetElem.getElem.arg_cont.revCDeriv_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (f : X → Cont) (idx : Idx) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => (f x)[idx]) = fun x =>\n let ydf := SciLean.revDerivProj K Idx f x;\n (ydf.1[idx], fun delem => ydf.2 idx delem)"} +{"name":"LeanColls.Indexed.set.arg_contelem.revCDerivUpdate_rule","declaration":"theorem LeanColls.Indexed.set.arg_contelem.revCDerivUpdate_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (cont : X → Cont) (idx : Idx) (elem : X → Elem) (hcont : SciLean.HasAdjDiff K cont) (helem : SciLean.HasAdjDiff K elem) : (SciLean.revDerivUpdate K fun x => LeanColls.Indexed.set (cont x) idx (elem x)) = fun x =>\n let cdc := SciLean.revDerivUpdate K cont x;\n let ede := SciLean.revDerivUpdate K elem x;\n (LeanColls.Indexed.set cdc.1 idx ede.1, fun dc dx =>\n let dci := dc[idx];\n let dc := LeanColls.Indexed.set dc idx 0;\n ede.2 dci (cdc.2 dc dx))"} +{"name":"LeanColls.Indexed.ofFn.arg_cont.revCDeriv_rule","declaration":"theorem LeanColls.Indexed.ofFn.arg_cont.revCDeriv_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (f : X → Idx → Elem) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => LeanColls.Indexed.ofFn (f x)) = fun x =>\n let fdf := SciLean.revDeriv K f x;\n (LeanColls.Indexed.ofFn fdf.1, fun dcont => fdf.2 fun idx => dcont[idx])"} +{"name":"GetElem.getElem.arg_cont.HasSemiAdjoint_rule_simple","declaration":"theorem GetElem.getElem.arg_cont.HasSemiAdjoint_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.SemiInnerProductSpace K Elem] (idx : Idx) : SciLean.HasSemiAdjoint K fun cont => cont[idx]"} +{"name":"LeanColls.Indexed.ofFn.arg_cont.HasSemiAdjoint_rule_simple","declaration":"theorem LeanColls.Indexed.ofFn.arg_cont.HasSemiAdjoint_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.SemiInnerProductSpace K Elem] : SciLean.HasSemiAdjoint K fun f => LeanColls.Indexed.ofFn f"} +{"name":"LeanColls.Indexed.set.arg_contelem.revCDerivProjUpdate_rule","declaration":"theorem LeanColls.Indexed.set.arg_contelem.revCDerivProjUpdate_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (cont : X → Cont) (idx : Idx) (elem : X → Elem) (hcont : SciLean.HasAdjDiff K cont) (helem : SciLean.HasAdjDiff K elem) : (SciLean.revDerivProjUpdate K Idx fun x => LeanColls.Indexed.set (cont x) idx (elem x)) = fun x =>\n let cdc := SciLean.revDerivProjUpdate K Idx cont x;\n let ede := SciLean.revDerivUpdate K elem x;\n (LeanColls.Indexed.set cdc.1 idx ede.1, fun i delem dx => if i = idx then ede.2 delem dx else cdc.2 i delem dx)"} +{"name":"LeanColls.Indexed.ofFn.arg_cont.revCDerivProj_rule","declaration":"theorem LeanColls.Indexed.ofFn.arg_cont.revCDerivProj_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (f : X → Idx → Elem) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K Idx fun x => LeanColls.Indexed.ofFn (f x)) = fun x =>\n let fdf := SciLean.revDerivProj K Idx f x;\n (LeanColls.Indexed.ofFn fdf.1, fdf.2)"} +{"name":"LeanColls.Indexed.ofFn.arg_f.IsSmoothLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.ofFn.arg_f.IsSmoothLinearMap_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.Vec K Elem] (idx : Idx) : SciLean.IsSmoothLinearMap K fun f => LeanColls.Indexed.ofFn f"} +{"name":"LeanColls.Indexed.set.arg_contelem.IsContinuousLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.set.arg_contelem.IsContinuousLinearMap_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [NormedAddCommGroup Elem] [NormedSpace K Elem] {idx : Idx} : SciLean.IsContinuousLinearMap K fun x =>\n match x with\n | (cont, elem) => LeanColls.Indexed.set cont idx elem"} +{"name":"LeanColls.Indexed.set.arg_cont.semiAdjoint_rule_simple","declaration":"theorem LeanColls.Indexed.set.arg_cont.semiAdjoint_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.SemiInnerProductSpace K Elem] (idx : Idx) : (SciLean.semiAdjoint K fun x =>\n match x with\n | (cont, elem) => LeanColls.Indexed.set cont idx elem) =\n fun cont' => (LeanColls.Indexed.set cont' idx 0, cont'[idx])"} +{"name":"LeanColls.Indexed.set.arg_contelem.revCDeriv_rule","declaration":"theorem LeanColls.Indexed.set.arg_contelem.revCDeriv_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (cont : X → Cont) (idx : Idx) (elem : X → Elem) (hcont : SciLean.HasAdjDiff K cont) (helem : SciLean.HasAdjDiff K elem) : (SciLean.revDeriv K fun x => LeanColls.Indexed.set (cont x) idx (elem x)) = fun x =>\n let cdc := SciLean.revDeriv K cont x;\n let ede := SciLean.revDerivUpdate K elem x;\n (LeanColls.Indexed.set cdc.1 idx ede.1, fun dc =>\n let dci := dc[idx];\n let dc := LeanColls.Indexed.set dc idx 0;\n ede.2 dci (cdc.2 dc))"} +{"name":"LeanColls.Indexed.set.arg_contelem.revCDerivProj_rule","declaration":"theorem LeanColls.Indexed.set.arg_contelem.revCDerivProj_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (cont : X → Cont) (idx : Idx) (elem : X → Elem) (hcont : SciLean.HasAdjDiff K cont) (helem : SciLean.HasAdjDiff K elem) : (SciLean.revDerivProj K Idx fun x => LeanColls.Indexed.set (cont x) idx (elem x)) = fun x =>\n let cdc := SciLean.revDerivProj K Idx cont x;\n let ede := SciLean.revDeriv K elem x;\n (LeanColls.Indexed.set cdc.1 idx ede.1, fun i delem => if i = idx then ede.2 delem else cdc.2 i delem)"} +{"name":"LeanColls.Indexed.ofFn.arg_f.IsLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.ofFn.arg_f.IsLinearMap_rule_simple {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] {R : Type} [CommSemiring R] [AddCommGroup Elem] [Module R Elem] : IsLinearMap R fun f => LeanColls.Indexed.ofFn f"} +{"name":"LeanColls.Indexed.set.arg_contelem.IsLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.set.arg_contelem.IsLinearMap_rule_simple {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] {R : Type} [CommSemiring R] [AddCommGroup Elem] [Module R Elem] (idx : Idx) : IsLinearMap R fun x =>\n match x with\n | (cont, elem) => LeanColls.Indexed.set cont idx elem"} +{"name":"LeanColls.Indexed.ofFn.arg_cont.semiAdjoint_rule_simple","declaration":"theorem LeanColls.Indexed.ofFn.arg_cont.semiAdjoint_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.SemiInnerProductSpace K Elem] : (SciLean.semiAdjoint K fun f => LeanColls.Indexed.ofFn f) = fun cont idx => cont[idx]"} +{"name":"GetElem.getElem.arg_cont.revCDerivUpdate_rule","declaration":"theorem GetElem.getElem.arg_cont.revCDerivUpdate_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (f : X → Cont) (idx : Idx) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => (f x)[idx]) = fun x =>\n let ydf := SciLean.revDerivProjUpdate K Idx f x;\n (ydf.1[idx], fun delem dx => ydf.2 idx delem dx)"} +{"name":"LeanColls.Indexed.set.arg_cont.HasSemiAdjoint_rule_simple","declaration":"theorem LeanColls.Indexed.set.arg_cont.HasSemiAdjoint_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.SemiInnerProductSpace K Elem] (idx : Idx) : SciLean.HasSemiAdjoint K fun x =>\n match x with\n | (cont, elem) => LeanColls.Indexed.set cont idx elem"} +{"name":"GetElem.getElem.arg_cont.IsSmoothLinearMap_rule_simple","declaration":"theorem GetElem.getElem.arg_cont.IsSmoothLinearMap_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [SciLean.Vec K Elem] (idx : Idx) : SciLean.IsSmoothLinearMap K fun xs => xs[idx]"} +{"name":"LeanColls.Indexed.get.arg_cont.IsLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.get.arg_cont.IsLinearMap_rule_simple {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] {R : Type} [CommSemiring R] [AddCommGroup Elem] [Module R Elem] (idx : Idx) : IsLinearMap R fun xs => xs[idx]"} +{"name":"GetElem.getElem.arg_cont.semiAdjoint_rule_simple","declaration":"theorem GetElem.getElem.arg_cont.semiAdjoint_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [DecidableEq Idx] [SciLean.SemiInnerProductSpace K Elem] (idx : Idx) : (SciLean.semiAdjoint K fun cont => cont[idx]) = fun elem => SciLean.oneHot idx elem"} +{"name":"LeanColls.Indexed.ofFn.arg_cont.revCDerivUpdate_rule","declaration":"theorem LeanColls.Indexed.ofFn.arg_cont.revCDerivUpdate_rule {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] {X : Type} [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpace K Elem] (f : X → Idx → Elem) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => LeanColls.Indexed.ofFn (f x)) = fun x =>\n let fdf := SciLean.revDerivUpdate K f x;\n (LeanColls.Indexed.ofFn fdf.1, fun dcont dx => fdf.2 (fun idx => dcont[idx]) dx)"} +{"name":"LeanColls.Indexed.ofFn.arg_f.IsContinuousLinearMap_rule_simple","declaration":"theorem LeanColls.Indexed.ofFn.arg_f.IsContinuousLinearMap_rule_simple {K : Type} [RCLike K] {Cont : Type} {Idx : outParam Type} {Elem : outParam Type} [SciLean.ArrayType Cont Idx Elem] [SciLean.IndexType Idx] [NormedAddCommGroup Elem] [NormedSpace K Elem] : SciLean.IsContinuousLinearMap K fun f => LeanColls.Indexed.ofFn f"} diff --git a/scilean-declarations/SciLean.Data.ArrayType.jsonl b/scilean-declarations/SciLean.Data.ArrayType.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Data.Curry.jsonl b/scilean-declarations/SciLean.Data.Curry.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..5d71c2754353bab85deec2d62c5a2cb265e7c852 --- /dev/null +++ b/scilean-declarations/SciLean.Data.Curry.jsonl @@ -0,0 +1,31 @@ +{"name":"SciLean.FunNArgs.uncurry","declaration":"def SciLean.FunNArgs.uncurry (n : Nat) {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} [self : SciLean.FunNArgs n F Xs Y] : F → Xs → Y"} +{"name":"SciLean.FunNArgs","declaration":"class SciLean.FunNArgs (n : Nat) (F : Sort u_1) (Xs : outParam (Sort u_2)) (Y : outParam (Sort u_3)) : Sort (max (max (max 1 u_1) u_2) u_3)"} +{"name":"SciLean.UncurryN.uncurry","declaration":"def SciLean.UncurryN.uncurry (n : Nat) {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} [self : SciLean.UncurryN n F Xs Y] : F → Xs → Y"} +{"name":"SciLean.instCurryNHAddNatInstHAddInstAddNatOfNatProdForAll","declaration":"instance SciLean.instCurryNHAddNatInstHAddInstAddNatOfNatProdForAll {n : Nat} {X : Type u_1} {Xs : Type u_2} {Y : Sort u_3} {F : outParam (Sort u_4)} [c : outParam (SciLean.CurryN n Xs Y F)] : SciLean.CurryN (n + 1) (X × Xs) Y (X → F)"} +{"name":"SciLean.instUncurryNHAddNatInstHAddInstAddNatOfNatForAllProd","declaration":"instance SciLean.instUncurryNHAddNatInstHAddInstAddNatOfNatForAllProd {n : Nat} {X : Type u_1} {Y : Sort u_2} {Xs' : outParam (Type u_3)} {Y' : outParam (Sort u_4)} [c : SciLean.UncurryN n Y Xs' Y'] : SciLean.UncurryN (n + 1) (X → Y) (X × Xs') Y'"} +{"name":"SciLean.instCurryAllHAddNatInstHAddInstAddNatOfNatProdForAll","declaration":"instance SciLean.instCurryAllHAddNatInstHAddInstAddNatOfNatProdForAll {n : Nat} {X : Type u_1} {Xs : Type u_2} {Y : Sort u_3} {F : outParam (Sort u_4)} [c : outParam (SciLean.CurryAll n Xs Y F)] : SciLean.CurryAll (n + 1) (X × Xs) Y (X → F)"} +{"name":"SciLean.CurryAll.mk","declaration":"ctor SciLean.CurryAll.mk {n : Nat} {Xs : Sort u_1} {Y : Sort u_2} {F : outParam (Sort u_3)} (curry : (Xs → Y) → F) : SciLean.CurryAll n Xs Y F"} +{"name":"SciLean.FunNArgs.is_equiv","declaration":"def SciLean.FunNArgs.is_equiv {n : Nat} {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} [self : SciLean.FunNArgs n F Xs Y] : SciLean.FunNArgs.curry n ∘ SciLean.FunNArgs.uncurry n = id ∧ SciLean.FunNArgs.uncurry n ∘ SciLean.FunNArgs.curry n = id"} +{"name":"SciLean.instUncurryNOfNatNatInstOfNatNatForAll","declaration":"instance SciLean.instUncurryNOfNatNatInstOfNatNatForAll {X : Sort u_1} {Y : Sort u_2} : SciLean.UncurryN 1 (X → Y) X Y"} +{"name":"SciLean.CurryAll","declaration":"class SciLean.CurryAll (n : Nat) (Xs : Sort u_1) (Y : Sort u_2) (F : outParam (Sort u_3)) : Sort (max (max (max 1 u_1) u_2) u_3)"} +{"name":"SciLean.CurryAll.curry","declaration":"def SciLean.CurryAll.curry (n : Nat) {Xs : Sort u_1} {Y : Sort u_2} {F : outParam (Sort u_3)} [self : SciLean.CurryAll n Xs Y F] : (Xs → Y) → F"} +{"name":"SciLean.instCurryNOfNatNatInstOfNatNatForAll","declaration":"instance SciLean.instCurryNOfNatNatInstOfNatNatForAll {X : Sort u_1} {Y : Sort u_2} : SciLean.CurryN 1 X Y (X → Y)"} +{"name":"SciLean.CurryN","declaration":"class SciLean.CurryN (n : Nat) (Xs : Sort u_1) (Y : Sort u_2) (F : outParam (Sort u_3)) : Sort (max (max (max 1 u_1) u_2) u_3)"} +{"name":"SciLean.UncurryAll.mk","declaration":"ctor SciLean.UncurryAll.mk {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} (uncurry : F → Xs → Y) : SciLean.UncurryAll F Xs Y"} +{"name":"SciLean.curryAll","declaration":"def SciLean.curryAll {Xs : outParam (Sort u_1)} {Y : outParam (Sort u_2)} {F : outParam (Sort u_3)} (n : Nat) (f : Xs → Y) [outParam (SciLean.CurryAll n Xs Y F)] : F"} +{"name":"SciLean.CurryN.curry","declaration":"def SciLean.CurryN.curry (n : Nat) {Xs : Sort u_1} {Y : Sort u_2} {F : outParam (Sort u_3)} [self : SciLean.CurryN n Xs Y F] : (Xs → Y) → F"} +{"name":"SciLean.uncurryN","declaration":"def SciLean.uncurryN {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} (n : Nat) (f : F) [SciLean.UncurryN n F Xs Y] : Xs → Y"} +{"name":"SciLean.instFunNArgsOfNatNatInstOfNatNatForAll","declaration":"instance SciLean.instFunNArgsOfNatNatInstOfNatNatForAll {X : Sort u_1} {Y : Sort u_2} : SciLean.FunNArgs 1 (X → Y) X Y"} +{"name":"SciLean.curryN","declaration":"def SciLean.curryN {Xs : outParam (Sort u_1)} {Y : outParam (Sort u_2)} {F : outParam (Sort u_3)} (n : Nat) (f : Xs → Y) [outParam (SciLean.CurryN n Xs Y F)] : F"} +{"name":"SciLean.instUncurryAllForAllProd","declaration":"instance SciLean.instUncurryAllForAllProd {X : Type u_1} {Y : Sort u_2} {Xs' : outParam (Type u_3)} {Y' : outParam (Sort u_4)} [c : SciLean.UncurryAll Y Xs' Y'] : SciLean.UncurryAll (X → Y) (X × Xs') Y'"} +{"name":"SciLean.CurryN.mk","declaration":"ctor SciLean.CurryN.mk {n : Nat} {Xs : Sort u_1} {Y : Sort u_2} {F : outParam (Sort u_3)} (curry : (Xs → Y) → F) : SciLean.CurryN n Xs Y F"} +{"name":"SciLean.FunNArgs.curry","declaration":"def SciLean.FunNArgs.curry (n : Nat) {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} [self : SciLean.FunNArgs n F Xs Y] : (Xs → Y) → F"} +{"name":"SciLean.UncurryAll.uncurry","declaration":"def SciLean.UncurryAll.uncurry {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} [self : SciLean.UncurryAll F Xs Y] : F → Xs → Y"} +{"name":"SciLean.uncurryAll","declaration":"def SciLean.uncurryAll {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} (f : F) [SciLean.UncurryAll F Xs Y] : Xs → Y"} +{"name":"SciLean.UncurryAll","declaration":"class SciLean.UncurryAll (F : Sort u_1) (Xs : outParam (Sort u_2)) (Y : outParam (Sort u_3)) : Sort (max (max (max 1 u_1) u_2) u_3)"} +{"name":"SciLean.FunNArgs.mk","declaration":"ctor SciLean.FunNArgs.mk {n : Nat} {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} (uncurry : F → Xs → Y) (curry : (Xs → Y) → F) (is_equiv : curry ∘ uncurry = id ∧ uncurry ∘ curry = id) : SciLean.FunNArgs n F Xs Y"} +{"name":"SciLean.UncurryN.mk","declaration":"ctor SciLean.UncurryN.mk {n : Nat} {F : Sort u_1} {Xs : outParam (Sort u_2)} {Y : outParam (Sort u_3)} (uncurry : F → Xs → Y) : SciLean.UncurryN n F Xs Y"} +{"name":"SciLean.instCurryAllOfNatNatInstOfNatNatForAll","declaration":"instance SciLean.instCurryAllOfNatNatInstOfNatNatForAll {X : Sort u_1} {Y : Sort u_2} : SciLean.CurryAll 1 X Y (X → Y)"} +{"name":"SciLean.UncurryN","declaration":"class SciLean.UncurryN (n : Nat) (F : Sort u_1) (Xs : outParam (Sort u_2)) (Y : outParam (Sort u_3)) : Sort (max (max (max 1 u_1) u_2) u_3)"} +{"name":"SciLean.instFunNArgsHAddNatInstHAddInstAddNatOfNatForAllProd","declaration":"instance SciLean.instFunNArgsHAddNatInstHAddInstAddNatOfNatForAllProd {n : Nat} {X : Type u_1} {Xs : Type u_2} {Y : Sort u_3} {F : Sort u_4} [fn : SciLean.FunNArgs n F Xs Y] : SciLean.FunNArgs (n + 1) (X → F) (X × Xs) Y"} +{"name":"SciLean.instUncurryAllForAll","declaration":"instance SciLean.instUncurryAllForAll {X : Sort u_1} {Y : Sort u_2} : SciLean.UncurryAll (X → Y) X Y"} diff --git a/scilean-declarations/SciLean.Data.DataArray.DataArray.jsonl b/scilean-declarations/SciLean.Data.DataArray.DataArray.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..077dbdb84a83a022fd6325264935fb86263a8b8e --- /dev/null +++ b/scilean-declarations/SciLean.Data.DataArray.DataArray.jsonl @@ -0,0 +1,45 @@ +{"name":"SciLean.DataArrayN.toList","declaration":"def SciLean.DataArrayN.toList {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (xs : SciLean.DataArrayN α ι) : List α"} +{"name":"SciLean.DataArrayN.set","declaration":"def SciLean.DataArrayN.set {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (xs : SciLean.DataArrayN α ι) (i : ι) (xi : α) : SciLean.DataArrayN α ι"} +{"name":"SciLean.instSizeDataArrayN","declaration":"instance SciLean.instSizeDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.Size (SciLean.DataArrayN α ι)"} +{"name":"SciLean.instArrayTypeNotationDataArrayN","declaration":"instance SciLean.instArrayTypeNotationDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : SciLean.ArrayTypeNotation (SciLean.DataArrayN α ι) ι α"} +{"name":"SciLean.DataArrayN","declaration":"structure SciLean.DataArrayN (α : Type) [pd : SciLean.PlainDataType α] (ι : Type) [SciLean.IndexType ι] : Type"} +{"name":"SciLean.DataArray.set","declaration":"def SciLean.DataArray.set {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) (i : Fin arr.size) (val : α) : SciLean.DataArray α"} +{"name":"SciLean.instToMultisetWithIdxDataArrayNProd","declaration":"instance SciLean.instToMultisetWithIdxDataArrayNProd {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.ToMultiset (LeanColls.Indexed.WithIdx (SciLean.DataArrayN α ι)) (ι × α)"} +{"name":"SciLean.instReadOnlyWithIdxDataArrayNProd","declaration":"instance SciLean.instReadOnlyWithIdxDataArrayNProd {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.MultiBag.ReadOnly (LeanColls.Indexed.WithIdx (SciLean.DataArrayN α ι)) (ι × α)"} +{"name":"SciLean.DataArrayN.modify","declaration":"def SciLean.DataArrayN.modify {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (xs : SciLean.DataArrayN α ι) (i : ι) (f : α → α) : SciLean.DataArrayN α ι"} +{"name":"SciLean.DataArray","declaration":"structure SciLean.DataArray (α : Type) [pd : SciLean.PlainDataType α] : Type"} +{"name":"SciLean.DataArray.byteData","declaration":"def SciLean.DataArray.byteData {α : Type} [pd : SciLean.PlainDataType α] (self : SciLean.DataArray α) : ByteArray"} +{"name":"SciLean.DataArray.swap","declaration":"def SciLean.DataArray.swap {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) (i : Fin arr.size) (j : Fin arr.size) : SciLean.DataArray α"} +{"name":"SciLean.instFoldDataArrayN","declaration":"instance SciLean.instFoldDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.Fold (SciLean.DataArrayN α ι) α"} +{"name":"SciLean.DataArrayN.reshape","declaration":"def SciLean.DataArrayN.reshape {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (x : SciLean.DataArrayN α ι) (κ : Type) [SciLean.IndexType κ] (hs : SciLean.IndexType.card κ = SciLean.IndexType.card ι) : SciLean.DataArrayN α κ"} +{"name":"SciLean.DataArray.h_size","declaration":"def SciLean.DataArray.h_size {α : Type} [pd : SciLean.PlainDataType α] (self : SciLean.DataArray α) : SciLean.PlainDataType.bytes pd self.size ≤ ByteArray.size self.byteData"} +{"name":"SciLean.DataArray.get","declaration":"def SciLean.DataArray.get {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) (i : Fin arr.size) : α"} +{"name":"SciLean.DataArray.capacity","declaration":"/-- Capacity of an array. The return type is `Squash Nat` as the capacity is is just an implementation detail and should not affect semantics of the program. -/\ndef SciLean.DataArray.capacity {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) : Squash ℕ"} +{"name":"SciLean.DataArray.reserve","declaration":"/-- Makes sure that `arr` fits at least `n` elements of `α` -/\ndef SciLean.DataArray.reserve {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) (capacity : ℕ) : SciLean.DataArray α"} +{"name":"SciLean.instMembershipProdWithIdxDataArrayN","declaration":"instance SciLean.instMembershipProdWithIdxDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : Membership (ι × α) (LeanColls.Indexed.WithIdx (SciLean.DataArrayN α ι))"} +{"name":"SciLean.instLawfulIndexedDataArrayNInstIndexedDataArrayN","declaration":"instance SciLean.instLawfulIndexedDataArrayNInstIndexedDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.LawfulIndexed (SciLean.DataArrayN α ι) ι α"} +{"name":"ByteArray.mkArray","declaration":"def ByteArray.mkArray (n : ℕ) (v : UInt8) : ByteArray"} +{"name":"SciLean.DataArray.mkEmpty","declaration":"def SciLean.DataArray.mkEmpty {α : Type} [pd : SciLean.PlainDataType α] (capacity : ℕ) : SciLean.DataArray α"} +{"name":"SciLean.DataArrayN.toListIdx","declaration":"def SciLean.DataArrayN.toListIdx {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (xs : SciLean.DataArrayN α ι) : List (ι × α)"} +{"name":"SciLean.instPlainDataType","declaration":"instance SciLean.instPlainDataType {Cont : Type} {ι : Type} {α : Type} [SciLean.ArrayType Cont ι α] [SciLean.IndexType ι] [Inhabited α] [pd : SciLean.PlainDataType α] : SciLean.PlainDataType Cont"} +{"name":"SciLean.instSizeWithIdxDataArrayN","declaration":"instance SciLean.instSizeWithIdxDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.Size (LeanColls.Indexed.WithIdx (SciLean.DataArrayN α ι))"} +{"name":"SciLean.DataArray.drop","declaration":"def SciLean.DataArray.drop {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) (k : ℕ) : SciLean.DataArray α"} +{"name":"SciLean.DataArrayN.h_size","declaration":"def SciLean.DataArrayN.h_size {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (self : SciLean.DataArrayN α ι) : SciLean.IndexType.card ι = self.data.size"} +{"name":"SciLean.instReadOnlyDataArrayN","declaration":"instance SciLean.instReadOnlyDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.MultiBag.ReadOnly (SciLean.DataArrayN α ι) α"} +{"name":"SciLean.DataArrayN.mk","declaration":"ctor SciLean.DataArrayN.mk {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (data : SciLean.DataArray α) (h_size : SciLean.IndexType.card ι = data.size) : SciLean.DataArrayN α ι"} +{"name":"SciLean.DataArray.mk","declaration":"ctor SciLean.DataArray.mk {α : Type} [pd : SciLean.PlainDataType α] (byteData : ByteArray) (size : ℕ) (h_size : SciLean.PlainDataType.bytes pd size ≤ ByteArray.size byteData) : SciLean.DataArray α"} +{"name":"SciLean.DataArray.ext","declaration":"/-- Extensionality of DataArray\n\nCurrently this is inconsistent, we need to turn DataArray into quotient!\n-/\ntheorem SciLean.DataArray.ext {α : Type} [pd : SciLean.PlainDataType α] (d : SciLean.DataArray α) (d' : SciLean.DataArray α) (h : d.size = d'.size) : (∀ (i : Fin d.size), SciLean.DataArray.get d i = SciLean.DataArray.get d' (h ▸ i)) → d = d'"} +{"name":"SciLean.DataArray.reverse","declaration":"def SciLean.DataArray.reverse {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) : SciLean.DataArray α"} +{"name":"SciLean.DataArrayN.curry","declaration":"def SciLean.DataArrayN.curry {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] {κ : Type} [SciLean.IndexType κ] [Inhabited α] (x : SciLean.DataArrayN α (ι × κ)) : SciLean.DataArrayN (SciLean.DataArrayN α κ) ι"} +{"name":"SciLean.instMembershipDataArrayN","declaration":"instance SciLean.instMembershipDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : Membership α (SciLean.DataArrayN α ι)"} +{"name":"SciLean.instIndexedDataArrayN","declaration":"instance SciLean.instIndexedDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.Indexed (SciLean.DataArrayN α ι) ι α"} +{"name":"SciLean.instArrayTypeDataArrayN","declaration":"instance SciLean.instArrayTypeDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : SciLean.ArrayType (SciLean.DataArrayN α ι) ι α"} +{"name":"SciLean.instToStringDataArray","declaration":"instance SciLean.instToStringDataArray {α : Type} [pd : SciLean.PlainDataType α] [ToString α] : ToString (SciLean.DataArray α)"} +{"name":"SciLean.DataArrayN.data","declaration":"def SciLean.DataArrayN.data {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (self : SciLean.DataArrayN α ι) : SciLean.DataArray α"} +{"name":"SciLean.instFoldWithIdxDataArrayNProd","declaration":"instance SciLean.instFoldWithIdxDataArrayNProd {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.Fold (LeanColls.Indexed.WithIdx (SciLean.DataArrayN α ι)) (ι × α)"} +{"name":"SciLean.instToMultisetDataArrayN","declaration":"instance SciLean.instToMultisetDataArrayN {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] : LeanColls.ToMultiset (SciLean.DataArrayN α ι) α"} +{"name":"SciLean.DataArray.size","declaration":"def SciLean.DataArray.size {α : Type} [pd : SciLean.PlainDataType α] (self : SciLean.DataArray α) : ℕ"} +{"name":"SciLean.instGetElemDataArrayNatLtInstLTNatSize","declaration":"instance SciLean.instGetElemDataArrayNatLtInstLTNatSize {α : Type} [pd : SciLean.PlainDataType α] : GetElem (SciLean.DataArray α) ℕ α fun a i => i < a.size"} +{"name":"SciLean.DataArray.push","declaration":"def SciLean.DataArray.push {α : Type} [pd : SciLean.PlainDataType α] (arr : SciLean.DataArray α) (val : α) (k : optParam ℕ 1) : SciLean.DataArray α"} +{"name":"SciLean.DataArray.intro","declaration":"def SciLean.DataArray.intro {α : Type} [pd : SciLean.PlainDataType α] {ι : Type u_1} [SciLean.IndexType ι] (f : ι → α) : SciLean.DataArray α"} +{"name":"SciLean.DataArrayN.get","declaration":"def SciLean.DataArrayN.get {α : Type} [pd : SciLean.PlainDataType α] {ι : Type} [SciLean.IndexType ι] (xs : SciLean.DataArrayN α ι) (i : ι) : α"} diff --git a/scilean-declarations/SciLean.Data.DataArray.Operations.jsonl b/scilean-declarations/SciLean.Data.DataArray.Operations.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..7ca1b1530cd9adf53a2309b7ae19753eae841eda --- /dev/null +++ b/scilean-declarations/SciLean.Data.DataArray.Operations.jsonl @@ -0,0 +1,8 @@ +{"name":"SciLean.DataArrayN.mapIdxMono","declaration":"def SciLean.DataArrayN.mapIdxMono {I : Type} {X : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] (x : SciLean.DataArrayN X I) (f : I → X → X) : SciLean.DataArrayN X I"} +{"name":"SciLean.DataArrayN.min","declaration":"def SciLean.DataArrayN.min {I : Type} {X : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] [Min X] [Inhabited X] (x : SciLean.DataArrayN X I) : X"} +{"name":"SciLean.DataArrayN.mapMono","declaration":"def SciLean.DataArrayN.mapMono {I : Type} {X : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] (x : SciLean.DataArrayN X I) (f : X → X) : SciLean.DataArrayN X I"} +{"name":"SciLean.DataArrayN.reduceD","declaration":"def SciLean.DataArrayN.reduceD {I : Type} {X : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] (x : SciLean.DataArrayN X I) (f : X → X → X) (default : X) : X"} +{"name":"SciLean.DataArrayN.foldIdx","declaration":"def SciLean.DataArrayN.foldIdx {I : Type} {X : Type} {Y : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] (x : SciLean.DataArrayN X I) (f : I → Y → X → Y) (init : Y) : Y"} +{"name":"SciLean.DataArrayN.max","declaration":"def SciLean.DataArrayN.max {I : Type} {X : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] [Max X] [Inhabited X] (x : SciLean.DataArrayN X I) : X"} +{"name":"SciLean.DataArrayN.reduce","declaration":"def SciLean.DataArrayN.reduce {I : Type} {X : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] [Inhabited X] (x : SciLean.DataArrayN X I) (f : X → X → X) : X"} +{"name":"SciLean.DataArrayN.fold","declaration":"def SciLean.DataArrayN.fold {I : Type} {X : Type} {Y : Type} [SciLean.IndexType I] [SciLean.PlainDataType X] (x : SciLean.DataArrayN X I) (f : Y → X → Y) (init : Y) : Y"} diff --git a/scilean-declarations/SciLean.Data.DataArray.PlainDataType.jsonl b/scilean-declarations/SciLean.Data.DataArray.PlainDataType.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..44fb72a3b37ef49e5fe97d2ab56f0bc4e63d166b --- /dev/null +++ b/scilean-declarations/SciLean.Data.DataArray.PlainDataType.jsonl @@ -0,0 +1,47 @@ +{"name":"SciLean.MProd.bitTypeMProd","declaration":"def SciLean.MProd.bitTypeMProd {α : Type} {β : Type} (ta : SciLean.BitType α) (tb : SciLean.BitType β) : SciLean.BitType (MProd α β) ⊕ SciLean.ByteType (MProd α β)"} +{"name":"SciLean.instPlainDataTypeUInt64_1","declaration":"instance SciLean.instPlainDataTypeUInt64_1 : SciLean.PlainDataType UInt64"} +{"name":"SciLean.instPlainDataTypeUInt32","declaration":"instance SciLean.instPlainDataTypeUInt32 : SciLean.PlainDataType UInt32"} +{"name":"SciLean.instPlainDataTypeUInt8","declaration":"instance SciLean.instPlainDataTypeUInt8 : SciLean.PlainDataType UInt8"} +{"name":"SciLean.ByteType.h_size","declaration":"def SciLean.ByteType.h_size {α : Type} (self : SciLean.ByteType α) : 1 < self.bytes"} +{"name":"SciLean.ByteType.toByteArray","declaration":"/-- Write `a : α` from byte array `b` starting at the byte `i` -/\ndef SciLean.ByteType.toByteArray {α : Type} (self : SciLean.ByteType α) (b : ByteArray) (i : USize) (h : USize.toNat (i + self.bytes) ≤ ByteArray.size b) (a : α) : ByteArray"} +{"name":"USize.toUInt8","declaration":"def USize.toUInt8 (x : USize) : UInt8"} +{"name":"SciLean.PlainDataType.bytes","declaration":"def SciLean.PlainDataType.bytes {α : Type} (pd : SciLean.PlainDataType α) (n : ℕ) : ℕ"} +{"name":"UInt8.toUSize","declaration":"def UInt8.toUSize (x : UInt8) : USize"} +{"name":"SciLean.ByteType.fromByteArray_toByteArray_other","declaration":"/-- `toByteArray` does not affect other bytes -/\ndef SciLean.ByteType.fromByteArray_toByteArray_other {α : Type} (self : SciLean.ByteType α) (a : α) (b : ByteArray) (i : USize) (j : USize) (h : USize.toNat (i + self.bytes) ≤ ByteArray.size b) : j < i ∨ i + self.bytes ≤ j → ByteArray.uget (self.toByteArray b i h a) j ⋯ = ByteArray.uget b j ⋯"} +{"name":"SciLean.BitType.fromByte","declaration":"def SciLean.BitType.fromByte {α : Type} (self : SciLean.BitType α) (b : UInt8) : α"} +{"name":"SciLean.BitType.fromByte_toByte","declaration":"def SciLean.BitType.fromByte_toByte {α : Type} (self : SciLean.BitType α) (a : α) : self.fromByte (self.toByte a) = a"} +{"name":"SciLean.Prod.bitTypeProd","declaration":"def SciLean.Prod.bitTypeProd {α : Type} {β : Type} (ta : SciLean.BitType α) (tb : SciLean.BitType β) : SciLean.BitType (α × β) ⊕ SciLean.ByteType (α × β)"} +{"name":"SciLean.Prod.bitTypeByteTypeProd","declaration":"def SciLean.Prod.bitTypeByteTypeProd {α : Type} {β : Type} (ta : SciLean.BitType α) (tb : SciLean.ByteType β) : SciLean.ByteType (α × β)"} +{"name":"SciLean.ByteType.bytes","declaration":"def SciLean.ByteType.bytes {α : Type} (self : SciLean.ByteType α) : USize"} +{"name":"SciLean.BitType.toByte","declaration":"def SciLean.BitType.toByte {α : Type} (self : SciLean.BitType α) (a : α) : UInt8"} +{"name":"SciLean.Fin.bitType","declaration":"def SciLean.Fin.bitType (n : ℕ) : n ≤ 256 → SciLean.BitType (Fin n)"} +{"name":"SciLean.PlainDataType.capacity","declaration":"/-- How many `α` can fit into a buffer with `byteNum` bytes -/\ndef SciLean.PlainDataType.capacity {α : Type} (pd : SciLean.PlainDataType α) (byteNum : ℕ) : ℕ"} +{"name":"SciLean.instPlainDataTypeFin","declaration":"instance SciLean.instPlainDataTypeFin (n : ℕ) : SciLean.PlainDataType (Fin n)"} +{"name":"SciLean.Fin.byteSize","declaration":"def SciLean.Fin.byteSize (n : ℕ) : ℕ"} +{"name":"SciLean.ByteType.fromByteArray","declaration":"/-- Read `a : α` from byte array `b` starting at the byte `i` -/\ndef SciLean.ByteType.fromByteArray {α : Type} (self : SciLean.ByteType α) (b : ByteArray) (i : USize) (h : USize.toNat (i + self.bytes) ≤ ByteArray.size b) : α"} +{"name":"SciLean.Fin.bitSize","declaration":"/-- Number of bits necessary to store `Fin n` -/\ndef SciLean.Fin.bitSize (n : ℕ) : ℕ"} +{"name":"SciLean.Prod.byteTypeBitTypeProd","declaration":"def SciLean.Prod.byteTypeBitTypeProd {α : Type} {β : Type} (ta : SciLean.ByteType α) (tb : SciLean.BitType β) : SciLean.ByteType (α × β)"} +{"name":"SciLean.MProd.byteTypeMProd","declaration":"def SciLean.MProd.byteTypeMProd {α : Type} {β : Type} (ta : SciLean.ByteType α) (tb : SciLean.ByteType β) : SciLean.ByteType (MProd α β)"} +{"name":"SciLean.instPlainDataTypeUInt64","declaration":"instance SciLean.instPlainDataTypeUInt64 : SciLean.PlainDataType UInt64"} +{"name":"SciLean.MProd.bitTypeByteTypeMProd","declaration":"def SciLean.MProd.bitTypeByteTypeMProd {α : Type} {β : Type} (ta : SciLean.BitType α) (tb : SciLean.ByteType β) : SciLean.ByteType (MProd α β)"} +{"name":"SciLean.Fin.byteType","declaration":"def SciLean.Fin.byteType (n : ℕ) : 256 < n → SciLean.ByteType (Fin n)"} +{"name":"SciLean.instPlainDataTypeBool","declaration":"instance SciLean.instPlainDataTypeBool : SciLean.PlainDataType Bool"} +{"name":"SciLean.instPlainDataTypeUInt16","declaration":"instance SciLean.instPlainDataTypeUInt16 : SciLean.PlainDataType UInt16"} +{"name":"SciLean.Float.byteType","declaration":"def SciLean.Float.byteType : SciLean.ByteType Float"} +{"name":"SciLean.PlainDataType.btype","declaration":"def SciLean.PlainDataType.btype {α : Type} [self : SciLean.PlainDataType α] : SciLean.BitType α ⊕ SciLean.ByteType α"} +{"name":"SciLean.instPlainDataTypeFloat","declaration":"instance SciLean.instPlainDataTypeFloat : SciLean.PlainDataType Float"} +{"name":"SciLean.instPlainDataTypeMProd","declaration":"instance SciLean.instPlainDataTypeMProd {α : Type} {β : Type} [ta : SciLean.PlainDataType α] [tb : SciLean.PlainDataType β] : SciLean.PlainDataType (MProd α β)"} +{"name":"SciLean.Bool.bitType","declaration":"def SciLean.Bool.bitType : SciLean.BitType Bool"} +{"name":"SciLean.BitType.bits","declaration":"def SciLean.BitType.bits {α : Type} (self : SciLean.BitType α) : UInt8"} +{"name":"SciLean.BitType.mk","declaration":"ctor SciLean.BitType.mk {α : Type} (bits : UInt8) (h_size : bits ≤ 8) (fromByte : UInt8 → α) (toByte : α → UInt8) (fromByte_toByte : ∀ (a : α), fromByte (toByte a) = a) : SciLean.BitType α"} +{"name":"SciLean.ByteType.mk","declaration":"ctor SciLean.ByteType.mk {α : Type} (bytes : USize) (h_size : 1 < bytes) (fromByteArray : (b : ByteArray) → (i : USize) → USize.toNat (i + bytes) ≤ ByteArray.size b → α) (toByteArray : (b : ByteArray) → (i : USize) → USize.toNat (i + bytes) ≤ ByteArray.size b → α → ByteArray) (toByteArray_size : ∀ (b : ByteArray) (i : USize) (h : USize.toNat (i + bytes) ≤ ByteArray.size b) (a : α),\n ByteArray.size (toByteArray b i h a) = ByteArray.size b) (fromByteArray_toByteArray : ∀ (a : α) (b : ByteArray) (i : USize) (h : USize.toNat (i + bytes) ≤ ByteArray.size b)\n (h' : USize.toNat (i + bytes) ≤ ByteArray.size (toByteArray b i h a)), fromByteArray (toByteArray b i h a) i h' = a) (fromByteArray_toByteArray_other : ∀ (a : α) (b : ByteArray) (i j : USize) (h : USize.toNat (i + bytes) ≤ ByteArray.size b),\n j < i ∨ i + bytes ≤ j → ByteArray.uget (toByteArray b i h a) j ⋯ = ByteArray.uget b j ⋯) : SciLean.ByteType α"} +{"name":"SciLean.ByteType.fromByteArray_toByteArray","declaration":"/-- we can recover `a` from bytes -/\ndef SciLean.ByteType.fromByteArray_toByteArray {α : Type} (self : SciLean.ByteType α) (a : α) (b : ByteArray) (i : USize) (h : USize.toNat (i + self.bytes) ≤ ByteArray.size b) (h' : USize.toNat (i + self.bytes) ≤ ByteArray.size (self.toByteArray b i h a)) : self.fromByteArray (self.toByteArray b i h a) i h' = a"} +{"name":"SciLean.PlainDataType","declaration":"/-- This rougly corresponds to Plain Old Data(POD)/Passive Data known from OOP\n\nwiki: https://en.wikipedia.org/wiki/Passive_data_structure\n\nWe distinguish between two main types of POD. `BitType` a type that is smaller or equal to a byte and `ByteType` that takes up multiple bytes. The main motivation is an efficient storage of `Array Bool` where `Bool` takes up only a single bit, so we can fit 8 bools into a single byte and achieve significant memore reduction.\n\nPotentially surprising edge case is array of fixed length, i.e. the type `{a : Array α // a.size = n}`. It is `PlainDataType` if `α` is `PlainDataType`. However, `Array α` is not `PlainDataType`, even if `α` is `PlainDataType`, as it does not have a fixed byte size.\n-/\nclass SciLean.PlainDataType (α : Type) : Type"} +{"name":"SciLean.BitType.h_size","declaration":"def SciLean.BitType.h_size {α : Type} (self : SciLean.BitType α) : self.bits ≤ 8"} +{"name":"SciLean.BitType","declaration":"structure SciLean.BitType (α : Type) : Type"} +{"name":"SciLean.PlainDataType.mk","declaration":"ctor SciLean.PlainDataType.mk {α : Type} (btype : SciLean.BitType α ⊕ SciLean.ByteType α) : SciLean.PlainDataType α"} +{"name":"SciLean.ByteType.toByteArray_size","declaration":"/-- `toByteArray` does not modify ByteArray size -/\ndef SciLean.ByteType.toByteArray_size {α : Type} (self : SciLean.ByteType α) (b : ByteArray) (i : USize) (h : USize.toNat (i + self.bytes) ≤ ByteArray.size b) (a : α) : ByteArray.size (self.toByteArray b i h a) = ByteArray.size b"} +{"name":"SciLean.Prod.byteTypeProd","declaration":"def SciLean.Prod.byteTypeProd {α : Type} {β : Type} (ta : SciLean.ByteType α) (tb : SciLean.ByteType β) : SciLean.ByteType (α × β)"} +{"name":"SciLean.MProd.byteTypeBitTypeMProd","declaration":"def SciLean.MProd.byteTypeBitTypeMProd {α : Type} {β : Type} (ta : SciLean.ByteType α) (tb : SciLean.BitType β) : SciLean.ByteType (MProd α β)"} +{"name":"SciLean.ByteType","declaration":"structure SciLean.ByteType (α : Type) : Type"} +{"name":"SciLean.instPlainDataTypeProd","declaration":"/-- Product of `PlainDataType` is `PlainDataType`\n\n**Instance diamond:** This instance is currently prefered over `instPlainDataTypeEnumtype`.\n\nThis instance makes a diamond together with `instPlainDataTypeEnumtype` when `α` and `β` are `Enumtype`. Using this instance is less computationally intensive when writting and reading from `DataArra` but it consumes more memory. The `instPlainDataTypeEnumtype` is doing the exact opposite.\n\nExample: `Fin (2^4+1) × Fin (2^4-1)`\n\nAs Product:\nThe type `Fin (2^4+1)` needs 5 bits.\nThe type `Fin (2^4-1)` needs 4 bits.\nThus `Fin (2^4+1) × Fin (2^4-1)` needs 9 bits, thus 2 bytes, as `instPlainDataTypeProd`\n\nAs Enumtype:\n`Fin (2^4+1) × Fin (2^4-1) ≈ Fin (2^8-1)`\nThe type `Fin (2^8-1)` needs 8 bits thus only a single byte as `instPlainDataTypeEnumtype`\n\n-/\ninstance SciLean.instPlainDataTypeProd {α : Type} {β : Type} [ta : SciLean.PlainDataType α] [tb : SciLean.PlainDataType β] : SciLean.PlainDataType (α × β)"} diff --git a/scilean-declarations/SciLean.Data.DataArray.VecN.jsonl b/scilean-declarations/SciLean.Data.DataArray.VecN.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..0174f3213a45673b47558a75d6177725c1599367 --- /dev/null +++ b/scilean-declarations/SciLean.Data.DataArray.VecN.jsonl @@ -0,0 +1,31 @@ +{"name":"SciLean.Vec3.instPlainDataTypeVec3","declaration":"instance SciLean.Vec3.instPlainDataTypeVec3 : SciLean.PlainDataType SciLean.Vec3"} +{"name":"SciLean.Vec3.instArrayTypeVec3FinOfNatNatInstOfNatNatFloat","declaration":"instance SciLean.Vec3.instArrayTypeVec3FinOfNatNatInstOfNatNatFloat : SciLean.ArrayType SciLean.Vec3 (Fin 3) Float"} +{"name":"SciLean.Vec2.instArrayTypeVec2FinOfNatNatInstOfNatNatFloat","declaration":"instance SciLean.Vec2.instArrayTypeVec2FinOfNatNatInstOfNatNatFloat : SciLean.ArrayType SciLean.Vec2 (Fin 2) Float"} +{"name":"SciLean.Vec2.get","declaration":"def SciLean.Vec2.get (v : SciLean.Vec2) (i : Fin 2) : Float"} +{"name":"SciLean.Vec2.instStructTypeVec2FinOfNatNatInstOfNatNatFloat","declaration":"instance SciLean.Vec2.instStructTypeVec2FinOfNatNatInstOfNatNatFloat : SciLean.StructType SciLean.Vec2 (Fin 2) fun x => Float"} +{"name":"SciLean.Vec2.x","declaration":"def SciLean.Vec2.x (self : SciLean.Vec2) : Float"} +{"name":"SciLean.Vec3.«termV[_,_,_]»","declaration":"def SciLean.Vec3.«termV[_,_,_]» : Lean.ParserDescr"} +{"name":"SciLean.Vec2.mk","declaration":"ctor SciLean.Vec2.mk (x : Float) (y : Float) : SciLean.Vec2"} +{"name":"SciLean.Vec3.instIndexedVec3FinOfNatNatInstOfNatNatFloat","declaration":"instance SciLean.Vec3.instIndexedVec3FinOfNatNatInstOfNatNatFloat : LeanColls.Indexed SciLean.Vec3 (Fin 3) Float"} +{"name":"SciLean.Vec2.instIndexedVec2FinOfNatNatInstOfNatNatFloat","declaration":"instance SciLean.Vec2.instIndexedVec2FinOfNatNatInstOfNatNatFloat : LeanColls.Indexed SciLean.Vec2 (Fin 2) Float"} +{"name":"SciLean.instInhabitedVec2","declaration":"instance SciLean.instInhabitedVec2 : Inhabited SciLean.Vec2"} +{"name":"SciLean.Vec3.set","declaration":"def SciLean.Vec3.set (v : SciLean.Vec3) (i : Fin 3) (vi : Float) : SciLean.Vec3"} +{"name":"SciLean.Vec2","declaration":"structure SciLean.Vec2 : Type"} +{"name":"SciLean.instInhabitedVec3","declaration":"instance SciLean.instInhabitedVec3 : Inhabited SciLean.Vec3"} +{"name":"SciLean.Vec2.set","declaration":"def SciLean.Vec2.set (v : SciLean.Vec2) (i : Fin 2) (vi : Float) : SciLean.Vec2"} +{"name":"SciLean.Vec3.get","declaration":"def SciLean.Vec3.get (v : SciLean.Vec3) (i : Fin 3) : Float"} +{"name":"SciLean.Vec3.mk","declaration":"ctor SciLean.Vec3.mk (x : Float) (y : Float) (z : Float) : SciLean.Vec3"} +{"name":"SciLean.Vec2.«termV[_,_]»","declaration":"def SciLean.Vec2.«termV[_,_]» : Lean.ParserDescr"} +{"name":"SciLean.Vec3.instStructTypeVec3FinOfNatNatInstOfNatNatFloat","declaration":"instance SciLean.Vec3.instStructTypeVec3FinOfNatNatInstOfNatNatFloat : SciLean.StructType SciLean.Vec3 (Fin 3) fun x => Float"} +{"name":"SciLean.Vec3.intro","declaration":"def SciLean.Vec3.intro (f : Fin 3 → Float) : SciLean.Vec3"} +{"name":"SciLean.det2","declaration":"def SciLean.det2 (A : SciLean.Vec2 → SciLean.Vec2) : Float"} +{"name":"SciLean.det3","declaration":"def SciLean.det3 (A : SciLean.Vec3 → SciLean.Vec3) : Float"} +{"name":"SciLean.Vec3.y","declaration":"def SciLean.Vec3.y (self : SciLean.Vec3) : Float"} +{"name":"SciLean.Vec2.intro","declaration":"def SciLean.Vec2.intro (f : Fin 2 → Float) : SciLean.Vec2"} +{"name":"SciLean.Vec2.normalize","declaration":"def SciLean.Vec2.normalize (v : SciLean.Vec2) : SciLean.Vec2"} +{"name":"SciLean.Vec3.z","declaration":"def SciLean.Vec3.z (self : SciLean.Vec3) : Float"} +{"name":"SciLean.Vec2.y","declaration":"def SciLean.Vec2.y (self : SciLean.Vec2) : Float"} +{"name":"SciLean.Vec2.instPlainDataTypeVec2","declaration":"instance SciLean.Vec2.instPlainDataTypeVec2 : SciLean.PlainDataType SciLean.Vec2"} +{"name":"SciLean.Vec3.normalize","declaration":"def SciLean.Vec3.normalize (v : SciLean.Vec3) : SciLean.Vec3"} +{"name":"SciLean.Vec3.x","declaration":"def SciLean.Vec3.x (self : SciLean.Vec3) : Float"} +{"name":"SciLean.Vec3","declaration":"structure SciLean.Vec3 : Type"} diff --git a/scilean-declarations/SciLean.Data.DataArray.jsonl b/scilean-declarations/SciLean.Data.DataArray.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/scilean-declarations/SciLean.Data.Fin.jsonl b/scilean-declarations/SciLean.Data.Fin.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..cfc89d65ece01f0b68e57b763bf60e751cfdd2a6 --- /dev/null +++ b/scilean-declarations/SciLean.Data.Fin.jsonl @@ -0,0 +1 @@ +{"name":"Fin.toFloat","declaration":"def Fin.toFloat {n : Nat} (i : Fin n) : Float"} diff --git a/scilean-declarations/SciLean.Data.Function.jsonl b/scilean-declarations/SciLean.Data.Function.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..11537e1259179c1ecb216e6cc885bcea302843e7 --- /dev/null +++ b/scilean-declarations/SciLean.Data.Function.jsonl @@ -0,0 +1,9 @@ +{"name":"Function.modify_same","declaration":"theorem Function.modify_same {α : Sort u} {β : α → Sort v} [DecidableEq α] (a : α) (g : β a → β a) (f : (a : α) → β a) : Function.modify f a g a = g (f a)"} +{"name":"Function.reduceD","declaration":"def Function.reduceD {α : Type u_1} {ι : Type u_2} [SciLean.IndexType ι] (f : ι → α) (op : α → α → α) (default : α) : α"} +{"name":"Function.reduceMD","declaration":"/-- TODO: needs beter implementation but that requires refining EnumType and Index\n-/\ndef Function.reduceMD {α : Type u_1} {ι : Type u_2} [SciLean.IndexType ι] {m : Type u_1 → Type u_4} [Monad m] (f : ι → α) (op : α → α → m α) (default : α) : m α"} +{"name":"Function.Inverse","declaration":"def Function.Inverse {β : Sort u_1} {α : Sort u_2} (g : β → α) (f : α → β) : Prop"} +{"name":"Function.foldl","declaration":"def Function.foldl {α : Sort u_1} {β : Type u_2} {ι : Type u_3} [SciLean.IndexType ι] (f : ι → α) (op : β → α → β) (init : β) : β"} +{"name":"Function.foldlM","declaration":"def Function.foldlM {α : Sort u_1} {β : Type u_2} {ι : Type u_3} [SciLean.IndexType ι] {m : Type u_2 → Type u_2} [Monad m] (f : ι → α) (op : β → α → m β) (init : β) : m β"} +{"name":"Function.reduce","declaration":"def Function.reduce {α : Type u_1} {ι : Type u_2} [SciLean.IndexType ι] [Inhabited α] (f : ι → α) (op : α → α → α) : α"} +{"name":"Function.modify","declaration":"/-- Similar to `Function.update` but `g` specifies how to change the value at `a'`. -/\ndef Function.modify {α : Sort u} {β : α → Sort v} [DecidableEq α] (f : (a : α) → β a) (a' : α) (g : β a' → β a') (a : α) : β a"} +{"name":"Function.modify_noteq","declaration":"theorem Function.modify_noteq {α : Sort u} {β : α → Sort v} [DecidableEq α] {a : α} {a' : α} (h : a ≠ a') (g : β a' → β a') (f : (a : α) → β a) : Function.modify f a' g a = f a"} diff --git a/scilean-declarations/SciLean.Data.IndexType.jsonl b/scilean-declarations/SciLean.Data.IndexType.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..875e6093bc8ca4f707acb49403db0ce53b68f0f5 --- /dev/null +++ b/scilean-declarations/SciLean.Data.IndexType.jsonl @@ -0,0 +1,12 @@ +{"name":"IndexType.reduceD","declaration":"def IndexType.reduceD {ι : Type v} [SciLean.IndexType ι] {α : Type u_2} (f : ι → α) (op : α → α → α) (default : α) : α"} +{"name":"IndexType.sum","declaration":"def IndexType.sum {ι : Type v} [SciLean.IndexType ι] {α : Type u} [Zero α] [Add α] (f : ι → α) : α"} +{"name":"IndexType.argValMax","declaration":"def IndexType.argValMax {X : Type u_1} {I : Type u_2} [SciLean.IndexType I] [Inhabited I] (f : I → X) [LT X] [(x x' : X) → Decidable (x < x')] : I × X"} +{"name":"IndexType.product","declaration":"def IndexType.product {α : Type u_1} [One α] [Mul α] {ι : Type u_2} [SciLean.IndexType ι] (f : ι → α) : α"} +{"name":"IndexType.reduce","declaration":"def IndexType.reduce {ι : Type v} [SciLean.IndexType ι] {α : Type u_2} [Inhabited α] (f : ι → α) (op : α → α → α) : α"} +{"name":"IndexType.unexpandProduct","declaration":"def IndexType.unexpandProduct : Lean.PrettyPrinter.Unexpander"} +{"name":"IndexType.«term∑_,_»","declaration":"def IndexType.«term∑_,_» : Lean.ParserDescr"} +{"name":"IndexType.unexpandSum","declaration":"def IndexType.unexpandSum : Lean.PrettyPrinter.Unexpander"} +{"name":"IndexType.«term∏_,_»","declaration":"def IndexType.«term∏_,_» : Lean.ParserDescr"} +{"name":"IndexType.argMax","declaration":"def IndexType.argMax {X : Type u_1} {I : Type u_2} [SciLean.IndexType I] [Inhabited I] (f : I → X) [LT X] [(x x' : X) → Decidable (x < x')] : I"} +{"name":"IndexType.instDecidableForAll","declaration":"instance IndexType.instDecidableForAll {ι : Type v} [SciLean.IndexType ι] (P : ι → Prop) [(i : ι) → Decidable (P i)] : Decidable (∀ (i : ι), P i)"} +{"name":"IndexType.reduceMD","declaration":"def IndexType.reduceMD {ι : Type v} [SciLean.IndexType ι] {α : Type u_2} {m : Type u_2 → Type u_3} [Monad m] (f : ι → α) (op : α → α → m α) (default : α) : m α"} diff --git a/scilean-declarations/SciLean.Data.ListN.jsonl b/scilean-declarations/SciLean.Data.ListN.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..66dcbaaea9816975108308509f7442d12459f17e --- /dev/null +++ b/scilean-declarations/SciLean.Data.ListN.jsonl @@ -0,0 +1,19 @@ +{"name":"ListN.add_elemwise","declaration":"theorem ListN.add_elemwise {α : Type u_1} {n : Nat} [Add α] (x : α) (y : α) (xs : ListN α n) (ys : ListN α n) : ListN.cons x xs + ListN.cons y ys = ListN.cons (x + y) (xs + ys)"} +{"name":"ListN.toArray","declaration":"def ListN.toArray {α : Type u_1} {n : Nat} (l : ListN α n) : Array α"} +{"name":"ListN","declaration":"inductive ListN (α : Type u) : Nat → Type u"} +{"name":"ListN.instMulListN","declaration":"instance ListN.instMulListN {α : Type u_1} {n : Nat} [Mul α] : Mul (ListN α n)"} +{"name":"ListN.instAddListN","declaration":"instance ListN.instAddListN {α : Type u_1} {n : Nat} [Add α] : Add (ListN α n)"} +{"name":"ListN.unexpandListNCons","declaration":"def ListN.unexpandListNCons : Lean.PrettyPrinter.Unexpander"} +{"name":"ListN.cons","declaration":"ctor ListN.cons {α : Type u} {n : Nat} (x : α) (xs : ListN α n) : ListN α (n + 1)"} +{"name":"ListN.nil","declaration":"ctor ListN.nil {α : Type u} : ListN α 0"} +{"name":"ListN.toList","declaration":"def ListN.toList {α : Type u_1} {n : Nat} (l : ListN α n) : List α"} +{"name":"ListN.unexpandListNNil","declaration":"def ListN.unexpandListNNil : Lean.PrettyPrinter.Unexpander"} +{"name":"ListN.instToStringListN","declaration":"instance ListN.instToStringListN {α : Type u_1} {n : Nat} [ToString α] : ToString (ListN α n)"} +{"name":"ListN.div_elemwise","declaration":"theorem ListN.div_elemwise {α : Type u_1} {n : Nat} [Div α] (x : α) (y : α) (xs : ListN α n) (ys : ListN α n) : ListN.cons x xs / ListN.cons y ys = ListN.cons (x / y) (xs / ys)"} +{"name":"ListN.instDivListN","declaration":"instance ListN.instDivListN {α : Type u_1} {n : Nat} [Div α] : Div (ListN α n)"} +{"name":"ListN.toArray.go","declaration":"def ListN.toArray.go {α : Type u_1} {m : Nat} (a : Array α) (l : ListN α m) : Array α"} +{"name":"ListN.instSubListN","declaration":"instance ListN.instSubListN {α : Type u_1} {n : Nat} [Sub α] : Sub (ListN α n)"} +{"name":"ListN.mul_elemwise","declaration":"theorem ListN.mul_elemwise {α : Type u_1} {n : Nat} [Mul α] (x : α) (y : α) (xs : ListN α n) (ys : ListN α n) : ListN.cons x xs * ListN.cons y ys = ListN.cons (x * y) (xs * ys)"} +{"name":"ListN.map₂","declaration":"def ListN.map₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {n : Nat} (op : α → β → γ) (l : ListN α n) (l' : ListN β n) : ListN γ n"} +{"name":"ListN.«term[_]'»","declaration":"/-- Notation for list literals with list lenght in its type. -/\ndef ListN.«term[_]'» : Lean.ParserDescr"} +{"name":"ListN.sub_elemwise","declaration":"theorem ListN.sub_elemwise {α : Type u_1} {n : Nat} [Sub α] (x : α) (y : α) (xs : ListN α n) (ys : ListN α n) : ListN.cons x xs - ListN.cons y ys = ListN.cons (x - y) (xs - ys)"} diff --git a/scilean-declarations/SciLean.Data.StructType.Algebra.jsonl b/scilean-declarations/SciLean.Data.StructType.Algebra.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..13a6b71fb2acbae587a569d2f29ec5de92966033 --- /dev/null +++ b/scilean-declarations/SciLean.Data.StructType.Algebra.jsonl @@ -0,0 +1,55 @@ +{"name":"SciLean.instTestFunctionsRecSumType","declaration":"instance SciLean.instTestFunctionsRecSumType {I : Type u_1} {EI : I → Type u_2} {J : Type u_3} {FJ : J → Type u_2} [(i : I) → SciLean.TestFunctions (EI i)] [(j : J) → SciLean.TestFunctions (FJ j)] (i : I ⊕ J) : SciLean.TestFunctions (Sum.rec EI FJ i)"} +{"name":"SciLean.instVecStructDefault","declaration":"instance SciLean.instVecStructDefault (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.Vec K X] : SciLean.VecStruct K X Unit fun x => X"} +{"name":"SciLean.oneHot_inl","declaration":"theorem SciLean.oneHot_inl {E : Type u_1} {I : Type u_4} {EI : I → Type u_5} [SciLean.StructType E I EI] {F : Type u_2} {J : Type u_3} {FJ : J → Type u_5} [SciLean.StructType F J FJ] [(i : I) → Zero (EI i)] [Zero F] [(j : J) → Zero (FJ j)] [SciLean.ZeroStruct F J FJ] [DecidableEq I] [DecidableEq J] (i : I) (xi : EI i) : SciLean.oneHot (Sum.inl i) xi = (SciLean.oneHot i xi, 0)"} +{"name":"SciLean.SMulStruct","declaration":"class SciLean.SMulStruct (K : Type u_1) (X : Type u_2) (I : Sort u_3) (XI : I → Type u_4) [SciLean.StructType X I XI] [SMul K X] [(i : I) → SMul K (XI i)] : Prop"} +{"name":"SciLean.ZeroStruct.mk","declaration":"ctor SciLean.ZeroStruct.mk {X : Type u_1} {I : Sort u_2} {XI : I → Type u_3} [SciLean.StructType X I XI] [Zero X] [(i : I) → Zero (XI i)] (structProj_zero : ∀ (i : I), SciLean.structProj 0 i = 0) : SciLean.ZeroStruct X I XI"} +{"name":"SciLean.instVecRecSumType","declaration":"instance SciLean.instVecRecSumType (K : Type u_1) [RCLike K] {I : Type u_2} {EI : I → Type u_3} {J : Type u_4} {FJ : J → Type u_3} [(i : I) → SciLean.Vec K (EI i)] [(j : J) → SciLean.Vec K (FJ j)] (i : I ⊕ J) : SciLean.Vec K (Sum.rec EI FJ i)"} +{"name":"SciLean.SMulStruct.mk","declaration":"ctor SciLean.SMulStruct.mk {K : Type u_1} {X : Type u_2} {I : Sort u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [SMul K X] [(i : I) → SMul K (XI i)] (structProj_smul : ∀ (i : I) (k : K) (x : X), SciLean.structProj (k • x) i = k • SciLean.structProj x i) : SciLean.SMulStruct K X I XI"} +{"name":"SciLean.StructType.structMake.arg_f.IsLinearMap_rule","declaration":"theorem SciLean.StructType.structMake.arg_f.IsLinearMap_rule (K : Type u_1) [RCLike K] {I : Type u_3} {X : Type u_5} {XI : I → Type u_4} [SciLean.StructType X I XI] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] {W : Type u_2} [SciLean.Vec K W] (f : W → (i : I) → XI i) (hf : IsLinearMap K f) : IsLinearMap K fun w => SciLean.structMake (f w)"} +{"name":"SciLean.instInnerRecSumType","declaration":"instance SciLean.instInnerRecSumType (K : Type u_1) {I : Type u_2} {EI : I → Type u_3} {J : Type u_4} {FJ : J → Type u_3} [(i : I) → Inner K (EI i)] [(j : J) → Inner K (FJ j)] (i : I ⊕ J) : Inner K (Sum.rec EI FJ i)"} +{"name":"SciLean.SemiInnerProductSpaceStruct.testFun_structProj","declaration":"def SciLean.SemiInnerProductSpaceStruct.testFun_structProj {K : Type u_1} {X : Type u_2} {I : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [RCLike K] [SciLean.IndexType I] [SciLean.LawfulIndexType I] [SciLean.SemiInnerProductSpace K X] [(i : I) → SciLean.SemiInnerProductSpace K (XI i)] [self : SciLean.SemiInnerProductSpaceStruct K X I XI] (x : X) : SciLean.TestFunction x ↔ ∀ (i : I), SciLean.TestFunction (SciLean.structProj x i)"} +{"name":"SciLean.AddStruct.mk","declaration":"ctor SciLean.AddStruct.mk {X : Type u_1} {I : Sort u_2} {XI : I → Type u_3} [SciLean.StructType X I XI] [Add X] [(i : I) → Add (XI i)] (structProj_add : ∀ (i : I) (x x' : X), SciLean.structProj (x + x') i = SciLean.structProj x i + SciLean.structProj x' i) : SciLean.AddStruct X I XI"} +{"name":"SciLean.inner_oneHot_eq_inner_structProj","declaration":"theorem SciLean.inner_oneHot_eq_inner_structProj (K : Type u_4) [RCLike K] {I : Type u_3} {X : Type u_1} {XI : I → Type u_2} [SciLean.StructType X I XI] [SciLean.IndexType I] [SciLean.LawfulIndexType I] [DecidableEq I] [(i : I) → SciLean.SemiInnerProductSpace K (XI i)] [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpaceStruct K X I XI] (i : I) (xi : XI i) (x : X) : ⟪x, SciLean.oneHot i xi⟫_K = ⟪SciLean.structProj x i, xi⟫_K"} +{"name":"SciLean.VecStruct.structProj_continuous","declaration":"def SciLean.VecStruct.structProj_continuous {K : Type u_1} {X : Type u_2} {I : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [RCLike K] [SciLean.Vec K X] [(i : I) → SciLean.Vec K (XI i)] [self : SciLean.VecStruct K X I XI] : Continuous fun x i => SciLean.structProj x i"} +{"name":"SciLean.ZeroStruct","declaration":"class SciLean.ZeroStruct (X : Type u_1) (I : Sort u_2) (XI : I → Type u_3) [SciLean.StructType X I XI] [Zero X] [(i : I) → Zero (XI i)] : Prop"} +{"name":"SciLean.StructType.structMake_zero","declaration":"theorem SciLean.StructType.structMake_zero (K : Type u_4) [RCLike K] {I : Type u_2} {X : Type u_1} {XI : I → Type u_3} [SciLean.StructType X I XI] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] : (SciLean.structMake fun x => 0) = 0"} +{"name":"SciLean.instSemiInnerProductSpaceRecSumType","declaration":"instance SciLean.instSemiInnerProductSpaceRecSumType (K : Type u_1) [RCLike K] {I : Type u_2} {EI : I → Type u_3} {J : Type u_4} {FJ : J → Type u_3} [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [(j : J) → SciLean.SemiInnerProductSpace K (FJ j)] (i : I ⊕ J) : SciLean.SemiInnerProductSpace K (Sum.rec EI FJ i)"} +{"name":"SciLean.instSemiInnerProductSpaceStructProdSumRecTypeInstStrucTypeProdInstIndexTypeSumInstLawfulIndexTypeSumInstIndexTypeSumInstSemiInnerProductSpaceProdInstSemiInnerProductSpaceRecSumType","declaration":"instance SciLean.instSemiInnerProductSpaceStructProdSumRecTypeInstStrucTypeProdInstIndexTypeSumInstLawfulIndexTypeSumInstIndexTypeSumInstSemiInnerProductSpaceProdInstSemiInnerProductSpaceRecSumType (K : Type u_1) [RCLike K] {E : Type u_2} {I : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] {F : Type u_5} {J : Type u_6} {FJ : J → Type u_4} [SciLean.StructType F J FJ] [SciLean.SemiInnerProductSpace K E] [SciLean.SemiInnerProductSpace K F] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [(j : J) → SciLean.SemiInnerProductSpace K (FJ j)] [SciLean.IndexType I] [SciLean.LawfulIndexType I] [SciLean.IndexType J] [SciLean.LawfulIndexType J] [SciLean.SemiInnerProductSpaceStruct K E I EI] [SciLean.SemiInnerProductSpaceStruct K F J FJ] : SciLean.SemiInnerProductSpaceStruct K (E × F) (I ⊕ J) fun t => Sum.rec EI FJ t"} +{"name":"SciLean.instSubRecSumType","declaration":"instance SciLean.instSubRecSumType {I : Type u_1} {EI : I → Type u_2} {J : Type u_3} {FJ : J → Type u_2} [(i : I) → Sub (EI i)] [(j : J) → Sub (FJ j)] (i : I ⊕ J) : Sub (Sum.rec EI FJ i)"} +{"name":"SciLean.instAddStructDefault","declaration":"instance SciLean.instAddStructDefault {X : Type u_1} [Add X] : SciLean.AddStruct X Unit fun x => X"} +{"name":"SciLean.SemiInnerProductSpaceStruct.inner_structProj","declaration":"def SciLean.SemiInnerProductSpaceStruct.inner_structProj {K : Type u_1} {X : Type u_2} {I : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [RCLike K] [SciLean.IndexType I] [SciLean.LawfulIndexType I] [SciLean.SemiInnerProductSpace K X] [(i : I) → SciLean.SemiInnerProductSpace K (XI i)] [self : SciLean.SemiInnerProductSpaceStruct K X I XI] (x : X) (x' : X) : ⟪x, x'⟫_K = ∑ i, ⟪SciLean.structProj x i, SciLean.structProj x' i⟫_K"} +{"name":"SciLean.StructType.structProj.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.StructType.structProj.arg_x.CDifferentiable_rule (K : Type u_1) [RCLike K] {I : Type u_5} {X : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] {W : Type u_2} [SciLean.Vec K W] (x : W → X) (i : I) (hx : SciLean.CDifferentiable K x) : SciLean.CDifferentiable K fun w => SciLean.structProj (x w) i"} +{"name":"SciLean.instSMulRecSumType","declaration":"instance SciLean.instSMulRecSumType (K : Type u_1) {I : Type u_2} {EI : I → Type u_3} {J : Type u_4} {FJ : J → Type u_3} [(i : I) → SMul K (EI i)] [(j : J) → SMul K (FJ j)] (i : I ⊕ J) : SMul K (Sum.rec EI FJ i)"} +{"name":"SciLean.VecStruct","declaration":"class SciLean.VecStruct (K : Type u_1) (X : Type u_2) (I : Type u_3) (XI : I → Type u_4) [SciLean.StructType X I XI] [RCLike K] [SciLean.Vec K X] [(i : I) → SciLean.Vec K (XI i)] : Prop"} +{"name":"SciLean.instSemiInnerProductSpaceStructUnitInstStructTypeDefaultInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnit","declaration":"instance SciLean.instSemiInnerProductSpaceStructUnitInstStructTypeDefaultInstIndexTypeUnitInstLawfulIndexTypeUnitInstIndexTypeUnit (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] : SciLean.SemiInnerProductSpaceStruct K X Unit fun x => X"} +{"name":"SciLean.instTopologicalSpaceRecSumType","declaration":"instance SciLean.instTopologicalSpaceRecSumType {I : Type u_1} {EI : I → Type u_2} {J : Type u_3} {FJ : J → Type u_2} [(i : I) → TopologicalSpace (EI i)] [(j : J) → TopologicalSpace (FJ j)] (i : I ⊕ J) : TopologicalSpace (Sum.rec EI FJ i)"} +{"name":"SciLean.SMulStruct.structProj_smul","declaration":"def SciLean.SMulStruct.structProj_smul {K : Type u_1} {X : Type u_2} {I : Sort u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [SMul K X] [(i : I) → SMul K (XI i)] [self : SciLean.SMulStruct K X I XI] (i : I) (k : K) (x : X) : SciLean.structProj (k • x) i = k • SciLean.structProj x i"} +{"name":"SciLean.instUniformSpaceRecSumType","declaration":"instance SciLean.instUniformSpaceRecSumType {I : Type u_1} {EI : I → Type u_2} {J : Type u_3} {FJ : J → Type u_2} [(i : I) → UniformSpace (EI i)] [(j : J) → UniformSpace (FJ j)] (i : I ⊕ J) : UniformSpace (Sum.rec EI FJ i)"} +{"name":"SciLean.VecStruct.structMake_continuous","declaration":"def SciLean.VecStruct.structMake_continuous {K : Type u_1} {X : Type u_2} {I : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [RCLike K] [SciLean.Vec K X] [(i : I) → SciLean.Vec K (XI i)] [self : SciLean.VecStruct K X I XI] : Continuous fun f => SciLean.structMake f"} +{"name":"SciLean.StructType.structMake.arg_f.CDifferentiable_rule","declaration":"theorem SciLean.StructType.structMake.arg_f.CDifferentiable_rule (K : Type u_1) [RCLike K] {I : Type u_3} {X : Type u_5} {XI : I → Type u_4} [SciLean.StructType X I XI] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] {W : Type u_2} [SciLean.Vec K W] (f : W → (i : I) → XI i) (hf : SciLean.CDifferentiable K f) : SciLean.CDifferentiable K fun w => SciLean.structMake (f w)"} +{"name":"SciLean.AddStruct.structProj_add","declaration":"def SciLean.AddStruct.structProj_add {X : Type u_1} {I : Sort u_2} {XI : I → Type u_3} [SciLean.StructType X I XI] [Add X] [(i : I) → Add (XI i)] [self : SciLean.AddStruct X I XI] (i : I) (x : X) (x' : X) : SciLean.structProj (x + x') i = SciLean.structProj x i + SciLean.structProj x' i"} +{"name":"SciLean.oneHot_inr","declaration":"theorem SciLean.oneHot_inr {E : Type u_1} {I : Type u_4} {EI : I → Type u_5} [SciLean.StructType E I EI] {F : Type u_2} {J : Type u_3} {FJ : J → Type u_5} [SciLean.StructType F J FJ] [Zero E] [(i : I) → Zero (EI i)] [SciLean.ZeroStruct E I EI] [(j : J) → Zero (FJ j)] [DecidableEq I] [DecidableEq J] (j : J) (xj : FJ j) : SciLean.oneHot (Sum.inr j) xj = (0, SciLean.oneHot j xj)"} +{"name":"SciLean.add_oneHot_eq_structModify","declaration":"theorem SciLean.add_oneHot_eq_structModify (K : Type u_2) [RCLike K] {I : Type u_3} {X : Type u_1} {XI : I → Type u_4} [SciLean.StructType X I XI] [DecidableEq I] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] (i : I) (xi : XI i) (x : X) : x + SciLean.oneHot i xi = SciLean.structModify i (fun xi' => xi' + xi) x"} +{"name":"SciLean.instSMulStructDefault","declaration":"instance SciLean.instSMulStructDefault (K : Type u_1) [RCLike K] {X : Type u_2} [SMul K X] : SciLean.SMulStruct K X Unit fun x => X"} +{"name":"SciLean.oneHot.arg_xi.IsLinearMap_rule_simple","declaration":"theorem SciLean.oneHot.arg_xi.IsLinearMap_rule_simple (K : Type u_1) [RCLike K] {I : Type u_4} {X : Type u_3} {XI : I → Type u_2} [SciLean.StructType X I XI] [DecidableEq I] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] (i : I) : IsLinearMap K fun xi => SciLean.oneHot i xi"} +{"name":"SciLean.inner_oneHot_eq_inner_proj'","declaration":"theorem SciLean.inner_oneHot_eq_inner_proj' (K : Type u_4) [RCLike K] {I : Type u_3} {X : Type u_1} {XI : I → Type u_2} [SciLean.StructType X I XI] [SciLean.IndexType I] [SciLean.LawfulIndexType I] [DecidableEq I] [(i : I) → SciLean.SemiInnerProductSpace K (XI i)] [SciLean.SemiInnerProductSpace K X] [SciLean.SemiInnerProductSpaceStruct K X I XI] (i : I) (xi : XI i) (x : X) : ⟪SciLean.oneHot i xi, x⟫_K = ⟪xi, SciLean.structProj x i⟫_K"} +{"name":"SciLean.SemiInnerProductSpaceStruct","declaration":"class SciLean.SemiInnerProductSpaceStruct (K : Type u_1) (X : Type u_2) (I : Type u_3) (XI : I → Type u_4) [SciLean.StructType X I XI] [RCLike K] [SciLean.IndexType I] [SciLean.LawfulIndexType I] [SciLean.SemiInnerProductSpace K X] [(i : I) → SciLean.SemiInnerProductSpace K (XI i)] : Prop"} +{"name":"SciLean.VecStruct.mk","declaration":"ctor SciLean.VecStruct.mk {K : Type u_1} {X : Type u_2} {I : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [RCLike K] [SciLean.Vec K X] [(i : I) → SciLean.Vec K (XI i)] [toZeroStruct : SciLean.ZeroStruct X I XI] [toAddStruct : SciLean.AddStruct X I XI] [toSMulStruct : SciLean.SMulStruct K X I XI] (structProj_continuous : Continuous fun x i => SciLean.structProj x i) (structMake_continuous : Continuous fun f => SciLean.structMake f) : SciLean.VecStruct K X I XI"} +{"name":"SciLean.instVecStructProd","declaration":"instance SciLean.instVecStructProd (K : Type u_1) [RCLike K] {E : Type u_2} {I : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] {F : Type u_5} {J : Type u_6} {FJ : J → Type u_4} [SciLean.StructType F J FJ] [SciLean.Vec K E] [SciLean.Vec K F] [(i : I) → SciLean.Vec K (EI i)] [(j : J) → SciLean.Vec K (FJ j)] [SciLean.VecStruct K E I EI] [SciLean.VecStruct K F J FJ] : SciLean.VecStruct K (E × F) (I ⊕ J) fun t => Sum.rec EI FJ t"} +{"name":"SciLean.instZeroRecSumType","declaration":"instance SciLean.instZeroRecSumType {I : Type u_1} {EI : I → Type u_2} {J : Type u_3} {FJ : J → Type u_2} [(i : I) → Zero (EI i)] [(j : J) → Zero (FJ j)] (i : I ⊕ J) : Zero (Sum.rec EI FJ i)"} +{"name":"SciLean.instAddStructProd","declaration":"instance SciLean.instAddStructProd {E : Type u_1} {I : Type u_2} {EI : I → Type u_3} [SciLean.StructType E I EI] {F : Type u_4} {J : Type u_5} {FJ : J → Type u_3} [SciLean.StructType F J FJ] [Add E] [Add F] [(i : I) → Add (EI i)] [(j : J) → Add (FJ j)] [SciLean.AddStruct E I EI] [SciLean.AddStruct F J FJ] : SciLean.AddStruct (E × F) (I ⊕ J) fun t => Sum.rec EI FJ t"} +{"name":"SciLean.SemiInnerProductSpaceStruct.mk","declaration":"ctor SciLean.SemiInnerProductSpaceStruct.mk {K : Type u_1} {X : Type u_2} {I : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [RCLike K] [SciLean.IndexType I] [SciLean.LawfulIndexType I] [SciLean.SemiInnerProductSpace K X] [(i : I) → SciLean.SemiInnerProductSpace K (XI i)] [toVecStruct : SciLean.VecStruct K X I XI] (inner_structProj : ∀ (x x' : X), ⟪x, x'⟫_K = ∑ i, ⟪SciLean.structProj x i, SciLean.structProj x' i⟫_K) (testFun_structProj : ∀ (x : X), SciLean.TestFunction x ↔ ∀ (i : I), SciLean.TestFunction (SciLean.structProj x i)) : SciLean.SemiInnerProductSpaceStruct K X I XI"} +{"name":"SciLean.StructType.structProj.arg_x.IsLinearMap_rule","declaration":"theorem SciLean.StructType.structProj.arg_x.IsLinearMap_rule (K : Type u_1) [RCLike K] {I : Type u_5} {X : Type u_3} {XI : I → Type u_4} [SciLean.StructType X I XI] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] {W : Type u_2} [SciLean.Vec K W] (x : W → X) (i : I) (hx : IsLinearMap K x) : IsLinearMap K fun w => SciLean.structProj (x w) i"} +{"name":"SciLean.ZeroStruct.structProj_zero","declaration":"def SciLean.ZeroStruct.structProj_zero {X : Type u_1} {I : Sort u_2} {XI : I → Type u_3} [SciLean.StructType X I XI] [Zero X] [(i : I) → Zero (XI i)] [self : SciLean.ZeroStruct X I XI] (i : I) : SciLean.structProj 0 i = 0"} +{"name":"SciLean.add_oneHot_eq_structModify'","declaration":"theorem SciLean.add_oneHot_eq_structModify' (K : Type u_2) [RCLike K] {I : Type u_3} {X : Type u_1} {XI : I → Type u_4} [SciLean.StructType X I XI] [DecidableEq I] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] (i : I) (xi : XI i) (x : X) : SciLean.oneHot i xi + x = SciLean.structModify i (fun xi' => xi + xi') x"} +{"name":"SciLean.instSemiHilbertRecSumType","declaration":"instance SciLean.instSemiHilbertRecSumType (K : Type u_1) [RCLike K] {I : Type u_2} {EI : I → Type u_3} {J : Type u_4} {FJ : J → Type u_3} [(i : I) → SciLean.SemiHilbert K (EI i)] [(j : J) → SciLean.SemiHilbert K (FJ j)] (i : I ⊕ J) : SciLean.SemiHilbert K (Sum.rec EI FJ i)"} +{"name":"SciLean.instAddRecSumType","declaration":"instance SciLean.instAddRecSumType {I : Type u_1} {EI : I → Type u_2} {J : Type u_3} {FJ : J → Type u_2} [(i : I) → Add (EI i)] [(j : J) → Add (FJ j)] (i : I ⊕ J) : Add (Sum.rec EI FJ i)"} +{"name":"SciLean.AddStruct","declaration":"class SciLean.AddStruct (X : Type u_1) (I : Sort u_2) (XI : I → Type u_3) [SciLean.StructType X I XI] [Add X] [(i : I) → Add (XI i)] : Prop"} +{"name":"SciLean.StructType.structMake.arg_f.IsLinearMap_rule_simple","declaration":"theorem SciLean.StructType.structMake.arg_f.IsLinearMap_rule_simple (K : Type u_1) [RCLike K] {I : Type u_2} {X : Type u_4} {XI : I → Type u_3} [SciLean.StructType X I XI] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] : IsLinearMap K fun f => SciLean.structMake f"} +{"name":"SciLean.instSMulStructProd","declaration":"instance SciLean.instSMulStructProd (K : Type u_1) [RCLike K] {E : Type u_2} {I : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] {F : Type u_5} {J : Type u_6} {FJ : J → Type u_4} [SciLean.StructType F J FJ] [SMul K E] [SMul K F] [(i : I) → SMul K (EI i)] [(j : J) → SMul K (FJ j)] [SciLean.SMulStruct K E I EI] [SciLean.SMulStruct K F J FJ] : SciLean.SMulStruct K (E × F) (I ⊕ J) fun t => Sum.rec EI FJ t"} +{"name":"SciLean.oneHot.arg_xi.IsLinearMap_rule","declaration":"theorem SciLean.oneHot.arg_xi.IsLinearMap_rule (K : Type u_1) [RCLike K] {I : Type u_5} {X : Type u_4} {XI : I → Type u_3} [SciLean.StructType X I XI] [DecidableEq I] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] {W : Type u_2} [SciLean.Vec K W] (i : I) (xi : W → XI i) (hxi : IsLinearMap K xi) : IsLinearMap K fun w => SciLean.oneHot i (xi w)"} +{"name":"SciLean.oneHot.arg_xi.CDifferentiable_rule","declaration":"theorem SciLean.oneHot.arg_xi.CDifferentiable_rule (K : Type u_1) [RCLike K] {I : Type u_5} {X : Type u_4} {XI : I → Type u_3} [SciLean.StructType X I XI] [DecidableEq I] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] {W : Type u_2} [SciLean.Vec K W] (i : I) (xi : W → XI i) (hxi : SciLean.CDifferentiable K xi) : SciLean.CDifferentiable K fun w => SciLean.oneHot i (xi w)"} +{"name":"SciLean.instZeroStructProd","declaration":"instance SciLean.instZeroStructProd {E : Type u_1} {I : Type u_2} {EI : I → Type u_3} [SciLean.StructType E I EI] {F : Type u_4} {J : Type u_5} {FJ : J → Type u_3} [SciLean.StructType F J FJ] [Zero E] [Zero F] [(i : I) → Zero (EI i)] [(j : J) → Zero (FJ j)] [SciLean.ZeroStruct E I EI] [SciLean.ZeroStruct F J FJ] : SciLean.ZeroStruct (E × F) (I ⊕ J) fun t => Sum.rec EI FJ t"} +{"name":"SciLean.StructType.structProj.arg_x.IsLinearMap_rule_simple","declaration":"theorem SciLean.StructType.structProj.arg_x.IsLinearMap_rule_simple (K : Type u_1) [RCLike K] {I : Type u_4} {X : Type u_2} {XI : I → Type u_3} [SciLean.StructType X I XI] [(i : I) → SciLean.Vec K (XI i)] [SciLean.Vec K X] (i : I) : IsLinearMap K fun x => SciLean.structProj x i"} +{"name":"SciLean.instNegRecSumType","declaration":"instance SciLean.instNegRecSumType {I : Type u_1} {EI : I → Type u_2} {J : Type u_3} {FJ : J → Type u_2} [(i : I) → Neg (EI i)] [(j : J) → Neg (FJ j)] (i : I ⊕ J) : Neg (Sum.rec EI FJ i)"} +{"name":"SciLean.instZeroStructDefault","declaration":"instance SciLean.instZeroStructDefault {X : Type u_1} [Zero X] : SciLean.ZeroStruct X Unit fun x => X"} diff --git a/scilean-declarations/SciLean.Data.StructType.Basic.jsonl b/scilean-declarations/SciLean.Data.StructType.Basic.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..9ff20d433f9dda8e92cac65c99119d99672bb232 --- /dev/null +++ b/scilean-declarations/SciLean.Data.StructType.Basic.jsonl @@ -0,0 +1,22 @@ +{"name":"SciLean.StructType.structProj","declaration":"def SciLean.StructType.structProj {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} [self : SciLean.StructType X I XI] (x : X) (i : I) : XI i"} +{"name":"SciLean.StructType.right_inv","declaration":"def SciLean.StructType.right_inv {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} [self : SciLean.StructType X I XI] : Function.RightInverse SciLean.structProj SciLean.structMake"} +{"name":"SciLean.StructType.mk","declaration":"ctor SciLean.StructType.mk {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} (structProj : X → (i : I) → XI i) (structMake : ((i : I) → XI i) → X) (structModify : (i : I) → (XI i → XI i) → X → X) (left_inv : Function.LeftInverse structProj structMake) (right_inv : Function.RightInverse structProj structMake) (structProj_structModify : ∀ (i : I) (f : XI i → XI i) (x : X), structProj (structModify i f x) i = f (structProj x i)) (structProj_structModify' : ∀ (i j : I) (f : XI i → XI i) (x : X), i ≠ j → structProj (structModify i f x) j = structProj x j) : SciLean.StructType X I XI"} +{"name":"SciLean.StructType.structMake_structProj","declaration":"theorem SciLean.StructType.structMake_structProj {X : Sort u_1} {I : Sort u_2} {XI : I → Sort u_3} [SciLean.StructType X I XI] (x : X) : (SciLean.structMake fun i => SciLean.structProj x i) = x"} +{"name":"SciLean.oneHot","declaration":"def SciLean.oneHot {X : Sort u_1} {I : Sort u_2} {XI : I → Type u_3} [SciLean.StructType X I XI] [DecidableEq I] [(i : I) → Zero (XI i)] (i : I) (xi : XI i) : X"} +{"name":"SciLean.StructType.structProj_structModify","declaration":"def SciLean.StructType.structProj_structModify {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} [self : SciLean.StructType X I XI] (i : I) (f : XI i → XI i) (x : X) : SciLean.structProj (SciLean.structModify i f x) i = f (SciLean.structProj x i)"} +{"name":"SciLean.StructType.structModify","declaration":"def SciLean.StructType.structModify {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} [self : SciLean.StructType X I XI] (i : I) (f : XI i → XI i) (x : X) : X"} +{"name":"SciLean.StructType.instStrucTypeArrow","declaration":"instance SciLean.StructType.instStrucTypeArrow (E : Type u_1) (I : Type u_2) (J : Type u_3) (EI : I → Type u_4) [SciLean.StructType E I EI] [DecidableEq J] : SciLean.StructType (J → E) (J × I) fun ji => EI ji.2"} +{"name":"SciLean.StructType.structProj_structMake","declaration":"theorem SciLean.StructType.structProj_structMake {X : Sort u_2} {I : Sort u_3} {XI : I → Sort u_1} [SciLean.StructType X I XI] (f : (i : I) → XI i) (i : I) : SciLean.structProj (SciLean.structMake f) i = f i"} +{"name":"SciLean.StructType.structProj_structModify'","declaration":"def SciLean.StructType.structProj_structModify' {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} [self : SciLean.StructType X I XI] (i : I) (j : I) (f : XI i → XI i) (x : X) : i ≠ j → SciLean.structProj (SciLean.structModify i f x) j = SciLean.structProj x j"} +{"name":"SciLean.structExt","declaration":"theorem SciLean.structExt {X : Sort u_2} {I : Sort u_3} {XI : I → Sort u_1} [SciLean.StructType X I XI] (x : X) (x' : X) : (∀ (i : I), SciLean.structProj x i = SciLean.structProj x' i) → x = x'"} +{"name":"SciLean.StructType.structMake","declaration":"def SciLean.StructType.structMake {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} [self : SciLean.StructType X I XI] (f : (i : I) → XI i) : X"} +{"name":"SciLean.StructType.instStrucTypeProd","declaration":"instance SciLean.StructType.instStrucTypeProd {I : Type u_1} {E : Type u_2} {EI : I → Sort u_3} {F : Type u_4} {J : Type u_5} {FJ : J → Sort u_3} [SciLean.StructType E I EI] [SciLean.StructType F J FJ] : SciLean.StructType (E × F) (I ⊕ J) fun t => Sum.rec EI FJ t"} +{"name":"SciLean.StructType.instStructTypeDefault","declaration":"/-- Every type is `StructType` with `Unit` as index set.\n\nThe motivation behind this instance is that type like `X×(Y×Z)` should have `StructType`\ninstance that the type has three components. Such instance is defines inductively\nand this is the base case of this induction, the inductive step is `instStrucTypeProd`.\n-/\ninstance SciLean.StructType.instStructTypeDefault {α : Sort u_1} : SciLean.StructType α Unit fun x => α"} +{"name":"SciLean.StructType.instStrucTypeArrowSimple","declaration":"instance SciLean.StructType.instStrucTypeArrowSimple (E : Type u_1) (J : Type u_2) [DecidableEq J] : SciLean.StructType (J → E) J fun x => E"} +{"name":"SciLean.StructType","declaration":"class SciLean.StructType (X : Sort u_1) (I : Sort u_2) (XI : outParam (I → Sort u_3)) : Sort (max (max (max 1 u_1) u_2) u_3)"} +{"name":"SciLean.StructType.oneHot_unit","declaration":"theorem SciLean.StructType.oneHot_unit {X : Type u_1} [Zero X] (x : X) : SciLean.oneHot () x = x"} +{"name":"SciLean.StructType.structProj_oneHot","declaration":"theorem SciLean.StructType.structProj_oneHot {X : Sort u_3} {I : Sort u_1} {XI : I → Type u_2} [SciLean.StructType X I XI] {i : I} [DecidableEq I] [(i : I) → Zero (XI i)] (xi : XI i) : SciLean.structProj (SciLean.oneHot i xi) i = xi"} +{"name":"SciLean.StructType.structProj_oneHot'","declaration":"theorem SciLean.StructType.structProj_oneHot' {X : Sort u_3} {I : Sort u_1} {XI : I → Type u_2} [SciLean.StructType X I XI] [DecidableEq I] [(i : I) → Zero (XI i)] (i : I) (j : I) (xi : XI i) (h : i ≠ j) : SciLean.structProj (SciLean.oneHot i xi) j = 0"} +{"name":"SciLean.StructType.instStrucTypePiSimple","declaration":"instance SciLean.StructType.instStrucTypePiSimple (I : Type u_1) (E : I → Type u_2) [DecidableEq I] : SciLean.StructType ((i : I) → E i) I E"} +{"name":"SciLean.StructType.left_inv","declaration":"def SciLean.StructType.left_inv {X : Sort u_1} {I : Sort u_2} {XI : outParam (I → Sort u_3)} [self : SciLean.StructType X I XI] : Function.LeftInverse SciLean.structProj SciLean.structMake"} +{"name":"SciLean.StructType.instStrucTypePi","declaration":"instance SciLean.StructType.instStrucTypePi (I : Type u_1) (E : I → Type u_2) (J : I → Type u_3) (EJ : (i : I) → J i → Type u_4) [(i : I) → SciLean.StructType (E i) (J i) (EJ i)] [DecidableEq I] : SciLean.StructType ((i : I) → E i) ((i : I) × J i) fun ij => EJ ij.fst ij.snd"} diff --git a/scilean-declarations/SciLean.Lean.Array.jsonl b/scilean-declarations/SciLean.Lean.Array.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..ae47977cabe6b615aaea7fa7f5336eafbeebfc73 --- /dev/null +++ b/scilean-declarations/SciLean.Lean.Array.jsonl @@ -0,0 +1,11 @@ +{"name":"Array.joinrM","declaration":"def Array.joinrM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [Monad m] [Inhabited β] (xs : Array α) (map : α → m β) (op : β → β → m β) : m β"} +{"name":"Array.colexOrd","declaration":"/-- Ordering by size then by colexicographical ordering(right to left).\n-/\ndef Array.colexOrd {α : Type u_1} [Ord α] (as : Array α) (bs : Array α) : Ordering"} +{"name":"Array.mergeSplit","declaration":"def Array.mergeSplit {α : Type u_1} (ids : Array (Nat ⊕ Nat)) (bs : Array α) (cs : Array α) [Inhabited α] : Array α"} +{"name":"Array.splitIdx","declaration":"/-- Splits array into two based on function p. It also returns indices that can be used to merge two array back together.\n-/\ndef Array.splitIdx {α : Type} (as : Array α) (p : Fin (Array.size as) → α → Bool) : Array α × Array α × Array (Nat ⊕ Nat)"} +{"name":"Array.joinlM","declaration":"def Array.joinlM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [Monad m] [Inhabited β] (xs : Array α) (map : α → m β) (op : β → β → m β) : m β"} +{"name":"Array.lexOrd","declaration":"/-- Ordering by size then by lexicographical ordering(left to right).\n-/\ndef Array.lexOrd {α : Type u_1} [Ord α] (as : Array α) (bs : Array α) : Ordering"} +{"name":"Array.riffle","declaration":"def Array.riffle {α : Type u_1} (xs : Array α) (ys : Array α) : Array α"} +{"name":"Array.splitM","declaration":"def Array.splitM {α : Type} {m : Type → Type u_1} [Monad m] (as : Array α) (p : α → m Bool) : m (Array α × Array α)"} +{"name":"Array.partitionIdxM","declaration":"def Array.partitionIdxM {α : Type} {m : Type → Type u_1} [Monad m] (as : Array α) (p : Fin (Array.size as) → α → m Bool) : m (Array α × Array α × Array (Nat ⊕ Nat))"} +{"name":"Array.joinl","declaration":"def Array.joinl {β : Type u_1} {α : Type u_2} [Inhabited β] (xs : Array α) (map : α → β) (op : β → β → β) : β"} +{"name":"Array.joinr","declaration":"def Array.joinr {β : Type u_1} {α : Type u_2} [Inhabited β] (xs : Array α) (map : α → β) (op : β → β → β) : β"} diff --git a/scilean-declarations/SciLean.jsonl b/scilean-declarations/SciLean.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391